microrheology and spatial heterogeneity of staphylococcus
TRANSCRIPT
1
Microrheology and Spatial Heterogeneity of Staphylococcus
aureus Biofilms Modulated by Hydrodynamic Shear and
Biofilm-Degrading Enzymes
And
The Interaction of Cationic Peptide G3 on Bacterial
Membranes Investigated using 3-dimensional Single Particle
Tracking and Solid-State Nuclear Magnetic Resonance
A thesis submitted to
The University of Manchester for the degree of
Doctor of Philosophy in the Faculty of Science and Engineering
2019
Jack W. Hart
Department of Physics and Astronomy
2
This page is left intentionally blank.
3
List of Contents
List of Figures ........................................................................................................ 7
Chapter 1 .............................................................................................................. 7
Chapter 2 .............................................................................................................. 7
Chapter 3 .............................................................................................................. 9
Chapter 4 ............................................................................................................ 12
List of Tables ........................................................................................................ 14
Chapter 3 ............................................................................................................ 14
Chapter 4 ............................................................................................................ 14
List of Publications .............................................................................................. 15
Abstract ................................................................................................................ 16
Declaration ........................................................................................................... 17
Copyright and ownership of intellectual property rights ................................... 18
Acknowledgements .............................................................................................. 19
Chapter 1. Introduction ....................................................................................... 20
1.1 Bacteria structure .......................................................................................... 20
1.1.1. Bacterial membranes............................................................................. 20
1.1.2. Gram-positive and Gram-negative cell envelopes .................................. 21
1.2. Bacterial biofilms ........................................................................................ 25
1.2.1. Definition of a ‘biofilm’ ......................................................................... 25
1.2.2. Biofilm formation and composition ........................................................ 25
1.2.3. Biofilms as a prevalent clinical issue ..................................................... 26
1.2.4. Biofilms as a prevalent societal issue .................................................... 28
1.2.5. Beneficial biofilms ................................................................................. 28
1.3. Antimicrobial peptides ................................................................................. 29
1.3.1. Antibiotic resistance .............................................................................. 29
1.3.2. The antimicrobial peptide, G3 ............................................................... 29
1.3.3. Physical mechanisms of peptide-induced cell membrane disruption ...... 32
1.4. Current state of biofilm research .................................................................. 34
1.5. Thesis outline .............................................................................................. 34
Chapter 2. Microrheology and spatial heterogeneity of Staphylococcus aureus
biofilms modulated by hydrodynamic shear and biofilm-degrading enzymes .. 36
2.1. Introduction and Chapter Aims .................................................................... 36
2.1.1. Biofilm material properties and response to hydrodynamic shear .......... 36
2.1.2. Aims of Chapter .................................................................................... 37
4
2.2. Theory ......................................................................................................... 38
2.2.1. Einstein’s derivation of the diffusion equation ....................................... 38
2.2.2. Mean squared displacement from the solution to the diffusion equation 40
2.2.3. Langevin equation and Brownian motion .............................................. 42
2.2.4. Complex shear modulus ........................................................................ 44
2.2.5. Relating the mean square displacement to the creep compliance ........... 45
2.2.6. Caveat for complex fluids ...................................................................... 47
2.2.7. Ripley’s K-function................................................................................ 47
2.3. Methodology ............................................................................................... 49
2.3.1. Bacteria preparation ............................................................................. 49
2.3.2. Biofilm cultivation ................................................................................. 49
2.3.3. Optical brightfield microscopy .............................................................. 52
2.3.4. Particle tracking ................................................................................... 52
2.3.5. Ripley’s K-function................................................................................ 55
2.4. Results and Discussion ................................................................................ 56
2.5. Conclusion .................................................................................................. 71
Chapter 3. Spatial distribution and diffusive dynamics of antimicrobial cationic
peptide G3 in gram-positive Staphylococcus aureus and gram-negative
Escherichia coli using 3-dimensional stochastic optical reconstruction
microscopy and fluorescence tracking. ............................................................... 72
3.1. Aims of Chapter and Introduction ................................................................ 72
3.2. Theory ......................................................................................................... 75
3.2.1. Definition of a complex electric field ..................................................... 75
3.2.2 Free-space propagation ......................................................................... 76
3.2.3. Simple lens configurations..................................................................... 78
3.2.4. Field intensity propagation ................................................................... 80
3.2.5. 2-dimensional optical resolution limit ................................................... 82
3.2.6. 3-dimensional point spread function and adaptive optics ....................... 86
3.2.7. Origin of fluorescence ........................................................................... 90
3.2.8. Fluorophore ‘blinking’ .......................................................................... 91
3.3. Methods....................................................................................................... 93
3.3.1. Fluorescence microscope apparatus ...................................................... 93
3.3.2. Adaptive optics calibration .................................................................... 94
3.3.3. Bacteria sample preparation ................................................................. 95
3.3.4. Stochastic optical reconstruction microscopy (STORM) and image
analysis using ThunderSTORM ....................................................................... 96
5
3.3.5. Bacteria identification using average shifted histogram super-resolution
image and parametric equation fitting ............................................................ 98
3.3.6. Normalising localisation position within different bacteria using the
major radius fraction ...................................................................................... 99
3.3.7. Characterising localisation spatial distributions through Monte Carlo
simulations ................................................................................................... 101
3.3.8. Combining individual localisations temporally into single ‘blinks’ ...... 104
3.3.9. Characterising fractional Brownian motion with DeepExponent ......... 105
3.3.10. Radial dependence of step size and anomalous exponent using the
structural similarity index (SSIM) ................................................................. 105
3.4. Results and Discussion .............................................................................. 107
3.4.1. Spatial distribution of fluorescence blinks within cells ......................... 107
3.4.2. Blink trajectories ................................................................................. 110
3.5. Conclusion ................................................................................................ 119
Chapter 4. Interactions between cationic peptide G3 and 𝒅54-DMPC/DMPG
bilayers explored by 31P and 2H solid state nuclear magnetic resonance ......... 120
4.1. Introduction and Aims of Chapter .............................................................. 120
4.2. Theory ....................................................................................................... 120
4.2.1. Spin states inside a magnetic field ....................................................... 120
4.2.2. Bloch equations ................................................................................... 122
4.2.3. Chemical shift anisotropy and magic angle spinning ........................... 131
4.2.4. Quadrupolar interactions for spin-1 nuclei.......................................... 135
4.3. Methodology ............................................................................................. 137
4.3.1. Sample preparation ............................................................................. 137
4.3.2. 31P and 2H solid-state nuclear magnetic resonance data acquisition.... 137
4.3.3. Data analysis and de-Pakeing ............................................................. 139
4.4. Results and Discussion .............................................................................. 141
4.4.1. 31P static solid-state nuclear magnetic resonance ................................ 141
4.4.2. 31P magic angle spinning solid-state nuclear magnetic resonance ....... 142
4.4.3. 31P 𝑇1 relaxation time ........................................................................ 144
4.4.4. 31P 𝑇2 relaxation time ........................................................................ 146
4.4.5. 2H quadrupole order parameters ......................................................... 148
4.5. Conclusion ................................................................................................ 150
Chapter 5. Conclusion ....................................................................................... 151
5.1. Summary of Chapter 2 results and timeline of future related work ............. 151
5.2. Summary of Chapter 3 results and timeline of future related work ............. 153
6
5.3. Summary of Chapter 4 results and timeline of future related work ............. 154
References .......................................................................................................... 156
Word count: 40176
7
List of Figures
Chapter 1
Figure 1.1. Scanning electron microscope images of ATCC 25923 Staphylococcus
aureus (Scale bars: a) = 5 µm and b) = 1 µm).
Figure 1.2. Schematic diagram representing generic gram-positive (left) and gram-
negative (right) cell envelopes (not to scale). Middle inset shows the cross-linking
mechanism forming the crystal structure of the peptidoglycan layer. Abbreviations
used: IM = inner membrane; PS = periplasmic space; PL = peptidoglycan layer; HL
= hairy layer; WTA = wall teichoic acid; LTA = lipoteichoic acid; AP = attached
protein (by ionic or covalent bonding); CI = cytoplasmic interior; IMP = integral
membrane protein; OM = outer membrane; OMP = outer membrane protein; LPS =
lipopolysaccharides; NAG = N-acetylglucosamine; NAM = N-acetylmuramic acid.
Figure 1.3. Schematic drawings of G(IIKK)3-NH2 (G3) in the 𝛼-helical conformation
adopted near bacterial membranes. a) Schiffer−Edmundson wheel projection, where
the angular rotation between adjacent amino acids is fixed at 100° (K = lysine,
I = isoleucine, G = glycine). b) Three-dimensional render of atomic positions colour-
coded by element (white = hydrogen, grey = carbon, blue = nitrogen, red = oxygen).
Cartoon ‘ribbon’ representation also shown in red. Scale bar is 7.5 Å. c) Three-
dimensional render emphasising bonds and omitting hydrogen. Amino acids are
colour-coded (blue = lysine, green = isoleucine, white = glycine). Scale bar is 3 Å.
Figure 1.4. Models of amphipathic peptide-induced membrane permeabilization. a)
Barrel-stave model – peptides are inserted perpendicular to the membrane surface
such that the hydrophobic face of the peptide faces towards the fatty acid chains and
the hydrophilic faces point inwards. b) Carpet model – transient peptide association
parallel to the membrane surface leads to an accumulation of minor defects that
ultimately disintegrate the membrane. c) Toroidal model – curvature induced by
peptide interaction leads to bending of the phospholipid leaflets until a critical point
where they form a pore.
Chapter 2
Figure 2.1. Schematic diagram of the chemostat used to grow the bacterial biofilms
(a flow cell fed with tryptic soya broth by a peristaltic pump) mounted around a
tracking microscope with a vibration isolation unit to reduce erroneous signals. A
closed system was employed to maintain sterility.
Figure 2.2. Example data from the tracking analysis. a) A brightfield microscopy
image of a S.aureus biofilm grown in the flow cell after an incubation time of 3
hours at 37˚C, at an elevation of 5 µm from the flow cell bottom surface. b) A
magnified section from 2a) with an overlay showing individual bacteria that have
been identified, their respective radii and tracks over 1000 frames, equivalent to 1
second. Each colour represents a unique bacterium that has been identified and
tracked. c) An enlarged rendering of an example ‘track’ constructed from the
displacements of a single bacterium position between adjacent frames, showing the
sub-pixel localisation precision attainable with the fitting protocol. d) All mean
square displacements, (MSDs, ⟨∆𝑟2(𝑡)⟩), shown as a function of time interval
corresponding to all bacteria identified in a). The scale bars are equal to 10 µm.
8
Figure 2.3. Creep compliance results a) At a reference time point of 10 ms, the mean
creep compliance was calculated for S. aureus biofilms after 4 hours of sustained
flow at 37 ˚C at incremental heights (represented by colour) above the flow cell
surface, subject to varying hydrodynamic shears. b), c) and d) show probability
distributions for compliances at reference time, 𝑡𝑟𝑒𝑓 = 10 ms for a height of 15 µm
subject to different hydrodynamic shears; 0 mPa (stationary), 1 mPa and 10 mPa.
Figure 2.4. Number of bacteria found in a biofilm as a function of time at a series of
hydrodynamic shear rates, namely a) 0, b) 1mPa and c) 10 mPa. Different colours
represent height above the attachment surface in the flow cell. Similar proportions of
bacteria were found at all shear rates. Within errors, bacteria proliferation was
approximately the same for the 1 mPa and 10 mPa but was reduced for static
biofilms.
Figure 2.5. Identified bacteria radii distributions across all experiments at all times
and heights (indicated by colour) and hydrodynamic shear regimes, namely a)
stationary, b) 1 mPa and c) 10 mPa. The insets in each graph show the mean (and
standard error) bacteria radius corresponding to a given height within the biofilm. As
can be seen, the maximum variation in radius size is ~6%, indicating no significant
contribution to the creep compliances calculated using 2.35.
Figure 2.6. The characteristic creep compliance at a single reference time interval
(𝑡𝑟𝑒𝑓 = 10 ms) plotted as a function of time for the entirety of the experiment (6
hours) with a) no flow, b) 1 mPa shear stress and c) 10 mPa shear stress. Heights in 5
µm intervals are shown as different coloured lines. The biofilms are seen to be softer
the further they are away from the surface of attachment in both no flow and 10 mPa
conditions. Under flow conditions the bacteria appear to be more securely fastened
to the surfaces and the compliances are lower (harder, with larger shear moduli).
Figure 2.7. a) Rescaled Ripley K-function analysis as a function of height for biofilm
grown after 5 hours at 37˚C subject to different hydrodynamic regimes, namely 0
mPa (stationary), 1 mPa and 10 mPa. Rescaled Ripley K-function analysis for
bacteria within a biofilm at the bottom layer (i.e. 0 µm) (b) and 5 µm above surface
of flow cell (c) over all time points and for all hydrodynamic regimes. A general
decrease in the value of the rescaled Ripley K-function for stationary conditions
suggest that the bacteria within the biofilm become more spatial homogeneous as the
biofilm matures.
Figure 2.8. Rescaled Ripley K-function for all time points (6 hours) for biofilms
three different hydrodynamic growth regimes, namely a) no flow, b) 1 mPa shear
stress and c) 10 mPa shear stress. The significant increase in the values of the
rescaled Ripley-K as a function of height (represented by different colours in 5 µm
intervals) indicates spatial clustering vertically during all time points of the biofilm
growth.
Figure 2.9. Comparative examples of bacteria distributions colour coded by creep
compliance of each bacteria for biofilms grown under 10 mPa hydrodynamic shear
stress for 5 hours at 37 ˚C at the surface of the flow cell (a) and 15 µm above the
surface (b). In b), a circle is shown with an arbitrary radius, r originating from a
randomly selected bacterium to convey the bacteria selected within a certain radius.
c) and d) show the normalised Ripley-K function corresponding to a) and b)
respectively, to demonstrate the extent of clustering at a radius, r = 10 µm.
9
Figure 2.10. Characteristic creep compliance as a function of bacteria density for all
time points, heights and hydrodynamic regimes, with a black fit line showing an
inverse linear correlation. The inset shows the same mean creep compliances plotted
as a function of the rescaled Ripley-K function, with a black fit line displaying a
positive linear correlation.
Figure 2.11. A rendering of the bacteria distributed within a biofilm subject to no
hydrodynamic shear after 6 hours of growth at 37˚C. Bacteria positions are extracted
from tracking data and colour coded based on height above surface in 5 µm height
intervals. Grey outlines represent bacteria in the spaces between focal planes. The
arrows indicate the direction of shear flow. Tapered columns can be seen with their
bases on the surface of attachment.
Figure 2.12. The mean compliances calculated at a characteristic time (10 ms)
plotted after 5 hours for two different enzymes (proteinase K and DNase-1) and the
control (just TSB) with no hydrodynamic shear. Biofilms grown in the presence of
proteinase K exhibited much larger creep compliances, characteristic of softer
biofilms. Biofilms, grown in the presence of DNase-1 showed a slight decrease in
creep compliance compared to no enzyme present.
Figure 2.13. Full time course results over 6 hours for biofilm characteristic creep
compliances at a reference time interval of 10 ms when in a) a 60 µg/mL solution of
proteinase K and b) a 100 µg/mL solution of DNase-1. Hydrodynamic shear on the
biofilm was 1 mPa. Height increments of 5 µm away from the flow cell surface are
represented as different colours.
Chapter 3
Figure 3.1. Representation of secondary point sources emanating spherical Huygen’s
wavelets from every point of the object plane, ��0(0), which is assumed to be a
perfect aperture (i.e. the boundaries do not influence the wavelets). The field at the
object plane, ��1(𝑧) can be thought of as the super-position of all wavelets as
described in Equation 3.3. 𝑅 and 휃 represent the distance and angle the Huygen’s
wavelets make with the imaging plane. The Fresnel approximation extends this by
assuming that the small angle approximation can be applied (see Equation 3.4).
Figure 3.2. Lens setup for a 4𝑓 imaging system typically used in microscopy. Fields
propagate from the object plane, ��0 to the image plane, ��1 through an effective
aperture, described by the function 𝑃(𝜉). The separation between the object plane
and the first lens is equal to the first lens’ focal length, 𝑓1 and likewise, the
separation between the image plane and the second lens is the second lens’ focal
length, 𝑓2.
Figure 3.3. Normalised intensity distributions at the imaging plane, ��1 caused by a-b)
a single incoherent point source (see Equation 3.25), c-d) two incoherent point
sources with transverse separation, Δ𝜌 that equals the Rayleigh criterion for
minimum resolution and e-f) transverse separation less than the Rayleigh criterion.
The 4𝑓 imaging system is assumed to have unit magnification, described by lens
focal length, 𝑓 and lens aperture diameter, 𝐷. All point sources depicted are assumed
to emit light with wavelength, 𝜆. Figures on the same row show the same intensity
distribution but from different viewing angles (isometric and top-down).The colour,
representing intensity, is normalised by the single point source maximum.
10
Figure 3.4. Normalised 3-dimensional intensity distribution associated with a 4𝑓
imaging system for an incoherent point source that is axially defocused by Δ𝑧, calculated by assuming a Gaussian aperture. Intensity profile calculated using
Equation 3.26.
Figure 3.5. Transverse deformation caused by adaptive optics mirror on the 3-
dimensional intensity profile for an incoherent point source at axial defocus, Δ𝑧 equal to a) 100 nm, b) 0 nm and c) -100 nm (calculated using Equation 3.27).
Figure 3.6. Jablonski diagram showing the energy bands of a fluorophore. The
energy of a state is represented by the height on the figure. Solid arrows between
energy bands are representative of radiative transitions, whereas dotted lines are non-
radiative. Energy bands are labelled as: 𝑆0, ground state; 𝑆1 singlet excited state
(opposing spin pairs); 𝑇1, excited triplet state (same spin pairs) and 𝑇1∗, radical anion
state (also called the ‘dark’ state). Abosrption occurs when an electron absorbs the
energy of an incident photon and is promoted to a higher energy level. Internal
conversion and intersystem crossing both liberate energy through non-radiative
means, such as in vibrational and rotational energy states. Approximate timescles for
each process are shown, with the exception of transitioning from a dark state back to
the ground state, which must be achieved through UV pumping.
Figure 3.7. Schematic diagram of the setup that was used to perform 3-dimensional
fluorescence tracking and stochastic optical reconstruction microscopy experiments.
Not shown in the figure is the floating table the entire apparatus is mounted on the
reduce vibrations from external sources.
Figure 3.8. Adaptive optics calibration, where the mirror astigmatism was set at
0.6 µm. a) Raw images of a single fluorescent bead intensity profile at the focal
plane and 300 nm above and below the focal plane. b) Normalised 𝑥 and 𝑦 axial
ratios of the 2-dimensional fitted Gaussian integral to the raw images.
Figure 3.9. Approximate 3-dimensional ellipsoid fit of S. aureus bacteria. Ellipses
are initially fit to the 2-dimensional STORM average shifted histogram projection
using Equation 3.29, shown as pink outlines. The major and minor radii for each
ellipsoid are shown as red dotted lines. Extrapolation into the ellipsoids was
achieved by assuming the minor radius in the axial direction was equal the minor
radius in the planar direction. Equation 3.32 was used to find the final parametric
description of the ellipsoid.
Figure 3.10. Projection of method used to normalise localisation positions amongst
different cell ellipsoid sizes. A smaller (or larger) ellipsoid with the same
eccentricity and centre as the bacteria was generated such that the fluorescence
localisation intercepted the surface of the new ellipsoid. As the eccentricity is the
same, the position within the cell can be put in terms of the major radius fraction. For
the figure above, 𝑟𝑧0 and 𝑟𝑥0 are the initial major and minor radii and 𝑟𝑧 and 𝑟𝑥 are
the major and minor radii associated with the surface intercepting the fluorophore.
Figure 3.11. Monte Carlo simulations of localisations within a unit sphere that are
a) uniformly distributed with a cut-off at the membrane, b) normally distributed with
𝜇 = 1, 𝜎 = 0.1 and c) normally distributed with 𝜇 = 0, 𝜎 = 0.5. Probability density
histograms as a function of major radius fraction are included for each condition.
Figure 3.12. Probability density histogram of Monte Carlo localisations as a function
of major radius fraction weighted by the relative surface area associated to the sphere
11
with radius equal to the major radius fraction. The distribution of localisations were
uniformly distributed with a cut-off at the membrane, membrane weighted (normally
distributed with 𝜇 = 1, 𝜎 = 0.1) and radially weighted (normally distributed with
𝜇 = 0, 𝜎 = 0.5).
Figure 3.13. 2-dimensional histograms showing the relationship between major
radius fraction (i.e. location within cell) and fluorophore displacement (over 1 frame)
for a simulated dataset. Correlations between the variables is set at a) direct
correlation, b) 50% correlation and c) no correlation (i.e. random).
Figure 3.14. Fluorescence localisation position in 3-dimensional space acquired via
STORM over the course of 200,000 frames at 400 frames per second. A 2-
dimensional reconstructed STORM projection of all localisations is shown
underneath the actual positions. The colour of the localisations is representative of
the time the signal was acquired.
Figure 3.15. Normalised histograms of Cy3B blink position as a function of major
radius fraction for a peptide negative control, G3 in S. aureus and E. coli. a) shows
the initial output of the histogram, whereas b) shows the frequency density weighted
by the relative ellipsoid surface area associated with each major radius fraction
value.
Figure 3.16. Four example blink trajectories of G3-Cy3B molecules within a
S. aureus cell, captured using fluorescent single-particle tracking. Localisation errors
are attributed based on the least-squares fitting of an astigmatic Gaussian integral.
Positions are extracted every 2.5 ms.
Figure 3.17. Fluorescence blink trajectory mean square displacements for a) G3-
Cy3b in S. aureus, b) G3-Cy3b in E. coli and c) Cy3b only (G3-negative control) in
S. aureus. Mean square displacement magnitudes that decrease at larger time
intervals are indicative of shorter blinks as the statistical weight for larger time
intervals is smaller.
Figure 3.18. Histogram of mean 𝛼 values extracted from passing blink trajectories
through the neural net DeepExponent for Cy3B only (in S. aureus cells), G3-Cy3B
in E. coli cells and G3-Cy3B in S. aureus cells.
Figure 3.19. a) Normalised probability histogram showing fluorophore displacement
between adjacent frames for Cy3B only (in S. aureus cells), G3-Cy3B in E. coli cells
and G3-Cy3B in S. aureus cells. b) Examples of blink trajectories for G3-Cy3B in
S. aureus cells where colour represents the magnitude of the fluorophore
displacement.
Figure 3.20. 2-dimensional histogram showing relationship between neural-net
extracted 𝛼 values and corresponding major radius position for a) G3-Cy3b in
S. aureus, b) G3-Cy3b in E. coli and c) Cy3b only (G3-negative control) in
S. aureus.
Figure 3.21. 2-dimensional histogram showing relationship between fluorophore
displacement and corresponding major radius position for a) G3-Cy3b in S. aureus,
b) G3-Cy3b in E. coli and c) Cy3b only (G3-negative control) in S. aureus.
12
Chapter 4
Figure 4.1. Net magnetization, ��(𝑡) precession about a 𝑧-axis magnetic field, 𝐵𝑧 at
the Larmor frequency, 𝑤. For nuclear magnetic resonance experiments, the initial net
magnetization, ��0 at time, 𝑡 = 0, is the result of the energy difference between
nuclei spin states.
Figure 4.2. Net magnetization, ��(𝑡) in the lab reference frame before (a), during (b)
and after (c) a transverse magnetic field is applied. The length, 𝑡 and orientation, 𝐵𝑥
of the pulse is such that the net magnetization make a 𝜋 2⁄ rotation about the 𝑥-axis.
It is assumed that the pulse length is also much shorter than the relaxation constants
(i.e. 𝑡 ≪ 𝑇1,2). As it rotates, the net magnetization precesses about the 𝑧-axis with
angular frequency equal to the Larmor frequency, 𝜔.
Figure 4.3. Net magnetization, ��(𝑡) in the lab reference frame initially orientated
along the 𝑥-axis (a), relaxing with characteristic timescales, 𝑇1 and 𝑇2 (b) and
completely returned to thermal equilibrium (c) in the presence of a 𝑧-orientated
external magnetic field, 𝐵𝑧. As the net magnetization relaxes, it precesses about the
𝑧-axis with angular frequency equal to the Larmor frequency, 𝜔.
Figure 4.4. Visual representation of the chemical shift anisotropy experienced by a
nucleus in a molecule. The laboratory reference frame, (𝑥, 𝑦, 𝑧 – black arrows) is
centred on the nucleus. An ellipsoid represents the chemical shift tensor associated
with the electron density surrounding the nucleus. The principal axis system, PAS
(𝑥𝑃𝐴𝑆, 𝑦𝑃𝐴𝑆, 𝑧𝑃𝐴𝑆 – blue dash-dot arrows) is orientated with respect to the principal
components of the chemical shift tensor, namely 𝛿11, 𝛿22 and 𝛿33 (where by
definition 𝛿11 ≤ 𝛿22 ≤ 𝛿33). Transformations between the laboratory frame and the
PAS frame can be made using the polar and azimuthal angles, 휃 and 𝜑.
Figure 4.5. Chemical structure of phospholipids used for construction of model
membrane bilayers in ssNMR experiments (where 𝑑54-DMPC = deuterated
dimyristoylphosphatidylcholine, DMPG = dimyristoylphosphatidylglycerol). The
CD-index is shown for each associated carbon atom on the fatty acid chain of 𝑑54-
DMPC.
Figure 4.6. Static 31P nuclear magnetic resonance spectra corresponding to a 7:3
molar ratio of d54-DMPC/DMPG, with three concentrations of cationic peptide G3
in molar ratios (lipid:peptide) of 10:1, 20:1 and 50:1. Lineshapes are indicative of
lamellae bilayers.
Figure 4.7. Magic angle spinning 31P nuclear magnetic resonance spectra
corresponding to a 7:3 molar ratio of 𝑑54-DMPC/DMPG, with three concentrations
of cationic peptide G3 in molar ratios (lipid:peptide) of 10:1, 20:1 and 50:1.
Figure 4.8. Example time course at increasing delay increments for magic angle
spinning 31P nuclear magnetic resonance spectra on 7:3 molar d54-DMPC/DMPG
using an inversion pul se sequence to extract the 𝑇1 relaxation time constant.
Figure 4.9. Exponential fits of the reconstitution of the longitudinal magnetization
and extraction of the 𝑇1 relaxation times (shown with errors in inset).
Figure 4.10. Example of the decaying spectra associated with the loss in transversal
magnetization in a MAS 31P NMR time course.
13
Figure 4.11. Exponential fits of the decay of the transverse magnetization and
extraction of the 𝑇2 relaxation times (shown with errors in inset).
Figure 4.12. Example of the 2H NMR spectra before and after the ‘dePaking’
procedure was carried out to reorient to the correct axis (see Equation 4.43)
Figure 4.13. CD order parameters extracted from the frequency separation associated
with dual peaks on the 2H NMR spectra for each carbon-deuterium on 𝑑54-DMPC.
14
List of Tables
Chapter 3
Table 3.1. Structured similarity index (SSIM) between normalised 2-dimension
histograms of neural net extrapolated 𝛼-value and fluorophore step size against
major radius fraction (indicative of location within the cell) and randomised
matching of the same variables. All values are approaching 1, suggesting a high
degree of similarity with the randomly assigned variables, pointing towards no
positional correlation. Conditions tested included Cy3B only (in S. aureus cells), G3-
Cy3B in E. coli cells and G3-Cy3B in S. aureus cells.
Chapter 4
Table 4.1. Chemical shift anisotropies extracted from single model fitting of 31P
ssNMR spectra for 𝑑54-DMPC/DMPG bilayers (7:3 molar ratio) following exposure
to various concentrations of cationic peptide G3.
Table 4.2. Fit values for magic angle spinning 31P nuclear magnetic resonance
spectra shown in Figure 4.7 corresponding to a 7:3 molar ratio of d54-
DMPC/DMPG, with three concentrations of cationic peptide G3 in molar ratios
(lipid:peptide) of 10:1, 20:1 and 50:1. Phospholipid peaks were fit using a sum of
two Gaussians (see Equation 4.40).
15
List of Publications
Hart J.W.; Waigh T.A.; Lu J.R.; Roberts I.S., Microrheology and spatial
heterogeneity of Staphylococcus aureus biofilms modulated by hydrodynamic shear
and biofilm-degrading enzymes, Langmuir 2019, 35(9), 3553-61.
16
Abstract
Antibiotic resistance is fast becoming a global health crisis, with the increase
in resistant bacteria outpacing the generation and translation of new antibiotics.
Bacteria at interfaces can produce a self-made architecture called a biofilm that acts
as a physical barrier, significantly reducing the efficacy of antibiotics. The aim of
this part of the project was to assess the material properties of developing biofilms
non-invasively. Passive microrheology was used investigate biofilms produced by
Staphylococcus aureus under various hydrodynamic shears and when exposed to
different anti-biofilm enzymes. Biofilms grown under any shear stress were harder
(i.e. had a lower creep compliance) than equivalent biofilms grown in stationary
media. Furthermore, statistical analysis of the spatial arrangement of bacteria during
biofilm growth revealed clustering as a function of height away from the interface
surface.
The cationic peptide G3 has been shown to be a potential antimicrobial peptide,
however the exact mechanism of action is unknown. Two methods were used to
probe the interaction of the peptide with the bacteria membrane. Firstly, a novel 3-
dimensional tracking method incorporating photoswitchable fluorophores and an
adaptive optics-based super-resolution imaging technique was used to investigate the
spatial distribution of G3 following exposure in Staphylococcus aureus and
Escherichia coli. No preferential localisation of G3 could be observed (i.e. G3 was
homogeneously distributed within the cell) for both bacteria. Diffusion kinetics were
approximated from short trajectories using a newly developed neural net software
package; however no spatial dependences were observed suggesting a weak binding
to the membrane. To further investigate the strength of the interaction between G3
and the cell membrane, solid state nuclear magnetic resonance was employed. Using
a model phospholipid system, increased disorder (i.e. spatial fluctuations) was
observed in bilayers exposed to G3, suggesting a transient interaction with the
membrane. From this and the tracking data, we hypothesise that G3 transiently
interacts broadly with the entire membrane to cause accumulative strain disruption.
No significant clustering or decreases in order parameters suggesting a strong
binding with the membrane were observed, however no positive control was
assessed in this study.
17
Declaration
No portion of the work referred to in the thesis has been submitted in support
of an application for another degree or qualification of this or any other university or
other institute of learning.
18
Copyright and ownership of intellectual property rights
1. The author of this thesis (including any appendices and/or schedules to this thesis)
owns certain copyright or related rights in it (the “Copyright”) and s/he has given
The University of Manchester certain rights to use such Copyright, including for
administrative purposes.
2. Copies of this thesis, either in full or in extracts and whether in hard or electronic
copy, may be made only in accordance with the Copyright, Designs and Patents Act
1988 (as amended) and regulations issued under it or, where appropriate, in
accordance Presentation of Theses Policy You are required to submit your thesis
electronically Page 11 of 25 with licensing agreements which the University has
from time to time. This page must form part of any such copies made.
3. The ownership of certain Copyright, patents, designs, trademarks and other
intellectual property (the “Intellectual Property”) and any reproductions of copyright
works in the thesis, for example graphs and tables (“Reproductions”), which may be
described in this thesis, may not be owned by the author and may be owned by third
parties. Such Intellectual Property and Reproductions cannot and must not be made
available for use without the prior written permission of the owner(s) of the relevant
Intellectual Property and/or Reproductions.
4. Further information on the conditions under which disclosure, publication and
commercialisation of this thesis, the Copyright and any Intellectual Property and/or
Reproductions described in it may take place is available in the University IP Policy
(see http://documents.manchester.ac.uk/DocuInfo.aspx?DocID=24420), in any
relevant Thesis restriction declarations deposited in the University Library, The
Library’s regulations (see http://www.library.manchester.ac.uk/about/regulations/)
and in The University’s policy on Presentation of Theses.
19
Acknowledgements
I would like to thank the Centre for Doctoral Training in Regenerative
Medicine for providing me with the opportunity to carry out this PhD. Funding for
this project was made possible by the Medical Research Council and the Engineering
and Physical Sciences Research Council.
I will forever be indebted to my supervisors, Dr. Thomas Waigh, Professor Jian Lu
and Professor Ian Roberts for their continual reassurance and guidance throughout
the PhD.
For their gracious hospitability and facilitation of the solid-state nuclear magnetic
resonance experiments during my time at the University of Melbourne, I would like
to thank Professor Frances Separovic, Dr. Marc Antoine Sani and all members of the
Bio21 Group. For the opportunity to perform a variety of experimental work within
their group, I would like to thank Professor Dick Strugnell, Leila Jebeli and all the
group members in the Peter Doherty Institute.
From the midnight chats about my lacklustre coding ability to the creation of the
successful, but ultimately short-lived, biotech start-up Neurodynamics, I would like
to thank my fellow PhD cohort students; Dr. Iwan Roberts, Dr. Cian Vyas,
Dr. Nathaniel Ng and Dr. Aljona Kolmogorova. Similar late-night discussions
(which often veered into the domain of recent Thai politics) were shared with Dr.
Sorasak Phanphak, one of several members of the Biological Physics department.
For their continual help, emotional support and insightful technical know-how
throughout my time at Manchester, I would like to thank Dr. Pantelis Georgiades,
Dr. Henry Cox and Dr. Ruiheng Li.
Finally, I would like to thank my friends and family. Their steadfast encouragement
and sacrifice through the peaks and troughs of the past four years are the only reason
I was able to finish my PhD. This work is dedicated to their unwavering belief and
support.
20
Chapter 1. Introduction
The following introductory chapter serves as a prerequisite overview of the
overarching subjects included in this thesis; namely bacteria structure, bacterial
biofilms, and antibacterial peptides as a means of counteracting the growing problem
of antibiotic resistance. At the end of the chapter, an outline will be presented
describing the content of the thesis in context.
1.1 Bacteria structure
1.1.1. Bacterial membranes
Bacterial membranes are semi-permeable barriers that separate the interior
cytoplasm from the extracellular environment. By volume, the primary constituent of
a bacterial membrane is a bilayer formed from amphipathic phospholipids that self-
assemble in water. Phospholipids encompass a broad range of lipids composed of a
hydrophilic ‘head’ containing a polar phosphoric acid derivative and a pair of
hydrophobic long-chain fatty acids connected by a glycerol. When in water, entropic
forces drive the formation of bilayers, such that the hydrophobic ‘tails’ pack together
and are isolated from the surrounding water by the hydrophilic head groups.
The secondary constituent of the cell membrane by volume is a vast collection of
proteins. Permanently attached integral membrane proteins span the entirety of the
lipid bilayer, providing a huge range of functions. Two functions, for example,
include: the transportation of molecules into and out of the bacteria (sometimes via
large multi-protein complexes such as pores or channels) and energy transduction.
Molecular receptors, adhesion complexes and enzymes are also found in the
membrane structure. Cholesterol and carbohydrates make up a minor percentage of
the membrane volume 1.
The ‘fluid-mosaic’ model describes the structure of the bilayer and embedded
protein complexes as a two-dimensional fluid 2. Due to geometrical restrictions,
lateral and axial rotational diffusion of phospholipids within a bilayer are relatively
high, having reported lateral diffusion coefficients of ~3x10-9 cm2s-1 (based on
average bacteria size, one can infer that phospholipids can diffuse the entire length of
the cell within ~1 second) 3. Diffusion of transmembrane proteins is much slower.
Whilst free diffusion still occurs, movement is constrained due to transient
21
confinement by obstacle clusters (i.e. cytoskeletal meshwork) or restricted
directional motion caused by attachment to the internal cytoskeleton 4.
1.1.2. Gram-positive and Gram-negative cell envelopes
Staphylococcus aureus are a prevalent gram-positive, non-motile, ellipsoidal
bacteria with an average major radius of ~0.5 µm 5. Gram-positive bacterial species
(so named in reference to the ability to retain the crystal violet stain ) are
characterised by a unique cell wall arrangement containing only one phospholipid
bilayer membrane. Immediately beyond the cell membrane of S. aureus is an ~16 nm
thick gel-like matrix called the periplasm 6. The periplasmic space has a variety of
purposes, ranging from nutrient binding to enzymatic degradation of toxic
substances that have penetrated the outer layers 7. Encompassing this is a rigid 20-80
nm thick outer layer called the cell wall, composed of cross-linked polymers called
peptidoglycans, that contains sugars and amino acids 8. The stability of the
peptidoglycan layer is a result of the crystal lattice structure formed between
adjacent chains of 𝛽-(1,4) glycosidic bonded amino sugars - N-acetylglucosamine
(NAG) and N-acetylmuramic acid (NAM). In S. aureus, each NAM is attached to a
short tetra-peptide chain composed of amino acids (specically L-alanine, D-
glutamine, L-lysine and D-alanine) with a 5-glycine cross-link to form the crystal
structure 9.
Distributed throughout the peptidoglycan layer are large anionic polymers called
teichoic acids. Wall teichoic acids (WTAs) are covalently bonded to either the NAM
or D-alanine peptide in the peptidoglycan structure, whereas lipoteichoic acids
(LTAs) are bonded directly to the lipid head groups of the cell membrane
phospholipid bilayer 1. In total, WTAs account for up to 60% of the dry mass of the
S. aureus cell wall 10. Through immuno-gold nanoparticle labelling, it has been
found that up to 50% of the total number of WTAs extend outside the peptidoglycan
lattice as a protruding `hairy' layer 11-12.
In S. aureus, WTAs are formed of long chain repeats of ribitol phosphate groups
interspersed with ester-bound NAG and D-alanine branches. As the phosphate ion
contains an exposed electron double pair on the unbound oxygen atom it is
extremely negatively charged. To some extent, the overall charge of the WTA is
mediated by the positive ion on the attached D-alanine branch (formed during the
esterification process) 13. Also protruding from the peptidoglycan layer is a range of
22
ionically and covalently attached proteins, a subcategory of which are responsible for
biotic attachment (collectively referred to as `microbial surface components
recognizing adhesive matrix molecules' or MSCRAMMs) 1. Figure 1.1 depicts two
scanning electron microscope images of ATCC 25923 S.aureus at different scales,
highlighting their resemblance to smooth ellipsoids 14.
Figure 1.1. Scanning electron microscope images of ATCC 25923 Staphylococcus
aureus (Scale bars: a) = 5 µm and b) = 1 µm) 14.
Gram-negative bacteria species, such as Escherichia coli, have two concentric
bilayer membranes (inner and outer) separated by a significantly thinner
peptidoglycan layer that is only 2-7 nm thick 15. The terms gram-positive and gram-
negative refer to the ability to retain the stain crystal violet, which is determined by
the thickness of the peptidoglycan layer (i.e. gram-positive have a thick layer and so
retain the stain) 16. The composition and lattice-like structure of the peptidoglycan
layer in both gram-positive and gram-negative bacteria form protective physical
barriers, but are permeable to particles on the order of ~2 nm 17. A unique
component of the gram-negative outer membrane is the existence of large, negatively
charged complexes called lipopolysaccharides (LPS), which protrude out of the
outermost leaflet of the bilayer. LPS can be separated into three distinct components.
In order to entrench the complex into the outer membrane, the first component of
LPS is Lipid A; consisting of two phosphorylated, β(1→6) linked glucosamine units
surrounded by acyl chains. The second component, an oligosaccharide core, acts as
an intermediate connection between Lipid A and the final component, a repeating
glycan polymer called the O-antigen 18. The variation in LPS composition varies
dramatically not only between different gram-negative species, but also between
a) b)
23
strains of the same species 19. A schematic diagram of a typical gram-positive and
gram-negative cell envelope is shown in Figure 1.2, which also depicts the tetra-
peptide bonding of peptidoglycan molecules.
(Caption for figure overleaf)
Figure 1.2. Schematic diagram representing generic gram-positive (left) and gram-
negative (right) cell envelopes (not to scale). Middle inset shows the cross-linking
mechanism forming the crystal structure of the peptidoglycan layer. Abbreviations
used: IM = inner membrane; PS = periplasmic space; PL = peptidoglycan layer; HL
= hairy layer; WTA = wall teichoic acid; LTA = lipoteichoic acid; AP = attached
protein (by ionic or covalent bonding); CI = cytoplasmic interior; IMP = integral
membrane protein; OM = outer membrane; OMP = outer membrane protein; LPS =
lipopolysaccharides; NAG = N-acetylglucosamine; NAM = N-acetylmuramic acid.
24
25
1.2. Bacterial biofilms
1.2.1. Definition of a ‘biofilm’
Following attachment to a multi-phase interface, such as the glass coverslip
in a flow chamber, the outer bow of a container ship or a catheter surface in vivo,
many bacteria species produce a complex extracellular matrix of polymeric material
collectively called a biofilm 20. A number of benefits for the proliferating bacterial
community are provided by the biofilm, ranging from facilitating communication via
quorum-sensing 21 to acting as physical barriers against phagocytic cells and
antibiotics, leading to increases in antibiotic resistance by factors of 100 22.
Additional physiological roles for biofilms include nutrient reservoirs 23, water-
resisting protein ‘raincoats’ 24 and to expedite horizontal gene transfer between
cells 25. It is expected that this is just a small fraction of the roles performed by
biofilms, since it is one of the earliest biological structures formed by evolution with
fossilised biofilms dating back to 3.3-3.5 billion years ago 26.
1.2.2. Biofilm formation and composition
The process of biofilm development can be described in three steps;
attachment to an interface, maturation and dispersal. Following the initial adhesion
process (the mechanisms of which are determined by the interface properties); the
biofilm enters a maturation stage primarily characterized by the rapid production and
extrusion of extracellular polysaccharides, nucleic acids and proteinaceous
adhesins 27. For S. aureus biofilms, partially de-acetylated polymer residues of poly-
β-1-6-linked N-acetylglucosamine (PNAG) act as the major component of the
extracellular ‘slime’, shown under certain nutrient-rich conditions to make up 80-
85% of the biofilm dry weight 28.
PNAG (or synonymously termed polysaccharide intercellular adhesion, PIA)
synthesis originates from the ica operon (where an operon is a series of adjacent
genes in bacterial DNA that are controlled by a single promoter), which consists of
four genes: icaA, icaD, icaB and icaC. The icaA gene encodes for transcription and
subsequent translation of a transmembrane enzyme with N-acetylglucosaminyl
transferase activity, which can transform uridine diphosphate N-acetylglucosamine
(a nucleotide sugar synthesised elsewhere in the bacteria from fructose) into PIA 29.
Coexpression of icaD drastically increases the quantity of transferase enzyme by
catalysing the formation of shorter oligomers consisting of only approximately 20
26
monomers. Long-chained (120 monomer) PIA is subsequently formed by expression
of the icaC gene and extruded by the bacteria 30. During the de-acetylation process
by surface-attached protein IcaB (encoded from the icaB gene), the polymer
becomes positively charged (via acetyl group cleavage, forming a free-radical with
an unbound electron on the central carbon atom) leading to a strong electrostatic
interaction between the slime and bacterial surface (mediated by teichoic acids in the
cell wall) 27. Kropec et al. found that S. aureus mutants lacking the ability to produce
PIA (via the deletion of the ica locus) had significantly reduced virulence in
inoculated mice models, indicating biofilms lacking PIA were more susceptible to
innate host defences 31.
Different nutrient compositions, environmental conditions and bacteria strains can
produce biofilms where PIA is not the main component. A guinea pig model of
biofilm-related infection following inoculation with 109 isogenic ica-negative mutant
bacteria resulted in no decreased virulence compared to wild-type bacteria 32. O’Neil
et al. discovered that biofilm regulation was associated to methicillin susceptibility.
As such, methicillin-resistant S. aureus (MRSA) were capable of forming biofilms
through PIA-independent mechanisms, whereas methicillin-sensitive S. aureus ica
operon-deletion mutants could not grow stable biofilms 33. Proteinaceous cell-to-cell
adhesion by protein A (SpA), fibronectin-binding proteins (FnBPs), autolysins,
accumulation-associated proteins (Aaps) and biofilm-related proteins (Baps) have
been shown to substitute PIA in biofilm production 34.
A striking feature of mature S. aureus biofilms is the mushroom-like structure they
appear to develop prior to dispersal. A suggested purpose for this structure is to
allow passage of water and nutrients to all parts of the biofilm, rather than just the
exposed outer layers. Mechanistically, Mack et al. theorized that this formation may
be caused by differential expression of PIA or the expression of phenol-soluble
modulins (PSMs) that can lead to detachment of cell clusters 35. Periasamy et al.
corroborated this idea by identifying PSM expression as a prevalent mediator of
biofilm maturation, driven by a variation of quorum-sensing activity 36.
1.2.3. Biofilms as a prevalent clinical issue
Findings from the National Institute of Health concluded that ~80% of
chronic infections in the United States are caused by bacteria in the biofilm state, as
opposed to planktonic (i.e. free-floating) bacteria in the blood 37. Restricting the
27
scope of incidences to only MRSA, a 2011 study found that 19,000 people die each
year due to infection-related complications. Generally, 20% of patients will contract
a bacterial infection following surgery whilst still in hospital 38. As such, surgical
outcomes are severely hampered as a result of biofilm formation. To elaborate with
an example, following endoscopic sinus surgery to alleviate chronic rhinosinusitis,
54% of patients had persistent postoperative symptoms, on-going mucosal
inflammation and infection due to S. aureus biofilm formation for up to 12 months
post-surgery 28. Collectively, it is estimated that bacterial biofilms cost the United
States healthcare system approximately $10 billion each year 38.
Furthermore, a range of medical devices and implants have been shown to
deteriorate as a direct result of biofilm exposure. Examination using SEM revealed
60% of intravenous and intra-arterial catheters were contaminated with bacteria in
the form of an adherent biofilm 39, corroborating an observational study by
Bregenzer et al. that showed a linear correlation between length of catheterization
and risk of infection for up to 10 days 40. A similar investigation using SEM by the
same authors revealed heavy bacterial colonization of the metal tip inner surface and
internal wires of an endocardial pacemaker lead, despite patients being on heavy
antibiotic therapies of cloxacillin and rifampicin 41. A retrospective review by Park et
al. found that patients fitted with ventricular catheters had a significantly increased
risk of infection over 4 days of use, with 8.6% contracting life-threatening
ventriculitis (an inflammation of the ventricles in the brain) during an external
ventricular drain 42. The identification of S. aureus resident on an external ventricular
drain by polymerase chain reaction (PCR) analysis lead to the conclusion that
biofilms pose a significant risk of recurrent ventriculoperitoneal shunt failure 43.
Anwar et al. proposed a model to explain the antibiotic resistance displayed by
biofilms by studying the bacteria population dynamics within Pseudomonas
aeruginosa biofilms grown in a chemostat. They hypothesized that the outermost
layer of biofilms are populated with larger bacteria that have highly permeable
membranes. Antibiotics enter these sacrificial cells in much larger doses than
necessary to cause cell death, reducing the number of antibiotic molecules
penetrating to the inner layers of the biofilm. The polysaccharide matrix of the
biofilm can also immobilize antibiotic molecules 44.
28
1.2.4. Biofilms as a prevalent societal issue
Microbial biofilm formation also negatively impacts society in a number of
areas outside the field of medicine. Marine biofilms will readily form on ships,
increasing hydrodynamic drag up to 10% by introducing a turbulent velocity profile
caused by the biofilm's innate heterogeneity 45. The American Society of Naval
Architects and Marine Engineers extrapolated that an extra $75-$100 million was
spent yearly by the United States naval fleet on fuel to overcome the effect of marine
biofilm-induced drag 46. Bacterial biofilms have also been shown to inhabit brass
coupons exposed to a water-based cooling system in an oil refinery. Whilst no
corrosive damage was found, biofilm blockage requires continuous removal
(therefore producing an unwanted but necessary cost) 47. Biofilm formation has also
shown to introduce blockage of reverse osmosis and ultrafiltration membranes,
critically reducing their efficiency and working lifespan 48.
1.2.5. Beneficial biofilms
It would be amiss to state that all biofilms are inherently bad when several
beneficial applications are utilized on an industrial scale. One example is the use of
biofilms to treat wastewater. Microorganisms inside a biofilm attached to a reactor
system (such as submerged granular filters or turbulent mobile beds) can consume
harmful organic matter from contaminated water. For instance, ammonium can be
removed by bacteria-mediated oxidation 49. Extending this idea, bioremediation is a
process whereby biofilms can be engineered to clean up hazardous wastes such as oil
and heavy metals by means of metabolism and oxidation. Horizontal rotating tubular
bioreactors containing biofilms grown out of enriched cultures from heavy-metal
laden soils showed metal removal from contaminated water at measured rates
between 2.5 – 8.3 mg L-1 h-1 50. Similarly, bioleaching is a process whereby biofilms
can be used to aid in the extraction of metals from their respective ores by oxidation
and regeneration of chemical oxidants. A 2017 review by Watling found that
bioleaching accounts for the production of over 1 million tonnes of copper per
year 51.
29
1.3. Antimicrobial peptides
1.3.1. Antibiotic resistance
It would not be a hyperbole to say that antibiotic resistance has the potential
to become a significant global health crisis, if it is not already. A 2014 World Health
Organisation report predicted sobering increases in infection-related deaths due to
the inability to treat antibiotic-resistant bacteria. Up to 25,000 more deaths per year
in the European Union; 23,000 per year in the United States of America and 38,000
per year in Thailand have been predicted 52. Whilst the evolutionary pressure to
become resistant can be mediated by reducing antibiotic overuse, the pipeline of new
antibiotics is currently lagging behind the demand for new treatments 53. One
potential avenue is the use of antimicrobial peptides that damage bacterial
membranes directly through physical mechanisms.
1.3.2. The antimicrobial peptide, G3
A promising antimicrobial peptide that has clinical translation potential was
developed by Hu et al. in 2014 54. Abbreviated to ‘G3’, the short peptide amino acid
composition is given in full as G(IIKK)3I-NH2 (where G = glycine, I = isoleucine,
K = lysine and NH2 is an amino group). In aqueous environments, the peptide is
uncoiled due to electrostatic repulsion between adjacent lysine residues and
hydrogen-bonding between the peptide backbone and water molecules. However, in
negatively charged environments the peptide adopts an amphipathic 𝛼-helical
structure, consisting of a hydrophobic face of non-polar isoleucine groups and a
hydrophilic face of positively charged lysine residues. Figures 1.3a-c depict the G3
helical structure in three ways. Figure 1.3a is a Schiffer−Edmundson wheel
projection, where the angular rotation between adjacent amino acids is fixed at 100°,
emphasising the peptide amphipathicity. Figures 1.3b and 1.3c are three-dimensional
renderings of the atomic positions in G3, generated using the open-source software
Avogadro 55. Figure 1.3b shows all atoms in the peptide colour-coded by element
from a side-on view and Figure 1.3c exaggerates bonds within colour-coded amino
acids looking down the helix (hydrogen has been omitted from Figure 1.3c).
30
Figure 1.3. Schematic drawings of G(IIKK)3-NH2 (G3) in the 𝛼-helical conformation
adopted near bacterial membranes. a) Schiffer−Edmundson wheel projection, where
the angular rotation between adjacent amino acids is fixed at 100° (K = lysine,
I = isoleucine, G = glycine). b) Three-dimensional render of atomic positions colour-
coded by element (white = hydrogen, grey = carbon, blue = nitrogen, red = oxygen).
Cartoon ‘ribbon’ representation also shown in red. Scale bar is 7.5 Å. c) Three-
dimensional render emphasising bonds and omitting hydrogen. Amino acids are
colour-coded (blue = lysine, green = isoleucine, white = glycine). Scale bar is 3 Å.
The efficacy of G3 as a specific antimicrobial peptide was demonstrated by the
original designers and subsequently confirmed in another later study 56. Both groups
found the minimum inhibitory concentration of G3 required to inhibit proliferation
of gram-positive bacteria (Bacillus subtilis) was 2 ± 0.5 µM and gram-negative
a)
b)
c)
31
bacteria (E. coli) was 8 ± 0.2 µM, which is comparable to the antimicrobial activity
of prescription ampicillin 57. Furthermore, the cytotoxic activity of G3 was minimal
even at relatively high concentrations. For example, hemolysis of human erythrocyte
(red blood) cells was only 15% following exposure to 100 µM G3 and fluorescence
imaging revealed no uptake of G3 by model host mammalian cells (NIH 3T3) after 1
hour incubation 54. It has been hypothesized that the mechanism of action of G3 is
primarily membrane permeabilization and subsequent internal binding and
aggregation to intercellular components such as DNA. The current evidence
supporting this hypothesis is direct SEM imaging, which revealed damaged Bacillus
subtilis envelopes following exposure to G3.
32
1.3.3. Physical mechanisms of peptide-induced cell membrane disruption
Antimicrobial peptides can have significantly different chemical
compositions, but the mechanisms involved in cell membrane approach, binding,
disruption and the subsequent cell death are all underpinned by similar physical
models. Initial interactions between peptides and bacteria are dominated by
electrostatics. Positive cationic peptides containing lysine residues are attracted to
the net negative bacteria cell envelope (and membrane) of both gram-positive and
gram-negative bacteria. Experimental proof of this has been found multiple times; as
an example Peschel et al. found that by modifying the transcription of wall teichoic
acids in S. aureus such that D-alanine was omitted, resulting in increased surface
charge on the cell envelope, enhanced cationic peptide efficacy (shown by a tenfold
reduction in the minimum inhibitory concentration of peptide compared to wild type
bacteria) 58. It is common for peptides that are interacting with the membrane to
undergo a conformational phase change from an unravelled state to an ordered state,
most commonly an 𝛼-helix (although 𝛽-sheets are also possible). A proposed reason
for this state change is that the packing of phospholipids within the membrane has an
innate periodicity, such that interactions between polar head groups and charged
residues on the peptide promote a helical conformation 59.
There are three main models describing membrane permeabilization by 𝛼-helical
peptides: the barrel-stave model; the carpet model and the toroidal model. Figure
1.4a-c shows all three models diagrammatically. For all cases, membrane
permeabilization is ultimately a response to mitigate the strain induced on the curved
bilayer membrane due to the lateral expansion (i.e. doming) associated with
adsorption of the 𝛼-helical peptide. In the barrel-stave model, the hydrophobic
interaction is the dominating force, to the extent that peptides are inserted
perpendicular to the membrane surface normal. The insertion of multiple peptides
can result in a pore where the hydrophobic faces of the peptides points towards the
phospholipid acyl chains and the hydrophilic faces point inwards (i.e. point towards
the pore centre). Alternatively, in the carpet model, if the electrostatic interaction
between the cationic peptide residues and the phospholipid polar head groups
dominate, the peptide 𝛼-helices are held parallel to the membrane surface. In the
limiting case of a few peptides, the strain on the membrane is not disruptive.
However above a critical threshold concentration, enough tension accumulates
33
between the two leaflets of the bilayer to induce the formation of transient holes in
the membrane, which eventually results in complete disintegration. Alternatively, in
the toroidal mechanism, the membrane relieves the curvature-induced strain by
bending inwards. Again, if a critical concentration is reached, the bending continues
until a pore that is lined with peptides forms 60.
Figure 1.4. Models of amphipathic peptide-induced membrane permeabilization. a)
Barrel-stave model – peptides are inserted perpendicular to the membrane surface
such that the hydrophobic face of the peptide faces towards the fatty acid chains and
the hydrophilic faces point inwards. b) Carpet model – transient peptide association
parallel to the membrane surface leads to an accumulation of minor defects that
ultimately disintegrate the membrane. c) Toroidal model – curvature induced by
peptide interaction leads to bending of the phospholipid leaflets until a critical point
where they form a pore.
c)
a) Phospholipid bilayer
(cell membrane)
b) Amphipathic peptide
Hydrophobic
face
Hydrophilic face
34
1.4. Current state of biofilm research
The importance of understanding biofilms and their role in promoting
antibiotic resistance has been developing over the past several decades. Whilst the
material properties of biofilms at a macroscopic level have been investigated since
the early 1970’s using parallel plate rheometers, studies working at the microscopic
level are still ongoing. With the advancement of processing power, computer vision-
based techniques are at the forefront of biofilm studies. Advanced, long time course
3-dimensional imaging techniques from the Drescher laboratory in the Max Planck
Institute for Terrestrial Microbiology & Philipps-Universität Marburg, Germany are
at the forefront of biofilm biophysics. A confocal microscopy technique matched
with identification through machine learning has revealed the structural dynamics of
bacterial biofilms in terms of nematic structure parameters 61. Using this technique,
the group have also created a novel way of tracking and categorizing collective
motions of bacteria within biofilms, as well as the degradation of the biofilm matrix
in response to antibiotics 62. In relation to the work presented here, the Drescher
group also use novel imaging techniques and cell-based simulations to assess the
impact flow has on individual bacteria orientation and dynamics when subject to
high shear stresses 63.
1.5. Thesis outline
The thesis can be broadly split into two separate investigations. As an
understanding of the mechanical properties of bacterial biofilms can lead to effective
removal and prevention strategies, Chapter 2 characterises the material properties
and bacteria spatial distributions within S. aureus biofilms using passive
microrheology. To expand on previous work, the biofilm system was subject to
various perturbations, including increased hydrodynamic shear (through various flow
rates) and biofilm-degrading enzymes (namely proteinase-K and DNase-1).
Chapters 3 and 4 investigate the antibacterial mechanisms of the cationic peptide G3.
To elucidate on the mechanism of peptide action on bacteria, in Chapter 3 the spatial
distributions and dynamics of G3 in S.aureus and E. coli was explored using
stochastic optical reconstruction microscopy (STORM) and fluorescence tracking. In
Chapter 4, the interaction between different molar concentrations of G3 and bacterial
membranes (modelled using constructed phospholipid bilayers) was probed using
35
static and magic angle spinning (MAS) 31P and 2H solid-state nuclear magnetic
resonance (ssNMR).
Each chapter includes all relevant theoretical background and experimental
methodology, as well as a brief introduction reviewing relevant details of related
work. Chapter 5 concludes the thesis by summarising the main results and outlines a
timeline of future possible work.
36
Chapter 2. Microrheology and spatial heterogeneity of
Staphylococcus aureus biofilms modulated by
hydrodynamic shear and biofilm-degrading enzymes
2.1. Introduction and Chapter Aims
2.1.1. Biofilm material properties and response to hydrodynamic shear
In 1999, a study undertaken by Stoodley et al. probed the response of a flow-
cell bound biofilm to changes in the hydrodynamic forces by altering the media flow
speed. Biofilm morphology was shown to change dramatically between laminar and
turbulent flow regimes. Under laminar flow, biofilms developed into elongated
‘streamers’ (defined as having a length-to-width ratio of 8) after 22 days of
maturation. Under turbulent flow conditions after the same amount of time, biofilms
formed into circular clusters (with length-to-width ratios less than 2). Furthermore,
the authors also found that laminar flow conditions produced higher biofilm surface
coverages, saturating at > 80% the area of the chemostat, whereas turbulent-regime
biofilms saturated at only 60% 64.
With the intent of investigating the effect of flow conditions on bacterial adhesion to
abiotic surfaces, Foka et al. measured the changes in ica operon expression in a
clinical isolate of Staphylococcus epidermis when exposed to different flow
velocities. By performing real-time PCR on bacteria exposed to 0.92 mL/min and
36.75 mL/min, the group was able to directly analyse the expression of icaA and
icaD (which as stated previously corresponds to PIA production). It was found that
the level of expression increased by up to a factor of 9 in bacteria exposed to higher
flow rates, indicating a distinct increase in PIA production 65.
Passive microrheology, the technique of tracking tracers (usually inert microspheres)
embedded within a medium was used by Rogers et al. to investigate the viscoelastic
properties of biofilms produced by Staphylococcus aureus grown in a single-speed,
parallel plate flow chamber. Analysis of the mean square displacements of the
bacteria themselves revealed minor initial variations in the biofilm creep compliance
as a function of height that eventually homogenised to a lower value. This would
indicate that biofilms gradually harden during maturation, but results were
inconsistent and not repeated 66.
37
2.1.2. Aims of Chapter
As an understanding of the viscoelastic properties of bacterial biofilms can
lead to more effective removal and prevention strategies, the aims of this Chapter are
to characterise the material properties of S. aureus biofilms during the initial 6 hours
of growth at increasing heights above the attachment surface. To have minimal
disruption to the biofilm as it grows, passive microrheology shall be used in the same
manner as Rogers et al. 66. As inserting inert spheres into the biofilm could introduce
new surfaces for the bacteria to attach to and grow on, the bacteria themselves will
be used as tracers. To perturb the biofilms, different hydrodynamic shears will be
induced by increasing the flow rate of the growth media passing through the
chemostat. Broad spectrum biofilm-degrading enzymes will also be introduced to see
the relative contribution of proteins and extracellular DNA to the overall viscoelastic
properties of the biofilm.
Furthermore, using Ripley’s K-function, a statistical analysis of the spatial
distributions of bacteria within the biofilm as it grows will be investigated. The
spatial statistics of the biofilm will then be correlated to the viscoelastic properties.
Different hydrodynamic shears will also be employed in this part of the study, to see
if there is any rearrangement of bacteria within the biofilm as a response to flow rate.
38
2.2. Theory
The following sections outline the theoretical derivations for the analysis
methods used in this Chapter. Firstly, two parallel methods will be used to attain
comparable equations for the mean square displacement of a diffusing particle. One
approach will follow Einstein’s statistical mechanics-based methodology, exploiting
the moments of the particle position distribution. The other approach will be more
grounded in a real system, equating the forces acting on a spherical probe in a fluid
and using the velocity correlation to find the mean square displacement. Comparing
the results of the two lines of thought will result in an analytic expression for the
diffusion coefficient, which underpins the theory of Brownian motion. Following
this, the concepts introduced will be extended to incorporate viscoelasticity. Finally,
in Section 2.2.5 a link between the creep compliance of a viscoelastic medium and
the measurable displacements of a probe immersed in said medium (forming the
experimental basis of passive microrheology).
Secondly, in Section 2.2.7 an introduction will be made to Ripley’s K-function; a
statistical tool that can be used to investigate and quantify the spatial clustering of a
distribution of particles.
2.2.1. Einstein’s derivation of the diffusion equation
Consider an ensemble of independent particles in 1-dimension situated at 𝑥0
at time, 𝑡 = 0. For each increment of time, 𝑑𝑡, that passes, each particle changes
position by a displacement, ∆. The quantity of ∆ is random, but associated with the
probability density function, 𝜑(∆) where by definition,
∫ 𝜑(∆)𝑑∆∞
−∞
= 1 .
Following an arbitrary time interval, 𝑡 + 𝜏, the number of particles between positions
𝑥 and 𝑥 + 𝑑𝑥 can be found using a density function, 𝜌(𝑥, 𝑡), such that,
𝜌(𝑥, 𝑡 + 𝜏)𝑑𝑥 = ∫ 𝜌(𝑥 + ∆, 𝑡)𝜑(∆)𝑑∆∞
−∞
𝑑𝑥
Eq. 2.1
Eq. 2.2
39
which follows the intuition that the density of particles after the time interval is
determined by the probability of a particle having a certain displacement. As 𝜏 and ∆
are infinitesimally small, we can use a Taylor series to expand the probability density
functions, namely
𝜌(𝑥, 𝑡 + 𝜏) = 𝜌(𝑥, 𝑡) + 𝜏𝛿𝜌(𝑥, 𝑡)
𝛿𝑡
and
𝜌(𝑥 + ∆, 𝑡) = 𝜌(𝑥, 𝑡) + ∆𝛿𝜌(𝑥, 𝑡)
𝛿𝑥+ (
∆2
2!)𝛿2𝜌(𝑥, 𝑡)
𝛿𝑥2
where only non-vanishing terms have been included. Subbing these into Equation
2.2 results in,
𝜌(𝑥, 𝑡) + 𝜏𝛿𝜌(𝑥, 𝑡)
𝛿𝑡
= 𝜌(𝑥, 𝑡)∫ 𝜑(∆)𝑑∆∞
−∞
+𝛿𝜌(𝑥, 𝑡)
𝛿𝑥∫ ∆𝜑(∆)𝑑∆∞
−∞
+𝛿2𝜌(𝑥, 𝑡)
𝛿𝑥2∫ (
∆2
2)𝜑(∆)𝑑∆
∞
−∞
.
From the definition of the probability density function (see Equation 2.1), the first
terms on both sides of the equation above cancel out. Furthermore, the second term
on the right-hand side of Equation 2.5 can also be cancelled if we assume no
directional bias in the displacement function, i.e. 𝜑(∆) = 𝜑(−∆).
Compiling the remaining terms results in the familiar diffusion equation,
𝛿𝜌(𝑥, 𝑡)
𝛿𝑡= 𝐷
𝛿2𝜌(𝑥, 𝑡)
𝛿𝑥2 where 𝐷 =
1
𝜏∫ (
∆2
2)𝜑(∆)𝑑∆
∞
−∞
where 𝐷 is called the diffusion coefficient 67.
Eq. 2.3
Eq. 2.4
Eq. 2.5
Eq. 2.6
40
If it is assumed that the initial form of the density function can be described by a
Gaussian pulse, then a solution to Equation 2.6 is,
𝜌(𝑥, 𝑡) =1
√4𝜋𝐷𝑡exp (
−(𝑥 − 𝑥0)2
4𝐷𝑡) .
2.2.2. Mean squared displacement from the solution to the diffusion equation
The mean square displacement, 𝑀𝑆𝐷(𝑡) is the averaged displacement of a
particle from an arbitrary initial position over a certain time interval, 𝑡, or
𝑀𝑆𝐷(𝑡) = ⟨(𝑥(𝑡) − 𝑥0)2⟩ = ⟨𝑥2⟩ + 𝑥0
2 − 2𝑥0⟨𝑥⟩
where ⟨𝑥⟩ is the mean position and ⟨𝑥2⟩ is the variance of positions (where temporal
notation has been dropped for clarity). In statistical mechanics, the mean and
variance are also the first and second moments of a function admitted by the
corresponding probability density function. For the position, 𝑥, it is clear that the
probability density function is of the same form as Equation 2.7, which from this
point on will be represented by the notation 𝑃(𝑥, 𝑡). To find the moments, we first
introduce the characteristic function, 𝐶(𝑘),
𝐶(𝑘) = ∫ exp(𝑖𝑘𝑥)𝑃(𝑥, 𝑡)𝑑𝑥∞
−∞
which can be interpreted as the Fourier transform of the probability density function.
Considering the series expansion of the exponential term, this becomes
𝐶(𝑘) = ∫ 𝑃(𝑥, 𝑡)𝑑𝑥∞
−∞
+ 𝑖𝑘∫ 𝑥. 𝑃(𝑥, 𝑡)𝑑𝑥∞
−∞
+(𝑖𝑘)2
2!∫ 𝑥2∞
−∞
. 𝑃(𝑥, 𝑡)𝑑𝑥 +⋯
which simplifies to become,
𝐶(𝑘) = 1 + 𝑖𝑘𝜇1 +(𝑖𝑘)2
2𝜇2 +⋯
Eq. 2.7
Eq. 2.8
Eq. 2.9
Eq. 2.10
Eq. 2.11
41
or even more succinctly,
𝐶(𝑘) = ∑(𝑖𝑘)𝑛
𝑛!. 𝜇𝑛
∞
𝑛=0
where we have defined 𝜇𝑛 as the 𝑛th moment (this can be seen by looking at the
integrals in Equation 2.10 and the standard definitions of moments 68). Therefore,
subbing Equation 2.7 into Equation 2.9,
𝐶(𝑘) =1
√4𝜋𝐷𝑡∫ exp(𝑖𝑘𝑥) exp (
−(𝑥 − 𝑥0)2
4𝐷𝑡)
∞
−∞
which can be expressed (by completing the square and using properties of
exponentials) as,
𝐶(𝑘) =1
√4𝜋𝐷𝑡∫ exp [−(
1
4𝐷𝑡)𝑥2 + (
1
4𝐷𝑡) (2𝑥0 + 4𝑖𝐷𝑡𝑘)𝑥 + (
−𝑥02
4𝐷𝑡)] 𝑑𝑥
∞
−∞
.
Knowing the standard solution for the area under a Gaussian,
∫ exp(−𝑎𝑥2 + 𝑏𝑥 + 𝑐) 𝑑𝑥∞
−∞
= √𝜋
𝑎exp (
𝑏2
4𝑎+ 𝑐)
it can be shown that Equation 2.14 has solution,
𝐶(𝑘) = exp(𝑖𝑘𝑥0 − 𝑘2𝐷𝑡) .
At this point we introduce the cumulant generating function by taking the natural
logarithm of Equation 2.12, such that,
ln (𝐶(𝑘)) = ∑(𝑖𝑘)𝑛
𝑛!𝜅𝑛
∞
𝑛=0
Eq. 2.12
Eq. 2.13
Eq. 2.14
Eq. 2.15
Eq. 2.16
Eq. 2.17
42
where we define the first and second cumulants, 𝜅1,2 by their relation to the
distributions moments as,
𝜅1 = 𝜇1 and 𝜅2 = 𝜇2 − 𝜇12 .
This allows for the extrapolation of cumulants from Equation 2.16 simply by taking
the natural logarithm and comparing the powers of 𝑖𝑘, leading to,
𝜅1 = 𝑥0 → 𝜇1 = 𝑥0 and 𝜅2 = 2𝐷𝑡 → 𝜇2 = 2𝐷𝑡 + 𝑥02 .
Inserting the moments back into the definition of the 𝑀𝑆𝐷(𝑡) (Equation 2.8) gives
the final result,
𝑀𝑆𝐷(𝑡) = 2𝐷𝑡 .
If instead the particle was travelling in 3-dimensions, all of which were statistically
independent of each other (i.e. any displacement in 𝑥 has no inference on motion in
𝑦 or 𝑧), then we accommodate for the increase in the number of degrees of freedom,
𝑛 by including the term,
𝑀𝑆𝐷(𝑡) = ⟨(𝑟(𝑡) − 𝑟0)2⟩ = ⟨∆𝑟2(𝑡)⟩ = 2𝑛𝐷𝑡
where 𝑟(𝑡)2 = 𝑥(𝑡)2 + 𝑦(𝑡)2 + 𝑧(𝑡)2 and 𝑟0 is the particle’s initial position 69.
2.2.3. Langevin equation and Brownian motion
An alternative approach to the problem of diffusion is to instead consider the
forces acting on a moving spherical probe immersed in a fluid. Intuition suggests
that the dominant force acting on the probe is a frictional drag proportional to (but
acting in the opposite direction of) the probes velocity, 𝑣(𝑡). However, a secondary
net force must also be in action as the probe’s velocity does not tend to zero over
long time periods. To accommodate for this generally, we can say that the probe
experiences a random force, 𝛿𝐹(𝑡), the time average of which, ⟨𝛿𝐹(𝑡)⟩ = 0.
Eq. 2.18
Eq. 2.19
Eq. 2.20
Eq. 2.21
43
The general Langevin equation then can be written as,
𝑚𝑑𝑣(𝑡)
𝑑𝑡= −𝜉𝑣(𝑡) + 𝛿𝐹(𝑡) where 𝜉 = 6𝜋휂𝑎
where 𝑚 is the probe mass, 휂 is the fluid viscosity, 𝑎 is the probe radius and 𝜉 is the
Stoke’s frictional drag coefficient (where the definition above is valid for spherical
probes). As the Langevin equation is a linear, 1st order, non-homogeneous
differential equation, it can be solved via an integrating factor to give,
𝑣(𝑡) = 𝑣(0) exp (−𝜉𝑡
𝑚) + ∫ (
𝛿𝐹(𝑡)
𝑚) exp (
−𝜉(𝑡 − 𝑡′)
𝑚)𝑑𝑡′ .
𝑡
0
The first term in Equation 2.23 is the exponential decay of the initial velocity, 𝑣(0)
caused by the drag experienced in the fluid. The second term corresponds to the
additional velocity provided by the random forces acting on the probe. In order to
find the mean square displacement, we must first find the velocity correlation
function at thermal equilibrium. Using the equipartition theory, which states that the
average kinetic energy available to the sphere is equal to the thermal energy of the
fluid, can be used to derive the equilibrium correlation expression,
⟨𝑣(𝑡)𝑣(0)⟩𝑒𝑞 =𝑛𝑘𝐵𝑇
𝑚exp (
−𝜉𝑡
𝑚)
where 𝑘𝐵 is Boltzmann’s constant, 𝑇 is the fluid temperature and 𝑛 is equal to the
number of translational degrees of freedom (or dimensions, as in Section 2.2.2). By
expressing the mean square displacement, ⟨∆𝑟2(𝑡)⟩ in terms of the velocity
correlation, namely,
⟨∆𝑟2(𝑡)⟩ = 2𝑡∫ ⟨𝑣(𝑡′)𝑣(0)⟩𝑒𝑞𝑑𝑡′𝑡
0
− 2∫ 𝑡′. ⟨𝑣(𝑡′)𝑣(0)⟩𝑒𝑞𝑑𝑡′𝑡
0
we can sub in Equation 2.24 and solve the integrals to find,
⟨∆𝑟2(𝑡)⟩ = 2𝑛𝑘𝐵𝑇
𝜉[𝑡 −
𝑚
𝜉+ (𝑚
𝜉)exp (
−𝜉𝑡
𝑚)] .
Eq. 2.22
Eq. 2.23
Eq. 2.24
Eq. 2.25
Eq. 2.26
44
At long timescales, the first term in Equation 2.26 dominates and so an expression
for the mean square displacement similar to Equation 2.20 remains. Equating the
reduced form of Equation 2.26 and Equation 2.20 reveals an expression for the
diffusion coefficient, 𝐷, for a spherical ball in a viscous fluid,
𝐷 =𝑘𝐵𝑇
𝜉=𝑘𝐵𝑇
6𝜋휂𝑎
which is known as the Stokes-Einstein formula 70. The details of Equation 2.27
underpin the mechanisms of Brownian motion; the rate of movement of a spherical
probe in a fluid at thermal equilibrium is determined by the thermal energy from the
fluid (through the transfer of kinetic energy via collisions between the probe and
fluid molecules), but weighted by the probe’s frictional drag.
2.2.4. Complex shear modulus
Before using the theoretical groundwork described in the previous section to
extract information about a material’s properties from an experiment, we must first
introduce some commonplace notation. Firstly, consider a material experiencing an
oscillatory shear strain, 휀(𝜔) at an angular frequency, 𝜔. The resultant stress
induced on the material, 𝜎(𝜔) permits the definition of the shear modulus, 𝐺(𝜔),
such that,
𝐺(𝜔) =𝜎(𝜔)
휀(𝜔)=𝜎0sin (𝜔𝑡 + 𝛿)
휀0sin (𝜔𝑡)
where 𝛿 is the phase difference between the applied stress and strain. In purely
elastic materials, the stress and strain occur in phase with each other, whereas in
purely viscous materials, the strain lags behind the stress with 𝛿 = 𝜋 2⁄ radians. For
viscoelastic materials, we can characterise the purely elastic component of the shear
modulus, 𝐺′ and the purely viscous component of the shear modulus, 𝐺′′ as
𝐺′ =𝜎0휀0cos(𝛿) and 𝐺′′ =
𝜎0휀0sin(𝛿)
Eq. 2.27
Eq. 2.28
Eq. 2.29
45
which are also referred to as the storage modulus and loss modulus respectively.
Using complex notation, we can define the complex shear modulus, 𝐺∗(𝜔) as,
𝐺∗(𝜔) = 𝐺′ + 𝑖𝐺′′ .
Furthermore, the complex shear viscosity, 휂∗(𝜔) of the material can be found by
with the ratio between shear stress and the rate of change of the shear strain, which
can be linked to the complex shear modulus by the relation,
휂∗(𝜔) =𝐺∗(𝜔)
𝑖𝜔 .
2.2.5. Relating the mean square displacement to the creep compliance
The Langevin equation in Equation 2.22 can be applied to complex fluids by
considering that the forces associated with drag are related to past drag interactions.
This time-dependent drag "memory" is due to energy stored in the elastic component
of the complex fluid. As such, the temporal autocorrelation of the random forces
acting on the probe are now also time-dependent on a similar memory function.
Mason et al. build on this by using the assumption that the memory function is
proportional to the viscosity. By implementing Equation 2.31, the material shear
modulus can be related to the mean square displacement of the particle by,
��(𝑠) =𝑘𝐵𝑇
𝜋𝑎𝑠⟨∆��2(𝑠)⟩
where 𝑘𝐵 is Boltzmann’s constant, 𝑇 is the temperature of the medium, 𝑎 is the
radius of a tracked spherical tracer that has an associated mean squared
displacement, ⟨∆𝑟2⟩. Any quantities marked with a ( ) represent the unilateral
Laplace transform of that quantity with Laplace frequency, 𝑠. An inverse Fourier
transform (into the time domain) followed by a unilateral Laplace transform (into the
𝑠 frequency domain) is required to directly convert the definition of the shear
modulus given in Equation 2.30 to the quantity in Equation 2.32. To find the right-
hand side of Equation 2.32, Mason et al. started at a generalized version of the
Langevin equation, similar to the one depicted in Equation 2.22 (see Section 2.2.3).
Eq. 2.30
Eq. 2.31
Eq. 2.32
46
A major difference when accounting for a material that possesses viscoelastic
properties is that energy is stored within the material rather than being dissipated as
in a viscous fluid. In terms of the Langevin equation, this energy storage results in
temporal correlations between what are, in a purely diffusive system, random forces
acting on the probe 71.
To link the Laplace transform of the shear modulus in Equation 2.32 to the time
domain mean square displacement, Xu et al. first defined the time domain shear
stress, 𝜎(𝑡) and shear strain, 휀(𝑡) as,
𝜎(𝑡) = −∫ 𝐺(𝑡 − 𝑡′)휀(𝑡′)𝑑𝑡′𝑡
0
and 휀(𝑡) = ∫ 𝐽(𝑡 − 𝑡′)��(𝑡′)𝑑𝑡′𝑡
0
where 𝐽(𝑡) is the creep compliance, or the strain response to an incident stress.
Explicitly, the definitions in Equation 2.33 relate the stress and strain responses to
instantaneously applied stresses and strains. Mathematically these can be described
with a Heaviside function, 𝐻(𝑡), (with time derivative equal to the Dirac delta
function, 𝛿(𝑡)) such that 𝜎(𝑡) = 𝜎0𝐻(𝑡) and 𝜎(𝑡) = 𝜎0𝐻(𝑡)). In the case of an
elastic solid experiencing deformation in the Hookean regime where stress is
proportional to strain, 𝐺(𝑡) = 𝐺0 and 𝐽(𝑡) = 1 𝐺0⁄ . A viscous fluid on the other
hand has viscosity, 휂 equal to the stress divided by the rate of strain, leading to
𝐺(𝑡) = 휂𝛿(𝑡) and 𝐽(𝑡) = 1 휂⁄ 𝐻(𝑡). Equating the Laplace transforms of
Equation 2.33 uncovers the relation,
𝑠𝐺(𝑠)𝐽(𝑠) = 1
and a simple substitution into Equation 2.33 followed by an inverse Laplace
transform (assuming 𝐽(0) = 0 for viscoelastic fluids) yields,
𝐽(𝑡) = 𝜋𝑎
𝑘𝐵𝑇⟨∆𝑟2(𝑡)⟩
which links the mean square displacement of a spherical probe immersed in a
medium to the viscoelastic creep compliance of the medium 72.
Eq. 2.33
Eq. 2.34
Eq. 2.35
47
2.2.6. Caveat for complex fluids
Whilst Equations 2.20 and 2.26 hold true for particles that are freely
diffusing, in many scenarios the time dependence of the 𝑀𝑆𝐷 is not linear, but
described by a power law with exponent, 𝛼, such that,
𝑀𝑆𝐷(𝑡) ∝ 𝑡𝛼 .
In practical terms, when 𝛼 < 1 the particle is said to be subdiffusive, whereby the
particle exhibits extended periods of confined retention, possibly due to physical
enclosure or the viscoelastic properties of the surrounding environment 73-74.
Similarly, when 𝛼 > 1 the particle exhibits instances of consistent motion in a
particular direction (akin to a Lévy walk) and is said to be superdiffusive. This type
of regime can be seen when particles are part of an active transport mechanism, or
subject to flow 75-76.
2.2.7. Ripley’s K-function
In order to interpret and quantise the spatial distributions of bacteria in
biofilms, we now introduce Ripley’s K-function, a statistical tool that probes the
degree of clustering in an arrangement of particles. Consider a distribution of 𝑁 non-
overlapping hard-shell particles placed within a square region of interest with area,
𝐴𝑅𝑂𝐼 . If we draw a circle of radius 𝑟 centred on a randomly selected particle (hereon
referred to as the 𝑖th particle), we can count all other particles enclosed by the circle
using an indicator function, 𝐼. If the Euclidean separation between the 𝑖th and 𝑗th
particle, 𝑟𝑖𝑗 is less than 𝑟 then 𝐼 = 1, or otherwise 𝐼 = 0. Summing over all particle-
pair possibilities and iterating across all potential circle centres, we arrive at the un-
normalised Ripley K-function for increasing circle radius values 77,
𝐾(𝑟) = ∑ ∑ 𝐼(𝑟𝑖𝑗 < 𝑟)
𝑁
𝑗=1,𝑗≠𝑖
𝑁
𝑖=1
where 𝐼 = {1, 𝑟𝑖𝑗 < 𝑟
0, 𝑟𝑖𝑗 > 𝑟 .
Two normalisation terms are required to accommodate for observational boundaries.
The first is a term to account for the area of the encompassing circle that is outside
the region of interest. To extrapolate an estimate for the true number of particles
enclosed by the circle but are lost by the bounds of the region of interest, a weighting
Eq. 2.37
Eq. 2.36
48
term equal to the total area of the circle divided by the section of the circle inside the
region of interest (with area, 𝐴𝑖𝑛) is required 78. A second normalisation term is
required to account for the finite size of the region of interest and the number of
particles contained within it. Hence the normalised Ripley K-function is defined
as 78,
𝐾(𝑟) = 𝐴𝑅𝑂𝐼
𝑁(𝑁 − 1)∑𝜋𝑟2
𝐴𝑖𝑛
𝑁
𝑖=1
∑ 𝐼(𝑟𝑖𝑗 < 𝑟)
𝑁
𝑗=1,𝑗≠𝑖
where 𝐼 = {1, 𝑟𝑖𝑗 < 𝑟
0, 𝑟𝑖𝑗 > 𝑟 .
Eq. 2.38
49
2.3. Methodology
2.3.1. Bacteria preparation
All biofilm experiments were carried out with the S. aureus clinical strain,
ATCC 25923. To ensure bacteria concentrations were consistent across all
experiments, 0.5 mL batches of a 3-hour culture grown in tryptic soy broth, TSB
(Sigma-Aldrich, Gillingham, UK) were frozen in a sterile 25% glycerol solution. For
every experimental repeat, a separate sample was thawed, centrifuged for 5 minutes
at 5000 rpm and re-suspended in fresh sterile TSB twice before incubation at 37˚C
for 30 minutes. Prior to the deposition of the bacteria in the flow cell, 0.1 mL of a 10-
5 dilution was plated on TSB agar (Sigma-Aldrich, Gillingham, UK) and the
subsequent colonies were counted after an overnight incubation. From this, the
number of colony forming units per mL (CFU/mL) was calculated. All bacteria
inoculants used in this study contained 5(±0.2) x107 CFU/mL (data extracted from
three sample plates).
2.3.2. Biofilm cultivation
Biofilms were grown in a chemostat, as shown in Figure 2.1. The chemostat
primarily consisted of an IBI 3-channel flow cell (purchased through Sigma-Aldrich,
Gillingham, UK) that was modified to operate in conjunction with an Ismatec
REGLO ICC digital peristaltic pump (Cole-Parmer, USA). Each channel had
dimensions equal to 1 mm x 4 mm x 40 mm (height, ℎ, width, 𝑤, and length
respectively). To sustain incubation temperatures, a Grant JB Aqua 18 Plus water
bath was maintained at 37˚C, which housed a reservoir of media that was sealed with
aluminium foil pierced with an air filter. Media was pumped out of the reservoir into
the flow cell through a bubble trap to prevent the passage of bubbles into the cell,
which would disrupt the fluid flow consistency and biofilm growth due to the huge
shearing induced at the liquid/air interface. The microscope and flow cell were
encapsulated in a custom-made incubator that was sustained at 37˚C using an Air-
ThERM ATX (World Precision Instruments Ltd, USA), coupled to a thermometer
with a feedback control loop. All waste media was collected in a separate container.
50
51
For each experiment, the chemostat was sterilised by first pumping through with a
3% Virkon-water solution for 12 hours, then evacuated entirely of liquid and
pumped through with a 5% Decon-water solution for a further 3 hours. Following
this, the chemostat was again evacuated of liquid and finally flushed with autoclaved
deionised water to ensure no sterilisation chemicals remained in the tubing or the
flow cell. Despite the proficiency of this technique to adequately remove all bacteria
from the interior of the flow cell, regular replacements of the flow cell chamber and
tubing were made to ensure sterility.
Before inoculation with bacteria, the flow cell was primed with an initial passage of
media. Biofilm development was initiated by injecting a sufficient dose of the
bacterial culture (as described in Section 2.3.1) to fill the entire volume of the flow
cell. Any bubbles that had formed during this step were removed by vigorous
shaking, before the bacteria were left to deposit onto the surfaces of the flow cell for
1 hour. Any planktonic bacteria remaining in the cell rapidly left the flow chamber
when the flow was started. Hydrodynamic shear stresses, 𝜎ℎ𝑦𝑑 were calculated from
the pump flow rates in the parallel plate microfluidic chamber using,
𝜎ℎ𝑦𝑑 = 6𝑄휂𝑇𝑆𝐵𝑤ℎ2
where 𝑄 is the media flow rate and 휂𝑇𝑆𝐵 is the dynamic viscosity of the TSB media,
which is assumed to be equal to water 79. For Equation 2.39 to be valid, it is assumed
that the media flowing through the chamber can be described as an incompressible,
Newtonian fluid that has a laminar flow profile and no-slip boundary conditions at
the chamber walls. In reality, some of these criteria may not be satisfied (for
example, the flow profile may be more accurately described by a plug flow due to
the action of the peristaltic pump or alternatively the presence of bacteria on the
chamber walls could interfere with the no slip boundary condition). This calculation
was included to estimate the required flow rate to achieve order-of-magnitude
changes in shear stress within the chamber, which was directly observed. Flow rates
were set at 0.15 and 1.50 mL/min to generate approximate hydrodynamic shear
stresses of 1 and 10 mPa respectively (although calculated for the centre of the
chamber), with a comparative dataset being produced for no flow (categorised as
‘stationary’). Experiments at 37˚C were conducted for 6 hours, which allowed for
Eq. 2.39
52
significant proliferation of the bacteria. Anti-biofilm enzymes were added to the
initial sterile TSB. To target the main components of the biofilm extracellular matrix,
proteinase K from Tritirachium album and Deoxyribonuclease 1 (DNase-1) from
bovine pancreas (purchased through Sigma-Aldrich, Gillingham, UK) were added to
final concentrations of 60 µg/mL and 100 µg/mL respectively 80-81. Proteinase K is a
broad-spectrum enzyme that digests extracellular proteins in the biofilm by cleaving
the peptide bond next to the carboxylic group of hydrophobic amino acids. DNase-1
was chosen as it selectively cleaves extracellular DNA, which is believed to be an
important structural component of the Staphylococcal biofilm 81.
2.3.3. Optical brightfield microscopy
The flow cell and incubator chamber were mounted on an Olympus IX70
inverted microscope fitted with a 100x oil-immersion objective lens illuminated by a
pE-100 LED (CoolLED, UK). An AVI350M dynamic vibration isolation system
(Table Stable Ltd., Switzerland) was employed to counteract unwanted external
vibrations. Videos of the bacteria motion were captured every hour on a Photron
Fastcam PCI camera (Photron Ltd., Bucks, U.K.) operating in bright-field mode. All
videos were recorded at 1000 frames per second, over a field of view of 1024 x 1024
pixels (equivalent to ~116 µm2). As the biofilm grew, bacterial motion was recorded
at different heights with 5 ± 1 µm increments using the built-in micrometre focusing
scale. A large gap (relative to the diameter of an individual bacterium, ~1 µm) was
used to ensure bacteria from adjacent height layers were not recorded. The pump and
air provider were temporarily turned off whilst videos were recorded to avoid
vibrations that could detrimentally alter the particle tracks. The LED was turned to a
low-medium power setting and all external sources of light in the lab were turned off
during image capture to limit the amount of flicker appearing on videos due to AC
mains electric input. The camera occasionally measured harmonics in the oscillatory
circuit used to modulate the power of the LED (the duty ratio of LEDs is modulated,
rather than the voltage, to vary the effective power, otherwise the spectral balance of
the LED can change), which was not intended for such fast camera applications. This
intermittent problem was corrected for using Fourier filtering.
2.3.4. Particle tracking
Individual bacteria were tracked with a MATLAB-based software package
called PolyParticleTracker, which employs a polynomial-fit, Gaussian-weight
53
algorithm to distinguish particles from the background noise 82. The software is
particularly effective with biofilms, because the polynomial fit to the background
makes it relatively robust to changes in background intensity (a constant threshold is
not used for identifying particles for the heterogeneous biofilms). Passive particle
tracking microrheology of the approximately spherical non-motile bacteria was used
based on the generalized Stokes-Einstein equation 83,71, 84. The mean-square
displacement, ⟨∆𝑟2(𝑡)⟩ was calculated using Equation 2.8 and converted to the shear
creep compliance, 𝐽(𝑡) using Equation 2.35 (the relationship is described in detail in
Section 2.2.5).
Figure 2.2a shows an example of a biofilm image which highlights some of the
identified bacteria and tracks (Figure 2.2b,c). Examples of the MSD signals as a
function of time interval are shown in Figure 2.2d. Creep compliances corresponding
to displacements of less than 10 nm over a time interval of 1 ms were considered to
be due to firmly attached bacteria, either to the microscope slide surface or the
surrounding biofilm, as any displacement at this magnitude would be
indistinguishable from noise at the limit of the camera resolution 82. If a bacterium
had a compliance value less than this resolution value, it was set to a value equal to
the noise limit. On Figure 2.2d, these would appear as level MSD plots that do not
increase in magnitude over time. If these signals were included, the distribution of
creep compliances would be weighted incorrectly to a lower value. As noise-limit
MSD signals appear approximately flat, they can be filtered out by extrapolating and
thresholding the log gradient. Erroneous vibrations occurring in the bacteria tracks at
frequencies greater than 50 Hz (likely caused by electrical noise in the LED) were
removed using a Fourier-based low-pass filter. If these signals were included on
Figure 2.2d, they would appear as normal MSD measurements swamped by an
oscillatory function corresponding to the noise frequency. The smooth monotonic
compliance curves meant that it was relatively easy to isolate the oscillatory noise
using the narrow band Fourier filter. To compare compliance values, a reference
time interval of 10 ms was used. The compliance values that were most
representative of the data sets were found by performing a log-normal fit to the
distribution at this reference time interval and the mean values were found. All data
is presented with the associated standard errors (on the assumption that the
movement of neighbouring bacteria is independent), produced by taking the standard
54
deviations from the log-normal fits and weighting by a factor of 1
√𝑁𝑏𝑎𝑐 (where 𝑁𝑏𝑎𝑐 is
equal to the number of successfully identified and tracked bacteria, which typically
was in the hundreds for each observation).
Figure 2.2. Example data from the tracking analysis. a) A brightfield microscopy
image of a S.aureus biofilm grown in the flow cell after an incubation time of 3
hours at 37˚C, at an elevation of 5 µm from the flow cell bottom surface. b) A
magnified section from 2a) with an overlay showing individual bacteria that have
been identified, their respective radii and tracks over 1000 frames, equivalent to 1
second. Each colour represents a unique bacterium that has been identified and
tracked. c) An enlarged rendering of an example ‘track’ constructed from the
displacements of a single bacterium position between adjacent frames, showing the
sub-pixel localisation precision attainable with the fitting protocol. d) All mean
square displacements, (MSDs, ⟨∆𝑟2(𝑡)⟩), shown as a function of time interval
corresponding to all bacteria identified in a). The scale bars are equal to 10 µm.
a)
b)
d)
c)
55
2.3.5. Ripley’s K-function
The spatial heterogeneity of the early-stages of biofilm growth was quantified
by calculating Ripley’s K-function, as described in Section 1.2.6. For bacteria that
are distributed randomly across the region of interest, the expected value for 𝐾(𝑟) is
E[𝐾(𝑟)] = 𝜋𝑟2. Significant clustering was indicated by dividing the calculated value
for 𝐾(𝑟) with the expected value associated with spatial randomness and subtracting
1. In this form, any positive value indicates clustering that is statistically significant
than what would be associated with randomness and likewise any negative value
would be associated with spatial autocorrelation (or ordering).
To visualise the change in the Ripley K-function as 𝑟 increase, plots of 𝐿(𝑟) − 𝑟
were plotted, where,
𝐿(𝑟) = √𝐾(𝑟)
𝜋
which has a value of 0 when bacteria are spaced completely randomly and is positive
when there is some degrees of clustering 85. As any one spatial distribution could
inadvertently exhibit a degree of clustering, upper and lower 97.5% critical values
for significance testing were computed by sampling 50 hard-shell Monte Carlo
simulations with the same number of bacteria of a fixed radius, 𝑎, in a region of
interest with equal area to the microscope field of view. In each simulation, circles
were assigned coordinates inside the region of interest using a uniform random
probability density function. To prevent overlapping, circles had to be separated by a
minimum of 2𝑎. Once 𝑁𝑏𝑎𝑐 circles had been placed, the K-function was calculated.
A total of 50 simulations were carried out for each field of view and the quantiles
were extracted through averaging.
Eq. 2.40
56
2.4. Results and Discussion
Figure 2.3a shows the mean creep compliance of all S. aureus bacteria in the
field of view at a reference time interval, 𝑡𝑟𝑒𝑓 of 10 ms as a function of height (in 5
μm increments) for biofilms grown under three different hydrodynamic shear
regimes after 4 hours. A growth period of 4 hours was chosen as it represents
significant biofilm proliferation compared to the initial number of bacteria following
deposition. This can be observed in Figure 2.4, which shows the average number of
bacteria for each hydrodynamic regime at each time point and each height.
Interestingly, the proliferation profile of bacteria grown under flow conditions
appears exponential, in comparison to the linear growth regime apparent in the static
condition. This could be due to reduced nutrient availability in the surrounding
environment caused by the lack of flow. Hydrodynamic shear stresses of 0 mPa
(stationary), 1 mPa, and 10 mPa were applied to the biofilm during development.
Error bars are presented as the standard deviation of a log-normal fit of a probability
density histogram containing all creep compliances observed in the field of view,
weighted by the square root of the number of bacteria (Figures 2.3b-d shows these
histograms for the biofilm-bound bacteria at a height of 15 μm at the different
hydrodynamic shears). Significant differences arose between equivalent heights
depending on the environmental flow conditions. For example, in a stationary
biofilm after 4 hours at 10 μm above the flow cell surface, the mean creep
compliance was 0.708 ± 0.014 Pa-1. For comparison, biofilms grown under shear
stresses of 1 mPa and 10 mPa had mean creep compliances of 0.247 ± 0.003 Pa-1 and
0.251 ± 0.004 Pa-1 respectively, indicating a biofilm ~3 times as rigid to a static
environment. Variations in the identified bacteria radii are not significant enough to
account for the differences in creep compliance observed throughout this experiment
(Figure 2.5 shows the distributions of bacteria radii measured for all time points and
heights for the three hydrodynamic regimes). Creep compliances for all time points
over the 6 hour observation period are shown in Figure 2.6, with similar
relationships occurring at all time points. Extrapolating the reciprocal limit of the
creep compliance allows us to compare the biofilms with the viscosity of water. As
such, biofilms grown under stationary conditions had a viscosity approximately
equivalent to twice that of water, whereas biofilms grown under the hydrodynamic
shears are approximately five times more viscous than water 86.
57
Figure 2.3. Creep compliance results a) At a reference time point of 10 ms, the mean
creep compliance was calculated for S. aureus biofilms after 4 hours of sustained
flow at 37 ˚C at incremental heights (represented by colour) above the flow cell
surface, subject to varying hydrodynamic shears. b), c) and d) show probability
distributions for compliances at reference time, 𝑡𝑟𝑒𝑓 = 10 ms for a height of 15 µm
subject to different hydrodynamic shears; 0 mPa (stationary), 1 mPa and 10 mPa.
a)
b) c)
d)
58
Figure 2.4. Number of bacteria found in a biofilm as a function of time at a series of
hydrodynamic shear rates, namely a) 0, b) 1mPa and c) 10 mPa. Different colours
represent height above the attachment surface in the flow cell. Similar proportions of
bacteria were found at all shear rates. Within errors, bacteria proliferation was
approximately the same for the 1 mPa and 10 mPa, but was reduced for static
biofilms.
a)
b)
c)
59
Figure 2.5. Identified bacteria radii distributions across all experiments at all times
and heights (indicated by colour) and hydrodynamic shear regimes, namely a)
stationary, b) 1 mPa and c) 10 mPa. The insets in each graph show the mean (and
standard error) bacteria radius corresponding to a given height within the biofilm. As
can be seen, the maximum variation in radius size is ~6%, indicating no significant
contribution to the creep compliances calculated using Equation 2.35.
a)
b)
c)
60
Figure 2.6. The characteristic creep compliance at a single reference time interval
(𝑡𝑟𝑒𝑓 = 10 ms) plotted as a function of time for the entirety of the experiment (6
hours) with a) no flow, b) 1 mPa shear stress and c) 10 mPa shear stress. Heights in 5
µm intervals are shown as different coloured lines. The biofilms are seen to be softer
the further they are away from the surface of attachment in both no flow and 10 mPa
conditions. Under flow conditions the bacteria appear to be more securely fastened
to the surfaces and the compliances are lower (harder, with larger shear moduli).
a)
b)
c)
61
All hydrodynamic regimes display increasing creep compliance as a function of
height (indicating biofilms are softer at greater heights), with stationary biofilms
exhibiting significantly larger compliances than either of the two flow regimes.
These results suggest that for S. aureus there is a response to higher shear stresses to
produce more rigid early-stage biofilms. This is in agreement with established results
for other bacteria species in the literature. Galy et al. showed using magnetic
microparticle actuation that the spatial distribution of creep compliance for an F
pilus-producing Escherichia coli biofilm grown after 24 hours is dependent on
height and inversely dependent on shear stress, corroborating the pattern and
magnitudes of results presented in this study 87. Fluorescent beads have also been
used to examine biofilm viscoelasticity through microrheology. A 2016 study by
Cao et al. found characteristic creep compliances an order of magnitude larger than
stated in this study but were limited to larger lag times due to the slow acquisition
speed of the confocal scanning microscope used in their experiment. Using their
technique, they found no significant difference in creep compliance between
increasing height layers, contrary to our results. However, it should be noted that
they were observing more mature biofilms (24 and 48 hours) that may be denser due
to extended proliferation 88. An advantage of our particle tracking technique is the
absence of physical perturbation caused by the addition of magnetic or fluorescent
beads to the biofilm during growth, which could act as abiotic surfaces for the
bacteria to attach to other than the surfaces of the flow cell.
Figure 2.7a shows the ratio of the Ripley K-function, 𝐾(𝑟), to the expected Ripley
K-function for complete spatial randomness, E[𝐾(𝑟)], when the clustering radius, 𝑟,
is equal to 10 μm for all three hydrodynamic growth regimes at incremental heights
after 5 hours. A biological interpretation of the rescaled Ripley’s K-function,
𝐾(𝑟)
E[𝐾(𝑟)]− 1, is the fractional difference in the number of bacteria that would be
expected within a circle centred on a random bacterium defined by the clustering
radius compared to the same total number of bacteria but randomly distributed. For
example, the data suggests that for any bacteria at a height of 10 μm above the
surface of the flow cell after 5 hours of growth when there is no flow present, one
would expect to find 32 ± 13% more bacteria present within a circle of radius of 10
μm when compared to a randomly distributed arrangement of the same total number
of bacteria. As the biofilm grows vertically, the degree of clustering becomes more
62
significant, indicating biofilm growth away from the surface occurs in narrowing
columns, rather than homogeneously. As an example, after 5 hours in a biofilm
subject to 10 mPa hydrodynamic shear in a 10 µm radius circle, our data suggests at
the surface there would be 3 ± 1% more bacteria present than a completely random
arrangement, whereas at 20 µm away from the surface, there would be 46 ± 19%
more bacteria than an equivalent number arranged randomly. This trend occurs
regardless of the hydrodynamic shear experienced by the biofilm.
Figures 2.7b and c show the rescaled Ripley K-function at the flow cell surface and
first height increment 5 μm above the surface for all shear regimes. For stationary
environment biofilms, the initial distribution of bacteria remains approximately
constant for the duration of the experiment, whereas the two flow regimes show a
decrease in the rescaled Ripley K-function value which is indicative of the bacteria
becoming more homogeneously organised within the biofilm. As S.aureus are non-
motile, this result suggests bacteria are preferentially growing horizontally rather
than vertically when under flow. Furthermore, the homogeneous distribution of
bacteria may be aiding the structural stability of the biofilm, as extracellular matrix is
being produced evenly at the base, rather than in localised clusters. The full dataset
of the rescaled Ripley K-function values over the course of the experiment for all
three hydrodynamic shear regimes are shown in Figure 2.8.
63
Figure 2.7. a) Rescaled Ripley K-function analysis as a function of height for biofilm
grown after 5 hours at 37˚C subject to different hydrodynamic regimes, namely 0
mPa (stationary), 1 mPa and 10 mPa. Rescaled Ripley K-function analysis for
bacteria within a biofilm at the bottom layer (i.e. 0 µm) (b) and 5 µm above surface
of flow cell (c) over all time points and for all hydrodynamic regimes. A general
decrease in the value of the rescaled Ripley K-function for stationary conditions
suggest that the bacteria within the biofilm become more spatial homogeneous as the
biofilm matures.
a)
b) c)
64
Figure 2.8. Rescaled Ripley K-function for all time points (6 hours) for biofilms
three different hydrodynamic growth regimes, namely a) no flow, b) 1 mPa shear
stress and c) 10 mPa shear stress. The significant increase in the values of the
rescaled Ripley-K as a function of height (represented by different colours in 5 µm
intervals) indicates spatial clustering vertically during all time points of the biofilm
growth.
a)
b)
c)
65
To elaborate on this further, Figures 2.9a and b shows the locations of bacteria after
5 hours in 10 mPa flow conditions at heights of 0 and 15 μm respectively, where the
bacteria have been colour coded based on their individual creep compliance value at
a reference time of 10 ms. Despite there being fewer bacteria at 15 µm height, it is
apparent visually that the distribution is not evenly spatially distributed, especially in
comparison to the bacteria at the surface. Figures 2.9c and d display the
corresponding L(r)-r functions with 97.5% and 2.5% quantiles from 50 Monte Carlo
simulations of randomly distributed hard-shell particles. Any positive value greater
than the error quantiles indicates clustering at that value of d (given in units of pixels
in these figures). A circle of r = 15 µm originating from a randomly chosen
bacterium is shown on Figure 2.9b, where the corresponding 𝐿(𝑟) − 𝑟 function
peaks, to illustrate the most statistically likely clustering size.
Figure 2.9. Comparative examples of bacteria distributions colour coded by creep
compliance of each bacteria for biofilms grown under 10 mPa hydrodynamic shear
stress for 5 hours at 37 ˚C at the surface of the flow cell (a) and 15 µm above the
surface (b). In b), a circle is shown with an arbitrary radius, r originating from a
randomly selected bacterium to convey the bacteria selected within a certain radius.
c) and d) show the normalised Ripley-K function corresponding to a) and b)
respectively, to demonstrate the extent of clustering at a radius, r = 10 µm.
a) b)
b) a)
c) d)
66
To combine the ideas of spatial heterogeneity and characteristic creep compliance,
Figure 2.10 shows the mean creeps at all time points and heights as a function of
bacteria density, with an inset showing the mean creeps as a function of rescaled
Ripley-K function. Linear fits are shown as black lines, revealing an inverse
relationship between creep and cell density, but a positive correlation between creep
and rescaled Ripley-K function. To quantify the correlation between the two values
in both graphs, the Spearman’s rank coefficient between creep and bacteria density
was found to be -0.64 (p < 0.001) and between creep and Ripley’s K function to be
0.41 (p < 0.001). Intuitively a greater number of adjacent bacteria would result in a
stiffer biofilm due to more overall cell-cell adhesion and shared extracellular
material. Moreover, the increase in creep associated with larger spatial heterogeneity
could be due to isolated columns of bacteria with no surrounding structural support.
Figure 2.10. Characteristic creep compliance as a function of bacteria density for all
time points, heights and hydrodynamic regimes, with a black fit line showing an
inverse linear correlation. The inset shows the same mean creep compliances plotted
as a function of the rescaled Ripley-K function, with a black fit line displaying a
positive linear correlation.
67
The early-stage biofilm structure investigated in this study may be the precursor to
the macro-scale features of mature biofilms that have been studied in the literature.
For example, Stoodley et al. showed that mature biofilms in laminar flow elongate
into long streamers, as opposed to circular clusters in turbulent flow conditions 64.
Three dimensional streamers are also present in motile bacteria (such as
Pseudomonas aeruginosa) biofilms, as demonstrated by Drescher et al., as a
response to unusual geometries, such as flow obstacles, gaps or corners 89. To
visualize the degree of clustering as a function of depth in the biofilm, Figure 2.11
shows a rendering of the segmented bacteria calculated at 5 µm increments within
the same sample for a static biofilm at the end of the observation period. Coloured
bacteria correspond to those resolved for the calculation of the creep compliances
and spatial statistical analysis (with out of focus bacteria shown as grey outlines).
Tapered columns can be seen at greater heights with broader bases attached to the
glass surface, reflecting the increase in spatial heterogeneity at the higher elevations
furthest from the surface.
Figure 2.11. A rendering of the bacteria distributed within a biofilm subject to no
hydrodynamic shear after 6 hours of growth at 37˚C. Bacteria positions are extracted
from tracking data and colour coded based on height above surface in 5 µm height
intervals. Grey outlines represent bacteria in the spaces between focal planes. The
arrows indicate the direction of shear flow. Tapered columns can be seen with their
bases on the surface of attachment.
116 µm
116 µm
68
Figure 2.12 shows the characteristic creep compliance value at a reference time of
10 ms for biofilm grown subject to a hydrodynamic shear of 1 mPa after 5 hours in
the presence of proteinase K and DNase-1. Biofilms formed in the presence of
proteinase K were unable to grow past 10 µm, suggesting the shear forces from the
surrounding flow overcame the intracellular attachment when a protein biofilm
component is removed. Addition of proteinase K results in a significant increase in
the compliance of the biofilm compared with the control (TSB alone). Bacterial
biofilm on the surface of the flow cell had a mean creep compliance of 0.369 ± 0.005
Pa-1 compared to the samples exposed to proteinase K, which had mean creep
compliance at the surface of 1.394 ± 0.017 Pa-1, indicating much softer viscoelastic
structures form. In contrast, the addition of DNase-1 at the same time point and
height reduced the mean compliance to 0.321 ± 0.006 Pa-1. This pattern was
observed for all heights and all time points. In their study on DNase-1 dependence
on biofilm coverage, Moormeier et al. found that there was no statistically
significant discrepancy between UAMS-1 S. aureus biofilms grown with and
without DNase-1 until after 6 hours, possibly because cell lysis is not induced until
after this time point 90-91. Characteristic creep compliances across all time points and
all heights subject to a hydrodynamic shear of 1 mPa in the presence proteinase K
and DNase-1 are shown in Figure 2.13.
69
Figure 2.12. The mean compliances calculated at a characteristic time (10 ms)
plotted after 5 hours for two different enzymes (proteinase K and DNase-1) and the
control (just TSB) with no hydrodynamic shear. Biofilms grown in the presence of
proteinase K exhibited much larger creep compliances, characteristic of softer
biofilms. Biofilms, grown in the presence of DNase-1 showed a slight decrease in
creep compliance compared to no enzyme present.
The elasticity gradient of the bacterial columns means there is not a single threshold
value of shear stress associated with S. aureus biofilm detachment. Instead a broad
range of elasticities and thus detachment stresses are presented by the biofilm, which
may help to insure both persistent colonisation of the chosen surface and subsequent
detachment events to colonise other surfaces or interfaces.
70
Figure 2.13. Full time course results over 6 hours for biofilm characteristic creep
compliances at a reference time interval of 10 ms when in a) a 60 µg/mL solution of
proteinase K and b) a 100 µg/mL solution of DNase-1. Hydrodynamic shear on the
biofilm was 1 mPa. Height increments of 5 µm away from the flow cell surface are
represented as different colours.
a)
b)
71
2.5. Conclusion
Passive-microrheology experiments on S. aureus biofilms show clear
evidence that the characteristic creep compliances become smaller under flow.
S. aureus therefore reacts to an increase in shear force by producing a more rigid
biofilm in which it is embedded, by up to a factor of 3 in relative compliance. A
statistical spatial analysis reveals early-stage biofilms grow vertically in column-like
arrangements. However, biofilms under flow grow more homogeneously at layers
closer to the surface over time, compared to biofilms grown in static conditions.
Coupled with the viscoelastic response of biofilms, this suggests a reinforcement of
the lower layers close to the attachment surface to prevent complete detachment of
the biofilm under shear stress. A vertical gradient of viscoelasticity and spatial
heterogeneity also facilitates biofilm dispersal, allowing loosely bound bacteria to be
removed from the biofilm structure while retaining an entrenched layer close to the
attachment surface. Treatment with proteinase K had a large effect on softening the
biofilms, whereas DNase-1 in contrast has the opposite effect, slightly hardening the
biofilms. This methodology could be extensively used to screen different anti-
biofilm compounds (as opposed to antibacterial compounds that directly kill
bacteria). It is thought that effective treatment of microbial infections associated with
biofilms will require mixed formulations targeted at both the bacteria (concentrated
antibiotics) and the biofilms (anti-biofilm molecules). Both DNases and proteinases
are primary candidates for anti-biofilm molecules in such formulations.
72
Chapter 3. Spatial distribution and diffusive dynamics of
antimicrobial cationic peptide G3 in gram-positive
Staphylococcus aureus and gram-negative Escherichia coli
using 3-dimensional stochastic optical reconstruction
microscopy and fluorescence tracking.
3.1. Aims of Chapter and Introduction
Whilst experimental studies have already proven the antibacterial efficacy of
G3, the actual mechanism leading to cell death is still in question. As stated in
Section 1.3.3, there are three models describing the stages following peptide
accumulation on the membrane surface that can lead to cell death; namely the
toroidal, carpet and barrel-stave models. The aim of the research presented in this
Chapter is to determine the model of peptide action by analysing the spatial
distributions of peptide when interacting with gram-positive and gram-negative
bacteria. Based on the arrangements of peptides required for each of these models to
occur, the spatial distributions should be distinguishable between the carpet model
and the two pore-forming models. Specifically, the pore-forming models in principle
should be made up of highly clustered regions of peptides associated to the pores,
whereas the carpet model should present peptides homogeneously distributed at
random across the bacteria surface. The secondary aim of this Chapter is to
investigate the diffusive dynamics of G3 when interacting with the bacteria
membrane using a similar methodology presented in Chapter 2. An initial hypothesis
would be to assume that the diffusivity of the peptides interacting with the
membrane should be reduced in comparison to the uninhibited diffusion experienced
in solution.
In order to investigate the spatial distribution of molecules within bacteria optically,
traditional fluorescence microscopy would not have been sufficient. As shall be
explained in detail in the Theory section of this Chapter, a consequence of the wave-
nature of light and the microscope setup is the diffraction resolution limit. When
imaging with optical wavelength light, this fundamental limit prevents the practical
identification of fluorescence markers with a precision less than 500 nm, which is on
the scale of the radius of most bacteria. To circumvent this limitation, stochastic
optical reconstruction microscopy (STORM) shall be employed. The inclusion of an
adaptive optics deformable mirror setup facilitates 3-dimensional STORM by
73
encoding astigmatism into the detected signal based on axial position (see Section
3.2.6 for full details). At the time of writing, only one study in the literature has
performed similar experiments using a deformable mirror adaptive optics setup to
track fluorescent proteins in 3-dimensions. Izeddin et al. claimed in their 2012 study
that they achieved axial resolutions of only 20 nm at axial depths of 800 nm inside
HeLa cells 92. As STORM uses intermittently blinking fluorophores to attain images,
tracking and the extraction of diffusive properties was only made possible by the
recent development of the software tool DeepExponent, which uses a feed-forward
neural network to characterise fractional Brownian motion and extrapolate power
law exponents from tracking data containing as few as 7 time points. DeepExponent
was created primarily by Daniel S. Han under the collective supervision of Nickolay
Korabel1, Runze Chen, Mark Johnston, Viki J. Allan, Sergei Fedotov and Thomas
A. Waigh. As of the time of writing, the software has not been published or made
publically available.
Two studies in the literature perform investigations with a similar tracking
methodology that will be implemented in this Chapter. Nelson and Schwartz used
fluorescence tracking of individual 𝛼-helical cationic peptides (specifically LL37
and melitten) on flat lipid bilayers constructed such that the inner leaflet consisted of
1,2-dioleoyl-sn-glycero-3-phos-phoethanolamin (DOPE) and the outer leaflet
consisted of monophosphoryl and diphosphoryl lipid A (extracted from Escherichia
coli). The authors found that the average diffusion coefficient for both peptides was
on the order of ~0.02 µm2s-1, much slower than reported diffusion coefficients for
membrane lipids. Furthermore, analysis of the step-size dynamics revealed that the
peptide motion exhibited two modes of diffusion: a highly restricted ‘crawling’ and
intermittent short-lived ‘hops’. It was concluded by the authors that this behaviour
could be explained by the peptide temporarily desorbing from the bilayer membrane
and freely diffusing in the surrounding solution. This result suggests that the
interaction between the peptide and membrane is transient, but detectable by single
molecule tracking 93.
The second study investigated the trajectory dynamics of 𝛼-helical antimicrobial
peptide Sushi 1 conjugated to fluorescent quantum dots when interacting with live
E. coli. The authors noted that when an extremely low concentration of peptide
interacted with bacteria (i.e. a single peptide per cell), the diffusion coefficient was
74
3.52 m2s-1, but when the concentration of peptides was increased to the equivalent
minimum inhibitory concentration level, the diffusion coefficient dropped to 0.04
m2s-1 after 5 minutes of exposure, and to 0.0067 m2s-1 after 10 minutes. Interestingly,
the same significant drop in the diffusion coefficient was not observed for another
antibiotic called polymixin B, suggesting that different mechanisms were involved in
cell lysis. The drop in diffusion coefficient was attributed to the formation of peptide
complexes of the membrane. To see what type of complexes, the authors also
directly imaged the bacteria after treatment using transmission electron microscopy
images but saw no evidence of pore formation and no disruption to the
lipopolysaccharide (LPS) network in the outer membrane. As LPS release in vivo
can trigger severe septic shock, this property of Sushi 1 is encouraging for it’s
potential clinical efficacy 94-95.
Whilst no direct studies have been performed detailing the surface distributions of
peptide through direct imaging, there have been investigations into the peptide
surface density required to induce cell death. Roversi et al. titrated fixed
concentrations of fluorophore-conjugated helical cationic peptide PMAP-23 with
increasing concentrations of E. coli until the minimum inhibitory concentration was
reached. By measuring the fluorescence emission, an estimation of the threshold
number of peptides per cell needed to induce death was reached using a volumetric
approximation of the cell shape and size. In conclusion, they found that cell death
only occurred when 106-107 peptides were bound to each cell, which (based on area)
could only be attributed to the carpet model 96.
75
3.2. Theory
In order to investigate the anti-bacterial mechanisms of G3 within
Staphylococcus aureus and E. coli, a fluorescence microscopy technique called
stochastic optical reconstruction microscopy (STORM) was employed. The choice
to use STORM was necessary due to the fundamental resolution limit of traditional
optical microscopy called the Rayleigh criterion. The aim of the following theory
sections is to elucidate the origin of the Rayleigh criterion in an idealised (i.e.
aberration-free) microscope setup via Fourier optics. Firstly, we shall define light as
monochromatic complex field and then characterise how it propagates through free
space, followed by a series of typical imaging setups. As imaging detectors measure
light intensity, the next step will be to characterise the propagation of incoherent
field intensity in a 3-dimensional coordinate system from which the resolution limit
naturally arises. As an adaptive optics setup was implemented to extract the axial
location of the fluorophores, we will also describe the effect the deformable mirror
has on the final detected image through the use of approximations. This section also
contains an explanation of the photo-physics behind the fluorescence emission of the
fluorophores, which we model as incoherent point sources.
3.2.1. Definition of a complex electric field
We begin by defining light as a field varying in space and time. To initially
simplify the following theory, we assume the light is monochromatic, such that it can
be described by only one temporal frequency, 𝜐 and therefore one associated
wavenumber, 𝜅 (where 𝜅 = (𝑛 𝑐⁄ )𝜐 =𝑛𝜆⁄ and 𝑛 is the refractive index of the
medium the light is propagating through, 𝜆 is the wavelength of the light and 𝑐 is the
speed of light in a vacuum). In Cartesian coordinates, we can describe the spherical
field, 𝐸(𝑟, 𝑡) as,
𝐸(𝑟, 𝑡) = 𝐸0exp (𝑖2𝜋(𝜐𝑡 ± 𝜅𝑟)
however, it will become more convenient to express the spatial component in terms
of the direction of propagation, 𝑧, and the corresponding orthogonal 2-dimensional
plane, ��. In this coordinate system, the wavevector is similarly separated into
propagation direction and transverse components, �� = (��⊥, 𝑘𝑧). Whilst we are
Eq. 3.1
76
dealing with monochromatic light, the time dependence is implicit and will be not be
included in the notation.
3.2.2 Free-space propagation
To begin with, we shall consider the propagation of the field from a plane at
𝑧 = 0, referred to as the object plane, to an arbitrary plane at 𝑧 > 0, referred to as the
image plane. We also define an ideal aperture on the object plane of size, 𝜌𝑎𝑝 which
bounds the field such that 𝐸(��0 > ��𝑎𝑝 , 0) = 0, but does not affect the field within
the aperture. If the field at the object plane, 𝐸(��0, 0) is known, then the field at the
imaging plane is given by the Rayleigh-Sommerfield diffraction integral,
𝐸(��, 𝑧) = −𝑖𝜅 ∫𝐸(��0, 0) (1 −1
𝑖2𝜋𝜅𝑅)exp(𝑖2𝜋𝜅𝑅)
𝑅cos (휃)𝑑2��0
where 𝑅 = √|�� − ��0|2 + 𝑧2 and cos(휃) = 𝑧 𝑅⁄ . To explore the ramifications of this
equation, we must consider two approximations. The first is when the separation
between planes is significantly larger than the wavelength of the field (i.e. 𝑅 ≫ 𝜆, or
𝜅𝑅 ≫ 1). Cancelling the vanishing terms in the equation above leads to,
𝐸(��, 𝑧) = −𝑖𝜅∫𝐸(��0, 0)exp(𝑖2𝜋𝜅𝑅)
𝑅cos (휃)𝑑2��0 .
Using the definition of a spherical monochromatic field in Equation 3.1, we can
qualitatively describe the field at the imaging plane as a superposition of secondary
spherical fields (called ‘Huygens wavelets’) emanating from every position within
the aperture on the object plane 97. A depiction of the secondary fields is shown in
Figure 3.1.
A further approximation, referred to as Fresnel diffraction, assumes that the
transverse variation of the fields during propagation between the two planes is very
small. This leads to us being able to use the small angle approximation on the
obliquity term, such that cos(휃) → 휃 and to expand 𝑅 as a Taylor series, such that
𝑅 → 𝑧 +|�� − ��0|
2
2𝑧⁄ where |�� − ��0|
2 = 𝜌2 − 2��. ��0 + 𝜌02.
Eq. 3.2
Eq. 3.3
77
Implementing these conditions to Equation 3.3 results in,
𝐸(��, 𝑧) = −𝑖𝜅
𝑧exp(𝑖2𝜋𝜅𝑧) exp (𝑖𝜋
𝜅
𝑧𝜌2)
×∫𝐸(��0, 0)exp (−𝑖2𝜋𝜅
𝑧��. ��0) exp (𝑖𝜋
𝜅
𝑧𝜌02) 𝑑2��0
which can be interpreted as a convolution between the field at 𝑧 = 0 and a free-space
propagator function 98.
Figure 3.1. Representation of secondary point sources emanating spherical Huygen’s
wavelets from every point of the object plane, ��0(0), which is assumed to be a
perfect aperture (i.e. the boundaries do not influence the wavelets). The field at the
object plane, ��1(𝑧) can be thought of as the super-position of all wavelets as
described in Equation 3.3. 𝑅 and 휃 represent the distance and angle the Huygen’s
wavelets make with the imaging plane. The Fresnel approximation extends this by
assuming that the small angle approximation can be applied (see Equation 3.4).
Eq. 3.4
78
3.2.3. Simple lens configurations
The next step is to introduce a perfect thin lens into the propagation path of
the wave, which is denoted with lateral position, 𝜉 and axial position, 𝑧𝜉 . On
transmitting through the lens, the effect on the wave can be described by the field
transmission factor, 𝑇(𝜉),
𝑇(𝜉) = 𝑃(𝜉)exp (−𝑖𝜋𝜅
𝑓𝜉2)
where 𝑓 is the focal length of the lens and 𝑃(𝜉) is called the pupil function. By
defining the separation between the lens and the object plane as 𝑠0(= 𝑧𝜉 − 𝑧0) and
the separation between the image plane and the lens as 𝑠1(= 𝑧1 − 𝑧𝜉) then the steps
describing the wave propagation are as follows:
1. Free space propagation from object plane to lens plane.
2. Transmission through lens.
3. Free space propagation from lens plane to image plane.
Invoking the Fresnel diffraction integral (Equation 3.4) and the transmission factor,
the field at the imaging plane is described by,
𝐸1(��1) = −𝜅2
𝑠0𝑠1exp (𝑖2𝜋𝜅(𝑠0 + 𝑠1))exp (𝑖𝜋
𝜅
𝑠1𝜌12) ×
∬𝐸0(��0) exp (−𝑖2𝜋𝜅𝜉. (��0𝑠0+��1𝑠1)) P(𝜉) exp(𝑖𝜋𝜅𝜉2 (
1
𝑠0+1
𝑠1−1
𝑓)) ×
exp (𝑖𝜋𝜅
𝑠0𝜌02) 𝑑2��0𝑑
2𝜉 .
To remove the second quadratic phase term, we can change the system to include
another lens, such that the field propagates through what is called a ‘4𝑓’ system,
which is commonly utilised in microscope lenses and shown in Figure 3.2. The two
lenses have individual focal lengths, 𝑓1 and 𝑓2 respectively. When arranged so that
the separation between the object plane and the first lens equals 𝑓1, the separation
between lenses is 𝑓1 + 𝑓2 and the separation between the second lens and the image
plane is 𝑓2 (as shown in Figure 3.2), the system can be approximated by a single
Eq. 3.5
Eq. 3.6
79
pupil function at the axial position 2𝑓1. The resultant field at the image plane is then
given by an equation similar to above without any quadratic phase terms,
𝐸1(��1) = −𝜅2
𝑓1𝑓2exp (𝑖4𝜋𝜅(𝑓1 + 𝑓2))∬𝐸0(��0)𝑃(𝜉)exp (−𝑖2𝜋𝜉. (
𝜅
𝑓2��1 +
𝜅
𝑓1��0))𝑑
2𝜉𝑑2��0
where we can see that the lens arrangement removes the quadratic phase terms and
the pupil function is now akin to an aperture. At this point we define the coherent
spread function, 𝐶𝑆𝐹(��) in terms of a generic focal length, 𝑓0 and transverse
location, �� as,
𝐶𝑆𝐹(��) = (𝜅
𝑓0)2
∫𝑃(𝜉)exp (𝑖2𝜋𝜅
𝑓0𝜉. ��) 𝑑2𝜉 = (
𝜅
𝑓0)2
ℱ𝜉[𝑃(𝜉)] (−𝜅
𝑓0��)
where the integral has been expressed as the Fourier transform of the pupil function.
We can also define the coherent transfer function, 𝐶𝑇𝐹(��⊥) as the Fourier transform
of the 𝐶𝑆𝐹(��), which intuitively returns a scaled pupil function,
𝐶𝑇𝐹(��⊥) = ∫𝐶𝑆𝐹(��) exp(−𝑖2𝜋��⊥. ��) 𝑑2�� = 𝑃 (
𝑓0𝜅��⊥)
Re-expressing Equation 3.7 in terms of the coherent spread function in Equation 3.8
results in,
𝐸1(��1) =1
𝑀exp (𝑖4𝜋𝜅(𝑓1 + 𝑓2))∫𝐸0(��0)𝐶𝑆𝐹 (
1
𝑀��1 − ��0) 𝑑
2��0
where 𝑀 = −𝑓2𝑓1⁄ is the magnification factor of the lens system. This equation
reveals a fundamental relationship at the core of Fourier optics, namely the field at
the image plane is the convolution of the field at the object plane and a propagator
function, the coherent spread function 97.
Eq. 3.7
Eq. 3.8
Eq. 3.9
Eq. 3.10
80
Figure 3.2. Lens setup for a 4𝑓 imaging system typically used in microscopy. Fields
propagate from the object plane, ��0 to the image plane, ��1 through an effective
aperture, described by the function 𝑃(𝜉). The separation between the object plane
and the first lens is equal to the first lens’ focal length, 𝑓1 and likewise, the
separation between the image plane and the second lens is the second lens’ focal
length, 𝑓2.
3.2.4. Field intensity propagation
Whilst the previous sections dealt with a monochromatic field, fluorescence
imaging involves the detection of light composed of many frequencies (collectively
called the frequency bandwidth). Furthermore, modern camera detectors cannot
measure the frequency-dependant fluctuations of a complex field, but can detect the
time-averaged intensity, 𝐼(��) given by
𝐼(��) = ⟨|𝐸(��, 𝑡)|⟩
where we shall take the assumption that averaging occurs over a time period where
the intensity remains constant, meaning the time dependence can once again not be
included in the notation. Before proceeding to the lens setup where Section 3.2.3 left
off, consider two fields at a generic plane. The mutual intensity, 𝐼𝑀𝑢𝑡(��, ��′), is
defined as the correlation between the two fields at two different arbitrary locations
on the plane,
𝐼𝑀𝑢𝑡(��, ��′) = ⟨𝐸(��)𝐸∗(��′)⟩
Eq. 3.11
Eq. 3.12
81
such that the actual intensity can be expressed as 𝐼(��) = 𝐼𝑀𝑢𝑡(��, ��′). At this point
we define a new coordinate system based on the mean, ��𝑐 and difference, ��𝑑 of the
two vectors, such that,
��𝑐 =1
2(�� + ��′) and ��𝑑 = �� − ��
′ .
Using the new coordinate system, the mutual intensity can be redefined as,
𝐼𝑀𝑢𝑡(��𝑐 , ��𝑑) = ⟨𝐸 (��𝑐 +1
2��𝑑)𝐸
∗ (��𝑐 −1
2��𝑑)⟩
Returning to Equation 3.10, we now extract the mutual intensity at the imaging plane
following field propagation through a 4𝑓 lens system (see Figure 3.2). The only
difference is that the exponential pre-factor is ignored as it has no relation to
imaging; otherwise we directly substitute the relevant equations to form,
𝐼𝑀𝑢𝑡,1(𝑀��1𝑐 , 𝑀��1𝑑) =1
𝑀2∬𝐶𝑆𝐹 (��1𝑐 +
1
2��1𝑑 − ��0𝑐 −
1
2��0𝑑) ×
𝐶𝑆𝐹∗ (��1𝑐 −1
2��1𝑑 − ��0𝑐 +
1
2��0𝑑) 𝐼𝑀𝑢𝑡(��0𝑐 , ��0𝑑)𝑑
2��0𝑐𝑑2��0𝑑 .
As the intensity at the mean planar location is equivalent to the mutual intensity
when the difference vector is zero (i.e. 𝐼(��𝑐) = 𝐼𝑀𝑢𝑡(��𝑐 , 0)), a simplification to
Equation 3.15 can be,
𝐼1(𝑀��1𝑐) =1
𝑀2∬𝐶𝑆𝐹 (��1𝑐 − ��0𝑐 −
1
2��0𝑑) ×
𝐶𝑆𝐹∗ (��1𝑐 − ��0𝑐 +1
2��0𝑑) 𝐼𝑀𝑢𝑡(��0𝑐 , ��0𝑑)𝑑
2��0𝑐𝑑2��0𝑑 .
As the fluorophore fluorescence emission is incoherent, we can approximate the
mutual intensity at the imaging plane as,
𝐼𝑀𝑢𝑡(��0𝑐 , ��0𝑑) →1
𝜅2𝐼0(��0𝑐)𝛿
2(��0𝑑)
Eq. 3.13
Eq. 3.14
Eq. 3.15
Eq. 3.16
Eq. 3.17
82
where 𝛿2 is the 2-dimensional delta function. The approximation in Equation 3.17
allows for the further simplification of Equation 3.16, leading to
𝐼1(𝑀��1𝑐) =1
𝑀2𝜅2∫|𝐶𝑆𝐹(��1𝑐 − ��0𝑐)|
2𝐼0(��0𝑐) 𝑑2��0𝑐 .
At this point we can introduce and define the point spread function, 𝑃𝑆𝐹(��),
𝑃𝑆𝐹(��) =1
𝜅2Ω0|𝐶𝑆𝐹(��)|2
where Ω0 is the pupil solid angle (i.e. the spatial integral of the square of the pupil
function weighted by the associated focal length squared, or more intuitively the area
seen by the pupil from the object plane). Subbing into Equation 3.18 yields,
𝐼1(𝑀��1𝑐) =Ω0𝑀2∫𝑃𝑆𝐹(��1𝑐 − ��0𝑐)𝐼0(��0𝑐) 𝑑
2��0𝑐
which has an approximate form equivalent to Equation 3.10. Therefore, the intensity
at the imaging plane caused by an incoherent light source at the object plane can be
thought of as a weighted convolution between the intensity at the object plane and
the point spread function, which is defined by the type of pupil in the lens setup 99.
3.2.5. 2-dimensional optical resolution limit
As stated previously, traditional microscopy lenses are set up in a 4𝑓
configuration. Explicitly, the pupil function can now be set as a circular aperture of
diameter, 𝐷, where
𝑃(𝜉) = { 1 𝜉 ≤ 𝐷 2⁄
0 𝜉 > 𝐷 2⁄ .
Eq. 3.18
Eq. 3.19
Eq. 3.20
Eq. 3.21
83
By making use of the spatial bandwidth, Δ𝜅⊥0 =𝜅𝐷
𝑓1⁄ , the coherent transfer
function can be derived using Equation 3.9,
𝐶𝑇𝐹(��⊥) = {1 𝜅⊥ ≤
1
2Δ𝜅⊥0
0 𝜅⊥ >1
2Δ𝜅⊥0
from which the coherent spread function can be derived via a Fourier transform
(Equation 3.8). Hence, using the standard result for the Fourier transform of a
Heaviside function, the coherent spread function is,
𝐶𝑆𝐹(𝜌) =𝜋
2Δ𝜅⊥0
2𝐽1(𝜋Δ𝜅⊥0𝜌)
𝜋Δ𝜅⊥0𝜌
where 𝐽1 is the 1st order cylindrical Bessel function. From here, the point spread
function can be derived using Equation 3.19 and a reformulation of the solid angle in
terms of the spatial bandwidth, namely Ω0 =𝜋4⁄ (Δ𝜅⊥0 𝜅⁄ )
2, which leads to
𝑃𝑆𝐹(𝜌) =𝜋
2Δ𝜅⊥0
2 (𝐽1(𝜋Δ𝜅⊥0𝜌)
𝜋Δ𝜅⊥0𝜌)2
which is also referred to as the Airy pattern 100. We are now in the position to
calculate the imaging intensity profile for an example 4𝑓 microscope system.
Consider the scenario where the intensity profile of a fluorescence emission at the
object plane can be described as an incoherent point source, such that
𝐼0(𝜌0) = 𝛿2(��0) and for simplicity the magnification factor, 𝑀 = −1 (and so 𝑓1 =
𝑓2 = 𝑓). Plugging all relevant definitions into Equation 3.20 yields the final intensity
profile at the imaging plane as,
𝐼1(𝜌) = Ω0𝑃𝑆𝐹(𝜌) = (𝜆
2𝑛𝜌)2
𝐽12 (𝜋𝑛𝐷𝜌
𝜆𝑓) .
A typical 100x lens has aperture diameter, 𝐷 = 6.4 × 10−3m and focal length, 𝑓 =
2 × 10−3 m, therefore when the fluorescence emission occurs at 560 nm, the
Eq. 3.22
Eq. 3.23
Eq. 3.24
Eq. 3.25
84
resulting image intensity profile is shown in Figures 3.3a and b. In 1879, Lord
Rayleigh defined the Rayleigh criterion for spatial resolution that states that two
adjacent point sources in the object plane are resolvable only if the distance between
them is greater than the distance corresponding to the first zero of the 1st order
Bessel function divided by 𝜋 101. From analytical calculations, the first zero of the 1st
order Bessel function can be approximated when 𝐽1(1.22), therefore from Equation
3.25, we can say that the minimum spatial resolution of the microscope, Δ𝜌𝑚𝑖𝑛 is
given by
Δ𝜌𝑚𝑖𝑛 = 1.22𝑓𝜆
𝐷
d
which is shown diagrammatically in Figure 3.3b and c. If the separation between
point sources is less than Δ𝜌𝑚𝑖𝑛 , the point source positions cannot be determined
(this is shown in Figures 3.3e and f).
Eq. 3.26
85
Figure 3.3. Normalised intensity distributions at the imaging plane, ��1 caused by a-b)
a single incoherent point source (see Equation 3.25), c-d) two incoherent point
sources with transverse separation, Δ𝜌 that equals the Rayleigh criterion for
minimum resolution and e-f) transverse separation less than the Rayleigh criterion.
The 4𝑓 imaging system is assumed to have unit magnification, described by lens
focal length, 𝑓 and lens aperture diameter, 𝐷. All point sources depicted are assumed
to emit light with wavelength, 𝜆. Figures on the same row show the same intensity
distribution but from different viewing angles (isometric and top-down). The colour,
representing intensity, is normalised by the single point source maximum.
a) b)
c) d)
e) f)
86
3.2.6. 3-dimensional point spread function and adaptive optics
In the previous sections it has been assumed that the object plane has always
been ‘in-focus’ (i.e. the object emitting light was positioned at the focal length of the
first lens in the 4𝑓 system). This however is an idealised scenario, especially in
biological experiments where fluorescence tracers are often out-of-focus above and
below the focal plane of the microscope. To accommodate for this mathematically,
we simply combine two previous field propagators in tandem. At the end of Section
3.2.2, it was shown that a field that had propagated through free-space could be
described by a convolution between the initial field and a free-space propagator,
which here will be denoted by 𝐻(��, 𝑧) (see Equation 3.4). In Section 3.2.3, the field
after propagation through a 4𝑓 lens system was described as the convolution
between the field at the imaging plane and the coherent spread function (see
Equation 3.20). Consider a light source that has a defocus given by Δ𝑧, which is
equivalent to the axial separation between the object plane of the lens system and the
source. The field at the imaging plane will therefore propagate through free space
over a distance of Δ𝑧 and then propagate through the 4𝑓 lens system, described
qualitatively as two convolutions,
𝐸(��1, 𝑧) = 𝐶𝑆𝐹(��1 − ��0)⨂[𝐻(��0, Δ𝑧)⨂𝐸(��0, 0)]
where ‘⨂’ represents a convolution. As shown in Section 3.2.4, when considering
the propagation of incoherent intensity, the same convolution forms are present. By
compiling the two convolutions into a 3-dimensional point spread function term, the
incoherent intensity at the imaging plane for an out-of-focus source is,
𝐼1(𝑀��1, 𝑀2𝑧) =
Ω0𝑀2∫𝑃𝑆𝐹(��1 − ��0, Δ𝑧)𝐼0(��0, 𝑧0) 𝑑
2��0 .
To avoid a laborious numerical integral that would have to be employed for a
circular aperture as in Section 3.2.5, we instead approximate a ‘Gaussian pupil’,
where now the pupil function is not a normalised rectangle function but rather,
𝑃(𝜉) = exp (−𝜉2
2𝑎𝜉2)
Eq. 3.26
Eq. 3.27
Eq. 3.28
87
where 𝑎𝜉 is the characteristic lens radius. By using the same assumptions as in
Section 3.2.5 (i.e. magnification equal to unity), the intensity profile for a defocused
point source is,
𝐼1(��1, Δ𝑧) =Ω0𝜋Δ𝜅⊥
2
(1 + 휁2)exp (
−𝜋2Δ𝜅⊥2𝜌2
(1 + 휁2))
where 휁 =𝜋Δ𝜅⊥
2𝑧2𝜅⁄ and the spatial bandwidth, Δ𝜅⊥ = 2𝜅
𝑎𝜉𝑓⁄ 98, p73. By using the
same microscope and imaging parameters used in Section 3.2.5, the normalised 3-
dimensional intensity plot for a point source is shown in Figure 3.4.
Figure 3.4. Normalised 3-dimensional intensity distribution associated with a 4𝑓
imaging system for an incoherent point source that is axially defocused by Δ𝑧, calculated by assuming a Gaussian aperture. Intensity profile calculated using
Equation 3.26.
The adaptive optics setup used in the experiments described in this thesis uses a
deformable mirror to induce a spatial phase change in the intensity profile at the
imaging plane similar to a cylindrical lens. In the derivations presented so far, there
Eq. 3.26
88
has been the implicit assumption of planar symmetry. The deformable mirror
introduces a transverse asymmetry as a function of 𝑧. A full derivation of the
resultant effect would have to include the shape of the deformable mirror, which was
regularly re-calibrated to accommodate for experimental aberrations (such as the
influence of using immersion oil on the lens and the refractive properties of the
coverslip) that have been ignored for brevity.
However, a qualitative understanding can be gained by introducing rudimental phase
terms to Equation 3.26 for the 3-dimensional point spread function 92. By defining
the transverse plane in terms of 𝑥 and 𝑦 (such that 𝜌2 = 𝑥2 + 𝑦2), the deformed
point spread function can be expressed as,
𝑃𝑆𝐹(𝑥, 𝑦, Δ𝑧) ∝ exp (−(𝑥
𝜔𝑥)2
) exp (−(𝑦
𝜔𝑦)
2
)
where 𝜔𝑥,𝑦 = (Δz ± ς
𝜍)2
where 𝜍 can be thought of as the ‘magnitude of astigmatism’ introduced by the
mirror deformation. This approximation is only applicable over short values of Δ𝑧,
typically of the order of ±500 nm (however the exponential intensity drop-off for
larger defocused sources would realistically prevent detection). The ramifications of
the adaptive optics setup can be seen in Figure 3.5, which shows the altered intensity
profile for a point source using the same optical setup used in Section 3.2.5 with the
arbitrary astigmatism set at 1 μm. The 𝑧-dependent transformation of the point
spread function allows for axial localisation.
Eq. 2.27
89
Figure 3.5. Transverse deformation caused by adaptive optics mirror on the 3-
dimensional intensity profile for an incoherent point source at axial defocus, Δ𝑧 equal to a) 100 nm, b) 0 nm and c) -100 nm (calculated using Equation 3.27).
a)
b)
c)
90
3.2.7. Origin of fluorescence
In the previous sections, all derived fields were modelled to have originated
from incoherent point sources. In practice, to generate a point source of light for
super-resolution microscopy and tracking experiments, fluorescent photoswitchable
probes called ‘fluorophores’ are used. At a fundamental level, fluorescent molecules
absorb energy from incident photons and release some of that energy by emitting
another photon.
A simplistic introduction to the quantum mechanical explanation of fluorescence is
to consider a fluorophore molecule inhabiting the lowest energy, ground state
denoted by 𝑆0. When in the ground state, one could imagine an electron pair
occupying the same molecular orbital, but (as a consequence of Pauli’s exclusion
principle) having opposing spin. The transition energy, Δ𝐸 required for an electron
to be excited into a higher energy orbital is determined by the fluorophore design. If
the fluorophore is exposed to light that has energy equal to the transition energy (i.e.
Δ𝐸 = ℎ𝜈, where ℎ is Planck’s constant and 𝜈 is the light frequency), then it is
possible for an electron to be excited into the higher energy singlet state, denoted by
𝑆1. ‘Singlet’ here refers to the spin multiplicity of the system when the total spin
angular momentum is equal to 0. The timescale of absorption is of the order of ~10 -15
seconds. Assuming no external stimulations, the molecule will liberate the energy as
another photon through a process called spontaneous emission, which as the name
suggests, is stochastic but can be described by a first-order rate equation. The typical
lifetime of an excited state for a Cy3B molecule has been shown to be up to the order
of 10-9 seconds, but can be altered by the surrounding environmental conditions 102.
The process of liberating energy by the emission of a photon during the electron’s
return to the ground state is referred to as fluorescence.
To extend the basic introduction given above, consider that each electronic energy
level is composed of many sub-levels associated with vibrational and rotational
energy states determined by the structuring of the molecule. Differences in
vibrational and rotational energy levels are on average significantly smaller than the
difference in electronic levels, resulting in energy bands rather than distinct levels.
Two consequences are immediately apparent. The first is an increase in the
bandwidth of photons that can be absorbed and emitted by the molecule. The second
is the facilitation of energy transfer within energy bands by non-radiative processes
91
called ‘internal conversion’, such as collisions with surrounding molecules. Heat
energy lost to the surroundings in this way results in the fluorescence emission
having lower energy (i.e. a larger wavelength) than the absorbed photon, making it
detectable by filtering 98,p256-259.
A further extension to consider is a collision during the non-radiative process called
‘intersystem crossing’ that can cause the spin state of the excited electron to flip. The
electron is now in an excited triplet band, 𝑇1 (so called because the total spin angular
momentum is now 1 and so the spin multiplicity is 3). Due to Pauli’s exclusion
principle, the electron in the triplet state must first change spin orientation before it
can de-excite to the ground state. Whilst in the triplet excited state, the molecule has
a higher affinity for chemical reactions. Reactions with reactive oxygen species can
change the electronic structure enough to permanently remove the molecules ability
to fluoresce in a process called ‘photobleaching’. The rate of photobleaching can be
mediated by performing experiments in an oxygen-scavenging buffer. Triplet states
can emit photons to return to the ground state (in a process called phosphorescence),
but the lifetime of the states is on the order of 10-3 seconds.
It is also possible for reactions to form radical anion states, denoted by 𝑇1∗, which
recover to the ground state via photo-dissociation after exposure to ultraviolet light
(called ‘UV pumping’). As this state has a relatively long lifetime on the order of 102
seconds but can be recovered to the ground state, it is referred to as the ‘dark’ state
and is crucial for stochastic optical reconstruction microscopy 100. The energy
structure of the fluorophore molecule is shown in a simplified Jablonski diagram,
shown in Figure 3.6, which also depicts the average lifetimes of each state.
3.2.8. Fluorophore ‘blinking’
Fluorophores that can exist in dark states as described in the previous section
facilitate ‘blinking’. If the fluorophore is initially in the ground state, incident
illumination at the absorption bandwidth will excite the molecule. Subsequent
liberation occurs on the nanosecond timescale as a fluorescent photon. Assuming
constant illumination, this process cycles until intersystem crossing causes the
fluorophore to enter the excited triplet state.
92
Figure 3.6. Jablonski diagram showing the energy bands of a fluorophore. The
energy of a state is represented by the height on the figure. Solid arrows between
energy bands are representative of radiative transitions, whereas dotted lines are non-
radiative. Energy bands are labelled as: 𝑆0, ground state; 𝑆1 singlet excited state
(opposing spin pairs); 𝑇1, excited triplet state (same spin pairs) and 𝑇1∗, radical anion
state (also called the ‘dark’ state). Absorption occurs when an electron absorbs the
energy of an incident photon and is promoted to a higher energy level. Internal
conversion and intersystem crossing both liberate energy through non-radiative
means, such as in vibrational and rotational energy states. Approximate timescales
for each process are shown, with the exception of transitioning from a dark state
back to the ground state, which must be achieved through UV pumping.
Therefore, a ‘blink’ is the continuous cycling of absorption and fluorescence prior to
entering the triplet or dark state, the probability of which is determined by the energy
structure of the fluorophore, as well as the buffer chemical composition (specifically
oxygen availability), temperature and pH. In stochastic optical reconstruction
microscopy, the duration of the blink is of no concern for imaging (i.e. localisations
associated with blinks lasting one frame of the camera are averaged). On the other
hand, single-particle tracking requires blinks to last longer than one frame. Several
authors have shown that the fluorescence intermittency (distribution of blink
lifetimes) obeys power-law dynamics 103-105.
93
3.3. Methods
3.3.1. Fluorescence microscope apparatus
All STORM and single particle tracking experiments were carried out on a
custom-built apparatus, a simplified version of which is shown in Figure 3.7. A
Laser QuantumTM Gem continuous wave 561 nm laser at an operating power of 90
mW was used to excite the Cy3B fluorophores into a fluorescence triplet state.
Concurrently, an OBISTM LX continuous wave 405 nm laser was used at a low
operating power of 0.5 mW to de-excite the fluorophores out of the dark state back
into the ground state (to extend fluorophore lifetime). Laser lines were directed
around the optical path with a combination of regular and dichroic mirrors. All
experiments were performed over similar timescales, so any volumetric heating
effect caused by the laser is consistent (it has been shown in the literature that the
maximum heating effect at the focal point is on the order of +1 K per 100 mW
continuous wave laser power 106). Following emission, both laser beams are directed
into an optical fibre that is being physically moved by a speaker oscillating at 10k
Hz. The physical movement of the fibre reduces the spatial coherence of the laser
beams, which would otherwise result in the production of unwanted interference
patterns at the illumination plane of the sample. The laser beams are then directed
into an OlympusTM IX-71 inverted microscope by means of a multiband filter (to
reduce back reflection from the sample entering the camera). The sample is mounted
on top of an OlympusTM 100x TIRF lens, which itself is mounted on a laser-
controlled self-correcting MadCityTM piezoelectric stage (which reduces drift in the
z-direction). Immersion oil is used to reduce light loss due to refraction caused by
the difference in refractive index between the lens setup and the sample. Although
not utilised in this experiment, the microscope mount is enclosed by a Solent
SolutionsTM incubator box. Fluorescence emissions are directed into a MicAOTM
3DSR adaptive optics box, before being collected by an ORCA-Flash 4.0 V2 CMOS
camera (82% peak quantum efficiency, 6.5 µm x 6.5 µm pixel size, 2.5 ms exposure
time). All image acquisition was performed using the software HCimage Live. To
improve image acquisition intensity, 2 x 2 pixel binning was utilized resulting in an
effective pixel scale of 130 nm. A field of view equal to 19.5 µm x 19.5 µm was
captured at 400 fps for a total of 200,000 frames (this was the maximum amount of
time that could be tracked as the imaging data was stored locally on the camera
94
memory before being transferred following capture). In an effort to reduce the
influence of outside vibrations, the entire microscope setup is constructed on top of a
NewportTM floating optical table.
Figure 3.7. Schematic diagram of the setup that was used to perform 3-dimensional
fluorescence tracking and stochastic optical reconstruction microscopy experiments.
Not shown in the figure is the floating table the entire apparatus is mounted on the
reduce vibrations from external sources.
3.3.2. Adaptive optics calibration
To calibrate the adaptive optics setup described in the apparatus in Figure
3.7, the astigmatism induced at various heights must first be recorded manually. In
order to achieve this, 10 µL of 200 nm diameter latex microspheres, amine-modified
polystyrene fluorescent orange (Sigma Aldrich, UK) were re-suspended in OxEA
imaging buffer at a final concentration of 100 µM. Fluorescent beads of a known
sub-diffraction radius are required to only observe the change to the point spread
function caused by the adaptive optics deformable mirror. After setting the mirror
astigmatism to +0.6 µm, images of the illuminated bead were taken at sequential 20
nm intervals up the z-axis over a total range of -300 nm to +300 nm (centred around
the actual bead location). As can be seen from Figure 3.8a, the adaptive optics
astigmatism results in a y-axis elongation of the point spread function when the focal
plane is below the central position of the bead and an x-axis elongation when the
focal plane is above the central bead position. The x and y axial ratios of the standard
deviations of a fitted 2-dimensional Gaussian integral are shown in Figure 3.8b,
95
which also shows that no astigmatism is found at the position where the bead is
actually located.
3.3.3. Bacteria sample preparation
The fluorophore used in this experiment was Cy3B, which was chosen as it
has a long lived blink duration (or intermittency lifetime) 107. To avoid excess
conjugation, the volume ratio used in the final solution of G3 and Cy3B was 1:1000.
In order to image G3 interactions with bacterial membranes on inactive cells, gram-
positive and gram-negative bacteria strains were prepared for single particle tracking
and STORM using the following procedure. ATCC 25923 S. aureus and ATCC
25922 E. coli were grown in 10 mL tryptic soy broth and M9 media respectively for
12 hours in a 37˚C shaking incubator, or until the stationary growth phase had been
reached. Eppendorfs containing 1 mL of the separate cultures were mixed with 50
µL of 2 mM concentration G3-Cy3B solution, ending up with a final peptide
concentration of 100 µM. Following thorough mixing for a minimum of 30 seconds,
the solution was span down in a SigmaTM 1-14 Microfuge centrifuge at 13,300 RPM
for 60 seconds. After removing the supernatant, the bacteria-peptide-dye pellet was
re-suspended in 1 mL of phosphate-buffered saline (PBS) solution. This step was
repeated three times in order to remove, free-floating G3-Cy3B molecules that would
otherwise oversaturate the imaging plane with unbound fluorescent signals.
Following the removal of the final supernatant, the bacteria pellet is re-suspended in
100 µL of PBS to increase the number concentration of bacteria by a factor of 10. A
50 µL volume is pipetted onto a clean coverslip which has been pre-treated with
Poly-L-Lysine, a polyamino acid that increases the adhesion strength between the
bacteria and the glass coverslip, as well as immobilising motile E.coli bacteria. After
10 minutes, sufficient numbers of bacteria are attached to the surface of the coverslip
to ensure a minimum of 50 individual bacteria per field of view. With the intent of
removing unattached bacteria from the environment, the area where the bacteria
were deposited was washed three times with 50 µL of PBS. When all liquid had been
effectively removed, 10 µL of OxEA imaging buffer was pipetted onto the bacteria
film, which stabilises the fluorophore photoswitching efficacy. Following this,
another coverslip was used to seal the bacteria in place, and reinforced with a
standard microscopy slide. Briefly, to produce a 10 mL stock solution of OxEA
96
buffer, the following components are simply mixed together: 7.7 mg of Cysteamine,
300 µL of OxyFluor, 2 mL of Sodium DL-lactate and 7.7 mL PBS.
3.3.4. Stochastic optical reconstruction microscopy (STORM) and image analysis
using ThunderSTORM
The methodology of STORM was originally conceived by a Harvard team
led by Rust, Bates and Zhuang in 2006 108. In essence, STORM reconstructs
structures by sampling many individual diffraction-limited fluorescence signals and
then compiling the localisations into a single image. The process is only possible
with sparsely labelled photoswitchable fluorophores, as single signals must be
resolved and fitted individually (see below). In this experiment, a 2-dimensional
STORM projection of the 3-dimensional localisations will be used to identify the
outlines of the bacteria. Fluorophore localisations were extracted from the raw data
using an ImageJ plugin called ThunderSTORM 109. With the initial input data
consisting of image stacks containing the diffraction-limited fluorescence blinks, the
analysis procedure within ThunderSTORM is as follows:
- Prior to processing, a median filter is applied by creating a composite image
of median pixel intensity values of each x,y pixel through the stack. This
median image is then subtracted from each image within the stack to remove
persistent artefacts (such as unwanted low-level autofluorescence).
- Next, each image is de-noised using a stationary wavelet transform using a
third order B-spline. Briefly, this method convolutes the image with various
dilated wavelet signals based upon a polynomial spline function and replaces
filter coefficients with zeros if below a given length scale (in essence
performing the same function as a bandpass filter). 110-111
- Bright spots on the fluorescence image are now approximately localised
using a local maximum method, whereby pixels with intensities greater than
the standard deviation of the background pixel noise are selected.
- Sub-pixel localisation is achieved by the fitting of a point-spread function
model to the identified pixels using maximum likelihood estimation. To
incorporate the 3D astigmatism cause by the adaptive optics setup, the point-
spread function model used was an integrated 2D Gaussian with xy-direction
standard deviation, 𝜎𝑥𝑦 and z-direction standard deviation, 𝜎𝑧 calibrated by
the fluorescent bead stack shown in Figure 3.8.
97
- This procedure is carried out for all images in the stack, resulting in an array
of fluorophore localisations with an associated error in the xy and z
directions. Erroneous localisations where the values of 𝜎𝑥𝑦 and 𝜎𝑧 were less
than 130 nm (the equivalent size of 1 pixel) were removed.
The goodness-of-fit for each localisation is directly proportional to the signal-to-
noise value, which in turn is proportional to the square root of the number of photons
emitted from the fluorophore during a blink. A visual output of the ThunderSTORM
analysis process is an average shifted histogram, ASH, which averages together
localisations into 2-dimensional bins. Localisation intensity and number density are
taken into account when generating the final value of each individual bin, but the
final image is weighted and shifted to accommodate the largest bin value. One
drawback of the ASH image is the lack of temporal information associated with
fluorescence intermittency of the fluorophores. This can result in several apparent
localisations occurring in the same or similar location, when in actuality a single
blink occurred for a longer duration than the exposure time of one frame. For this
reason, ASH images were only used for outlining bacteria positions.
Figure 3.8. Adaptive optics calibration, where the mirror astigmatism was set at
0.6 µm. a) Raw images of a single fluorescent bead intensity profile at the focal
plane and 300 nm above and below the focal plane. b) Normalised 𝑥 and 𝑦 axial
ratios of the 2-dimensional fitted Gaussian integral to the raw images.
a) b)
98
3.3.5. Bacteria identification using average shifted histogram super-resolution
image and parametric equation fitting
Rather than use a bacteria membrane dye, such as NileRed, to identify the
exact location of the bacterial membrane, the average shifted histogram output of the
ThunderSTORM analysis was used instead. Bacteria membranes on the ASH images
were manually identified using the multi-point selection tool built into ImageJ. The
approximate projection of the bacteria membrane was fitted with an ellipse by
minimising the 2D parametric equation using the boundary points of the form,
𝑎𝑥2 + 𝑏𝑥𝑦 + 𝑐𝑦2 + 𝑑𝑥 + 𝑒𝑦 + 𝑓 = 0
where 𝑎, 𝑏, 𝑐, 𝑑, 𝑒 and 𝑓 are constants, using a non-iterative least squares fitting
algorithm as described by Fitzgibbon et al. 112. Values of x and y corresponding to
Equation 3.29 equalling 0 give the outline of the best fitting ellipse for the points
selected, and therefore the outline of the bacteria projection in 2D. The major and
minor radii for each fitted ellipse, 𝑟𝑚𝑎𝑗 and 𝑟𝑚𝑖𝑛 were extracted using,
𝑟𝑚𝑎𝑗, 𝑟𝑚𝑖𝑛 = √2(𝑎𝑒2 + 𝑐𝑑2 − 𝑏𝑑𝑒 + (𝑏2 − 4𝑎𝑐)𝑓) ((𝑎 + 𝑐) ± √(𝑎 − 𝑐)2 + 𝑏2)
𝑏2 − 4𝑎𝑐
with the ellipse centre point, denoted by the coordinates 𝑥0 and 𝑦0, given by,
𝑥0 = 2𝑐𝑑 − 𝑏𝑒
𝑏2 − 4𝑎𝑐 and 𝑦0 =
2𝑎𝑒 − 𝑏𝑑
𝑏2 − 4𝑎𝑐
As the focal plane of the microscope was set to the approximate middle height of the
bacteria, a fair estimation for the bacteria shape could be inferred by projecting the
ellipse into the z-dimension to the extent of the minor radius value, 𝑟𝑚𝑖𝑛. This was
achieved by generating noisy points around the positions (𝑥0, 𝑦0, ±𝑟𝑚𝑖𝑛), and in
combination with the previously described ellipse points, fitting with a 3D
ellipsoidal parametric equation of the form:
𝐴𝑥2 + 𝐵𝑦2 + 𝐶𝑧2 + 2𝐷𝑥𝑦 + 2𝐸𝑥𝑧 + 2𝐹𝑦𝑧 + 2𝐺𝑥 + 2𝐻𝑦 + 2𝐼𝑧 + 𝐽 = 0
Eq. 3.29
Eq. 3.30
Eq. 3.31
Eq. 3.32
99
where 𝐴,𝐵, 𝐶, 𝐷, 𝐸, 𝐹, 𝐺, 𝐻, 𝐼, 𝐽 are constants. Now in 3-dimensional space, the new
ellipsoid major radius and minor radius were recalculated using the eigenvalues of
the algebraic form of the ellipsoid. Figure 3.9 shows multiple example ellipsoids
extracted from the ASH image, where the initial 2D ellipse outline is shown in pink
and the final corresponding ellipsoid, major radius and minor radius are shown in
red.
Figure 3.9. Approximate 3-dimensional ellipsoid fit of S. aureus bacteria. Ellipses
are initially fit to the 2-dimensional STORM average shifted histogram projection
using Equation 3.29, shown as pink outlines. The major and minor radii for each
ellipsoid are shown as red dotted lines. Extrapolation into the ellipsoids was
achieved by assuming the minor radius in the axial direction was equal the minor
radius in the planar direction. Equation 3.32 was used to find the final parametric
description of the ellipsoid.
3.3.6. Normalising localisation position within different bacteria using the major
radius fraction
To investigate the spatial distributions of G3-Cy3B localisations, a method
that normalises the radial distance of the localisation from the centre is required. By
definition, ellipsoids have a non-uniform radial distance from the centre to the
surface, unlike a sphere where the distance from the centre to the surface is the same
regardless of azimuthal or polar angle. To accommodate for this and the variation in
bacteria sizes, a normalised major radius fraction was developed.
To explain the method for finding the major radius fraction, consider a single
localisation found somewhere within the volume of the ellipsoid used to represent
100
the bacteria membrane. By maintaining the eccentricity of the membrane ellipsoid, 휀
defined by,
휀 = √1 −𝑟𝑚𝑖𝑛2
𝑟𝑚𝑎𝑗2 .
Progressively smaller ellipsoids are generated until the localisation intercepts the
surface of the new smaller ellipsoid. The major radius of the localisation ellipsoid,
𝑟𝑚𝑎𝑗𝑙𝑜𝑐 is divided by the membrane major radius value to give the major radius
fraction. Figure 3.10 shows the methodology described in this section with an
isometric projection.
Figure 3.10. Projection of method used to normalise localisation positions amongst
different cell ellipsoid sizes. A smaller (or larger) ellipsoid with the same
eccentricity and centre as the bacteria was generated such that the fluorescence
localisation intercepted the surface of the new ellipsoid. As the eccentricity is the
same, the position within the cell can be put in terms of the major radius fraction. For
the figure above, 𝑟𝑧0 and 𝑟𝑥0 are the initial major and minor radii and 𝑟𝑧 and 𝑟𝑥 are
the major and minor radii associated with the surface intercepting the fluorophore.
Eq. 3.33
101
3.3.7. Characterising localisation spatial distributions through Monte Carlo
simulations
Monte Carlo simulations were employed to assess the spatial distributions of
localisations within all cells, as simply plotting a normalised histogram of the major
radius fraction would be weighted incorrectly by the geometry of the cell. In other
words, localisations in the centre would be less common due to the fact that there is a
smaller surface area associated with the ellipsoid at smaller major radius fractions
and the finite size of the G3-Cy3B molecule.
To demonstrate different possible spatial distributions, 1x106 data points were
randomly generated under three different pre-determined criteria relative to the
guiding volume of a unit sphere (representative of an ellipsoid where 휀 = 0). The
first was a uniform distribution of data points confined only to the volume of the
sphere. In order to assure spatial uniformity, random values were generated from a
uniform distribution with bounds between 0 and 1. Generating said localisations
started with the following input variables: 𝑟𝑟𝑎𝑛𝑑 is a uniformly random radius
between 0 and 1, 휃𝑟𝑎𝑛𝑑 is a uniformly random angle in radians between 0 and 2π and
𝑢 is a uniformly random value between -1 and 1. From there, the Monte Carlo
localisation, 𝑀𝐶(𝑥, 𝑦, 𝑧) is generated by,
𝑀𝐶(𝑥, 𝑦, 𝑧) = (𝑟𝑟𝑎𝑛𝑑 cos(휃𝑟𝑎𝑛𝑑)√1 − 𝑢2 , 𝑟𝑟𝑎𝑛𝑑 sin(휃𝑟𝑎𝑛𝑑)√1 − 𝑢2 , 𝑟𝑟𝑎𝑛𝑑𝑢)
and the major radius fraction was subsequently calculated by finding the radial
distance of the point to the origin. Uniformly randomly distributed points within a
bounding cube of side length equal to 2 were also generated to include a consistent
‘noise’ signal. Figure 3.11a shows the cross section of all generated data points,
where colour is used to represent the radius fraction. Also included in the figure are
three histograms. The first histogram shows the normalised radius fraction frequency
density of the generated sphere-bound localisations, the second histogram shows the
normalised radius fraction frequency density of the ‘noise’ localisations and the third
histogram is a composite of the two. As can be seen in Figure 3.11a, despite having
an equal number density of localisations throughout the sphere, the histogram shows
an increasing trend towards a radius fraction of 1, highlighting the need to weight the
histogram values by the surface area of an equivalent sphere of radius equal to the
Eq. 2.34
102
radius fraction. Analytically, the radial distribution for uniformly distributed points
within a sphere is the same as the radial distribution of the volume of a sphere;
however points were generated in the way described above as to lend to the other
distributions required.
Two other radial distributions were generated by changing the distribution of radii
values from a uniform distribution to a normal distribution defined by the probability
density function,
𝑓(𝑥|𝜇, 𝜎) =1
𝜎√2𝜋𝑒−(𝑥−𝜇)2
2𝜎2 for 𝑥 ∈ ℝ
where 𝜇 is the mean and 𝜎 is the standard deviation. Following the same format as
Figure 3.11a, the normal distribution used in Figure 3.11b has 𝜇 = 1 and 𝜎 = 0.1, to
represent localisations favourably positioned around the unit sphere membrane.
Similarly, for Figure 3.11c the normal distribution has 𝜇 = 0 and 𝜎 = 0.5, to represent
localisations concentrated in the centre.
Eq. 3.35
103
Figure 3.11. Monte Carlo simulations of localisations within a unit sphere that are
a) uniformly distributed with a cut-off at the membrane, b) normally distributed with
𝜇 = 1, 𝜎 = 0.1 and c) normally distributed with 𝜇 = 0, 𝜎 = 0.5. Probability density
histograms as a function of major radius fraction are included for each condition.
a)
b)
c)
104
To remove the influence of available surface area described earlier, Figure 3.12
shows the histogram bin frequency densities divided by the relative surface area of
the equivalent ellipsoid with a radius equal to the radius fraction shown.
Localisations that are preferentially located on the membrane result in a distinct peak
at a radius fraction of 1. On the other hand, continually descending weighted
frequency densities indicate no bias for membrane localisation.
Figure 3.12. Probability density histogram of Monte Carlo localisations as a function
of major radius fraction weighted by the relative surface area associated to the sphere
with radius equal to the major radius fraction. The distribution of localisations were
uniformly distributed with a cut-off at the membrane, membrane weighted (normally
distributed with 𝜇 = 1, 𝜎 = 0.1) and radially weighted (normally distributed with
𝜇 = 0, 𝜎 = 0.5).
3.3.8. Combining individual localisations temporally into single ‘blinks’
As the fluorescence intermittency of the Cy3B fluorophore can result in a
single blink lasting longer than 2.5 ms (the exposure time of the camera), temporal
segmentation is required to isolate single ‘blinks’. Therefore, a ‘blink’ is defined as a
sequence of detectable fluorescence point spread functions that were successfully
fitted by the ThunderSTORM algorithm that occurred concurrently in time (i.e. over
adjacent frames) and had a positions that overlapped by both 𝜎𝑥𝑦 and 𝜎𝑧 (the
standard deviation of the 2D Gaussian integral in the xy and z planes). A potential
criticism of this method is that it cannot account for multiple nearby fluorophores
that coincidentally fluoresce within the time period of image acquisition. Cases like
this were minimised by having a low density of fluorophores, reducing the total
105
number of blinks over the experiment time course, and minimising the exposure time
of the camera to its lowest setting.
By analysing the displacement trajectories of individual blinks, the interaction the
G3-Cy3B molecule has with the cell components can be assessed. The analysis
protocol follows the same principles of the passive microrheology that was the main
methodology behind the work shown in Chapter 2. However, due to the small track
lengths, mean square displacement (MSD) values cannot be used due to their
intrinsically low statistical weight. Therefore, step size at each time point will be
used instead. Furthermore, as diffusion power law coefficients cannot be extracted
by taking the linear fit of the log MSD against log time interval (see Equation 2.36),
a newly developed neural net-based method described in Section 2.3.9 will be used.
3.3.9. Characterising fractional Brownian motion with DeepExponent
As a means to overcome the low statistical weight associated with MSDs
calculated from a small number of time intervals available, the anomalous
trajectories of G3-Cy3B molecules within the bacteria were instead analysed using a
newly developed deep learning feedforward neural network called DeepExponent.
Using simulated fractional Brownian trajectories as training data, DeepExponent is
capable of determining anomalous exponents with a mean absolute error of ~0.1,
directly from previously unseen trajectory data consisting of as few as 7 data points.
3.3.10. Radial dependence of step size and anomalous exponent using the structural
similarity index (SSIM)
Another investigation path was to determine if there was any spatial
correlation of G3-Cy3B step-size (displacement over 2.5 ms frame intervals),
extracted MSD at a reference time interval of 20 ms or anomalous exponents. For
example, if G3-Cy3B molecules had a stronger interaction with the bacterial
membrane than intercellular components, all of the values stated above would have a
bias of being smaller at major radius fractions close to 1. To test this hypothesis, 2D
histograms of the above quantities were plotted against major radius fraction and
compared quantitatively with 2D histograms of randomly correlated variables, but
maintain the same spatial distribution (i.e. each localisation is the same position, but
the associated variables are reassigned at random). Figure 3.13 shows three 2D
histograms with direct correlation, 50% correlation and no correlation (random
assignment) between fluorophore displacement (step size) and major radius fraction
106
proximity to the membrane (i.e. a major radius fraction of 1). The distribution of
fluorophore displacements was randomly generated to match the distribution of G3-
Cy3B localisations within S. aureus bacteria.
Pairs of 2D histograms can be compared quantitatively using the structural similarity
index (SSIM), which was originally developed to assess the loss of image quality
between an original reference and a compressed or distorted duplicate 113. For two
images with equal dimensions, the SSIM can be calculated between two windows of
size x and y, such that,
SSIM(𝑥, 𝑦) = (2𝜇𝑥𝜇𝑦 + 𝑐1)(2𝜎𝑥𝑦 + 𝑐2)
(𝜇𝑥2 + 𝜇𝑦2 + 𝑐1)(𝜎𝑥2 + 𝜎𝑦2 + 𝑐2)
where 𝜇𝑥,𝑦 is the average pixel value in x,y, 𝜎𝑥,𝑦2 is the variance of x,y, and 𝜎𝑥𝑦 is the
covariance of x and y.
Figure 3.13. 2-dimensional histograms showing the relationship between major
radius fraction (i.e. location within cell) and fluorophore displacement (over 1 frame)
for a simulated dataset. Correlations between the variables is set at a) direct
correlation, b) 50% correlation and c) no correlation (i.e. random).
Eq. 3.36
a) b)
c)
107
3.4. Results and Discussion
3.4.1. Spatial distribution of fluorescence blinks within cells
Three different conditions were investigated during this study; gram-positive
S. aureus and gram-negative E. coli exposed to G3-Cy3B at 100 µM concentrations
and a peptide-negative control of S. aureus exposed to unbound Cy3B at 100 µM
concentration. In total, 212 S. aureus cells in the peptide-negative control were
analysed, whereas 102 E. coli and 349 S. aureus cells were analysed following
exposure to G3-Cy3B. Figure 3.14 shows an example of all fluorescence
localisations detected over 200,000 frames with the fitted cell membrane shown as a
translucent red mask. The frame number at which each localisation was detected is
represented by colour. Also included at the bottom of the figure is the interpolated
STORM reconstruction, where all 3D localisations have been projected onto a 2D
plane. All localisations are shown in the figure without regard for combining
localisations into single blinks (as described in Section 3.3.8), but isolated regions of
points with similar colours (indicative of a fluorescence blink lasting multiple
frames) can still be seen. Across all conditions, localisations were only selected if the
associated uncertainty in xy < 25 nm and z < 100 nm. Combining localisations into
temporally resolved blinks, an order of magnitude lower uptake of fluorophore when
not conjugated to G3 was found, with only an average of 38 ± 2 blinks per cell for
the peptide-negative control. In comparison, 365 ± 25 blinks per E.coli cell and 651
± 22 blinks per S.aureus cell were found when Cy3B was conjugated to G3. This
result indicates that any increase in fluorophore retention within and around the
bacteria must be associated with the action of G3 rather than just fluorophore
permeation.
Figure 3.15a shows the spatial distribution of fluorophore localisations that were
attributed to blinks. As described in Section 3.3.6, radial positions were normalised
across all bacteria by measuring major radius fraction relative to fitted ellipsoid
surfaces at the supposed bacteria membranes. Although Figure 3.15a initially
appears to show a mean major radius fraction of ~0.8 for S.aureus with peptide and
~1 for E.coli with peptide and S.aureus without peptide, dividing by the relative
available surface area reveals minimal preference for any location within or
surrounding the bacteria. The frequency density weighted by relative ellipsoid
surface area is shown in Figure 3.15b. Comparing with the Monte Carlo simulations
108
shown in Figure 3.12, all three conditions appear to show a radially decaying density
of fluorophores, up until a major radius fraction of ~1, where the population
decreases substantially. Weighted frequency densities close to zero at major radius
fractions larger than 1.5 confirm that there were minimal free-floating or unbound
fluorophores. A higher membrane affinity would be revealed by a relative increase in
the weighted frequency around a major radius fraction of 1, but that is not seen in
any of the conditions.
Figure 3.14. Fluorescence localisation position in 3-dimensional space acquired via
STORM over the course of 200,000 frames at 400 frames per second. A 2-
dimensional reconstructed STORM projection of all localisations is shown
underneath the actual positions. The colour of the localisations is representative of
the time the signal was acquired.
109
Figure 3.15. Normalised histograms of Cy3B blink position as a function of major
radius fraction for a peptide negative control, G3 in S. aureus and E.coli. a) shows
the initial output of the histogram, whereas b) shows the frequency density weighted
by the relative ellipsoid surface area associated with each major radius fraction
value.
a)
b)
110
No preferential clustering was observed on the membranes of any bacteria. These
results support the hypothesis that the mechanism of action for G3 is best described
by the carpet model for both S. aureus and E. coli. The carpet model states that
peptide should be homogenously distributed across the bacteria membrane, with no
clustering that could be associated with the formation of pore-like complexes. As
there was no density preference at any radial distance from the centre of both gram-
positive and gram-negative bacteria, the interaction strength between the peptide and
the membrane and the peptide and the intercellular components must be of
comparable magnitude. If there was a stronger interaction with the membrane
specifically, the number density of detected G3 should appear more like the Monte
Carlo simulation for a weighted distribution (see Figure 3.12).
A caveat to this approach is that it may not be sensitive to clustering at different
radial layers if the average peptide density for clustering and homogeneous
distributions was approximately the same. Whilst the method shown here is
sufficient for determining the spatial distributions as a function of radius, and direct
observation of the average shifted histogram 2-dimensional projections suggest no
clustering, alternative methods may be required to quantify the volumetric
distributions. One approach could be to extend the method applied in Chapter 2,
namely Ripley’s K-function. As the function is intended for 2-dimensional
distributions, axial slices could be taken at incremental heights and analysed in the
same way as Chapter 1. Alternatively, the localisations only at the surface of
growing ellipsoids could be projected onto a two-dimensional surface using a
pseudo-cylindrical projection, but corrections would have to be incorporated to
account for distortion and periodic boundary conditions. A simpler approach would
be to extend Ripley’s K-function into the third dimension, whereby the Euclidean
separation was still calculated, but spheres of increasing radius would be used rather
circles.
3.4.2. Blink trajectories
Figure 3.16 shows an example S.aureus cell with 4 blink tracks shown and
the associated uncertainties at each time point. Figures 3.17a-c shows the MSDs
associated to all tracked Cy3B blinks for S.aureus and E.coli with peptide and
S.aureus without peptide. As can be seen from the figures, several MSDs decrease at
larger time intervals, highlighting the need for an alternative method to calculate the
111
anomalous diffusion exponent. A cause of a decreasing MSD can be attributed to a
small trajectory that ‘circles back’ in on itself. Consider an equally spaced trajectory
around an arbitrarily sized circle in the xy plane, with no movement in z. The
maximum MSD value will equal the circle diameter squared and will occur at a time
interval equal to half the total trajectory time. Subsequent larger time intervals will
have decreasing MSD values, culminating in the final MSD value at the largest time
interval being equal to the MSD at the smallest time interval. As stated in Section
3.3.4, erroneous signals attributed to camera noise are removed before further
classification. As with the tracking data in Chapter 2, systematic vibrational noise
would appear in the MSD as an apparent oscillatory function. As this does not
appear in the data, it can be assumed that vibrational sources of noise were not
encountered or occurred on timescales longer than that detected in this experiment.
Figure 3.16. Four example blink trajectories of G3-Cy3B molecules within a
S. aureus cell, captured using fluorescent single-particle tracking. Localisation errors
are attributed based on the least-squares fitting of an astigmatic Gaussian integral.
Positions are extracted every 2.5 ms.
112
Figure 3.17. Fluorescence blink trajectory mean square displacements for a) G3-
Cy3b in S. aureus, b) G3-Cy3b in E. coli and c) Cy3b only (G3-negative control) in
S. aureus. Mean square displacement magnitudes that decrease at larger time
intervals are indicative of shorter blinks as the statistical weight for larger time
intervals is smaller.
a)
b)
c)
113
Anomalous exponents associated with the blink trajectories were found using the
neural net DeepExponent. Figure 3.18 depicts histograms of the mean anomalous
exponent per track, 𝛼 for all three conditions. Following fitting with a Gaussian
distribution, it was found that Cy3B in S. aureus cells had a mean exponent of
0.41 ± 0.08, G3-Cy3B in E. coli cells had a mean exponent of 0.41 ± 0.08 and G3-
Cy3B in S. aureus cells had a mean exponent of 0.42 ± 0.09. Therefore, no
statistically significant difference between conditions was observed. Corroborating
with this evidence is the distribution of step sizes of all subsequent points in all
tracks, shown in Figure 3.19a, which is approximately the same for all conditions.
A limitation of the DeepExponent analysis method is that it assumes that the MSD
maintains the same power-law exponent at the infinite time limit. As some of the
MSD signals appear to 'level off' at larger time intervals, it may be possible that
some of the G3 trajectories are space-restricted, invalidating the limit assumed. The
exponents extrapolated here should be stated under the caveat of the assumption that
DeepExponent can investigate only the instantaneous dynamic behaviour of the G3
within the timescales measured.
Figure 3.18. Histogram of mean 𝛼 values extracted from passing blink trajectories
through the neural net DeepExponent for Cy3B only (in S. aureus cells), G3-Cy3B
in E. coli cells and G3-Cy3B in S. aureus cells.
114
An average 𝛼-value of ~0.41 strongly suggests the movement of G3 within the
bacteria is, broadly, sub-diffusive. However, as the peptide-negative control (i.e. just
the fluorophore) also had a similar 𝛼-value, the sub-diffusive motion may be a
consequence of entrapment inside the viscous cell interior.
Figure 3.19. a) Normalised probability histogram showing fluorophore displacement
between adjacent frames for Cy3B only (in S. aureus cells), G3-Cy3B in E. coli
cells and G3-Cy3B in S. aureus cells. b) Examples of blink trajectories for G3-Cy3B
in S. aureus cells where colour represents the magnitude of the fluorophore
displacement.
a)
b)
115
In order to investigate any correlative effect between radial position and movement
mechanics, several parameters were plotted against major radius fraction in 2-
dimensional histograms. Parameters chosen to test included MSD at a reference time
interval of 20 ms, anomalous exponent (𝛼) and fluorophore displacement over a
single 2.5 ms time step (referred to as step size). A visual representation of the
variation in step size for fluorophore tracks within a single example S.aureus cell can
be seen in Figure 3.19b, where colour is used to represent displacement in nm.
Figures 3.20a-c and 3.21a-c show normalised 2D histograms for all three conditions
(S. aureus with Cy3B, E. coli with G3-Cy3B and S. aureus with G3-Cy3B) of the
parameters 𝛼-value and fluorophore displacement step-size against the major radius
fraction respectively. To quantify these histograms, they were compared with
randomly correlated equivalents and the SSIM was calculated, which are collected in
Table 3.1. As can be seen, all SSIM values are approximately or greater than 0.9,
indicating a 90% similarity between the experimentally extracted values and
randomly correlated values. This result suggests that there is no correlation between
radial position and any of the parameters measured in this experimental setup.
Condition SSIM, radius against
alpha (compared with
random)
SSIM, radius against
step size (compared
with random)
Cy3B-only in S. aureus 0.9862 ± 0.0005 0.9784 ± 0.0007
G3-Cy3B in E. coli 0.953 ± 0.001 0.932 ± 0.002
G3-Cy3B in S. aureus 0.916 ± 0.002 0.892 ± 0.003
Table 3.1. Structured similarity index (SSIM) between normalised 2-dimension
histograms of neural net extrapolated 𝛼-value and fluorophore step size against
major radius fraction (indicative of location within the cell) and randomised
matching of the same variables. All values are approaching 1, suggesting a high
degree of similarity with the randomly assigned variables, pointing towards no
positional correlation. Conditions tested included Cy3B only (in S. aureus cells), G3-
Cy3B in E. coli cells and G3-Cy3B in S. aureus cells.
116
Figure 3.20. 2-dimensional histogram showing relationship between neural-net
extracted 𝛼 values and corresponding major radius position for a) G3-Cy3b in
S. aureus, b) G3-Cy3b in E. coli and c) Cy3b only (G3-negative control) in
S. aureus.
a)
b)
c)
117
Figure 3.21. 2-dimensional histogram showing relationship between fluorophore
displacement and corresponding major radius position for a) G3-Cy3b in S. aureus,
b) G3-Cy3b in E. coli and c) Cy3b only (G3-negative control) in S. aureus.
a)
b)
c)
118
These results further support the hypothesis that the mechanism of action for G3 can
be described by the carpet model. Despite having a range of up to two orders of
magnitude, the diffusion coefficient associated with the G3-Cy3b and Cy3b
movement can be extrapolated at a reference time of 10 ms using Equation 2.35 from
the Chapter 2. For all conditions, the mean diffusion coefficients were found to be
0.09 ± 0.01 µm2s-1, which is comparable in magnitude to the values found in the
literature for a transient interaction with the membrane 93, but an order of magnitude
larger than what was interpreted as the diffusion coefficient associated to peptides in
the pore formation 95. The lack of correlation between the fluorophore step-size and
radial position indicates that the magnitude of the interaction between the peptide
and the intercellular components was also comparable.
It may still be possible for the peptide to have a unique interaction with the
membrane if the timescale of the interaction is less than the framerate of the camera
used to acquire tracks. Whilst higher framerate cameras could be employed, the
trade-off is localisation precision, as a higher framerate necessitates a faster shutter
speed, which in turn collects fewer photons. A reduction in the number of photons
collected results in a greater uncertainty in the localisation. A solution to this
problem could be to conjugate brighter fluorophores or quantum dots to the peptide.
Furthermore, it may be possible that the interaction between the peptide and the
surface occurs immediately upon exposure, followed by the peptide inhabiting the
interior of the cell, masking any small clusters that may be apparent on the
membrane surface. However, one might expect to continue seeing small cluster
remnants on the membrane surface or at the least some morphological damage to cell
(which should have been apparent even optically).
119
3.5. Conclusion
In order to ascertain the mechanism of action of the cationic 𝛼-helical peptide
G3 in gram-positive and gram-negative bacteria, single-particle fluorescence
tracking in combination with 3-dimensional STORM was used to extrapolate the
spatial distributions and diffusive dynamics of the peptide. The axial positions of
peptides conjugated to photoswitchable fluorophores were found by inducing a
height-based astigmatism to the intensity profiles of the fluorescence signals.
Bacteria shape, and therefore membrane position, was found by using a projection of
a volumetric STORM image and elliptical fitting. G3-Cy3B had a homogeneous
distribution throughout both S. aureus and E. coli, as did the peptide negative
control. Whilst G3-Cy3B exhibited sub-diffusive motion and diffusion coefficients
similar to what has been found in the literature, no correlation to radial position was
identified. These results indicate no preferential binding to the membrane, at least on
the timescales used in this experimental setup, and so indicate that the mechanism of
action for G3 can be best described by the carpet model.
120
Chapter 4. Interactions between cationic peptide G3 and
𝒅54-DMPC/DMPG bilayers explored by 31P and 2H solid
state nuclear magnetic resonance
4.1. Introduction and Aims of Chapter
Solid-state nuclear magnetic resonance (ssNMR) is a method of probing the
interactions between molecules by exploiting the intrinsic angular momentum
property known as spin. The technique employed in this chapter is the only
methodology of measuring the interactions between molecules on the millisecond
and microsecond timescales within an ensemble collective. In this study, static and
magic angle spinning (MAS) 31P ssNMR and quadrupolar 2H spectra will be used to
investigate the interaction strength of cationic peptide G3 on constructed 𝑑54-
DMPC/DMPG phospholipid bilayers.
4.2. Theory
4.2.1. Spin states inside a magnetic field
The most important and fundamental concept of nuclear magnetic resonance
(NMR) is that fermions possess an intrinsic quantum angular momentum referred to
as ‘spin’. A system with non-zero spin, 𝑆, also has proportional magnetic dipole
moment, �� given by,
�� = 𝛾𝑆
where the proportionality constant, 𝛾 is the called the gyromagnetic ratio. A similar
proportionality exists between the angular momentum and magnetic dipole moment
in a classical system when a charged sphere is rotating. Therefore, the classical
angular momentum of a rotating sphere is analogous to quantum spin, the caveat
being the quantisation. 1H, consisting of only one proton, has spin quantum number,
𝑠 = 1 2⁄ and as such has total spin angular momentum, 𝑆 given by,
𝑆 = ℏ√𝑠(𝑠 + 1) = ℏ√3
2 .
Eq. 4.1
Eq.4 .2
121
The spin component along the 𝑧-axis, 𝑠𝑧, is quantised by the magnetic quantum
number, 𝑚𝑠, which can take values between –𝑠 and 𝑠 in integer steps (i.e. there are
2𝑠 + 1 possible values of 𝑚𝑠 for a given 𝑠). As the magnitude of 𝑠𝑧 is given by,
𝑠𝑧 = 𝑚𝑠ℏ
it can be inferred from Equation 4.1 that the z-component of the magnetic moment,
𝜇𝑧 = 𝛾𝑚𝑠ℏ. The two possible values of 𝑚𝑠 = ±12⁄ for 𝑠 = 1 2⁄ systems correspond
to two spin states often referred to as ‘spin-up’ and ‘spin-down’. In the absence of a
magnetic field, the two states are degenerate. However, if a magnetic field parallel to
the 𝑧-axis, 𝐵𝑧, is applied, the energy levels split according to the direction of the
magnetic moment (referred to as ‘Zeeman splitting’). The difference in energy
between the spin states, Δ𝐸, is therefore given by,
Δ𝐸 = −�� (𝑚𝑠 = −1
2) . �� + �� (𝑚𝑠 =
1
2) . �� = 𝛾ℏ𝐵𝑧
and it is also clear that the spin-up state has lower energy. As there are two possible
energy states for the spin system to be in, the relative probabilities of being in either
of the states can be calculated using a Boltzmann distribution. If the probabilities of
being in the higher energy or lower energy state are denoted by 𝑃(𝑚𝑠 = −12⁄ ) and
𝑃(𝑚𝑠 =12⁄ ) respectively then,
𝑃(𝑚𝑠 =12⁄ )
𝑃(𝑚𝑠 = −12⁄ )= exp (
Δ𝐸
𝑘𝐵𝑇) = exp (
𝛾ℏ𝐵𝑧𝑘𝐵𝑇
)
where 𝑘𝐵 is the Boltzmann constant and 𝑇 is the system temperature. NMR
experiments involve the measurement of the net magnetization, however to
demonstrate the sensitivity limitation, consider an ensemble of 1H nuclei (𝑠 = 1 2⁄ )
in an external magnetic field of magnitude 9.39 T (the typical central field strength
for commercial nuclear magnetic resonance machines). The gyromagnetic ratio for a
1H nucleus, 𝛾1𝐻 = 267.52 × 106 rad T-1 s-1, so for an experiment carried out at 310
K, the probability ratio from Equation 4.5 above is equal to 1.000062. Therefore at
Eq. 4.3
Eq. 4.4
Eq. 4.5
122
thermal equilibrium, if there are 1,000,000 1H nuclei in the magnetic field, the
overall net magnetization aligned parallel to the external magnetic field would only
come from 32 surplus nuclei in the lower energy state 114.
In NMR experiments, energy is transferred to the lower energy spin states by
photons with energy equal to Δ𝐸. Using the values from the population scenario
above, the excitation photons would need to have frequency, 𝜈 = 2𝜋𝛾𝐵𝑧 ≈ 10 GHz
(approximately in the radio regime). The term ‘resonance’ is used explicitly, as
photons with energy not equal to Δ𝐸 are not absorbed by the nuclei and consequently
do not influence the net magnetization. If the radio frequency (RF) pulse is applied
orthogonally to the external magnetic field, the net magnetization effectively rotates,
which can be mathematically explained using the Bloch equations (see Section
4.2.2).
It should be noted here that the idea of magnetic moments aligning with the external
magnetic field is a simplification in order to avoid a full quantum mechanical
explanation. In actuality, a single spin system exists as a superposition of both up
and down spin states. When an external magnetic field is applied, the weighting of
the two states changes resulting in a preferential bias towards the spin up state. It is
important to distinguish that a measurement of the net magnetization does not
perturb the system enough to collapse the individual nucleus superpositions into their
eigenstates (as opposed to experiments carried out with a Stern-Gerlach apparatus).
Furthermore, when a transverse RF pulse is applied to the two-state system, it will
undergo Rabi oscillations whereby the probability of finding the system in the higher
energy state oscillates with frequency equal to pulse. The overall effect is transversal
net magnetisation, not cancellation 115.
4.2.2. Bloch equations
As stated in the previous section, NMR experiments involve the
measurement of the net magnetization of a collection of nuclei rather than individual
nuclear moments. The net magnetization, �� is the sum of all individual magnetic
moments, such that
�� = ∑��𝑖
𝑁
𝑖=1
Eq. 4.6
123
where 𝑁 is the total number of nuclei in the ensemble. In the presence of an external
magnetic field, ��, the net magnetization experiences a torque which can be
expressed in terms of the rate of change of the net magnetization such that,
𝑑��
𝑑𝑡= 𝛾(�� × ��)
where the notation for time dependence is implicit and unless otherwise stated the
external magnetic field is fixed in the 𝑧-direction as stated previously
(i.e. �� = (0,0,𝐵𝑧)). On the assumption that the solution to Equation 4.7 will be
oscillatory, consider the matrix for a clockwise rotation about the 𝑧-axis, ��𝜔(𝑡), with
the form,
��𝜔(𝑡) = (cos (𝜔𝑡) sin (𝜔𝑡) 0−sin (𝜔𝑡) cos (𝜔𝑡) 0
0 0 1
)
where 𝜔 is the angular frequency of rotation (the time dependence will also be
omitted from the notation from now on). As we expect the net magnetization to be
rotating, we can set �� to be of the form �� = ��𝜔��0 where ��0 is the initial net
magnetization. Furthermore, the magnetic field can be set at as �� = ��𝜔��0 without
consequence as magnetic field is aligned with the 𝑧-axis. Subbing in the new
definitions into Equation 4.7,
𝑑��
𝑑𝑡− 𝛾(�� × ��) =
𝑑��𝜔𝑑𝑡 ��0 − 𝛾(��𝜔��0 × ��𝜔��0)
which can be immediately simplified by using the matrix identity,
(����) × (����) = (det ��)(��−1)𝑇(�� × ��) and realising that ��𝜔 is orthogonal, such that,
𝑑��
𝑑𝑡− 𝛾(�� × ��) =
𝑑��𝜔𝑑𝑡 ��0 − 𝛾��𝜔(��0 × ��0) .
Eq. 4.7
Eq. 4.8
Eq. 4.9
Eq. 4.10
124
By factoring out ��𝜔, the first term on the right hand side of Equation 4.10 can be
evaluated directly as,
(��𝜔)𝑇𝑑��𝜔𝑑𝑡 ��0 = (
cos(𝜔𝑡) −sin(𝜔𝑡) 0sin(𝜔𝑡) cos(𝜔𝑡) 00 0 1
)(−ωcos(𝜔𝑡) ω sin(𝜔𝑡) 0−ωcos(𝜔𝑡) −ω sin(𝜔𝑡) 0
0 0 0
)(
𝑀0𝑥𝑀0𝑦𝑀0𝑧
) = −𝜔(−𝑀0𝑦𝑀0𝑥0
)
which is equivalent to the cross product of the 𝑧-axis unit vector, �� and the initial
magnetization,
(��𝜔)𝑇𝑑��𝜔𝑑𝑡 ��0 = −𝜔(
𝑀0𝑥𝑀0𝑦0
) = 𝜔(�� × ��) .
As the external magnetic field is orientated in the 𝑧-axis only, the 𝑧-axis unit vector
can be expressed as �� = ��0 |��0|⁄ , and so Equation 4.9 is reconstituted as,
𝑑��
𝑑𝑡− 𝛾(�� × ��) = (
𝜔
|��0|− 𝛾) ��𝜔(��0 × ��0) .
Equation 4.13 above shows that �� = ��𝜔��0 is a solution to Equation 4.7 when 𝜔 =
𝛾|��0|, which is called the ‘Larmor frequency’. The rotation of the net magnetization
is referred to as ‘precession’ about the magnetic field and is shown in Figure 4.1.
Figure 4.1. Net magnetization, ��(𝑡) precession about a 𝑧-axis magnetic field, 𝐵𝑧 at
the Larmor frequency, 𝑤. For nuclear magnetic resonance experiments, the initial net
magnetization, ��0 at time, 𝑡 = 0, is the result of the energy difference between
nuclei spin states.
Eq. 4.11
Eq. 4.12
Eq. 4.13
125
In ensembles of nuclei, two additional relaxation terms must be included in the
equations of motion describing the net magnetization. It is convenient to first
segment the net magnetization into a longitudinal component, ��∥(= (0,0,𝑀𝑧)) and
transverse component, ��⊥(= (𝑀𝑥 , 𝑀𝑦, 0)). The first relaxation term is related to the
exponential recovery of the longitudinal component of the magnetization to the
thermal equilibrium value, 𝑀𝑧,𝑒𝑞, with occurs with characteristic time constant, 𝑇1
called the ‘spin-lattice’ relaxation time. To explain this phenomenon, consider that a
spin state of a nucleus inside a molecule experiences a complex magnetic field
caused by the vibrational and rotational motion of the surrounding atoms, called the
‘lattice’. Energy can be distributed between the spin state and the lattice, effectively
increasing the lattice temperature which is then dissipated as the system returns to
equilibrium. Therefore, the magnitude of 𝑇1 is highly dependent on the lattice
mobility.
The second relaxation effect is related to the exponential decay of the transverse
magnetization caused by the individual spin states experiencing magnetic field
inhomogeneities due to the surrounding chemistry anisotropy within the molecule,
which results in a loss of net transverse magnetization. The characteristic time
constant for the expoential transverse decay, 𝑇2 is called the ‘spin-spin’ relaxation
time. Incorporating the two relaxation terms into Equation 4.13 results in,
𝑑��
𝑑𝑡= 𝛾(�� × ��) −
1
𝑇1(��∥ − ��𝑒𝑞) −
1
𝑇2��⊥
where ��𝑒𝑞 is the net magnetization at thermal equilibrium (��𝑒𝑞 = (0,0,𝑀𝑧,𝑒𝑞).
Endeavours to find a solution to Equation 4.14 above are simplified by working in a
frame of reference that is rotating around the 𝑧-axis with angular frequency, 𝜓. This
is readily achieved by multiplying the net magnetization by the transpose of the
rotation matrix defined in Equation 4.8, such that
�� = ��𝜓𝑇��
Eq. 4.14
Eq. 4.15
126
where �� is the net magnetization in the rotating reference frame. In the rotating
frame of reference, the rate of change of net magnetization must be evaluated using
the chain rule, as the rotation matrix is implicitly a function of time, producing,
𝑑��
𝑑𝑡=𝑑��𝜓
𝑇
𝑑𝑡��𝜓�� + ��𝜓
𝑇 𝑑��
𝑑𝑡 .
A direct substitution of Equation 4.14 can be made in the second term on the right-
hand side of Equation 4.16 to yield,
��𝜓𝑇 𝑑��
𝑑𝑡= 𝛾 (�� × ��𝜓
𝑇��) −
1
𝑇1(��∥ − ��𝑒𝑞) −
1
𝑇2��⊥
where ��∥, ��⊥ and ��𝑒𝑞 are the longitudinal, transverse and equilibrium
magnetizations in the rotating frame of reference. By following the same direct
method as in Equation 4.13, the first term of the right-hand side of Equation 4.16 can
be found to be,
𝑑��𝜓𝑇
𝑑𝑡��𝜓�� = −𝜓(�� × ��) .
Therefore, after rearrangement, Equation 4.16 can be written as,
𝑑��
𝑑𝑡= 𝛾 (�� × (��𝜓
𝑇�� −
𝜓
𝛾��)) −
1
𝑇1(��∥ − ��𝑒𝑞) −
1
𝑇2��⊥ .
As there is always an external magnetic field parallel to the 𝑧-axis, the first term on
the right hand side of Equation 4.19 can be expressed as,
𝛾 (�� × (��𝜓𝑇�� −
𝜓
𝛾��)) = 𝛾 (�� × (��𝜓
𝑇Δ�� + (1 −
𝜓
𝛾|��0|) ��0))
where �� is the magnetic field component that is restricted to the transverse plane.
From the definition stated above, if the frame of reference rotates at the Larmor
Eq. 4.16
Eq. 4.17
Eq. 4.18
Eq. 4.19
Eq. 4.20
127
frequency, the second term in Equation 4.20 becomes zero. Hence, the final equation
of motion describing the net magnetization in a frame of reference rotating at the
Larmor frequency is,
𝑑��
𝑑𝑡= 𝛾 (�� × ��𝜔
𝑇Δ��) −
1
𝑇1(��∥ − ��𝑒𝑞) −
1
𝑇2��⊥ .
Two scenarios are of particular interest in NMR experiments, namely excitation by a
transverse RF pulses and relaxation to thermal equilibrium, which is what is actually
detected experimentally. Consider first a net magnetization that is wholly orientated
in the positive 𝑧 direction, such that ��0 = ��0 = (0,0,𝑀𝑧). An RF pulse incident on
the nuclei ensemble can be described mathematically by setting the transverse
magnetic field component, Δ�� = ��𝜔𝐵1(𝑡), where we shall define the incident
magnetic field as 𝐵1(𝑡) = |𝐵1(𝑡)|��. By assuming the pulse occurs on a timescale
much shorter than the relaxation timescales (i.e. 𝑡 ≪ 𝑇1,2), then Equation 4.21
becomes,
𝑑��
𝑑𝑡= 𝛾|𝐵1(𝑡)|(�� × ��) .
By noting the similarity between Equations 4.12 and 4.7, we can infer that a solution
to Equation 4.22 is,
�� = ��(휃)��0
where
��(휃) = (1 0 00 cos (휃) sin (휃)0 −sin (휃) cos (휃)
) when 휃 = 𝛾∫|𝐵1(𝑡′)|𝑑𝑡′
𝑡
0
where 휃 is referred to as the ‘pulse angle’, which is a function of pulse length, 𝑡.
Eq. 4.21
Eq. 4.22
Eq. 4.23
128
From the form of ��(휃), it becomes clear that a transverse pulse acts to rotate the net
magnetization around the axis of incidence. In the lab reference frame, the net
magnetization is therefore,
��(𝑡) = ��𝜔��(휃)��0 = (
𝑚0𝑧 sin(𝜔𝑡)sin (휃)
𝑚0𝑧 cos(𝜔𝑡)sin (휃)𝑚0𝑧cos (휃)
) .
As the magnitude of the pulse magnetic field is entrenched in the energy level
separation of the spin states, the angle the net magnetization rotates by is determined
by the pulse length. Figures 4.2a-c show the beginning, middle and end of a
𝜋2⁄ -pulse 116.
We now consider the relaxation of the magnetization and recovery to the equilibrium
value. In this scenario, we assume the transverse magnetic field component to be
equal to zero, such that Equation 4.21 becomes
𝑑��
𝑑𝑡= −
1
𝑇1(��∥ − ��𝑒𝑞) −
1
𝑇2��⊥
which can be solved easily by dimensional decoupling, such that
��∥ = ��0∥ exp (−𝑡
𝑇1) + ��𝑒𝑞 (1 − exp (−
𝑡
𝑇1))
��⊥ = ��0⊥ exp (−𝑡
𝑇2) .
In the lab reference plane, the magnetization therefore relaxes as,
��(𝑡) = ��𝜔�� =
(
(𝑚0𝑥 cos(𝜔𝑡) +𝑚0𝑦sin (𝜔𝑡))exp (−
𝑡
𝑇2)
(−𝑚0𝑥 sin(𝜔𝑡) +𝑚0𝑦cos (𝜔𝑡))exp (−𝑡
𝑇2)
𝑚0𝑧exp (−𝑡
𝑇1)
)
+(
00
|��𝑒𝑞| (1 − exp (−𝑡
𝑇1))) .
For an initial net magnetization wholly orientated along the 𝑥-axis, the subsequent
decay is illustrated in Figures 4.3a-c.
Eq. 4.24
Eq. 4.25
Eq.4.26
Eq. 4.27
129
Figure 4.2. Net magnetization, ��(𝑡) in the lab reference frame before (a), during (b)
and after (c) a transverse magnetic field is applied. The length, 𝑡 and orientation, 𝐵𝑥
of the pulse is such that the net magnetization make a 𝜋 2⁄ rotation about the 𝑥-axis.
It is assumed that the pulse length is also much shorter than the relaxation constants
(i.e. 𝑡 ≪ 𝑇1,2). As it rotates, the net magnetization precesses about the 𝑧-axis with
angular frequency equal to the Larmor frequency, 𝜔.
a)
b)
c)
130
Figure 4.3. Net magnetization, ��(𝑡) in the lab reference frame initially orientated
along the 𝑥-axis (a), relaxing with characteristic timescales, 𝑇1 and 𝑇2 (b) and
completely returned to thermal equilibrium (c) in the presence of a 𝑧-orientated
external magnetic field, 𝐵𝑧. As the net magnetization relaxes, it precesses about the
𝑧-axis with angular frequency equal to the Larmor frequency, 𝜔.
a)
b)
c)
131
Relaxation of the transverse magnetization emits an electromagnetic wave with
angular frequency equal to the Larmor frequency. If a metal wire coil is placed
parallel to the transverse plane, the electromagnetic wave will induce an
electromotive force, 𝐸𝑀𝐹(𝑡) given by,
𝐸𝑀𝐹(𝑡) = −𝑑𝜙(𝑡)
𝑑𝑡
where 𝜙(𝑡) is the flux of the electromagnetic wave through the surface defined by
the coil. As an example, when the coil is parallel to the 𝑦-axis, the flux is
proportional to the component of the net magnetization given in Equation 4.27 in the
𝑥 direction. Therefore,
𝐸𝑀𝐹(𝑡) ∝ |��0| (1
𝑇2cos(𝜔𝑡) + 𝜔 sin(𝜔𝑡)) exp (−
𝑡
𝑇2)
which is referred to as the ‘free induction decay’ (FID). A Fourier transform of the
FID yields peaks at the resonant frequencies associated with photons that facilitate
spin state excitation and is the typical output of a NMR experiment 117.
4.2.3. Chemical shift anisotropy and magic angle spinning
When defining the spin-lattice timescale, 𝑇1 in Section 4.2.2, it was briefly
mentioned that the nuclei experience a complex composite magnetic field rather than
just the external field in isolation. In fact, it is the variations in the magnetic field
experienced by the nuclei that facilitate the majority of NMR experiments. As the
valence electrons surrounding the nuclei are negatively charged, the presence of the
external magnetic field induces a Lorentz force causing a circular motion in the
transverse plane. The Biot-Savart law states that moving charges (akin to a current)
generate a perpendicular magnetic field. The combination of the anti-parallel
magnetic fields can be expressed as an effective magnetic field, ��𝑒𝑓𝑓 given by,
��𝑒𝑓𝑓 = ��0(𝐼 − ��)
where 𝐼 is a 3×3 unit matrix and �� is the nuclear shielding tensor. From Equation
4.4, the energy separation between spin states is proportional to the incident
Eq. 4.28
Eq. 4.29
Eq. 4.30
132
magnetic field, which is now given by Equation 4.27. Consequently, the resonant
frequency, 𝜈 of the photon needed to excite the spin states is shifted as a result of the
effective magnetic field such that,
𝜈 = (𝛾
2𝜋) |��𝑒𝑓𝑓| .
To standardise the magnitude of the frequency difference caused by electronic
shielding, it is often expressed relative to the absolute resonant frequency of a
uniform sample, 𝜈𝑟𝑒𝑓 . Therefore, the chemical shift, Δ𝛿 can be defined as,
Δ𝛿 = (𝜈 − 𝜈𝑟𝑒𝑓)
𝜈𝑟𝑒𝑓 .
As the external field is fixed parallel to the 𝑧-axis, all terms in the nuclear shielding
tensor are zero, with the exception of 𝜎𝑧𝑧, as the circular motion of the electrons is
perpendicular to the external field. By following the previous three equations, it can
be concluded that the resultant chemical shift is inversely proportional to shielding
experienced by the nuclei. In other words, the greater the shielding experienced by
the nuclei, the lower the effective magnetic field and so the lower the energy
separation between spin states. Therefore, the excitation photon resonant frequency
is also lower and hence the chemical shift is smaller. Likewise, a larger chemical
shift is associated with a smaller degree of shielding 118.
The value of 𝜎𝑧𝑧, and therefore Δ𝛿, is dependent on the electron density distribution
surrounding the nuclei, which is a function of the molecular atomic structure and the
orientation of the molecule with respect to the external field. As the chemical shift
tensor often does not align with any particular axis, it is convenient to define the
‘principal axis’ coordinate system, which is orientated along the chemical shift
principal components, 𝛿11, 𝛿22 and 𝛿33. The designation of the principal components
follows the convention that 𝛿11 ≥ 𝛿22 ≥ 𝛿33 119. To illustrate this, consider a
molecule in the laboratory frame centred on a spin-1 2⁄ nuclei as depicted in Figure.
4.4, where the chemical shift tensor is represented by an ellipsoid. In the principal
axis reference frame, denoted by 𝑥𝑃𝐴𝑆, 𝑦𝑃𝐴𝑆 and 𝑧𝑃𝐴𝑆, the principal components are
shown as the parameters defining the ellipsoid surface. The non-zero chemical shift
Eq. 4.31
Eq. 4.32
133
tensor component in the laboratory frame, 𝛿𝑧𝑧 can be extracted from the principal
axis system components through the relation 120,
𝛿𝑧𝑧 = (sin(휃) cos(𝜑))2𝛿11 + (sin(휃) sin(𝜑))
2𝛿22 + (cos(휃))2𝛿33
where 휃 and 𝜑 are the polar and azimuthal angles relating the laboratory frame to the
principal axis. By defining the average isotropic chemical shift, 𝛿𝑖𝑠𝑜 as,
𝛿𝑖𝑠𝑜 =(𝛿11 + 𝛿22 + 𝛿33)
3
and the asymmetry parameter, 휂 as,
휂 =(𝛿22 − 𝛿11)
𝛿33
then 𝛿𝑧𝑧 can be rearranged to 121,
𝛿𝑧𝑧 = 𝛿𝑖𝑠𝑜(3 cos2(휃) − 1 − 휂 sin2(휃) cos(2𝜑)) .
In solution nuclear magnetic resonance experiments, all polar angles are averaged by
molecular tumbling. However, in solid-state NMR experiments the chemical shift
anisotropy is persistent, resulting in broad peaks in the Fourier transform of the free
induction decay. As an example, nuclei that have axially symmetric principal
components (i.e. 𝛿11 ≥ 𝛿22 = 𝛿33) produce static spectra that have characteristic
intense transverse shielding edges at small chemical shifts and shallow longitudinal
shielding edges at higher chemical shifts. The 31P static spectra of membranes, where
the principal components are axially symmetric due to the local molecular
environment, have this distinct shape.
Eq. 4.33
Eq. 4.34
Eq. 4.35
Eq. 4.36
134
Chemical shift anisotropies can be mitigated to an extent by ‘magic angle spinning’
(MAS). If the entire sample is rotated at a sufficient angular frequency and
orientated at the magic angle, 휃𝑚𝑎𝑔𝑖𝑐 , given by,
휃𝑚𝑎𝑔𝑖𝑐 = cos−1 (
1
√3) ≈ 54.74°
we can see by inspection of Equation 4.36 that the anisotropic chemical shift can be
approximated to be equal to the isotropic chemical shift. The resulting NMR spectra
are sharpened significantly to the single resonance associated with the isotropic
chemical shift 121. In other words, the magic angle is the angle of the sample relative
to the external magnetic field that reduces anisotropic interactions by effectively
averaging them all to zero, leaving only the isotropic interaction and resulting in a
significantly sharper spectrum.
Figure 4.4. Visual representation of the chemical shift anisotropy experienced by a
nucleus in a molecule. The laboratory reference frame, (𝑥, 𝑦, 𝑧 – black arrows) is
centred on the nucleus. An ellipsoid represents the chemical shift tensor associated
with the electron density surrounding the nucleus. The principal axis system, PAS
(𝑥𝑃𝐴𝑆, 𝑦𝑃𝐴𝑆, 𝑧𝑃𝐴𝑆 – blue dash-dot arrows) is orientated with respect to the principal
components of the chemical shift tensor, namely 𝛿11, 𝛿22 and 𝛿33 (where by
definition 𝛿11 ≤ 𝛿22 ≤ 𝛿33). Transformations between the laboratory frame and the
PAS frame can be made using the polar and azimuthal angles, 휃 and 𝜑.
Eq. 4.37
135
4.2.4. Quadrupolar interactions for spin-1 nuclei
In all previous sections, the nuclei in question is assumed to have a spin value
of 1 2⁄ leading to the two possible energy states, the net magnetization for an
ensemble and ultimately the NMR signal. For nuclei with spin equal to 1, there now
exists three possible energy states corresponding to the quantum numbers 𝑚𝑠 =
−1,0,1, for the nuclei to occupy in the presence of an external magnetic field (see
Equation 4.3). The most immediate effect, given the radio-frequency pulse is still
utilised, is that now two transitions for the spin state to undergo are possible (namely
𝑚𝑠 = −1 → 𝑚𝑠 = 0 and 𝑚𝑠 = 0 → 𝑚𝑠 = 1). This results in a dual-peak symmetric
NMR spectrum about zero.
The chemical shift anisotropy described in the previous section can be thought of as
a first-order perturbation on the Zeeman splitting that is dominant for certain spin-
12⁄ nuclei, such as 31P. Whilst the anisotropic chemical shift perturbation still occurs
for spin-1 nuclei, such as 2H, a more significant perturbation is related to the
quadrupolar splitting. A 2H nucleus possesses a quadrupole moment due to a non-
symmetrical distribution of electric charge which couples with the surrounding
electric field gradient (EFG). The resultant EFG is directionally dependent in the
same way as the chemical shift anisotropy in the previous section. Therefore, the
frequency separation between dual peaks, Δ𝜐 on an NMR spectrum is quoted directly
from Davis as,
Δ𝜐 = (3
4) (𝑒2𝑞𝑄
ℎ) (3 cos2(휃) − 1 − 휂 sin2(휃) cos(2𝜑))
where 𝑒2𝑞𝑄
ℎ⁄ is the quadrupole coupling constant, 휃 and 𝜑 are the polar and
azimuthal angle used in the transformation between the laboratory frame and the
principal axis system of the EFG and 휂 is the asymmetry parameter (defined in the
same way as Equation 4.31 except in terms of the EFG principal components) 122.
Due to molecular motion, it is suitable to evaluate orientation averages in terms of
order parameters. In the special case of a lipid bilayer orientated such that the bilayer
normal is parallel to the external magnetic field and when axial molecular
Eq. 4.38
136
orientations are symmetric, the order parameter tensor reduces to a single term, 𝑆𝐶𝐷 ,
given by,
𝑆𝐶𝐷 = (2
3) (
ℎ
𝑒2𝑞𝑄) ⟨Δ𝜐⟩ .
In practical terms, the order parameter is a measure of the nucleus mobility in
relation to its position within the molecule. Smaller order parameters suggest the
nucleus experiences increased motion, or more disorder and likewise larger order
parameters indicate reduced mobility 123.
Eq. 4.39
137
4.3. Methodology
4.3.1. Sample preparation
Mixed lipid bilayers were prepared by initially measuring out 35 mg of a 7:3
molar ratio mixture of deuterated dimyristoylphosphatidylcholine (𝑑54-DMPC) with
dimyristoylphosphatidylglycerol (DMPG). The chemical structures of both lipids are
shown in Figure 4.5. This combination of two lipids was chosen as a model as it
replicates the net negative electrostatics associated with bacterial membranes 124. The
dry lipid mixture was co-stabilised by dissolving in 500 μL of 1:3 molar ratio
methanol and chloroform solvent. The solution was then dried using compressed N2
until a thick gel formed. Trace amounts of solvent were removed by placing the
sample in a high vacuum for 1 hour. Following this, the sample was re-suspended in
ultra pure water and mixed with various concentrations of cationic peptide, G3 (the
full description is G(IIKK)3I-NH2 using the standard notation for amino acids;
K = lysine, I = isoleucine and G = Glycine). A total of three lipid-to-peptide molar
ratios were investigated; 10:1, 20:1 and 50:1. A peptide-negative control was also
produced as a reference to measure the effects of the peptide against. The samples
were lyophilized for 12 hours, before being re-suspended in a limiting volume of
20 mM tris(hydroxymethyl)aminomethane (TRIS) buffer with 100 mM salt (to assist
solubility), resulting in a final pH = 7.4 and hydration level of 65%. At this stage, the
samples were highly viscous, resembling a thick translucent paste. The final lipid-
peptide mixture was then flash freeze-thawed multiple times (using liquid nitrogen
and a 30°C water bath) to ensure sample homogeneity, centrifuged and inserted into
a 5 mm NMR rotor.
4.3.2. 31P and 2H solid-state nuclear magnetic resonance data acquisition
All solid-state nuclear magnetic resonance experiments were carried out on a
Bruker AscendTM 400 DNP spectrometer at 37°C. Static proton-decoupled 31P NMR
spectra were obtained at an operating frequency of 162 MHz and averaged over 2048
scans. The spectra spanned a frequency range of ± 62,458 Hz (± 385.54 ppm) with
an effective spectral resolution of ± 7.63 Hz (± 0.047 ppm). Magic angle spinning
31P NMR spectra were also gathered, using a spinning frequency of 8000 Hz (which
was a sufficient frequency to remove sidebands) and averaged over 128 scans. 𝑇1
relaxation times were measured using inversion recovery pulse sequences (i.e. a 180°
pulse followed by a 90° pulse after increasing delay increments between 0.05 and 4
138
seconds). 𝑇2 relaxation times were measured with a Hahn spin-echo pulse sequence
(i.e. a 90° pulse followed by a 180° inversion pulse to remove the spin
inhomogeneous dephasing, resulting in an echo relative to the initial signal). The
delay times were also varied, however on a much shorter time scale between 1 and
40 ms. Both relaxation spectra were averaged over 32 repeated scans at each delay
time. Deuterium static spectra were obtained at an operating frequency of 61 MHz
using a quadrupolar-echo pulse sequence and averaged over 32768 scans. For both
experiments, the equivalent magnetic field strength was approximately 9.4 T.
Figure 4.5. Chemical structure of phospholipids used for construction of model
membrane bilayers in ssNMR experiments (where 𝑑54-DMPC = deuterated
dimyristoylphosphatidylcholine, DMPG = dimyristoylphosphatidylglycerol). The
CD-index is shown for each associated carbon atom on the fatty acid chain of 𝑑54-
DMPC.
139
4.3.3. Data analysis and de-Pakeing
Chemical shift anisotropies for static 31P NMR spectra were extracted using
the DMFIT program by fitting a ‘CSA Static’ model 125. Only a single model was fit
to the acquired data to get an estimation of the chemical shift anisotropy despite the
signal being a convolution of two spectra (corresponding to the two-phospholipid
system). For the MAS 31P spectra, individual peaks were deconvoluted using the
multi-fitter tool in OriginPro, whereby the spectra was fitted as the sum of two
Gaussian peaks defined by,
𝑓(𝛿) =∑𝐴𝑖
𝑤𝑖√𝜋 2⁄exp(−2
(𝛿 − 𝛿𝑐,𝑖)2
𝑤𝑖2 )
2
𝑖=1
where 𝐴𝑖 is the area under the peak, 𝑤𝑖 is the peak width and Δ𝛿𝑐,𝑖 is the peak centre.
The same fitting was used to identify the relative peak heights (corresponding to
intensities) in order to calculate the relaxation times 𝑇1 and 𝑇2. As 180° inversion
was used for 𝑇1 measurement, the resulting peak heights 𝐼(𝜏) at their respective
delay time, 𝜏 are directly proportional to the longitudinal magnetization (see
Equation 4.26). Hence, the time course of the recovering spectra was fitted in
OriginPro according to,
𝐼(𝜏) = 1 − 2 exp (−𝜏
𝑇1)
where the intensities have been normalised by the intensity at thermal equilibrium.
Similarly, 𝑇2 relaxation times were extracted by fitting the normalised peak
intensities of the Hahn-echo measurements at increasing delay time intervals using,
𝐼(𝜏) = exp (−𝜏
𝑇2)
which again from comparison to Equation 4.26 can be seen to represent the
magnitude of the transverse magnetization.
Eq. 4.40
Eq. 4.41
Eq. 4.42
140
2H spectra NMR spectra were numerically deconvoluted (referred to as ‘dePaked’) in
MatLab using the algorithm proposed by McCabe and Wassall 126, summarised as
transforming the spectra, 𝑓(𝛿), such that 127,
𝑓(−2𝛿) ∝ √|𝛿|ℱ[𝐹𝐼𝐷(𝑡)√𝑡]
where ℱ[ ] represents the Fourier transform, and 𝐹𝐼𝐷(𝑡) is the free induction decay
(i.e. the induced NMR signal). The deconvolution has the effect of orientating the
bilayer normal to be perpendicular to the external magnetic field (akin to a flat
bilayer rather than a semi-spherical vesicle. The signal contrast increases due to the
molecules being numerically aligned with the external magnetic field and therefore
producing a more distinct signal. Carbon-deuterium chain order parameters, 𝑆𝐶𝐷,𝑖
were calculated from the dePaked quadrupolar splitting, Δ𝛿𝑖 (where 𝑖 refers to the 𝑖th
carbon in the acyl chain of the d54-DMPC phospholipid) using Equation 4.39, where
the quadrupolar splitting constant equals 167 kHz for carbon-deuterium bonds.
Solid-state NMR experiments are primarily used to measure the relative change in
order experienced by molecules experiencing a perturbation. As such, reduction or
increases in the width of the 31P chemical shift anisotropy revealed through static or
MAS experiments correspond directly to increases or decreases in the orientational
order of the molecule respectively. This is directly caused by a relative change in the
surrounding electronic environment within the ensemble of molecules in the bilayer
membrane, which will appears as broadening in the spectrum. Similarly, relative
increases or decreases in the quadrupole splitting correspond to an increase or
decrease in the 2H order (or position fluctuation). The key difference between the
two measurements is the time period over which the disorder occurs; for 31P
experiments the frequency regime is of the order of 105 Hz, whereas for 2H
experiments the regime is in the 103 Hz regime.
Eq. 4.43
141
4.4. Results and Discussion
4.4.1. 31P static solid-state nuclear magnetic resonance
Figure 4.6 shows the static 31P NMR powder patterns corresponding to
unoriented d54-DMPC/DMPG bilayers (7:3 molar ratio) exposed to four
concentrations of cationic peptide G3. The general shape of all the powder patterns is
indicative of a lamellae structure, containing a high-intensity high-field edge
corresponding to axial symmetry 128. By fitting with a single ‘static-CSA’ model in
DMFit, an approximation for the effective chemical shift anisotropy, Δ𝛿 was found
for each peptide concentration and summarised in Table 4.1. The fit of the 31P static
NMR spectra for the peptide-negative d54-DMPC/DMPG bilayer control showed an
effective Δ𝛿 = 38.06 ppm, in agreement with values for lipid bilayer systems found
in the literature. As an example, Gehman et al. found that the 31P chemical shift
anisotropy for DMPC/DMPG bilayers in a 2:1 molar ratio was 41.5 ppm 129.
The chemical shift anisotropies are summarised in Table 4.1, and show a decrease in
Δ𝛿 for 10:1 lipid-to-peptide concentration but an increase for the 20:1 lipid-to-
peptide ratio. These results imply that a 10:1 lipid-to-peptide concentration increases
head group mobility, whereas a 20:1 concentration decreases head group mobility.
Whilst this may represent a true result, it is more likely to have been caused by
erroneous model fitting, as only a single line-shape was fit to the spectra, when in
reality they are a convolution of multiple line-shapes (corresponding to the two lipid
components). Dave et al. demonstrate multiple line-shape fitting to static 31P ssNMR
spectra in 1-palmitoyl-2-oleoyl-sn-glycero-phosphocholine (POPC) bilayers when
exposed to 4% dilutions of transmembrane protein phospholamban (PLB) at
different temperatures. Their fitting protocol involved the simulation and summation
of different idealised spectra; an approach that could be implemented in future
experiments 130.
142
Lipid:Peptide Chemical shift anisotropy,
Δ𝛿, ppm
Difference from control,
ppm
Control 25.37 0
10:1 23.42 -1.95
20:1 26.65 +1.28
50:1 25.32 -0.05
Table 4.1. Chemical shift anisotropies extracted from single model fitting of 31P
ssNMR spectra for 𝑑54-DMPC/DMPG bilayers (7:3 molar ratio) following exposure
to various concentrations of cationic peptide G3.
4.4.2. 31P magic angle spinning solid-state nuclear magnetic resonance
To investigate the influence of G3 on the DMPC and DMPG phospholipids
independently, the isotropic chemical shift corresponding to the two-phospholipid
head-groups was found using MAS NMR at a spinning frequency of 8000 Hz. The
resulting powder patterns are shown in Figure 4.7, which show two distinct peaks
characteristic of the two lipid components in the bilayer. Individual peak chemical
shifts and relative intensities are shown in Table 4.2. Extrapolation of the chemical
shifts and relative intensities was performed using multiple-peak Gaussian fitting (all
of which had adjusted-R2 values exceeding 0.99, justifying their ability to accurately
fit the spectra).
Intensity ratios (i.e. the height of the 𝑑54-DMPC peak divided by the height of the
DMPG peak) for all peptide concentrations were within 17% of 7/3 (≈ 2.33). The
proximity of the intensity ratios to the molar ratios used to construct the bilayers
indicates the preparation procedure was performed carefully and that minimal lipid
degradation occurred during the experiment. Differences in the MAS isotropic shift
reveal a more consistent effect of the G3 on the model membranes than the static
NMR results. At all concentrations, a shift of at least 20 Hz was observed, indicating
the head group mobility was significantly increased by the presence of the peptides.
An increase in the head group mobility could be caused by a reorientation of the
lipids due to membrane restructuring, such as a lateral phase separation or greater
spacing between head groups 131.
143
Lipid:
peptide
𝑑54-DMPC DMPG
Intensity
ratio 𝛿 (31P),
ppm
FWHM,
ppm
Intensity
(height)
𝛿 (31P),
ppm
FWHM,
ppm
Intensity
(height)
Control -69.80 93.46 915700 80.89 120.41 409000 2.24
10:1 -102.50 63.25 311500 52.22 81.36 134200 2.32
20:1 -101.72 95.54 548500 49.50 107.85 268100 2.05
50:1 -90.97 90.78 491400 49.87 100.92 245300 2.00
Table 4.2. Fit values for magic angle spinning 31P nuclear magnetic resonance
spectra shown in Figure 4.7 corresponding to a 7:3 molar ratio of d54-
DMPG/DMPC, with three concentrations of cationic peptide G3 in molar ratios
(lipid:peptide) of 10:1, 20:1 and 50:1. Phospholipid peaks were fit using a sum of
two Gaussians (see Equation 4.40).
Figure 4.6. Static 31P nuclear magnetic resonance spectra corresponding to a 7:3
molar ratio of d54-DMPG/DMPC, with three concentrations of cationic peptide G3
in molar ratios (lipid:peptide) of 10:1, 20:1 and 50:1. Lineshapes are indicative of
lamellae bilayers.
144
Figure 4.7. Magic angle spinning 31P nuclear magnetic resonance spectra
corresponding to a 7:3 molar ratio of 𝑑54-DMPG/DMPC, with three concentrations
of cationic peptide G3 in molar ratios (lipid:peptide) of 10:1, 20:1 and 50:1.
4.4.3. 31P 𝑇1 relaxation time
Figure 4.8 depicts an example spectra time course used to extract the 𝑇1
relaxation time, where the time intervals on the x-axis correspond to the delay
intervals used in between the inversion 180° pulse and transverse 90°pulse. Relative
intensities are normalised by the final time point intensity, which was assumed to
correspond to the intensity associated with the thermal equilibrium longitudinal
magnetization. Normalised intensities corresponding to the MAS ssNMR 𝑑54-
DMPG peak were extracted at each delay interval for each lipid-to-peptide ratio and
used to plot Figure 4.9 (although both peaks result in the same exponential
relationship). Using Equation 4.41, the relaxation time constant, 𝑇1 was found and
summarised in the subplot within Figure 4.9. All fits shown on Figure 4.9 had
adjusted-R2 values of greater 0.95. All 𝑇1 times found were consistent with values in
the literature for lipid bilayers. For example, Grage et al. found that the 𝑇1 relaxation
time for POPC/POPE bilayers (where POPE is an abbreviation for 1-palmitoyl-2-
oleoyl-sn-glycero-3-phosphoethanolamine) was 0.698 seconds 132. The only
145
statistically significant change to the 𝑇1 relaxation time was observed for the 10:1
lipid-to-peptide ratio, which showed a relative increase compared to the peptide-
negative control bilayer of 75 ± 10 ms. As stated in Section 4.2.2, 𝑇1 is dependent on
the local fluctuations in the magnetic field surrounding each nucleus. Therefore, it is
sensitive to perturbations in the individual lipid dynamics at the nanosecond
timescale 133. An increase in 𝑇1 corresponds to a reduction in the mobility of lipids,
as the spin-states cannot as effectively transfer energy to the surrounding lattice.
Figure 4.8. Example time course at increasing delay increments for magic angle
spinning 31P nuclear magnetic resonance spectra on 7:3 molar d54-DMPG/DMPC
using an inversion pul se sequence to extract the 𝑇1 relaxation time constant.
Figure 4.9. Exponential fits of the reconstitution of the longitudinal magnetization
and extraction of the 𝑇1 relaxation times (shown with errors in inset).
𝐼(𝜏)
/𝐼𝑒𝑞
146
4.4.4. 31P 𝑇2 relaxation time
Figure 4.10 depicts an example spectra time course used to extract the 𝑇2
relaxation time, where the time intervals on the x-axis corresponds to the delay
intervals used in between the initial transverse pulse and the inversion 180° pulse.
Relative intensities are normalised to the initial time point intensity, associated to the
maximum transverse magnetization. Normalised intensities corresponding to the
𝑑54-DMPG peak were extracted at each time point for each lipid-to-peptide ratio
and used to plot Figure 4.11. Using Equation 4.42, the relaxation time constant, 𝑇2
was found and summarised in the subplot within Figure 4.11. All fits shown on
Figure 4.11 had adjusted-R2 values that were greater 0.98, which is reflected in the
magnitude of 𝑇2 relaxation time errors. 𝑇2 values were also consistent with those
found in the literature. For example, in the same study referenced previously, Grage
et al. found 𝑇2 times for POPC/POPE bilayers to be 13.3 ms 132. As the 𝑇2 relaxation
time measures the motion of the bilayer surface on the millisecond timescale, the
results presented here would suggest that a lipid-to-peptide ratio of 10:1 decreases
the bilayer motion, whereas a 20:1 ratio increases bilayer motion. The 𝑇2 relaxation
time for the lipid-to-peptide ratio of 50:1 was not statistically significant from the
peptide-negative control.
147
Figure 4.10. Example of the decaying spectra associated with the loss in transversal
magnetization in a MAS 31P NMR time course.
Figure 4.11. Exponential fits of the decay of the transverse magnetization and
extraction of the 𝑇2 relaxation times (shown with errors in inset).
𝐼(𝜏)
/𝐼𝑒𝑞
148
4.4.5. 2H quadrupole order parameters
By exploiting the quadrupolar perturbations to the Zeeman splitting
associated with spin-1 nuclei, 2H NMR spectra were used to analyse the order
parameters associated with carbon-deuterium (CD) bonds in the fatty acid chain of
𝑑54-DMPC (which are shown explicitly in the chemical structure in Figure 4.5).
Following acquisition, spectra were numerically ‘dePaked’ to correct for orientation.
The 2H spectrum for 𝑑54-DMPC/DMPG following exposure to G3 in a lipid-to-
peptide ratio before and after dePakeing is shown in Figure 4.12. By fitting the peaks
with Gaussians (in the same manner as the 31P spectra peaks), the frequency
separation between opposing peaks was extracted. Using Equation 4.39, the
individual CD order parameters were calculated and plotted in Figure 4.40. Whilst
the difference in CD order parameter between the peptide-negative control and 50:1
lipid-to-peptide ratio was statistically insignificant, the CD order parameters for 20:1
ratio was significantly lower for all but one of the CD pairs in the acyl chain. A
lower order parameter is indicative of increased mobility of the methylene segment.
Therefore we can infer that G3 increases disorder throughout the hydrophobic core
of the bilayer, which can be attributed to peptide insertion into the bilayer 134.
Alternatively, the previous MAS 31P data showed a distinct interaction between the
peptide and the phospholipid head-group region, which could be interpreted as an
increase in the spacing between phospholipids within the bilayer. As the bilayer area
increases, the total bilayer volume must be maintained, which forces a thinning of
the bilayer that is detected by increased disorder of the fatty acid CD pairs 135. Broad
deformations in the bilayer structure, such as this potential thinning, supports the
hypothesis that the mechanism of action of G3 is best described by the carpet model.
149
Figure 4.12. Example of the 2H NMR spectra before and after the ‘dePaking’
procedure was carried out to reorient to the correct axis (see Equation 4.43)
Figure 4.13. CD order parameters extracted from the frequency separation associated
with dual peaks on the 2H NMR spectra for each carbon-deuterium on 𝑑54-DMPC.
150
4.5. Conclusion
The mechanism of action for G3 acting on phospholipid bilayers was
investigated using a variety of ssNMR techniques. Static 31P ssNMR results were
inconsistent, as were the results for the characteristic relaxation times 𝑇1 and 𝑇2.
However, MAS 31P ssNMR spectra strongly suggested that G3 increases the
mobility of phospholipid head groups, even at low concentrations. Furthermore,
quadrupolar 2H spectra indicated that G3 increases the disorder of CD pairs deep
within the hydrophobic region. Therefore, we conclude that the mechanism of action
for G3 on model phospholipid bilayers can be best described by the carpet model,
whereby the membrane degradation is characterised by the thinning of the bilayer
until disintegration. It should be noted that due to time limitations no positive control
could be investigated (i.e. a peptide with a known membrane mechanism of action).
Such data would have been crucial in this analysis as all measurements are carried
out relative to a negative control, which in the case of this experiment was the
membrane in the absence of any peptide. Whilst in isolation ssNMR data of the kind
presented here cannot definitively determine a mechanism of action, future work
could incorporate high resolution visualisation techniques such as atomic force
microscopy or scanning electron microscopy to directly observe the effect the
peptide on the membrane surface.
151
Chapter 5. Conclusion
5.1. Summary of Chapter 2 results and timeline of future related work
In Chapter 2, the main aim was to probe the mechanical response of
Staphylococcus aureus biofilms using passive microrheology, as a better
understanding of the material properties of biofilms could lead to better methods of
biofilm removal and prevention. Passive microrheology was selected as the most
suitable technique to probe the viscoelasticity of the biofilms without disturbing the
bacteria as it involves non-invasively tracking the bacteria using high-speed
brightfield microscopy.
Two fundamental results become apparent when considering the characteristic creep
compliances as a function of height away from the attachment surface and as a
function of shear stress. When subject to increasing hydrodynamic shear, which was
induced by increasing the velocity of the growth media through the chemostat, the
characteristic creep compliances became smaller by up to a factor of three when
compared to biofilms grown in stationary media. This is a clear indication that
S. aureus biofilms respond to higher shear forces by producing more rigid biofilms.
Furthermore, biofilms have larger creep compliance at distances further away from
the attachment surface, suggesting biofilms exhibit a vertical gradient of
viscoelasticity. Both of these results are in good agreement with individual pieces of
evidence in the literature for different bacterial biofilms.
An analysis of the spatial arrangements of bacteria within biofilms using Ripley’s K-
function as a statistical tool found that early-stage biofilms grew preferentially
upwards away from the flow cell surface in column-like structures. When the
biofilms were grown in static conditions, the spatial clustering associated with
column-like growth continued for the duration of the experiment (6 hours).
However, when the biofilm was grown under even weak flow the magnitude of
clustering decreased and the spatial distribution of bacteria became more
homogenous. When taken in context with the viscoelastic response of biofilms to
flow, this evidence suggests a reinforcement of the lower layers close to the
attachment surface to prevent complete detachment of the biofilm under shear stress.
One could argue that the vertical gradient of viscoelasticity seen in the experiments
undertaken here, coupled with spatial distributions, facilitate the dispersal of biofilms
152
as it enters maturation. Loosely bound bacteria in the top layers are more readily
removed, whilst a more rigid, entrenched layer close to the surface remains.
Treatment with proteinase K showed significant softening of the biofilm in
comparison to DNase-1 (which actually showed a slight hardening) and control
biofilms. Therefore, successive treatments of proteinase K and high-shear flushes
could be adequate to remove S. aureus biofilms.
Short term extensions to this direction of study are numerous. If the mature biofilm
components could be extracted and distilled from the bacteria non-destructively, it
would be possible to quantifiably identify the proteins being affected by the
proteinase-K treatment through techniques such as sodium dodecyl sulfate–
polyacrylamide gel electrophoresis (SDS-PAGE, which separates proteins based by
their molecular masses) and then by mass spectrometry. Variants of this have already
been implemented on bacterial biofilms through the use of a high salt-content
(1.5 M) wash, which the authors of the technique claim remove extracellular
components of the biofilm without inducing cell lysis 136.
Furthermore, as biofilm composition is also dependent on nutrient availability, the
influence of different growth medias on the material properties of biofilms could be
investigated. A dramatic example of unprecedented biofilm composition in the
literature is an experiment carried out by Schwartz et al., who found that biofilms
grown in extremely nutrient-limited conditions contained micron-scale amyloid
fibres composed of phenol soluble modulins. Investigating the mechanical response
of biofilms to varying media compositions more closely replicates the conditions
biofilms face outside of the lab.
In the long term, a more ambitious, novel experimental technique to explore would
be the brightfield tracking of S. aureus biofilms in 3-dimensions. A drawback of the
methodology used in this study was the limitation of imaging at 5 µm intervals. A
continuous axial imaging method could reveal information at all heights. In the
approximation that S. aureus bacteria are spherical, then their intensity profiles under
incoherent illumination could be modelled in the same way as polystyrene
microspheres. Two groups have developed models describing a brightfield
equivalent to the point-spread function using a completely theoretical derivation 137
and an experimentally attained model by axially scanning through fixed microbeads
138. In principle, the biofilm could be imaged by rapid axial scanning using a
153
piezoelectric stage (although corrections would have to be made for sinusoidal drift
and noise). Then the 3-dimensional intensity stack could be deconvoluted using the
equivalent brightfield point spread function each cycle to reveal the positions of the
bacteria in 3-dimensional space. With a sufficiently fast camera, 3-dimensional
passive microrheology could be realistically achievable. Furthermore, if biofilm
columns were grown in static conditions and then subject to hydrodynamic shear, the
deflection of the biofilm could be acquired and a full description of the relative
stresses and strains of bacteria within the biofilm could be calculated (something that
has only currently been done with immersed boundary fluid dynamic simulations
139).
5.2. Summary of Chapter 3 results and timeline of future related work
In Chapter 3, the spatial distributions and diffusive dynamics of the cationic
peptide G3 inside S. aureus and Escherichia coli bacteria were investigated using a
combination of 3-dimensional stochastic optical reconstruction microscopy
(STORM) and single particle tracking, facilitated by adaptive optics. By extracting
the approximate position of the bacteria membrane using parametric ellipsoid fitting
of the projected STORM image, the number density of G3 conjugated to Cy3B was
shown to have no localisation preference at any normalised radial distance.
Individual fluorescence blinks were recorded, and the trajectories were used to find
the diffusive power law exponents using a newly developed neural net. G3-Cy3B
pairs showed sub-diffusive motion by having a characteristic 𝛼-value of ~0.4,
however unconjugated Cy3B alone also had an exponent of this magnitude,
suggesting the sub-diffusive motion was dominated by entrapment within the cell,
rather than localised entrapment due to interactions with the membrane or
intercellular materials. Encouragingly, the diffusion coefficients of G3-Cy3B pairs
calculated from the data obtained matched previously found results for similar
experimental setups (albeit in 2-dimensions). When correlated to radial position
within the cell, no dependence was found for either trajectory step size or 𝛼-
eexponent, which was confirmed be comparisons with both Monte Carlo simulations
and artificially assigned values. As no preferential location was found on the cell, we
conclude that this evidence supports the carpet model of peptide action, as no regions
of clustered peptides were identified.
154
As stated in the Chapter results section, the most immediate course of action moving
forward is to check the localised spatial distributions by developing a 3-dimensional
Ripley K-function that could accommodate for the non-uniform boundary conditions
associated with the bacteria cell. Otherwise, another possible method to enhance the
experimental technique without significant alteration would be to repeat with a lower
quantity of fluorophores that are have longer blink times. Whilst sacrificing cell
position precision, longer blinkers and therefore longer trajectories would result in
mean square displacements (and so, 𝛼-exponents and diffusion coefficients) with
more statistical weighting at longer time intervals.
5.3. Summary of Chapter 4 results and timeline of future related work
In Chapter 4, 31P and 2H static and magic angle spinning solid state nuclear
magnetic resonance (MAS ssNMR) experiments were carried out to probe the
interaction of increasing concentration of G3 on model lipid bilayers constructed
from a 7:3 molar ratio of 𝑑54-DMPC/DMPG. Whilst static 31P ssNMR experiments
did not identify a consistent trend, MAS 31P ssNMR provided strong evidence that
G3 increased the mobility of phospholipid head groups in the bilayer, consisitent
with the peptide causing lateral phase separation or increased membrane curvature.
This result was compounded by the dePaked 2H quadrupole splitting results, which
showed that at high concentrations, the peptide reduced the carbon-deuterium order
parameter for almost the entire length of the fatty acid chain on the phospholipid.
Taken in context with the previous result, this data suggests that the peptide
transiently interacts with the bilayer causing membrane thinning. An accumulation
of the thinning eventually leads to strains that cause disintegration of the membrane,
as described by the carpet model for the mechanism of peptide action.
An extension to the ssNMR work carried out here woud be to repeat all the
experiments but with live bacteria in the NMR apparatus. Whilst this has been done
in the literature, the differences between model phospholipid bilayers and actual
bacteria amount to numerous problems. First of all, natural membranes contain an
array of phospholipids which would limit the specificity of any experiments carried
out. Another problem is that living bacteria contain a lot of potential background
signal in the form of DNA and RNA, all of which could be spin-active, therefore
broadening the signal. Lastly, MAS ssNMR experiments can often last a long time
155
and the sample is subject to high rotational velocities (up to 8000 Hz), so keeping the
bacteria alive is a nontrivial problem 140.
156
References
1. Silhavy, T. J.; Kahne, D.; Walker, S., The bacterial cell envelope. Cold
Spring Harbor perspectives in biology 2010, 2 (5), a000414.
2. Nicolson, G. L., The Fluid—Mosaic Model of Membrane Structure: Still
relevant to understanding the structure, function and dynamics of biological
membranes after more than 40 years. Biochimica et Biophysica Acta (BBA)-
Biomembranes 2014, 1838 (6), 1451-1466.
3. Metcalf, T. N.; Wang, J. L.; Schindler, M., Lateral diffusion of phospholipids
in the plasma membrane of soybean protoplasts: evidence for membrane lipid
domains. Proceedings of the National Academy of Sciences 1986, 83 (1), 95-99.
4. Jacobson, K.; Sheets, E. D.; Simson, R., Revisiting the fluid mosaic model of
membranes. Science 1995, 268 (5216), 1441-1443.
5. Lund, V. A.; Wacnik, K.; Turner, R. D.; Cotterell, B. E.; Walther, C. G.;
Fenn, S. J.; Grein, F.; Wollman, A. J.; Leake, M. C.; Olivier, N., Molecular
coordination of Staphylococcus aureus cell division. Elife 2018, 7, e32057.
6. Matias, V. R.; Beveridge, T. J., Native cell wall organization shown by cryo-
electron microscopy confirms the existence of a periplasmic space in Staphylococcus
aureus. Journal of bacteriology 2006, 188 (3), 1011-1021.
7. Klein, D. W.; Prescott, L. M.; Harley, J., Microbiology. McGraw-Hill Higher
Education: Boston, USA, 2005.
8. Madigan, M. T.; Martinko, J. S.; Dunlap, P. V.; Clark, D. P., Brock Biology
of Microorganisms. Benjamin-Cummins Publishing Company: USA, 2009.
9. White, D., The physiology and biochemistry of prokaryotes. Oxford
University Press Inc.: 2007; Vol. 3rd Edition.
10. Brown, S.; Santa Maria Jr, J. P.; Walker, S., Wall teichoic acids of gram-
positive bacteria. Annual review of microbiology 2013, 67, 313-336.
11. Umeda, A.; Yokoyama, S.; Arizono, T.; Amako, K., Location of
peptidoglycan and teichoic acid on the cell wall surface of Staphylococcus aureus as
determined by immunoelectron microscopy. Microscopy 1992, 41 (1), 46-52.
12. Neuhaus, F. C.; Baddiley, J., A continuum of anionic charge: structures and
functions of D-alanyl-teichoic acids in gram-positive bacteria. Microbiol. Mol. Biol.
Rev. 2003, 67 (4), 686-723.
13. Sewell, E. W.; Brown, E. D., Taking aim at wall teichoic acid synthesis: new
biology and new leads for antibiotics. The Journal of antibiotics 2014, 67 (1), 43.
14. Zajmi, A.; Hashim, N. M.; Noordin, M. I.; Khalifa, S. A.; Ramli, F.; Ali, H.
M.; El-Seedi, H. R., Ultrastructural study on the antibacterial activity of artonin e
versus streptomycin against Staphylococcus aureus strains. PLoS One 2015, 10 (6).
15. Vollmer, W.; Seligman, S. J., Architecture of peptidoglycan: more data and
more models. Trends in microbiology 2010, 18 (2), 59-66.
16. Coico, R., Gram staining. Current protocols in microbiology 2006, (1), A.
3C. 1-A. 3C. 2.
157
17. Demchick, P.; Koch, A. L., The permeability of the wall fabric of Escherichia
coli and Bacillus subtilis. Journal of bacteriology 1996, 178 (3), 768-773.
18. Raetz, C. R.; Guan, Z.; Ingram, B. O.; Six, D. A.; Song, F.; Wang, X.; Zhao,
J., Discovery of new biosynthetic pathways: the lipid A story. Journal of lipid
research 2009, 50 (Supplement), S103-S108.
19. Raetz, C. R.; Whitfield, C., Lipopolysaccharide endotoxins. Annual review of
biochemistry 2002, 71 (1), 635-700.
20. Branda, S. S.; Chu, F.; Kearns, D. B.; Losick, R.; Kolter, R., A major protein
component of the Bacillus subtilis biofilm matrix. Molecular microbiology 2006, 59
(4), 1229-1238.
21. Yarwood, J. M.; Bartels, D. J.; Volper, E. M.; Greenberg, E. P., Quorum
sensing in Staphylococcus aureus biofilms. Journal of bacteriology 2004, 186 (6),
1838-1850.
22. Bjarnsholt, T.; Kirketerp‐Møller, K.; Kristiansen, S.; Phipps, R.; Nielsen, A.
K.; Jensen, P. Ø.; Høiby, N.; Givskov, M., Silver against Pseudomonas aeruginosa
biofilms. Apmis 2007, 115 (8), 921-928.
23. Flemming, H.-C.; Wingender, J.; Szewzyk, U.; Steinberg, P.; Rice, S. A.;
Kjelleberg, S., Biofilms: an emergent form of bacterial life. Nature Reviews
Microbiology 2016, 14 (9), 563.
24. Arnaouteli, S.; MacPhee, C. E.; Stanley-Wall, N. R., Just in case it rains:
building a hydrophobic biofilm the Bacillus subtilis way. Current opinion in
microbiology 2016, 34, 7-12.
25. Stalder, T.; Top, E., Plasmid transfer in biofilms: a perspective on limitations
and opportunities. NPJ biofilms and microbiomes 2016, 2, 16022.
26. Westall, F.; de Wit, M. J.; Dann, J.; van der Gaast, S.; de Ronde, C. E.;
Gerneke, D., Early Archean fossil bacteria and biofilms in hydrothermally-
influenced sediments from the Barberton greenstone belt, South Africa. Precambrian
Research 2001, 106 (1-2), 93-116.
27. Otto, M., Staphylococcal biofilms. In Bacterial biofilms, Springer: 2008; pp
207-228.
28. Singhal, D.; Foreman, A.; Bardy, J. J.; Wormald, P. J., Staphylococcus
aureus biofilms: Nemesis of endoscopic sinus surgery. The Laryngoscope 2011, 121
(7), 1578-1583.
29. Milewski, S.; Gabriel, I.; Olchowy, J., Enzymes of UDP‐GlcNAc
biosynthesis in yeast. Yeast 2006, 23 (1), 1-14.
30. Oliveira, A.; Cunha, M., Bacterial biofilms with emphasis on coagulase-
negative staphylococci. Journal of Venomous Animals and Toxins including Tropical
Diseases 2008, 14 (4), 572-596.
31. Kropec, A.; Maira-Litran, T.; Jefferson, K. K.; Grout, M.; Cramton, S. E.;
Götz, F.; Goldmann, D. A.; Pier, G. B., Poly-N-acetylglucosamine production in
Staphylococcus aureus is essential for virulence in murine models of systemic
infection. Infection and immunity 2005, 73 (10), 6868-6876.
32. Francois, P.; Tu Quoc, P. H.; Bisognano, C.; Kelley, W. L.; Lew, D. P.;
Schrenzel, J.; Cramton, S. E.; Götz, F.; Vaudaux, P., Lack of biofilm contribution to
158
bacterial colonisation in an experimental model of foreign body infection by
Staphylococcus aureus and Staphylococcus epidermidis. FEMS Immunology &
Medical Microbiology 2003, 35 (2), 135-140.
33. O'Neill, E.; Pozzi, C.; Houston, P.; Smyth, D.; Humphreys, H.; Robinson, D.
A.; O'Gara, J. P., Association between methicillin susceptibility and biofilm
regulation in Staphylococcus aureus isolates from device-related infections. Journal
of clinical microbiology 2007, 45 (5), 1379-1388.
34. Archer, N. K.; Mazaitis, M. J.; Costerton, J. W.; Leid, J. G.; Powers, M. E.;
Shirtliff, M. E., Staphylococcus aureus biofilms: properties, regulation, and roles in
human disease. Virulence 2011, 2 (5), 445-459.
35. Mack, D.; Fischer, W.; Krokotsch, A.; Leopold, K.; Hartmann, R.; Egge, H.;
Laufs, R., The intercellular adhesin involved in biofilm accumulation of
Staphylococcus epidermidis is a linear beta-1, 6-linked glucosaminoglycan:
purification and structural analysis. Journal of bacteriology 1996, 178 (1), 175-183.
36. Periasamy, S.; Joo, H.-S.; Duong, A. C.; Bach, T.-H. L.; Tan, V. Y.;
Chatterjee, S. S.; Cheung, G. Y.; Otto, M., How Staphylococcus aureus biofilms
develop their characteristic structure. Proceedings of the National Academy of
Sciences 2012, 109 (4), 1281-1286.
37. Janssens, J. C.; Steenackers, H.; Robijns, S.; Gellens, E.; Levin, J.; Zhao, H.;
Hermans, K.; De Coster, D.; Verhoeven, T. L.; Marchal, K.; Vanderleyden, J.; De
Vos, D. E.; De Keersmaecker, S. C., Brominated furanones inhibit biofilm formation
by Salmonella enterica serovar Typhimurium. Appl Environ Microbiol 2008, 74
(21), 6639-48.
38. Brady, R. A.; Graeme, A.; Leid, J. G.; Costerton, J. W.; Shirtliff, M. E.,
Resolution of Staphylococcus aureus biofilm infection using vaccination and
antibiotic treatment. Infection and immunity 2011, 79 (4), 1797-1803.
39. Marrie, T.; Costerton, J., Scanning and transmission electron microscopy of
in situ bacterial colonization of intravenous and intraarterial catheters. Journal of
Clinical Microbiology 1984, 19 (5), 687-693.
40. Bregenzer, T.; Conen, D.; Sakmann, P.; Widmer, A. F., Is routine
replacement of peripheral intravenous catheters necessary? Archives of internal
medicine 1998, 158 (2), 151-156.
41. Marrie, T. J.; Nelligan, J.; Costerton, J. W., A scanning and transmission
electron microscopic study of an infected endocardial pacemaker lead. Circulation
1982, 66 (6), 1339-1341.
42. Park, P.; Garton, H. J.; Kocan, M. J.; Thompson, B. G., Risk of infection
with prolonged ventricular catheterization. Neurosurgery 2004, 55 (3), 594-601.
43. Stoodley, P.; Braxton, E. E., Jr.; Nistico, L.; Hall-Stoodley, L.; Johnson, S.;
Quigley, M.; Post, J. C.; Ehrlich, G. D.; Kathju, S., Direct demonstration of
Staphylococcus biofilm in an external ventricular drain in a patient with a history of
recurrent ventriculoperitoneal shunt failure. Pediatr Neurosurg 2010, 46 (2), 127-32.
44. Anwar, H.; Strap, J.; Costerton, J., Establishment of aging biofilms: possible
mechanism of bacterial resistance to antimicrobial therapy. Antimicrobial agents and
chemotherapy 1992, 36 (7), 1347.
159
45. Schultz, M. P.; Swain, G. W., The influence of biofilms on skin friction drag.
Biofouling 2000, 15 (1-3), 129-139.
46. Haslbeck, E. G.; Bohlander, G. S. Microbial biofilm effects on drag-lab and
field; Naval surface warfare center carderock div annapolis MD: 1992.
47. Almeida, M.; De França, F., Biofilm formation on brass coupons exposed to
a cooling system of an oil refinery. Journal of Industrial Microbiology and
Biotechnology 1998, 20 (1), 39-44.
48. Goosen, M.; Sablani, S.; Al‐Hinai, H.; Al‐Obeidani, S.; Al‐Belushi, R.;
Jackson, a., Fouling of reverse osmosis and ultrafiltration membranes: a critical
review. Separation science and technology 2005, 39 (10), 2261-2297.
49. Capdeville, B.; Rols, J., Introduction to biofilms in water and wastewater
treatment. In Biofilms—Science and Technology, Springer: 1992; pp 13-20.
50. Zeiner, M.; Rezic, T.; Santek, B.; Rezic, I.; Hann, S.; Stingeder, G., Removal
of Cr, Mn, and Co from textile wastewater by horizontal rotating tubular bioreactor.
Environmental science & technology 2012, 46 (19), 10690-10696.
51. Watling, H., The bioleaching of sulphide minerals with emphasis on copper
sulphides—a review. Hydrometallurgy 2006, 84 (1-2), 81-108.
52. Organization, W. H., Antimicrobial resistance: global report on surveillance.
World Health Organization: 2014.
53. Boucher, H. W.; Talbot, G. H.; Benjamin Jr, D. K.; Bradley, J.; Guidos, R. J.;
Jones, R. N.; Murray, B. E.; Bonomo, R. A.; Gilbert, D.; America, I. D. S. o., 10×'20
progress—development of new drugs active against gram-negative bacilli: an update
from the Infectious Diseases Society of America. Clinical infectious diseases 2013,
56 (12), 1685-1694.
54. Hu, J.; Chen, C.; Zhang, S.; Zhao, X.; Xu, H.; Zhao, X.; Lu, J. R., Designed
antimicrobial and antitumor peptides with high selectivity. Biomacromolecules 2011,
12 (11), 3839-3843.
55. Hanwell, M. D.; Curtis, D. E.; Lonie, D. C.; Vandermeersch, T.; Zurek, E.;
Hutchison, G. R., Avogadro: an advanced semantic chemical editor, visualization,
and analysis platform. Journal of cheminformatics 2012, 4 (1), 17.
56. Chen, C.; Hu, J.; Zeng, P.; Chen, Y.; Xu, H.; Lu, J. R., High cell selectivity
and low-level antibacterial resistance of designed amphiphilic peptide G (IIKK) 3I-
NH2. ACS applied materials & interfaces 2014, 6 (19), 16529-16536.
57. Salmon, S. A.; Watts, J. L., Minimum inhibitory concentration
determinations for various antimicrobial agents against 1570 bacterial isolates from
turkey poults. Avian diseases 2000, 85-98.
58. Peschel, A.; Otto, M.; Jack, R. W.; Kalbacher, H.; Jung, G.; Götz, F.,
Inactivation of the dlt Operon inStaphylococcus aureus Confers Sensitivity to
Defensins, Protegrins, and Other Antimicrobial Peptides. Journal of Biological
Chemistry 1999, 274 (13), 8405-8410.
59. Yeaman, M. R.; Yount, N. Y., Mechanisms of antimicrobial peptide action
and resistance. Pharmacological reviews 2003, 55 (1), 27-55.
160
60. Toke, O., Antimicrobial peptides: new candidates in the fight against
bacterial infections. Peptide Science: Original Research on Biomolecules 2005, 80
(6), 717-735.
61. Jeckel, H.; Jelli, E.; Hartmann, R.; Singh, P. K.; Mok, R.; Totz, J. F.;
Vidakovic, L.; Eckhardt, B.; Dunkel, J.; Drescher, K., Learning the space-time phase
diagram of bacterial swarm expansion. Proceedings of the National Academy of
Sciences 2019, 116 (5), 1489-1494.
62. Díaz-Pascual, F.; Hartmann, R.; Lempp, M.; Vidakovic, L.; Song, B.; Jeckel,
H.; Thormann, K. M.; Yildiz, F. H.; Dunkel, J.; Link, H., Breakdown of Vibrio
cholerae biofilm architecture induced by antibiotics disrupts community barrier
function. Nature microbiology 2019, 4 (12), 2136-2145.
63. Pearce, P.; Song, B.; Skinner, D. J.; Mok, R.; Hartmann, R.; Singh, P. K.;
Jeckel, H.; Oishi, J. S.; Drescher, K.; Dunkel, J., Flow-induced symmetry breaking
in growing bacterial biofilms. Physical Review Letters 2019, 123 (25), 258101.
64. Stoodley, P.; Dodds, I.; Boyle, J. D.; Lappin‐Scott, H., Influence of
hydrodynamics and nutrients on biofilm structure. Journal of applied microbiology
1998, 85 (S1), 19S-28S.
65. Foka, A.; Katsikogianni, M. G.; Anastassiou, E. D.; Spiliopoulou, I.;
Missirlis, Y. F., The combined effect of surface chemistry and flow conditions on
Staphylococcus epidermidis adhesion and ica operon expression. European Cells
and Materials 2012, 24, 386-402.
66. Rogers, S.; Van Der Walle, C.; Waigh, T., Microrheology of bacterial
biofilms in vitro: Staphylococcus aureus and Pseudomonas aeruginosa. Langmuir
2008, 24 (23), 13549-13555.
67. Einstein, A., On the movement of small particles suspended in a stationary
liquid demanded by the molecular-kinetic theory of heart. Annalen Der Physik 1905,
17, 549-560.
68. Kenney, J. F., Keeping, E. S, Mathematics of Statistics, Pt. 2. 2nd ed.; Van
Nostrand: Princeton, NJ, 1951; pp 72-77.
69. Berg, H. C., Random walks in biology. Princeton University Press: 1993.
70. Zwanzig, R., Nonequilibrium statistical mechanics. Oxford University Press:
2001.
71. Mason, T. G.; Weitz, D. A., Optical measurements of frequency-dependent
linear viscoelastic moduli of complex fluids. Phys Rev Lett 1995, 74 (7), 1250-1253.
72. Xu, J.; Viasnoff, V.; Wirtz, D., Compliance of actin filament networks
measured by particle-tracking microrheology and diffusing wave spectroscopy.
Rheologica Acta 1998, 37 (4), 387-398.
73. Weiss, M.; Elsner, M.; Kartberg, F.; Nilsson, T., Anomalous subdiffusion is a
measure for cytoplasmic crowding in living cells. Biophysical journal 2004, 87 (5),
3518-3524.
74. Saxton, M. J., Anomalous subdiffusion in fluorescence photobleaching
recovery: a Monte Carlo study. Biophysical journal 2001, 81 (4), 2226-2240.
161
75. Dubkov, A. A.; Spagnolo, B.; Uchaikin, V. V., Lévy flight superdiffusion: an
introduction. International Journal of Bifurcation and Chaos 2008, 18 (09), 2649-
2672.
76. Bourgoin, M.; Ouellette, N. T.; Xu, H.; Berg, J.; Bodenschatz, E., The role of
pair dispersion in turbulent flow. Science 2006, 311 (5762), 835-838.
77. Ripley, B. D., Modelling spatial patterns. Journal of the Royal Statistical
Society. Series B (Methodological) 1977, 172-212.
78. Dixon, P. M., Ripley’s K function. Encyclopedia of environmetrics 2001, 3,
1796.
79. Nauman, E. A.; Risic, K. J.; Keaveny, T. M.; Satcher, R. L., Quantitative
assessment of steady and pulsatile flow fields in a parallel plate flow chamber.
Annals of biomedical engineering 1999, 27 (2), 194-199.
80. Shukla, S. K.; Rao, T. S., Dispersal of Bap-mediated Staphylococcus aureus
biofilm by proteinase K. The Journal of antibiotics 2013, 66 (2), 55.
81. Izano, E. A.; Amarante, M. A.; Kher, W. B.; Kaplan, J. B., Differential roles
of poly-N-acetylglucosamine surface polysaccharide and extracellular DNA in
Staphylococcus aureus and Staphylococcus epidermidis biofilms. Applied and
environmental microbiology 2008, 74 (2), 470-476.
82. Rogers, S. S.; Waigh, T. A.; Zhao, X.; Lu, J. R., Precise particle tracking
against a complicated background: polynomial fitting with Gaussian weight.
Physical Biology 2007, 4 (3), 220.
83. Waigh, T. A., Microrheology of complex fluids. Reports on Progress in
Physics 2005, 68 (3), 685-742.
84. Waigh, T. A., Advances in the microrheology of complex fluids. Rep Prog
Phys 2016, 79 (7), 074601.
85. Besag, J., Comments on Ripley’s paper: Royal Statistical Society. Journal
1977, 39, 193-195.
86. Mainardi, F.; Spada, G., Creep, relaxation and viscosity properties for basic
fractional models in rheology. The European Physical Journal Special Topics 2011,
193 (1), 133-160.
87. Galy, O.; Latour-Lambert, P.; Zrelli, K.; Ghigo, J. M.; Beloin, C.; Henry, N.,
Mapping of bacterial biofilm local mechanics by magnetic microparticle actuation.
Biophys J 2012, 103 (6), 1400-8.
88. Cao, H.; Habimana, O.; Safari, A.; Heffernan, R.; Dai, Y.; Casey, E.,
Revealing region-specific biofilm viscoelastic properties by means of a micro-
rheological approach. npj Biofilms and Microbiomes 2016, 2 (1), 5.
89. Drescher, K.; Shen, Y.; Bassler, B. L.; Stone, H. A., Biofilm streamers cause
catastrophic disruption of flow with consequences for environmental and medical
systems. Proceedings of the National Academy of Sciences 2013, 110 (11), 4345-
4350.
90. Moormeier, D. E.; Bose, J. L.; Horswill, A. R.; Bayles, K. W., Temporal and
stochastic control of Staphylococcus aureus biofilm development. MBio 2014, 5 (5),
e01341-14.
162
91. Mann, E. E.; Rice, K. C.; Boles, B. R.; Endres, J. L.; Ranjit, D.;
Chandramohan, L.; Tsang, L. H.; Smeltzer, M. S.; Horswill, A. R.; Bayles, K. W.,
Modulation of eDNA release and degradation affects Staphylococcus aureus biofilm
maturation. PloS one 2009, 4 (6), e5822.
92. Izeddin, I.; El Beheiry, M.; Andilla, J.; Ciepielewski, D.; Darzacq, X.;
Dahan, M., PSF shaping using adaptive optics for three-dimensional single-molecule
super-resolution imaging and tracking. Optics express 2012, 20 (5), 4957-4967.
93. Nelson, N.; Schwartz, D. K., Single-Molecule Resolution of Antimicrobial
Peptide Interactions with Supported Lipid A Bilayers. Biophysical journal 2018, 114
(11), 2606-2616.
94. Suffredini, A. F.; Fromm, R. E.; Parker, M. M.; Brenner, M.; Kovacs, J. A.;
Wesley, R. A.; Parrillo, J. E., The cardiovascular response of normal humans to the
administration of endotoxin. New England Journal of Medicine 1989, 321 (5), 280-
287.
95. Leptihn, S.; Har, J. Y.; Chen, J.; Ho, B.; Wohland, T.; Ding, J. L., Single
molecule resolution of the antimicrobial action of quantum dot-labeled sushi peptide
on live bacteria. BMC biology 2009, 7 (1), 22.
96. Roversi, D.; Luca, V.; Aureli, S.; Park, Y.; Mangoni, M. L.; Stella, L., How
many antimicrobial peptide molecules kill a bacterium? The case of PMAP-23. ACS
chemical biology 2014, 9 (9), 2003-2007.
97. Goodman, J. W., Introduction to Fourier optics. Roberts and Company
Publishers: 2005.
98. Mertz, J., Introduction to optical microscopy. Cambridge University Press:
2019.
99. Born, M.; Wolf, E., Principles of optics: electromagnetic theory of
propagation, interference and diffraction of light. Elsevier: 2013.
100. Novotny, L.; Hecht, B., Principles of nano-optics. Cambridge university
press: 2012.
101. Rayleigh, L., XV. On the theory of optical images, with special reference to
the microscope. The London, Edinburgh, and Dublin Philosophical Magazine and
Journal of Science 1896, 42 (255), 167-195.
102. Sanborn, M. E.; Connolly, B. K.; Gurunathan, K.; Levitus, M., Fluorescence
properties and photophysics of the sulfoindocyanine Cy3 linked covalently to DNA.
The Journal of Physical Chemistry B 2007, 111 (37), 11064-11074.
103. Frantsuzov, P.; Kuno, M.; Janko, B.; Marcus, R. A., Universal emission
intermittency in quantum dots, nanorods and nanowires. Nature Physics 2008, 4 (7),
519.
104. Frantsuzov, P. A.; Volkan-Kacso, S. n.; Janko, B. r., Universality of the
fluorescence intermittency in nanoscale systems: experiment and theory. Nano
letters 2013, 13 (2), 402-408.
105. Fu, Y.; Zhang, J.; Lakowicz, J. R., Reduced blinking and long-lasting
fluorescence of single fluorophores coupling to silver nanoparticles. Langmuir 2008,
24 (7), 3429-3433.
163
106. Peterman, E. J.; Gittes, F.; Schmidt, C. F., Laser-induced heating in optical
traps. Biophysical journal 2003, 84 (2), 1308-1316.
107. Xia, T.; Li, N.; Fang, X., Single-molecule fluorescence imaging in living
cells. Annual review of physical chemistry 2013, 64, 459-480.
108. Rust, M. J.; Bates, M.; Zhuang, X., Sub-diffraction-limit imaging by
stochastic optical reconstruction microscopy (STORM). Nat Methods 2006, 3 (10),
793-5.
109. Ovesný, M.; Křížek, P.; Borkovec, J.; Svindrych, Z.; Hagen, G. M.,
ThunderSTORM: a comprehensive ImageJ plug-in for PALM and STORM data
analysis and super-resolution imaging. Bioinformatics 2014, 30 (16), 2389-2390.
110. Holschneider, M.; Kronland-Martinet, R.; Morlet, J.; Tchamitchian, P., A
real-time algorithm for signal analysis with the help of the wavelet transform. In
Wavelets, Springer: 1990; pp 286-297.
111. Izeddin, I.; Boulanger, J.; Racine, V.; Specht, C.; Kechkar, A.; Nair, D.;
Triller, A.; Choquet, D.; Dahan, M.; Sibarita, J., Wavelet analysis for single
molecule localization microscopy. Optics express 2012, 20 (3), 2081-2095.
112. Fitzgibbon, A.; Pilu, M.; Fisher, R. B., Direct least square fitting of ellipses.
IEEE Transactions on pattern analysis and machine intelligence 1999, 21 (5), 476-
480.
113. Wang, Z.; Simoncelli, E. P.; Bovik, A. C. In Multiscale structural similarity
for image quality assessment, The Thrity-Seventh Asilomar Conference on Signals,
Systems & Computers, 2003, Ieee: 2003; pp 1398-1402.
114. Emsley, J. W.; Feeney, J.; Sutcliffe, L. H., High resolution nuclear magnetic
resonance spectroscopy. Elsevier: 2013; Vol. 2.
115. Hanson, L. G., Is quantum mechanics necessary for understanding magnetic
resonance? Concepts in Magnetic Resonance Part A: An Educational Journal 2008,
32 (5), 329-340.
116. Abragam, A.; Abragam, A., The principles of nuclear magnetism. Oxford
university press: 1961.
117. Liang, Z.-P.; Lauterbur, P. C., Principles of magnetic resonance imaging: a
signal processing perspective. SPIE Optical Engineering Press: 2000.
118. Mehring, M., High resolution NMR spectroscopy in solids. Springer Science
& Business Media: 2012; Vol. 11.
119. Mason, J., Conventions for the reporting of nuclear magnetic shielding (or
shift) tensors suggested by participants in the NATO ARW on NMR shielding
constants at the University of Maryland, College Park, July 1992. Solid state nuclear
magnetic resonance 1993, 2 (5), 285.
120. Saitô, H.; Ando, I.; Ramamoorthy, A., Chemical shift tensor–the heart of
NMR: insights into biological aspects of proteins. Progress in nuclear magnetic
resonance spectroscopy 2010, 57 (2), 181.
121. Separovic, F.; Naito, A., Advances in Biological Solid-State NMR: Proteins
and Membrane-Active Peptides. Royal Society of Chemistry: 2014.
164
122. Davis, J. H., The description of membrane lipid conformation, order and
dynamics by 2H-NMR. Biochimica et Biophysica Acta (BBA)-Reviews on
Biomembranes 1983, 737 (1), 117-171.
123. Man, P. P., Quadrupole couplings in nuclear magnetic resonance, general.
Encyclopedia of Analytical Chemistry: Applications, Theory and Instrumentation
2006.
124. Epand, R. F.; Schmitt, M. A.; Gellman, S. H.; Epand, R. M., Role of
membrane lipids in the mechanism of bacterial species selective toxicity by two α/β-
antimicrobial peptides. Biochimica et Biophysica Acta (BBA)-Biomembranes 2006,
1758 (9), 1343-1350.
125. Massiot, D.; Fayon, F.; Capron, M.; King, I.; Le Calvé, S.; Alonso, B.;
Durand, J. O.; Bujoli, B.; Gan, Z.; Hoatson, G., Modelling one‐and two‐dimensional
solid‐state NMR spectra. Magnetic resonance in chemistry 2002, 40 (1), 70-76.
126. McCabe, M. A.; Wassall, S. R., Fast-Fourier-transform depaking. Journal of
Magnetic Resonance, Series B 1995, 106 (1), 80-82.
127. Sani, M.-A.; Weber, D. K.; Delaglio, F.; Separovic, F.; Gehman, J. D., A
practical implementation of de-Pake-ing via weighted Fourier transformation. PeerJ
2013, 1, e30.
128. Facelli, J. C., Chemical shift tensors: Theory and application to molecular
structural problems. Progress in nuclear magnetic resonance spectroscopy 2011, 58
(3-4), 176.
129. Gehman, J. D.; Luc, F.; Hall, K.; Lee, T.-H.; Boland, M. P.; Pukala, T. L.;
Bowie, J. H.; Aguilar, M.-I.; Separovic, F., Effect of antimicrobial peptides from
Australian tree frogs on anionic phospholipid membranes. Biochemistry 2008, 47
(33), 8557-8565.
130. Dave, P. C.; Tiburu, E. K.; Damodaran, K.; Lorigan, G. A., Investigating
structural changes in the lipid bilayer upon insertion of the transmembrane domain of
the membrane-bound protein phospholamban utilizing 31P and 2H solid-state NMR
spectroscopy. Biophysical journal 2004, 86 (3), 1564-1573.
131. Sherman, P. J.; Separovic, F.; Bowie, J. H., The investigation of membrane
binding by amphibian peptide agonists of CCK2R using 31P and 2H solid-state
NMR. Peptides 2014, 55, 98-102.
132. Grage, S. L.; Sani, M.-A.; Cheneval, O.; Henriques, S. T.; Schalck, C.;
Heinzmann, R.; Mylne, J. S.; Mykhailiuk, P. K.; Afonin, S.; Komarov, I. V.,
Orientation and location of the cyclotide kalata B1 in lipid bilayers revealed by
solid-state NMR. Biophysical journal 2017, 112 (4), 630-642.
133. Dufourc, E. J.; Mayer, C.; Stohrer, J.; Althoff, G.; Kothe, G., Dynamics of
phosphate head groups in biomembranes. Comprehensive analysis using
phosphorus-31 nuclear magnetic resonance lineshape and relaxation time
measurements. Biophysical journal 1992, 61 (1), 42-57.
134. Booth, V.; Warschawski, D. E.; Santisteban, N. P.; Laadhari, M.; Marcotte,
I., Recent progress on the application of 2H solid-state NMR to probe the interaction
of antimicrobial peptides with intact bacteria. Biochimica et Biophysica Acta (BBA)-
Proteins and Proteomics 2017, 1865 (11), 1500-1511.
165
135. Petrache, H. I.; Dodd, S. W.; Brown, M. F., Area per lipid and acyl length
distributions in fluid phosphatidylcholines determined by 2H NMR spectroscopy.
Biophysical journal 2000, 79 (6), 3172-3192.
136. Chiba, A.; Sugimoto, S.; Sato, F.; Hori, S.; Mizunoe, Y., A refined technique
for extraction of extracellular matrices from bacterial biofilms and its applicability.
Microbial biotechnology 2015, 8 (3), 392-403.
137. Wang, Y.; Li, X.; Bi, S.; Zhu, X.; Liu, J., 3D micro-particle image modeling
and its application in measurement resolution investigation for visual sensing based
axial localization in an optical microscope. Measurement Science and Technology
2016, 28 (1), 015402.
138. Tadrous, P., A method of PSF generation for 3D brightfield deconvolution.
Journal of microscopy 2010, 237 (2), 192-199.
139. Alpkvist, E.; Klapper, I., Description of mechanical response including
detachment using a novel particle model of biofilm/flow interaction. Water science
and technology 2007, 55 (8-9), 265-273.
140. Overall, S.; Zhu, S.; Hanssen, E.; Separovic, F.; Sani, M.-A., In Situ
Monitoring of Bacteria under Antimicrobial Stress Using 31P Solid-State NMR.
International journal of molecular sciences 2019, 20 (1), 181.