Trajectory Studies of NMR Relaxation in Flexible ... - ACS Publications

18 Trajectory Studies of N M R Relaxation in Flexible Molecules RONALD M. LEVY Rutgers University, Department of Chemistry, New Brunswick, NJ 08903 MARTIN KARPLUS Harvard University, Department of Chemistry, Cambridge, MA 02138

Downloaded by PURDUE UNIV on April 8, 2016 | http://pubs.acs.org Publication Date: June 1, 1983 | doi: 10.1021/ba-1983-0204.ch018

Stochastic dynamics trajectories for alkanes in aqueous solution have been used to examine a variety of problems that arise in the interpretation of C-NMR relaxation experiments. Exact results for spin-lattice and spin-spin relaxation times, and nuclear Overhauser enhancement values obtained from these trajectories, have been employed to analyze the relaxation behavior of small alkanes and macromolecular side chains and to test the validity of simplified relaxation models for these systems. A molecular dynamics simulation of a protein has been used to demonstrate the effects of picosecond fluctuations on observed C spin-latticerelaxation times. It is shown how an increase in spin-lattice relaxation time can be related to order parameters for the picosecond motional averaging of the carbon-hydrogen dipolar interactions, and how the order parameters can be calculated from a protein molecular dynamics trajectory. The present work provides a firm theoretical foundation for the continuing effort to use NMR measurements for the experimental analysis of the dynamics of molecules with internal degrees of freedom. 13

13

N

U C L E A R M A G N E T I C R E S O N A N C E RELAXATION M E A S U R E M E N T S provide

an important probe of the dynamics of molecules because the spinlattice (T ) and spin-spin (T ) relaxation times and the nuclear Overhauser enhancement (NOE) factor (r|) all depend on the thermal motions. For protonated carbons, C N M R is particularly well suited for the study of dynamics because the relaxation is dominated by the fluctuating dipolar interactions between C nuclei and directly bonded protons. Applications of C N M R have been made to the dynamics of small molecules in solution (1-6), polymers (7-13), and molecules of biological interest including lipids (14-16) and proteins (17-23). Because the motions of L

2

1 3

1 3

1 3

0065-2393/83/0204-0445$07.00/0 © 1983 American Chemical Society

Haile and Mansoori; Molecular-Based Study of Fluids Advances in Chemistry; American Chemical Society: Washington, DC, 1983.

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MOLECULAR-BASED STUDY OF FLUIDS


molecules with many internal degrees of freedom (e.g., macromolecules) are complicated, the interpretation of NMR measurements for such systems is often not unique. Empirical rules have been developed to fit the relaxation data to the molecular tumbling time combined with internal segmental motions (6-8). Alternatively, the experimental results have been interpreted in terms of analytically tractable descriptions of the dynamics based on continuous diffusion (9-24), restricted diffusion (2528), and lattice jump models (24, 26, 29, 30). While it is usually possible to fit the experimental results in this way, the data in themselves generally are not sufficient to determine whether or not a model gives the correct description of the dynamics. A powerful method of testing relaxation models is provided by computer-generated trajectories that make use of realistic potentials to simulate the motion of the system of interest. From such trajectories, the time-dependent correlation functions can be determined and T T , and T| can be calculated. Thus the trajectory provides simultaneously a complete knowledge of the dynamics and exact values of the NMR parameters. It is now possible to proceed by using the calculated values of T T , and r\ as "experimental" quantities and fitting the various models to them. The resulting model dynamics can then be compared with the exact dynamics from the trajectory to determine how well the former corresponds to the latter. Also, the results of the trajectory studies may be compared directly with experiment without recourse to simplified theories. In this chapter we review our work, which employs the results of computer simulations for the analysis of experimental and theoretical N M R studies of the motions of flexible molecules. The systems considered include butane and heptane tumbling in aqueous solution and the short- and long-time dynamics of side chains attached to macromolecules. The first section of this chapter reviews the procedures used to generate the trajectories and outlines the methodology employed to extract the N M R parameters from them. Then the butane and heptane trajectories are used, in the next section, to evaluate the relaxation times for these alkane chains in the motional narrowing limit, and the results are compared with experiment and with simplified models. The following section analyzes the dynamics of a side chain protruding from a macromolecule and employs the heptane trajectories to critically examine a lattice jump model for the relaxation. In the final section we make use of a full molecular dynamics simulation of a protein to demonstrate the effects of fast protein motions on observed C - N M R relaxation times. l9

2

1 ?

2

13

Methodology In this section we outline the method used for obtaining the alkane trajectories and present the procedure for determining NMR relaxation


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447

parameters from these trajectories. We show how trajectory results can be applied to study simplified dynamical models that have been used to interpret N M R relaxation data. For the analysis of C spin lattice relaxation in proteins we show how an increase in the relaxation time is related to order parameters for the picosecond motional averaging of the C , H dipolar interactions, and how these order parameters are calculated from a molecular dynamics trajectory. Diffusive Langevin Dynamics of Model Alkanes. The equation of motion descriptive of Brownian particles is the Langevin equation (31) 1 3


1 3

where m-v^ and are the mass, velocity, and friction constant of the ith particle and F is the systematic force acting on the fth particle due to the potential of mean force; F in general depends on the coordinates of all the particles. The stochastic term A (t) represents the randomly fluctuating force on the particle due to the solvent. In the diffusive regime, the particle momenta relax to equilibrium much more rapidly than the displacements. With the assumption that the forces are slowly varying, i.e., that it is possible to take time steps that are large compared to the t

{

f

momentum relaxation time ^At »

Equation 1 can be integrated

to obtain the equation (31-33) for the displacement vector, ri

(2)

rlt + At) = rtf) + [FJWQAt + Ar (t) (

The term Ar((t) is the random displacement due to the stochastic force; it is chosen from a Gaussian distribution with zero mean and second moment 6DtAt, where D , the diffusion coefficient, is KT/^. Equation 2 is the equation used to calculate the trajectories of the hydrocarbon chains (34). An extended-atom model was introduced, in which the C H and C H units are represented as spheres of van der Waals radius 1.85 A; the bond lengths and angles of the alkane chains were set equal to 1.523 A and 111.3°, respectively. Each extended-atom group along the chain acted as a point center of frictional resistance with a Stokes* law friction constant ^ = 67rr|a. To determine r\, the viscosity of water at 25 °C (T) = 0.01 poise) was used and a was set equal to the extended-atom van der Waals radius; with these parameters, £ = 3.45 X 1 0 " g/s. This is to be compared with the value of ^ = 2.2 X 1 0 " g/s obtained from the experimental diffusion coefficient of methane in water at 25° C. POTENTIAL FUNCTION. The negative gradient of the potential of mean force is used for F, in Equations 1 and 2. This is a combination of f

3

2

4

9

9

American Chemical Society Library



448


the potential energy arising from the interactions among the particles composing the solute molecule and the effective potential due to the solvent molecules. The molecular potential was expressed as a sum of two types of terms; the first is a torsional potential energy for each of the dihedral angles, and the second consists of pairwise Lennard-Jones interactions between extended atoms separated by four or more bonds (35, 36). For butane there is one angular degree of freedom and only the associated torsional potential was used, but for heptane both torsional and nonbonded terms are required. The solvent contribution to the torsional nonbonded potential was obtained from the work of Pratt and Chandler (3), which expresses the potential due to the solvent as — kT ln (y) where y is the cavity distribution function. The differences in energy at 25 °C between the trans and gauche butane geometry in the vacuum and in the solvent are 0.70 kcal/mol and 0.16 kcal/mol, respectively. For the solvent effect on the nonbonded interaction, the cavity distribution function for two methane molecules dissolved in water was used (37). The solvent-modified potential has a slightly deeper energy minimum (— 0.38 versus — 0.30 kcal/ mol), which occurs at a smaller interparticle separation (3.0 versus 4.15 A); at larger separations the attraction is more quickly screened in the solvent. For all the systems considered, Equation 2 was integrated in Cartesian coordinates with the bond lengths and angles of the molecule constrained to the initial value by use of the S H A K E algorithm (35). In general, a correction term should be introduced when the Langevin equation is solved with constraints (38). This term was omitted from the present simulation, in which the effect is expected to be small (39). For the butane simulation, time steps of 0.005 ps and 0.05 ps were compared; for the other molecules, time steps between 0.025 and 0.05 ps were used. Butane trajectories were run for 90 ns (25 °C) and 20 ns (50 °C) on the solvent-modified potential surface and 20 ns (25 °C) on the vacuum molecular potential surface. Heptane trajectories on the solvent-modified surface (25 °C) were recorded for 20 ns without constraints and for 10 ns with three atoms held fixed. For 10 ns of the butane trajectory (0.05-ps time step), the required central processing unit time on an IBM 370/168 computer was 15 min. Exact Calculation of N M R Parameters. The NMR relaxation parameters probe angular correlation functions of the relaxing nucleus. These correlation functions and their Fourier transforms, the spectral densities, can be obtained directly from the alkane trajectories, which provide the complete chain dynamics in the diffusive limit. For the C nucleus in an alkane chain, the dynamical quantities of interest are the spherical polar coordinates (with respect to a laboratory frame) of the C - H internuclear vectors. The well-known relations between the N M R 1 3


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relaxation parameters T T , and T] and the spectral densities for the case of dipolar relaxation involving two different spin 1/2 nuclei (40) are LF

Y

=

2

2

i4/o(0) + / o ( « c

%

-

«H)+

3/t(«c)

+ B / K ) + 6/ (co + co )}


2

Is 7c

c

(4)

H

6/ (co + co ) - /Q(CO ~ co ) 2

_/ (w 0

c

-

c

H

c

H

(5)

w ) + S/^Wc) + 6/ (co + co )_ H

2

c

H

where y , y and a> , co are the gyromagnetic ratios and Larmor frequencies of the C and H nuclei, r is the C - H bond distance, and N is the number of protons directly bonded to the relaxing carbon nucleus. The spectral densities, / (co), are the Fourier transforms of the secondorder spherical harmonics, Y^(6, ) given by c

c

H

1 3

H

L

m

/,»

= [ O&WM)]

y% [e(0)cb(0)]> cos co* dt

(6)

2

Jo where the spherical-harmonic time correlation function is defined as the ensemble average

= \d*d$ \dvdv Y* (e, 4>) G(e,

t e', ;

o)Y£ (e'c|>') 2

(7)

Here G(6, ((), f; 6', (j)', 0) is the conditional probability that a C H vector has spherical polar coordinates (6) at times t, given that they were equal to (6'()>') at time 0. The spherical-harmonic time correlation functions are obtained from the alkane trajectories by replacing the ensemble average (Equation 7) with a time average (41). Since the orientation of the laboratory z axis is arbitrary for freely rotating molecules, the angular correlations and their spectral densities are independent of subscript m. Once the correlation functions (Equation 7) have been evaluated from the time integral over the trajectory, the spectral densities are obtained by numerical Fourier transformation, as indicated in Equation 6. The resulting values of / (co) are then introduced into Equations 3-5 to determine the NMR parameters. m


450


Relaxation Models. The C - N M R spectra of low molecular weight species in solution have tumbling times that are much smaller than the reciprocal of the Larmor frequencies of current spectrometers. Consequently, the resonances of such systems are observed in the motional narrowing limit (OO T) < < 1, where the relaxation time T is the time integral of the angular correlation function 13

C

T -

2

f *l4>O) f

(15)

+

The d* terms are real reduced Wigner matrix elements, with (3 the complement of the rigid C C C bond angle, P(c|>f|cj>'0) is the conditional probability that a rotational angle is (f) at time t, given that it was ' at time 0, and P(') is the equilibrium probability of a ()>' state. Explicit expressions for the conditional probabilities are derived in terms of the eigenvalues and eigenvectors of the rate matrix (29, 41).


b

0

The lattice jump model is studied in some detail later in this chapter. The heptane trajectory is used to evaluate T T , and N O E for a model amino acid side chain with four internal rotational angles. Figure 1 shows the four rotational angles and indicates the nature of the three types of coordinate systems that are used to define the configurations of the chain; that is, the laboratory system, the macromolecular system with respect to which the overall tumbling is defined, and the four local coordinate frames associated with the internal degrees of freedom. For the analysis, the coordinate frame centered on C2 is assumed to be rigidly attached to the macromolecule, which is tumbling isotropically with relaxation time T . Only carbons C2 through C6 are considered, because the C7 methyl protons cannot be located from an extended-atom alkane model. The N M R parameters calculated exactly from the trajectory are compared with those obtained from the independent lattice jump model. For the latter, only the isomerization rate constants K and KQ for each of the rotational angles are required (34, 43). Another class of NMR relaxation problems for which excluded volume effects are important deals with the contribution of fast (picosecond) motions to the relaxation of amino acid side chains in the interior of proteins (21-29). In the last section of this chapter we review our use of a full molecular dynamics simulation of the protein pancreatic trypsin inhibitor (PTI) to demonstrate the effects of picosecond fluctuations on observed C T values. The protein trajectory has been used to evaluate the short time decay of internal correlation functions (Equation 14b), which determine the N M R relaxation (44-45). Because of the highly restricted nature of the motion in the protein interior, the internal correlation functions generally do not decay to zero. Instead, a plateau value l9

2

0

T

1 3

t


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LEVY AND KARPLUS

H Downloaded by PURDUE UNIV on April 8, 2016 | http://pubs.acs.org Publication Date: June 1, 1983 | doi: 10.1021/ba-1983-0204.ch018

453


H

H

H

H

H

H

H

T(ps)

-r(ps)

:2

C1-C2 C2-C3 C3-C4 C4-C5 C5-C6 Cr>-C7

(39.3) (46.7) (51.9) (50.3) (47.3) (37.3)

33.4 42.2 49.8 46.7 41.6 35.7

13.4 16.5 17.6 18.1 16.5 13.7

14.7 15.1 19.6 16.9 17.2 12.9

15.9 16.5 20.2 18.3 18.3 13.5

C2-H C3-H C4-H C5-H C6-H

(32.6) (31.5) (35.1) (33.8) (25.4)

29.1 29.4 33.1 33.4 23.3

12.6 13.9 15.1 13.4 9.9

11.7 11.3 11.6 12.3 12.9

11.9 13.5 13.3 12.5 12.7

a

a

Note: Relaxation time from least-squares fit of the relaxation to a single exponential over 10 ps, except as noted. Fit to a single exponential over 20 ps. a

II, respectively. Since the relaxation is not due to isotropic rigid body rotation, the single exponential fit is approximate. In Table III we compare the relaxation times obtained from a single exponential fit to the C 6 - H vector in heptane with the results from the time integral of the angular correlation function (Equation 8). For the complete dynamics (top row of results), the time integral of the correlation function gives a value for the relaxation time about 14% smaller than that estimated from a single exponentialfit.Such a difference is not important for the analysis described below. The relaxation times listed in Tables I and II demonstrate a number of important trends. They can be summarized as follows: 1. The relaxation times for these small alkanes in water, whose overall tumbling puts them in the motional narrowing limit, are typically between 5 and 50 ps. 2. The relaxation times increase as the chain length increases from butane to heptane. 3. The relaxation times increase from the ends of the heptane chain toward the center. 4. The relaxation of the YJ correlation functions are always slower than that of the Y functions, but the ratio of T /T is less than three. 2

M

/ = 1

/ = 2

5. The C - H internuclear vectors relax faster than the C - C vectors.


456


Table III. Time Integral of Angular Correlation Function of C 6 - H Vector and Exponential Relaxation Times

[Vg[e(0)cit JO

(ps)

Complete dynamics

0.68

Relaxation Time by Single Exponential Fit (ps) 9.9" 10.5 9.7* 11. T


C

Tumbling and internal motions uncoupled Tumbling relaxation only Internal relaxation only

0.58 1.21

(0.79) (0.84) (0.77) (0.93)

fc

13.0° (1.03) 15.8 (1.26) 39 (3.1) 40.r (3.2) C

—

c

f

Least-squares fit to a single exponential over 10 ps. Values in parentheses are l/4ir times the exponential fit to correspond to the time integral (Equation 8). Least-squares fit to a single exponential over 20 ps. Least-squares fit to a single exponential over 50 ps. a

h

0

d

The first three results are in accord with experiment; there are no data concerning the final two. From the C - H relaxation times listed in Tables I and II, the C spin-lattice relaxation times (T ) are calculated by means of Equation 9. Table IV lists the predicted C 7\ values for each of the methylene carbons of butane and heptane; the final column gives the ratio of T for carbon C2 and the ith internal carbon. Unfortunately, experimental measurements of these relaxation times for small alkanes in aqueous solution are not available. However, measurements of T values have been reported for neat liquid alkanes (10, 11), and some of these results are given in Table IV. Quantitative comparison of the heptane results shows that the T values calculated from the trajectories are shorter than the experimental results by a factor of six. One source of this difference is in the larger viscosity of the aqueous solution relative to the neat liquids. The viscosities of the neat alkanes at the experimental temperature of 40 °C are three to four times smaller than the viscosity used for the aqueous solution calculation (T| = 0.27 centipoise for hexane at 40 °C, T| = 0.34 centipoise for heptane at 40 °C, compared with TJ = 1 centipoise for aqueous solution). If we assume the trajectory results scale linearly with the viscosity (as they are expected to do in the diffusive limit), we find that the calculated relaxation times for heptane are a factor of 1.5-2 shorter than the experimental values. Differences of this order may be due to the limitations of the stochastic dynamics model (34); these include the neglect of the inertial term of the complete Langevin equa1 3

x

1 3

:

x

x


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Table IV. Alkane NMR Spin-Lattice Relaxation Times

Butane carbons C2, C3 Heptane carbons C2, C6 C3, C5 C4

Trajectory Results" 5.9

± 0.12

1.94 ± 0.18 1.82 ± 0.14 1.75 ± 0.24 Experimental Results "

—

1.07 1.11


1

Heptane carbons [CI, C7] C2, C6 C3, C5 C4 Eicosane carbons [CI, C20] C2, C19 C3, C18 C4, C17 Interior** a b c d

[10.9] 13.2 12.8 12.0

[0.81 ]

[3.6] 2.3 1.6 1.1 0.8

[0.43 ]

c

—

1.03 1.10 c

—

1.44 2.09 2.88

25 °C, y\ = 1 centipoise, aqueous solution. 39 °C, T| = 0.34 centipoise for neat heptane (10). Normalized to the same number of directly bonded protons. Individual values not obtained.

tion, the neglect of hydrodynamic interaction, and the use of an atomic friction coefficient obtained from the translational diffusion coefficient of a monomer unit in the Langevin equation for the alkane chain. In this regard, if the covalent radius (0.77 A) is used to obtain the monomer friction coefficient, as has been suggested (49), the calculated T values are increased by 2.4 and are closer to experimental values. It is important to note that, while the absolute values of the predicted T values are somewhat too short, the trajectory results reproduce the experimentally observed gradient in the values of T along the heptane chain. Test of Simplified Models. To explore whether relaxation in the motional narrowing limit can be divided into contributions from tumbling and segmental motions, the relaxation of the C6 carbon of heptane was analyzed. The molecular tumbling axis of the heptane molecule was defined by atoms C1-C2-C3. Both the tumbling correlation function and the correlation functions describing the internal relaxation were calculated directly from the trajectory. Figure 2 compares the decay of the second-order spherical harmonics of the C 6 - H internuclear vector calY

x

1


458


47T


C(t )

60

Air

C(t)

60

PICOSECONDS Figure 2. Relaxation of the second-order spherical harmonics

Trajectory Studies of NMR Relaxation in Flexible ... - ACS Publications

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