The Dynamics of Growing Biofilm

J. P. Keener (Utah),

with lots of help from

Nick Cogan (Tulane), Jack Dockery (Montana SU)and the NSF

UU The Power of MathematicsThe Excitement of Biology

Biofilms

Biofilm fouling of fiber filter

Placque on teethUU The Power of Mathematics

The Excitement of Biology

Some of the Interesting and Important Questions

P. Aeruginosa (on catheters, IV tubes, etc.)

•How do gels grow?

Mucus secretion (bronchial tubes, stomach lining)Colloidal suspensions of cells (cancer cell growth)

Gel Morphology (The cellular shape of sponges)

•Why are gels important?

Protective capability

High viscosity (low washout rate) -for drugs, acid protection

Biofilm formation in Pseudomonas Aeruginosa

Wild type Biofilm mutant Mutant with autoinducer

Heterogeneous structures

The Dynamics of Growing Biofilm

• Quorum Sensing:– What is it?– How does it work?

• Heterogeneous structures:

– How do these cells use polymer gel (EPS) for locomotion?

– What are the mechanisms of pattern (structure) formation?

– Why is polymer gel so effective as a protective environment?

Quorum Sensing in P. Aeruginosa

P. Aeruginosa

•Major cause of hospital infection in the US.•Major cause of deaths in intubated CF patients, and IV fed patients.

Quorum Sensing: The ability of a bacterial colony to sense its size and regulate its activity in response.

Examples: Vibrio fisheri, myxococcus, P. Aeruginasa

P. Aeruginosa in planktonic (non-colonized) form are non-toxic, but as a biofilm, they are highly toxic and well protected by the polymer gel in which they reside. However, they do not become toxic or begin to form polymer gel until the colony is of sufficient size to overwhelm the immune system. Before this, they cannot be detected bythe immune system.

Quorum sensing in P. Aeruginosa

Planktonic

Loosely Bound EPS secreting

“Wall Sensing” in P. Aeruginosa

Wall Sensing: The ability of bacteria to differentiate in response toContact with a wall (the substratum).

Planktonic Loosely Bound EPS secreting

The Dynamics of Growing Biofilm

• Quorum Sensing:– What is it?– How does it work?

• Heterogeneous structures:

– How do these cells use polymer gel for locomotion?

– What are the mechanisms of pattern (structure) formation?

– Why is polymer gel so effective as a protective environment?

The Biochemistry of Quorum Sensing

ALasR

lasR

lasI

LasR LasIA

3-oxo-C12-HSL

rsaL LasR

B

C4-HSL

RhrR B

RhlI

rhlI

GacA

Vfr

rhlR

RhlR

An ODE Model

Variable Concentration

R LasR

A 3-oxo-C12-HSL

P Dimer complex

L LasI

S RsaL

r lasR mRNA

l lasI mRNA

s rsaL mRNA

lasI

LasI

ALasR

lasRLasR A

3-oxo-C12-HSL

rsaL LasR

Modeling Biochemical Reactions with ODE’s

€

dPdt

=

€

k+ARRate of production

€

−k−PRate of degradation

€

dLdt

=

€

kl lRate of production

€

−KLLRate of degradation

€

dldt

=

€

VmaxPK l +P

€

−k−l l

€

l ⏐ → ⏐ LProduction of enzyme from mRNA

lasI

LasI

€

A+R← → ⏐ PBimolecular reaction

ALasR

LasR A

3-oxo-C12-HSL

€

P ⏐ → ⏐ lProduction of mRNA

€

dPdt

=kRARA−kPP,

dRdt

=−kRARA+kPP−kRR+k1r,

dAdt

=−kRARA+kPP+k2L−kAA,

dLdt

=k3l−klL,

dSdt

=k4s−kSS,

dsdt

=VsP

Ks +P−kss,

drdt

=VrP

Kr +P−krr+r0,

dldt

=VlP

Kl +P1

KS +S−kl l+l0.

The full system of differential equations for the biochemistry:

lasI

LasI

ALasR

lasRLasR A

3-oxo-C12-HSL

rsaL LasR

Modeling diffusion across cell membrane

Does this model exhibit quorum sensing?

Two ways to proceed:1) Numerical simulation (but few of the 22 parameters are known)2) Qualitative analysis

€

dAdt

+kRARA−kPP−k2L+kAA

€

dEdt

+kEE

A

E

where is the cell volume fraction.

€

ρ(

€

)

€

(1−ρ)(

€

)

€

ρ

€

ρ

€

1−ρ

€

=δ(E−A)

€

=δ(A−E)

Quasi-steady state analysis: Assume that all “fast” variables are in quasi-steady state

€

dRdt

=VRP

KR +P−kRR+R0,

dAdt

=VAP

KA +P+A0 −d(ρ)A,

P=kRARAkP

, d(ρ)=kA +δρ

(kE (1−ρ)

δ+kE (1−ρ)).

An even simpler model

Set R=A (which is not correct), to find

A Simple Model for Wall Sensing

€

∂V∂t

=G(V,A), V∈ℜ7

∂A∂t

=F(A,V)+d(E(x=L)−A),

∂E∂t

=DE

∂2E∂x2 −kEE+dδ(x−L)(A−E )(=0)

Cells are immobile, so internal variables A and V do not diffuse,but extracellular autoinducer E diffuses (rapidly) so

L

E

x

A Simple Model for Wall Sensing - cont’d

€

∂V∂t

=G(V,A), V∈ℜ7

∂A∂t

=F(A,V)−dp

1+pA,

p=μDd

(1+tanh(μDL))

We can solve for E(x=L) assuming a quasi-equilibrium, to find

Since p decreases as L 0, this has a chance of up-regulating for small L.

A Proper (PDE) Model

€

∂V∂t

=G(V,A), V∈ℜ7

∂A∂t

=F(A,V)+δρ

(E−A),

∂E∂t

=∇ 2E −kEE +δ

1−ρ(A−E),

Cells are immobile, so internal variables A and V do not diffuse,but extracellular autoinducer E diffuses so

on the domain , subject to the (Robin) boundary condition

€

Ω

€

n•D∇E +αE =0, on

€

∂Ω

€

Ω

€

∂Ω

Autoinducer with fixed colony size, variable density

Autoinducer concentration; fixed density, variable size colony

QuickTime™ and aAnimation decompressor

are needed to see this picture.

Heterogeneous Structures

•How do these cells use polymer gel for locomotion?

•What are the mechanisms of pattern (structure) formation?

•Why is polymer gel so effective as a protective environment?

A Hydrogel Primer

•What is a hydrogel?A tangled polymer network in solvent

•Examples of biological hydrogels:Micellar gelsJello (a collagen gel ~ 97% water)Extracellular matrixBlood clotMucin - lining the stomach, bronchial tubes, intestinesGycocalyx - lining epithelial cells of blood vesselsSinus secretions

•Other examples of gel-like structuresCell colonies - colloidal suspensionsGel morphology - sponges, jellyfish

Fibrin Network (Blood Clot)

A Hydrogel Primer - II

•Function of a biological hydrogelDecreased permeability to large moleculesStructural strength (for epithelial cell walls)Capture and clearance of foreign substancesDecreased resistance to sliding/glidingHigh internal viscosity (low washout)

•Important features of gelsUsually comprised of highly polyionic polymersOften exhibit large volumetric changes

eg. Highly compressed in secretory vessicle and expand rapidly and dramatically on releaseCan undergo volumetric phase transitions in response to ionic

concentrations (Ca++, H+), temperature, ..Volume is determined by combination of attractive and

repulsive forces:-- repulsive electrostatic, hydrophobic-- attractive, hydrogen binding, cross-linking

How gels grow

•Polymerization/deposition

(blood clots)

•Secretion

High Ca++

Ca++ Na+

Highly condensed Decondensed (swelled)

Interesting facts: - Diffusion coefficient can vary substantially as a function of Ca++ - Expansion does not occur in distilled water.

Modeling Biofilm Growth

€

ϑA two phase material with polymer volume fraction

€

∂ϑ∂t

+∇ •(Vnϑ )=gn Network Phase (EPS)

€

∂(1−ϑ )∂t

+∇ •(Vs(1−ϑ ))=0 Solute Phase

€

∂b∂t

+∇ •(Vnb) =gb Bacterial Concentration

€

∂(1−ϑ )u∂t

+∇ •((1−ϑ )(Vsu−D∇u) =gu Resource

Force Balance

Solute phase

€

hfϑ (1−ϑ )(Vn −Vs)

(solute-network friction)

€

η∇ •(ϑ n(∇Vn +∇VnT ))

(viscosity)

(pressure)

€

−(1−ϑ )∇p=0

Network phase

(pressure)

€

−ϑ∇p=0(solute-network friction)

€

−hfϑ (1−ϑ )(Vn −Vs)(osmosis)

€

−∇ψ(ϑ )

Incompressibility

€

∇ •(Vnϑ +Vs(1−ϑ ))=gn

€

∂ϑ∂t

+∇ •(Vnϑ )=gn

€

∂(1−ϑ )∂t

+∇ •(Vs(1−ϑ ))=0

Remark: Models of this form for the network phase have been used by lots of people (Dembo and He, Lubkin and Jackson, Wolgemuth, Oster, Mogilner, …)

Remark 2: A better model might keep track of Ca++ and Na+ concentrations as well

Osmotic Pressure

What is the meaning of the term

In some formulations

(osmosis)

€

−∇ψ(ϑ ) ?

€

ψ(ϑ )=F '(ϑ ) where

€

F(ϑ )

€

F(ϑ )

€

ψ(ϑ )

€

ϑ

€

ϑ

is the Free Energy

€

ψ '(ϑ )> 0 gives expansion (swelling)

< 0 gives contraction (deswelling){

€

ψ(ϑ )=kT(k1(1−θ)ln(1−θ)+χθ(1−θ)) From Flory-Huggins theory

Phase SeparationDouble-welled potentials give phase transitions and phase separation, similar to Cahn-Hilliard

€

ψ(ϑ )

€

ϑ

€

ϑ (x)

€

x

To maintain an edge,

€

ψ(ϑ ) must be of the form

€

ψ(ϑ )=ϑ 2G(ϑ )

€

ψ(ϑ )

€

ϑ0

0

0

€

ϑ (x)

€

x

(This is a theorem)

Movement by Swelling

QuickTime™ and aAnimation decompressor

are needed to see this picture.

Propulsion by Swelling

QuickTime™ and aAnimation decompressor

are needed to see this picture.

No gel; no swell

QuickTime™ and aAnimation decompressor

are needed to see this picture.

Heterogeneous Structures

•How do these cells use polymer gel for locomotion?

•What are the mechanisms of pattern (structure) formation?

•Why is polymer gel so effective as a protective environment?

“Limited Resource” Instability

Remark: This instability does not occur in a resource rich environment.

Channeling

A Pouiseille flow

Channeling

Channeling

Conclusions

• Quorum Sensing:– What is it? A biochemical switch– How does it work? Extracellular autoinducer fails to diffuse away.

• Heterogeneous structures:

– How do these cells use polymer gel for locomotion? Expansion by swelling

– What are the mechanisms of pattern (structure) formation?• Limited resource fingering• Friction driven channeling• Others?(Is sloughing related to a Kelvin-Helmholz

instability?)– Why is EPS so effective as a protective environment?

• Restricted pore size, chemical reactivity???

More Questions (than answers)

Why does P.Aeruginosa attack only immune-compromised individuals?(burn victims, post-surgery)

Why does P.Aeruginosa find a ready host in CF patients? (Decreased diffusion???)

What is it about the CFTR mutation in CF patients that makes their mucin so thick?

Is there cross talk between P. Aeruginaosa and other pathogens and if so, how?