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Relaxation in Biological Systems
Robert G. BryantUniversity of Virginia
Charlottesville, VA, USA
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Collaborators
• Jean-Pierre Korb• Pierre Levitz• Dominique Petit• Yanina Goddard• Yanina Goddard• Galina Diakova• Ken Victor• David Cafiso
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Simple Solutions
• Relaxation by rotational and translational motions of spin ½ nuclei: 1H, 13C, 15N, 31P
• Dipolar relaxation dominates at low magnetic field strengthsmagnetic field strengths
• Chemical shift anisotropy makes contribution at high field strengths– 13C, 15N, 31P
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Small Diamagnetic Molecules
• Translational correlation times are short, tens of ps.
• Rotational correlation times are tens of ps.• No dispersion usually detected.• No dispersion usually detected.
– Rare possibility of modulation of scalar couplings and very low field, e.g., H2
17O– Dispersion from molecular oxygen if not
removed
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Paramagnetic Effects• Contributions from oxygen at ambient
concentration easily detected.
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Paramagnetics
• Shift dispersions from translation and rotation into the MHz range
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2
2.5
3
3.5107 108 109 1010 1011 1012
CrDTPA_pH 5.8CrEDTA_pH 3.5CrEDTA_pH 4.50CrEDTA_pH 5.40CrDTPAam_pH 4.4CrDTPAam_pH 7.7CrEDTAam_pH 6.1CrEDTAam_pH 7.7CrEDTAam_pH 4.0
Rel
axat
ion
Rat
e C
onst
ant (
mM
-1,
sec
-1)
Electron Larmor Frequency (rad/sec)
0
0.5
1
1.5
104 105 106 107 108 109 1010
Rel
axat
ion
Rat
e C
onst
ant (
mM
-1,
sec
-1)
Proton Larmor Frequency (rad/sec)
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Protein Solution MRD
• Coupling to rotational motion of the protein – Exchange of whole water molecules– Exchange of protons
• Changes in rotation and translation at the interface are small: of order a factor of 3
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Koenig’s Early Data
S. H. Koenig, W. E. Schillinger, J. Biol. Chem., 244:3283-3289 (1969).
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0.5
0.4
0.3
0.2
0.20
0.16
0.12
0.08
108
109
1010
1011
1012
ωS (rad/s)
A
B
Protein Solution MRDD2O with oxygen
H2O with oxygen
D O without oxygen
0.60
0.55
0.50
0.45
0.40
0.35
104
105
106
107
108
109
1010
ωI (rad/s)
0.11
0.10
0.09
0.08
0.07
0.06
0.05
C
D
D2O without oxygen
H2O without oxygen
When dilute and no aggregation, the dispersion is Lorentzian
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Chemical exchange couples protein rotation to H 2O
τrot 10-50 ns
Internal Water Moleculesτlocal = 10-30 ns 1H2O gel
τexchange = 0.5-100 µs
-
+
−
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Effects of ExchangeDilute Solutions
• The mixing equation
1 1 1
1 1 boundo
bound exchange
P
T T T τ= +
+• In slow-exchange limit
1 1 1bound exchangeT T T τ+
1 1
1 1 boundo
exchange
P
T T τ= +
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Exchange Coupling• The exchange time limits the size of the
exchange contribution.
1 1 1
1 1 bound io
i sites bound i exchange i
P
T T T τ−
− − −
= ++∑
• Water lifetimes generally short compared with proton lifetimes– Proton lifetimes are pH dependent and generally 1st
order in [OH-].
• Water effects often dominate for large proteins or solids.
1 1 1bound i exchange i− −
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Direct Dipolar Couplings
• Bound protons couple to protein protons and each other by dipole-dipole interactions.
• Surface interactions between water and • Surface interactions between water and biomacromolecules are dipolar– Coupling modulated by rotation and
translation– Correlation times are short, tens of ps– Effective relaxation contributions are small– Logarithmic function of magnetic field strength
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0.4
0.5
0.6
0.7
0.8
Dispersion for BSA(300 uM)
Rel
axat
ion
Rat
e C
onst
ant
(1/s
) HOD Stretched fit for HOD Lorentzian for HOD DMSO Stretched fit for DMSO Lorenztian for DMSO Acetone Stretched fit for AcetoneLorentzian for Acetone
104
105
106
107
108
109
1010
0.1
0.2
0.3
0.4
Rel
axat
ion
Rat
e C
onst
ant
(1/s
)
1H Larmor Frequency (rad/sec)
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Protein Proton Dispersion
• Low field rate linear in rotational correlation time.– For small protein, of order 100-200 s-1
• Proton relaxation is superposition of all • Proton relaxation is superposition of all rates at low resolution.
• System is not completely homogenized by rapid spin diffusion.
• Sample shuttle experiments are difficult.
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Protein Solution MRD
Examine the high Examine the high field region
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Protein Solution MRDAlbumin, lysozyme, ribonuclease A
H2O
D2O observe residual H
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2-D Translational Diffusion Relaxation Equation
2 21( ) log fJ
ω τω τ
+ ∝
11/ ( ) 4 (2 )T B J Jω ω= +
22 2
2
1( ) log f
ff
fs
Jω τ
ω ττ
ω ττ
+ ∝ +
1 1 1
1
( ) ( )b s
b s
P P
T T Tω ω= +
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Intermolecular Dipolar Coupling Modulated by Translation
30 ps
Water-1H –protein 1H dipolar coupling modulated by relative translational motion of the water at the protein interface
HO
DH
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Lysozyme Intermolecular Contribution
0.09
0.095
0.1
0.105
Lysozyme degassed 8/25/2006
-1
y = 0.03+m1*10^ 8*m2*10 -12*...
ErrorValue
0.0237950.54596m1
1.938131.242m2
NA6.0268e-6Chisq
NA0.99824R
Residual H in HOD; solution
0.065
0.07
0.075
0.08
0.085
10 100 1000
1/T
1, s-1
1H Larmor Frequency, MHz
31 ps
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Lysozyme in H2O
0.52
0.54
0.56
Lysozyme H2O, degassed 8_17_2006
-1
y = 0.28+m1*10^ 8*m2*10^-12*...
ErrorValue
0.320273.36m1
1.827614.544m2
NA0.00010289Chisq
NA0.99576R
0.42
0.44
0.46
0.48
0.5
40 60 80 100 300
1/T
1, s-1
1H Larmor Frequency, MHz
NA0.99576R
15 ps
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Intermolecular Dipolar Coupling Modulated by Translation at Surface
15 ps
2-particles move Relative translational diffusion constant is sum of each particle.
HO
HH
of each particle.
τ1 = 2τrelative= 30 ps20 ns
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Charge Distribution: Ribonuclease A
pH = 7.5All residues were assumed to have their usual pKa i n water and were modified to appropriate protonated states accordingly. The prot ein was set with Conolly atomic radii, CVFF formal charge parameters, and the dielectric c onstant of 2.0. The solvent was set with 1.4 Å radius, zero ionic strength and the dielectri c constant of 80.0. The solute was centered, filling 70% of the grids in each dimensio n, in an 83 ×83×83 grid box of 0.816 Å per grid point. Finite difference methods were used to calculate potential functions providing full coulomb boundary conditions.
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C(t)
+
-
H HO
Rotational Correlation: Long times
t
Orientationalbias
3 ps
15 ps
-
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Water-Surface Dynamics
The translational correlation time for water at the protein water at the protein interface is 30 ps–fast but still slower than in the bulk.
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0.3
0.4
0.5107 108 109 1010 1011 1012
Hb HOD Paramagnetic 0907
1, s-1
HOD Paramagnetic Contribution
τf = 6.4 psτs = 2.6 ns
0
0.1
0.2
104 105 106 107 108 109 1010
1/T
1
Radial Frequency, s -1
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40
50
60
R1_4.5%gelBSA, s-1
Immobilization Effects
11
1( )
( )bA C R
Tω ω
ω−= + =
Power Law
Observing water protons
0
10
20
30
0.01 0.1 1 10 100
R1_4.5%gelBSAR1_7%gelBSAR1_10%gelBSAR1_15%gelBSAR1_20%gelBSAR1_30%gelBSAR1_40%gelBSA
Proton Larmor Frequency, MHz
1/T
1, s
R. G. Bryant, D. Mendelson, C. C. Lester, Magn. Reson. Med. 21, 117-126 (1991).
Power Law
Lorentzian
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14N Level Crossing
70
80
90
100
BSADIPS121504
1/T
1, s-1
30
40
50
60
70
2.4 2.6 2.8 3 3.2 3.4
1/T
Proton Larmor Frequency, MHz
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Experimental Results
11
1( )
( )bA C R
Tω ω
ω−= + =
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Experimental Results
11
1( )
( )bA C R
Tω ω
ω−= + =
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Non-rotating protein:Solid spin population
Water: Liquid population
Shirley &Bryant, JACS104:2910 (1982)
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Relaxation Coupling
Dry and D2O same, dominated by relaxation of CH3
H2O adds new minimumH2O adds new minimum
H2O at low T adds to relaxation load of CH3
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Spin-coupling of the solid population is general.
Similar to a transferred NOE only the solid population is usually not observed directly.
Effects modeled well by 2-spin-population model.
Line-shape is subject to saturation effects, as usual in cw NMR.
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1,
1, 0( )
WP W WPWW
WP WPPPP
R R RMM
R RMRM
F Fω
+ − = − − +
&
&
Relaxation Coupling
eqPeqW
m Number of protein protonsF
m Number of water protons= =
R Wm Number of water protons
R1,P
Lattice
WaterSolidprotein
RWP
R1,W
R1,
slow
(s-1
)
RWP=P0kex
R1,P
Frequency (MHz)
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Backbone Origin of Power Law Polyglycine
ν0.80
Polyalanine
T1 =Aν0.77
Lysozyme et BSA
10-3
10-2
10-1
100
BSALysozyme
T 1 (
s)
ν0.78T =A
C
N
H
C
HH
O
C
n
T1 =Aν0.80
C
N
H
C
CH3H
O
C
n
R. Kimmich
Dynamics ∈∈∈∈ Primary structure
C
N
H
C
RH
O
C
n
n
J.-P. Korb et R.G. Bryant
10-4
10-2 10-1 100 101 102
Frequency (MHz)
νT1=A
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Basis of Spin-Fracton Model
• Kimmich: dipolar couplings modulated by backbone fluctuations.
• Vibrational model for local fluctuations– Structural disturbance propagates along backbone– Protein is not uniform in 3-space– Protein is not uniform in 3-space
• Distribution of mass is fractal df ~2.4-3• Vibrational distribution of states is not standard
• Fluctuation is localized and scale is related to the frequency (wavelength)
1 1; 1.3sd dsdω ω− −≠
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Note the proximity of protons and their connectivity by dipole-dipole couplings.
Protons not uniformly placed in space.
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Spin-phonon Relaxation
Spin Lattice
1
2n0
nδHdip
Direct process
)()()( pdip
eqpdip
pdip HHH δ+=
Spin displacements in localized normal modes
s
f
Propagation of displacement
along the backbone d
Spatial distribution
of protons d
⇒
⇒
modes
1
T1(ω0 ) ∝ A kT ω0
−2( )ω0dS −1( )ω0
2 d s / d f( )density of
states
localization
Spin LatticeLocal Structural Fluctuation /
( )s fd d
l ωωΩ ∝
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<NH
> R 2.5∝
Lysozyme
Compute df based on crystal structure of protein as the slope of log-log plot of the number of protons within a radius r from a reference proton.R
100
101
102
103
Fractal Dimension
The dimensionality d explored by the disturbance affects the density of vibrational states σ(ω) that determines relaxation efficiency.
101 10
Distance from a reference proton, Å
lα
ω < ωα
ω > ωα
σ(ω0 ) = 3Ndω0
d−1
Ω d
with σ ω( )0
Ω
∫ dω = 3Nd→→→→ds
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The reduction in effective dimensionality caused by the 1-dimensional character of the primary protein structure increases the density of vibrational states, particularly in the 10 -20
10 -14
(d)/
(3N
) 1.33
1
021
1( ) Bk T
Tσ ω∝
Ω
Ω
Constrained Propagation
For d~1, at low frequency, enhancement of σ by 18 orders of magnitude. Localization of the disturbance is also required.
states, particularly in the low frequency range.
10 -32
10 -26
10 -20
10 -2 101 104 10 7
σ(d
)/(3
N)
Frequency (MHz)
3
2
d
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Theory - Experiment
0.6
0.8b
max = 0.783
2
21
1 3 1 14 1
4 2
b
B s o b ok TM d
T E Eν ν
ωπ− = +
hh
2000
2500
3000
BSA dry
Lysozyme dry
C
N
H
C
O
C
RH
RHO
N
H
CR H
Polypeptide backbone
C
n
Mode Amide II Ev// =1560 cm-1
I.R.
Stretched modes
b = 3 −2ds
d f
− ds
0
0.2
0.4
1.5 2 2.5 3
b
df
bmin
=
0
500
1000
1500
2000
10-2 10-1 100 101 1021/
T1
(s-1)
Frequency (MHz)
ω0-0.79
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C-C~350 kJ/molH-bond~20 kJ/mol in gasvdW~2 kJ/mol
Temperature Dependence
102
103
BSA
R1 0.1 MHzR1 0.2 MHzR1 0.3 MHzR1 0.01MHzR1 0.03 MHzR1 0.05 MHzR1 0.08 MHZR1 30 MHzR1 10 MHzR1 20 MHZR1 5MHz
Low frequencyR1 = kT
Propagation along stiff connections
vdW
( )( )
( )( ) / 22 21
1cos( tan( ))
1f
bB fB
fbb
k T Tk TA B bA
T T
τωτ
ω ω ω τ∝ +
+
100
101
150 200 250 300 350
R1 5MHz
Temperature (°C)
High frequencyR1 = kT exp(Ea/RT)
(K)
Chain Side-chain
/0
Ea RTc c eτ τ=
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Low Temperature Serum Albumin
102
103
, s-1
101
10-2 10-1 100 101
R1 @ T=-118CR1 @ 25C
1/T
1, s
Frequency, MHz
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Model Peptides
polyglycine and polyalanine, 183 KBiopolymers, 2007
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0.6
0.7
0.8
0.9
1/T
1, s-1
0.6
0.7
0.8
0.9MRD_1H_BSA10%gel_060525
1/T
1, s-1
Gel ∝ω-0.8
Solution ∝lnω
BSA BSA
High Field MRD
0.4
0.5
0 50 100 150 200 250 300 350
Proton Larmor Frequency, MHz
0.4
0.5
0 100 200 300 400 500 600
Proton Larmor Frequency, MHz
•Constant composition•Glutaraldehyde cross-link•Relaxation changes when rotational state of the protein changes
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1000
104
BSA_D2O_15%gel_043007_NaCl_D2O_061107 11:47:09 AM 6 /15/2007
R1_BSA_gel_D2O_043007R1_BSA_dry
1/T
1, s
-1
1H BSA Dry and in 15% gel in D2O
1
10
100
0.001 0.01 0.1 1 10 100
1/T
1, s
-1
Proton Larmor Frequency, MHz
No 1H in solventobserve protein 1H
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Conclusion
• The parameters of the protein dynamics model, df and ds, do not change with hydration.
• The hydration does not substantially change the underlying polypeptide dynamics.
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1000
104
BSA_dry_D2O_gel
R1_BSA_gel_D2O_043007R1_15%BSA_gel_NaCl_D2O_061107R1_BSA_dry
, s-1
15% BSA gel in D2O
Dry BSA
15% BSA in H2O
1H Relaxation in BSA Gels
1
10
100
0.001 0.01 0.1 1 10 100
1/T
1, s
1H Larmor Frequency, MHz
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10
100
BSA_gels_+ Lorentzian
R1_20%gelBSA, s
-1y = 0+m1*10 3*(m0*6.285*10^...
ErrorValue
0.0204721.8798m1
0.0225930.21842m3
3.081426.087m4
NA9.6653Chisq
NA0.99889R
BSA Gel + Lorentzian
0.1
1
0.001 0.01 0.1 1 10 100 1000
1/T
1, s
1H Larmor Frequency, MHz
26 ns
Motion of water inside the protein
Consistent with dielectric relaxation
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Coupled and Uncoupled Dyanmics
1000
104
Coupled Dynamics
-1 100
1000
104
Decoupled Dynamics
-1
COUPLED UNCOUPLED
0.1
1
10
100
0.01 0.1 1 10 100 1000
1/T
1, s-1
Larmor Frequency, MHz
0.01
0.1
1
10
100
0.01 0.1 1 10 100 1000
1/T
1, s-1
Larmor Frequency, MHz
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Uncoupled Motion: Water In a Protein
No rigid dynamical coupling to protein motions,magnetic dipole-dipole coupling remains.
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Coupled Motion: Water In a Protein
H-bond mediated coupling to protein motions
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BSA Gel 10%, 20%
101
Rel
axat
ion
Rat
e
10-2
100
102
100
Proton Larmor Frequency, MHz
Rel
axat
ion
Rat
e
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Hydrated Albumin at 185 K
102
10332 g water/100 g BSA
Difference fit to Lorentzian with correlation time
10 -2 10 -1 10 0 10 1
Proton Larmor Frequency, MHz
101
100
correlation time of 92 ns.
Consistent with uncoupled local rotational motion.
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Summary• Simple relaxation model accounts for all the
known data– The protein protons relax by an efficient
mechanism that causes a power law in frequency.
• The water protons magnetically coupled to the protein spin relaxation. protein spin relaxation. – Magnetic coupling maps protein relaxation onto
water proton relaxation.
• Side-chains important only at high frequency, low T.
• Internal water dynamics important in middle frequency range (tens of ns).
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Phospholipids
0.07
0.08107 108 109 1010 1011 1012
POPC 121102S (water) 2/5/2004
POPC @ 40mM
40 mM POPCResidual H in D2O
Small contribution to
0.03
0.04
0.05
0.06
104 105 106 107 108 109 1010
1/T
1, s-1
Radial Frequency [rad/s]
Small contribution to relaxation rate
Linear in log of frequency
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Lipids
0.5
0.6
0.7
0.8
POPC:POPS (3:1) @ 25mM with 680mM sucrose by extrusion 041003
D2O buffer with 680mM sucrose by extrusion 041103
POPC:POPS (3:1) @ 25mM with 30uM dodecyl maltoside by extrusion 041403
107 108 109 1010 1011 1012
Rel
axat
ion
Rat
e C
onst
ant
(1/s
)electron ωωωω
S(rad/sec)
BtuB in POPC:POPS (3:1) LUVs Dispersions of Water ProbeBuffer: 100mM formate, 10mM HEPES, pH=6.8, in D
2O
Eliminated if
0
0.1
0.2
0.3
0.4
0.5
104 105 106 107 108 109 1010
Rel
axat
ion
Rat
e C
onst
ant
(1/s
)
1H ωωωωI(rad/sec)
Eliminated if sucrose scrubbed of metal ions
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The Magnetic Field and Temperature Dependences of Proton Spin-Lattice Relaxation in Proteins, Y. Goddard, J.-P. Korb, R. G. Bryant, J. Chem. Phys. 126, 175105/1 (2007).
Relaxation of Protons by Radicals in Rotationally Immobilized Proteins, J.-P. Korb, G. Diakova, Y. Goddard, J. Magn. Reson., 186, 176-181 (2007).
Nuclear Magnetic Relaxation Dispersion Study of the Dynamics in Solid Homopolypeptides, Y. Goddard, J.-P. Korb, R. G. Bryant, Biopolymers, 86, 148-154 (2007).
Structural and Dynamical Changes in Serum Albumin at Low Temperature: Examination of theGlass Transition, Y. Goddard, J.-P. Korb, R. G. Bryant, Biophysical Journal 91, 3841-3847 (2006).
Structural and Paramagnetic Relaxation of Protons in Rotationally Immobilized Proteins, J.-P. Structural and Paramagnetic Relaxation of Protons in Rotationally Immobilized Proteins, J.-P. Korb, G. Diakova, R. G. Bryant, J. Chem. Phys. 124, 134910/1-134910/6, 2006.
Magnetic Field Dependence of Proton Spin-lattice Relaxation of Confined Proteins. J.-P. Korb, R. G. Bryant, C. R. Physique 5 (2004) 349-359.
Noise and Functional Protein Dynamics. J.-P. Korb, R. G. Bryant, Biophysical Journal, 89, 2685-92 (2005).
The Magnetic Field Dependence of T1. J.-P. Korb, R. G. Bryant, Magn. Reson. Med. 48, 21-26 (2002).
The Physical Basis for the Magnetic Field Dependence of Proton Spin-Lattice Relaxation Rates in Proteins and Tissues. J.-P Korb, R. G. Bryant, J. Chem. Phys. 115, 10964-10974 (2001).
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Thank you!
Support: National Institutes of Health, University of Virginia, CNRS, Ecole Polytechnique