Relaxation in Biological Systems

Robert G. BryantUniversity of Virginia

Charlottesville, VA, USA

Collaborators

• Jean-Pierre Korb• Pierre Levitz• Dominique Petit• Yanina Goddard• Yanina Goddard• Galina Diakova• Ken Victor• David Cafiso

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

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

Paramagnetic Effects• Contributions from oxygen at ambient

concentration easily detected.

Paramagnetics

• Shift dispersions from translation and rotation into the MHz range

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)

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

Koenig’s Early Data

S. H. Koenig, W. E. Schillinger, J. Biol. Chem., 244:3283-3289 (1969).

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

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

-

+

−

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= +

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− −

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

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)

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.

Protein Solution MRD

Examine the high Examine the high field region

Protein Solution MRDAlbumin, lysozyme, ribonuclease A

H2O

D2O observe residual H

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= +

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

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

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

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

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.

C(t)

+

-

H HO

Rotational Correlation: Long times

t

Orientationalbias

3 ps

15 ps

-

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.

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

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

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

Experimental Results

11

1( )

( )bA C R

Tω ω

ω−= + =

Experimental Results

11

1( )

( )bA C R

Tω ω

ω−= + =

Non-rotating protein:Solid spin population

Water: Liquid population

Shirley &Bryant, JACS104:2910 (1982)

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

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.

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)

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

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− −≠

Note the proximity of protons and their connectivity by dipole-dipole couplings.

Protons not uniformly placed in space.

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 ∝

<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

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

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

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=

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

Model Peptides

polyglycine and polyalanine, 183 KBiopolymers, 2007

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

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

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.

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

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

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

Uncoupled Motion: Water In a Protein

No rigid dynamical coupling to protein motions,magnetic dipole-dipole coupling remains.

Coupled Motion: Water In a Protein

H-bond mediated coupling to protein motions

BSA Gel 10%, 20%

101

Rel

axat

ion

Rat

e

10-2

100

102

100

Proton Larmor Frequency, MHz

Rel

axat

ion

Rat

e

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.

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).

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

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

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).

Thank you!

Support: National Institutes of Health, University of Virginia, CNRS, Ecole Polytechnique