Heteronuclear Relaxation and Macromolecular Structure and Dynamics

Outline:

Note: refer to lecture on “Relaxation & nOe”

• Information Available from Relaxation Measurements

• Relaxation Mechanisms

• Relaxation Rates

• Experimental Methods

• Data Analysis

• Case Studies

1. Fushman, D., R. Xu, et al. (1999). "Direct determination of changes of interdomain orientation on ligation: Use of the orientational dependence of N-15 NMR relaxation in Abl SH(32)." Biochemistry 38(32): 10225-10230.

1. Eisenmesser, E. Z., D. A. Bosco, et al. (2002). "Enzyme dynamics during catalysis." Science 295(5559): 1520-1523.

1. Lee, A. L., S. A. Kinnear, et al. (2000). "Redistribution and loss of side chain entropy upon formation of a calmodulin-peptide complex." Nature Structural Biology 7(1): 72-77.

1. Ishima, R., D. I. Freedberg, et al. (1999). "Flap opening and dimer-interface flexibility in the free and inhibitor-bound HIV protease, and their implications for function." Structure with Folding & Design 7(9): 1047-55.

• References

Biomolecules are not static:

• rotational diffusion (c)• translational diffusion (D)• internal dynamics of backbone and sidechains (i)• degree of order for backbone and sidechains (S2)• conformational exchange (Rex)• interactions with other molecules (kon,koff)

Biomolecules are often not globular spheres:

• anisotropy (Dxx,Dyy,Dzz)

Structure/Dynamics Function

c

S

i

Rex

D

kon

koff

NMR Relaxation and Dynamics

NMR relaxation measurements provide information on structure and dynamics at a wide range of time scales that is site specific:

• Dynamics on Different Time Scales

time scale example experiment type

ns – ps bond librations lab frame relaxationreorientation of protein T1, T2

motions of protein main chainside chain rotations(case study #2)

us – ms rapid conformational exchange lineshape analysis(case study #4) rotating frame relax.

T1

ms – s interconversion of discrete magnetization exch. conformations

> s protein folding exchange ratesopening of 2o structures (H/D exchange)

• Structural Information from Relaxation

• anisotropy of overall shape (case study #1)

• distance information from cross-correlation relaxation

• Thermodynamics from Relaxation

• relationship to entropy (case study #3)

Relaxation

Bloch equations – introduce relaxation to account for return of magnetization to equilibrium state:

excite

relax

treat relaxation as a first order process:

dM/dt = M x B – R(M-Mo)

where

T1 (longitudinal or spin-lattice relaxation time) is the time constant used to describe rate at which Mz component of magnetization returns to equilibrium (the Boltzman distribution) after perturbation.

T2 (transverse or spin-spin relaxation time) is the time constant used to describe rate at which Mxy component of magnetization returns to equilibrium (completely dephased, no coherence) after perturbation.

R =

1/T2 0 0

0 1/T2 0

0 0 1/T1

so far, all we have is a time constant; is it possible to get a “picture” of what is causing relaxation?

• consider spontaneous emission of photon:

transition probability 1/3 = 10-20 s-1 for NMR

• consider stimulated emission:

the excited state couples to the EMF inducing transitions – this phenomenon is observed in optical spectroscopy (eg. lasers) but its effect is negligible in RF fields.

• in a historic paper, Bloembergen, Purcell and Pound (Phys. Rev. 73, 679-712 (1948)) found that relaxation is related to molecular motion (NMR relaxation time varied as a function of viscosity or temperature). They postulated that relaxation is caused by fluctuating fields caused by molecular motion.

RF photon

• relaxation is dependent on motion of molecule

• Zeeman interaction is independent of molecular motion therefore “local fields” exist that are orientation dependent and couple the magnetic moment with the external environment (the “lattice”)

• time dependence of interaction determines how efficiently the moment couples to the lattice

• it is the fluctuating “local fields” that induce transitions between energy levels of spins:

RF

source of local fields?

timescale of fluctuation?

Relaxation Mechanisms

The relaxation of a nuclear spin is governed by the fluctuations of local fields that result when molecules reorient in a strong external magnetic field. Although a variety of interactions exist that can give rise to a fluctuating local field, the dominant sources of local fields experienced by 15N and 13C nuclei in biomolecules are dipole-dipole interactions and chemical shift anisotropy: • Magnetic Dipole-Dipole Interaction - the dipolar interaction is a through-space coupling between two nuclear spins:

I

SrIS

The local field experienced by spin I is:

Hloc = Sh/r3IS ((3cos2 – 1)/2)

• Chemical Shift Anisotropy - the CSA interaction is due to the distribution of electrons surrounding the nucleus, and the local magnetic field generated by these electrons as they precess under the influence of the applied magnetic field. The effective field at the nucleus is:

Hloc = Ho(1-)

where Ho is the strength of the applied static magnetic field and is the orientationally dependent component of the CSA tensor.

Expressions for Relaxation Rates

The relaxation rate constants for dipolar, CSA and quadrupolar interactions are linear combinations of spectral density functions, J(). For example, one can derive the following equations for dipolar relaxation of a heteronucleus (i.e. 15N or 13C) by a proton

R1,N = 1/T1,N = (d2/4)[J(H-N) + 3J(N) + 6J(H+N)]

R2,N = 1/T2,N = (d2/8)[4J(0) + J(H-N) + 3J(N) + 6J(H) + 6J(H+N)]

NOE15N{1H} = 1 + (d2/4)(H/N) [6J(H+N) - J(H-N)] x T1,N

where d = (HN(h/8)/rHN3)

The J() terms are “spectral density” terms that tell us what frequency of motions are going to contribute to relaxation. They have the form

J() = c/(1+2c2)

and allow the motional characteristics of the system (the correlation time c) to be expressed in terms of the “power” available for relaxation at frequency :

J()

c

c

c

Measurement of Relaxation Rates

• spin lattice relaxation is measured using an inversion recovery sequence:

180

I

I = Io(1-2exp(-/T1))

• spin-spin relaxation is measured using a “spin echo” sequence (removes effect of field inhomogeneity):

90 180

I = Ioexp(-/T2)

I

Measurement of Relaxation Rates

The inversion-recovery sequence and spin-echo sequence can be incorporated into a 2D 1H-15N HSQC pulse sequence in order to measure 15N T1 and T2 for each crosspeak in the HSQC:

Experimental techniques for 15N (a) R1, (b) R2, and (c) {1H}15N NOE spin relaxation measurements using two-dimensional, proton-detected pulse sequences. R1 and R2 intensity decay curves are recorded by varying the relaxation period T in a series of two dimensional experiments. The NOE is measured by recording one spectrum with saturation of 1H magnetization and one spectrum without saturation.

Data Analysis

Analysis of the relaxation data provides dynamical parameters (amplitude and timescale of motion) for each bond vector under study and parameters related to the overall shape of the molecule (rotational diffusion tensor):

Dynamical parameters in proteins. (a) Overall rotational diffusion of the molecule is represented using an axially symmetric diffusion tensor for an ellipsoid of revolution. The diffusion constants are D|| for diffusion around the symmetry axis of the tensor and Dperp. for diffusion around the two orthogonal axes. For isotropic rotational diffusion, D|| = Dperp.. The equilibrium position of the ith N-H bond vector is located at an angle i with respect to the symmetry axis of the diffusion tensor. Picosecond-nanosecond dynamics of the bond vector are depicted as stochastic motions within a cone with amplitude characterized by S2 and time scale characterized by e. (b) The value of S2 is graphed as a function of (-) o calculated using Equation 22 for diffusion within a cone or (- - -) f calculated using Equation 23 with = 70.5° for the GAF (Gaussian Axial Fluctuation) model.from: Palmer, A. G. (2001). “NMR probes of molecular dynamics: Overview and comparison with other techniques.” Annual Review of Biophysics and Biomolecular Structure 30: 129.

“Model Free” analysis of relaxation based on Lipari, G. and A. Szabo “Model-Free Approach to the Interpretation of Nuclear Magnetic Resonance Relaxation in Macromolecules. 1. Theory and Range of Validity.” Journal of the American Chemical Society 104: 4546 (1982).

Internal dynamics characterized by: • internal correlation time, e

• spatial restriction of motion of bond vector, S2

S2 = 1 highly restricted S2 = 0 no restriction

• Rex, exchange contribution to T2

The spectral density terms in the relaxation equations are modified with terms representing internal dynamics and spatial restriction of bond vector:

J() ~ { S2c/(1+2c2) + (1-S2)/(1+22) }

where = ec/(e + c).

Analysis of relaxation data using software package (eg. Model-Free or DASHA) allows the dynamical parameters to be calculated:

Data Analysis

c

Rex,S2e

15N1H

measure:15N T115N T215N{1H} NOE

calculate relaxation data for a given c

recalculate by varying values of S2, e and Rex

Compare measured vs. calc. value

Defining Regions of Structure using NMR Relaxation Measurements

Case study #1

Red indicates chemical shift changes observed upon ligand binding

Case study #2

Case study #3

goal: measure effects of inhibitor binding on conformational fluctuations of HIV protease on s-ms timescale.

sample: 0.3mM protease dimer + DMP323 inhibitor

experiments: 1H and 15N T2 and T1 at 500MHz

Case study #4

result: inhibitor binding enhances dyanamics on the ms timescale of the -sheet interface, a region that stabilizes the dimeric structure of the protease (residues 95-98). Relaxation behavior of the flap (residues 48-55) indicates a transition from a slow dynamic equilibrium between semi-open conformations on the 100s timescale to a closed conformation upon inhibitor binding.

References

Palmer, A. G. (2001). “NMR probes of molecular dynamics: Overview and comparison with other techniques.” Annual Review of Biophysics and Biomolecular Structure 30: 129.

Palmer, A. G., C. D. Kroenke and J. P. Loria (2001). “Nuclear magnetic resonance methods for quantifying microsecond-to-millisecond motions in biological macromolecules.” Nuclear Magnetic Resonance of Biological Macromolecules, Pt B 339: 204.

Brutscher, B. (2000). “Principles and applications of cross-correlated relaxation in biomolecules.” Concepts in Magnetic Resonance 12(4): 207.

Engelke, J. and H. Ruterjans (1999). Recent Developments in Studying the Dynamics of Protein Structures from 15N and 13C Relaxation Time Measurements. Biological Magnetic Resonance. N. R. Krishna and L. J. Berliner. New York, Kluwer Academic/ Plenum Publishers. 17: 357-418.

Fischer, M. W. F., A. Majumdar and E. R. P. Zuiderweg (1998). “Protein NMR relaxation: theory, applications and outlook.” Progress in Nuclear Magnetic Resonance Spectroscopy 33(4): 207-272.

Daragan, V. A. and K. H. Mayo (1997). “Motional Model Analyses of Protein and Peptide Dynamics Using 13C and 15N NMR Relaxation.” Progress in Nuclear Magnetic Resonance Spectroscopy 31: 63-105.

Cavanagh, J., W. J. Fairbrother, A. G. Palmer and N. J. Skelton (1996). Protein NMR Spectroscopy: Principles and Practice, Academic Press.Chapter 5 “Relaxation and Dynamic Processes”

Nicholson, L. K., L. E. Kay and D. A. Torchia (1996). Protein Dynamics as Studied by Solution NMR Techniques. NMR Spectroscopy and Its Application to Biomedical Research. S. K. Sarkar.

Peng, J. W. and G. Wagner (1994). “Investigation of protein motions via relaxation measurements.” Methods in Enzymology 239: 563-96.

Wagner, G., S. Hyberts and J. W. Peng (1993). Study of Protein Dynamics by NMR. NMR of Proteins. G. M. Clore and A. M. Gronenborn, CRC Press: 220-257.

Mini Reviews:

Ishima, R. and D. A. Torchia (2000). “Protein dynamics from NMR.” Nature Structural Biology 7(9): 740-743.

Kay, L. E. (1998). “Protein dynamics from NMR.” Nature Structural Biology 5: 513-7.

Palmer, A. G., 3rd (1997). “Probing molecular motion by NMR.” Current Opinion in Structural Biology 7(5): 732-7.

Case Studies:

Fushman, D., R. Xu, et al. (1999). "Direct determination of changes of interdomain orientation on ligation: Use of the orientational dependence of N-15 NMR relaxation in Abl SH(32)." Biochemistry 38(32): 10225-10230.

Eisenmesser, E. Z., D. A. Bosco, et al. (2002). "Enzyme dynamics during catalysis." Science 295(5559): 1520-1523.

Lee, A. L., S. A. Kinnear, et al. (2000). "Redistribution and loss of side chain entropy upon formation of a calmodulin-peptide complex." Nature Structural Biology 7(1): 72-77.

Ishima, R., D. I. Freedberg, et al. (1999). "Flap opening and dimer-interface flexibility in the free and inhibitor-bound HIV protease, and their implications for function." Structure with Folding & Design 7(9): 1047-55.