MARDIGRAS-A procedure for matrix analysis of relaxation for discerning geometry of an aqueous structure

JOURNAL

OF MAGNETIC

RESONANCE

87,475-487

( 1990)

MARDIGRAS-A Procedurefor Matrix Analysis of Relaxation for Discerning Geometry of an AqueousStructure BRANDANA.BORGIAS*ANDTHOMAS

L. JAMES?

Department of Pharmaceutical Chemistry, University of California, San Francisco, California 94143 Received July 24, 1989; revised September 12, 1989 An accurate method for estimating distances from the two-dimensional nuclear Overhauser effect spectrum is described. The method entails a back-calculation of the relaxation matrix from the measured 2D NOE intensity matrix. While a straightforward backcalculation of distances will fail in the absence of all cross- and diagonal-peak intensities, the MARDIGRAS procedure supplements the observed intensities with calculated values from an arbitrary model and is less prone to the mathematical singularities encountered with an incomplete data set. Several model structures were used for generating the initial matrix of intensities for embedding the observed intensities, and the method is found to be relatively insensitive to the model structure. 0 1990 Academic PRESS,hc.

Qualitative proton-proton distance constraints obtained from two-dimensional nuclear Overhauser effect spectra have proven useful in determinations of molecular structure in noncrystalline environments by distance geometry (DG) and restrained molecular dynamics (r-MD) calculations ( l-3). While the results from such studies are gratifying, it is quite likely that refinements in structure can be achieved if more accurate distances are available. In the case of proteins, in particular we will want better defined structures at ligand binding sites (with and without ligand bound). In the case of nucleic acids, we are often interested in fairly subtle structural changes in the DNA helix which are sequence-dependent and, consequently, guide protein or drug recognition. These subtle variations demand detailed knowledge of the structure and, therefore, accurate internuclear distances. Use of a complete relaxation matrix approach offers the opportunity of determining solution structure with greater accuracy and resolution. Specifically, this paper describes a method for accurately estimating distances from the 2D NOE spectrum. CORMA

AND DISTANCE

CALCULATIONS

Accurate intensities(incorporating the effects of spin-diffusion and network relaxation effects) can readily be calculated for a known three-dimensional structure after diagonalizing the relaxation matrix. We call this procedure CORMA for complete relaxation matrix analysis (4, 5). A much more useful calculation, albeit technically * Current address: Cray Research, Inc.. 3 130 Crow Canyon Place, Suite 400, San Ramon, California 94583. t To whom correspondence should be addressed. 475

0022-2364190 $3.00 Copyright 0 1990 by Academic Press. Inc. All rights of reproduction in any form resewed.

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difficult, is the generation of distunces from the measured intensities ( 6, 7). Several difficulties arise when one tries to calculate distances directly from the intensity matrix. Typically there is insufficient information available, primarily due to unresolved cross and diagonal peaks, to effectively perform the calculation (8). Having a secondary impact on the calculation is the fact that a significant amount of the total magnetization is undetectable as it lies below the spectral noise level. Following the lead of an iterative method recently proposed for calculating the distances with the aid of rMD or DG (9, IO), we iteratively calculate the relaxation matrix based on the combined information of known structural constraints and the well-resolved experimental intensities. Distances are generated after the calculated and observed intensities and the resulting relaxation matrix are self-consistent. These distances can then be used to generate new structures via r-MD or DG. The “ideal” way to calculate distances is to directly transform the scaled intensities (mixing coefficients) from the experimental 2D NOE spectra into their associated dipole-dipole relaxation rates and then distances (6, 7). Equation [ l] shows the fundamental logarithmic relationship between the rates and mixing coefficients:

In Eq. [I], a is the matrix of mixing coefficients which are proportional to the measured 2D NOE intensities, 7, is the experimental mixing time, a (0) refers to the diagonal matrix of intensities for an experiment with mixing time zero, and R is the matrix of relaxation rates. The relatively simple expressions for the relaxation rates which obtain under conditions of isotropic tumbling will be assumed here (Eqs.

[2a,bl): R,;=Zq$(n,I,

1) 1 + 1.5 + 1+ 6 (or(.)2 4(07,)2 3 6 1 + (w7,)2 + 1 + 4(07,)2

PaI Pbl

The solution of Eq. [l] is typically performed by finding the eigenvalues of the mixing coefficient matrix and then performing the simple matrix multiplication: R = x(ln X/7,)x’.

131

With currently available computers (i.e., VAX, Sun, etc.), the time and memory requirements to solve Eq. [I] by finding the eigenvalues and vectors and performing the matrix multiplication of Eq. [ 31 are modest (minutes to hours for macromolecules ranging up to 1000 unique protons). Computer power is not the real problem here. Rather the paucity of usable information (2D NOE intensities) is the limiting factor in the calculation. Major problems are encountered when all peaks are not well resolved as we have previously shown (8).

GEOMETRY

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fixed-distance intensities r a// experimental ink

ratBS with ideal’ I,.,. ~--+3 l-7 Replace

constramed

Recalculate R(i,i) all new R(i,j)

Calculate R

yes i &/Input lo’ -L--T-/

from

new rates:

;X(-gx’ m distance

Replace rates with no I,, wth rates from model

FIG. 1. Flow chart of the MARDIGRAS algorithm. The program is a loop with a “forward” CORMA calculation to obtain the calculated mixing coefficients from either the model structure (first pass) or the iterated relaxation rates (all subsequent passes), and a “back-calculation” which generates the iterated relaxation rates after merging the experimental and calculated mixing coefficients. Along each branch, steps are taken to constrain the calculation wherever feasible and to ensure internal consistency among the intensities (or rates). The value for the convergence criterion, RR < 0.003, was chosen to correspond to the average irreducible error for any intensity (the S/N level for the trials was set at ltO.003). This value can be changed to fit the quality of the data.

Kaptein and co-workers have recently shown it feasible to substitute into the scaled matrix of observed intensities the corresponding mixing coefficients calculated for a reasonable model structure. This yields a full set of mixing coefficients a’ which can be successfully transformed to distances via Eqs. [ 11, [2a], and [2b]. Such an approach has been applied in an iterative manner to successfully generate solution structures (9, 10). This approach (called IRMA for iterative relaxation matrix approach) was initially applied within the context of an evolving structure in which DG or r-MD was used after successfully obtaining distances from the substituted matrix to generate a new structure that would then be cycled through the process again. The focus of this paper is to determine how well the distances can be determined from the relaxation matrix analysis alone, prior to generating new structures with the much more computationally intensive DG or r-MD algorithms. THE MARDIGRAS

ALGORITHM

A flow chart of the MARDIGRAS program is shown in Fig. 1. The central features of the algorithm are ( 1) the use of CORMA (4,5) to calculate a new matrix of mixing

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coefficients (a,) from the initial model structure and the evolving distance set, and ( 2 ) the back-calculation of the rate matrix via Eqs. [ l] - [ 3 ] from the embedded matrix of mixing coefficients (a’). Several important manipulations are incorporated into the algorithm. The user has the option to normalize the observed intensities I0 yielding zero mixing time mixing coefficients ao. While we believe that empirical normalization may be a better way to get accurate intensities, we incorporate two alternative normalization schemes in MARDIGRAS. The first is a normalization based on the average fit of all experimental intensities to the corresponding calculated intensities. The second normalization procedure depends only on intensities of constrained-distance cross peaks. The difference between these two routes may be small, and it is of little consequence if the model is close to the “true” structure. However, the first method may be preferred in cases where the fixed-distance intensities are known to be suspicious. Alternatively, if the model structure is far from the ideal, there is a risk of the normalization constant being substantially in error if all intensities are included in the normalization. After normalization, and during the iteration process, only cross-peak rates which have a corresponding observed intensity are allowed to change. Since the intensity matrix can be sparse, allowing all elements to be replaced at each cycle could result in spurious distance shifts that might compensate for the real changes demanded by the data. Where information about the structure is known with certainty, such as aromatic proton distances or geminal proton distances, this information should be used to help the process. Therefore, cross-peak rates that correspond to known distances are not allowed to change. These are always reset to their known values at every iteration. Similarly, if the iteration process yields results that are physically impossible, such distances if allowed to enter into the rate matrix calculation will distort other distances in the neighborhood. While both upper and lower bound violations could occur, the lower bound violation has the greatest impact. Cross-peak rates that correspond to unreasonably short contacts (arbitrarily set to 1.5 A) are reset to the minimum (assuming negative NOE values here) allowable value. Finally, after new crossrelaxation rates are assigned to the measured intensities and all other rates are reset to their initial values, the diagonal rate constants (Rii) are replaced by the appropriate sums (cf. Eq. [2a] ) based on the calculated and constrained cross-peak rates. This step ensures that the relaxation matrix is numerically self-consistent (though there may still be structural inconsistencies found in the corresponding distance matrix). After the error between calculated and observed intensities reaches a minimum value, the final distances are calculated and ranges are assigned to each value depending on the following criteria. An approximate (two-spin assumption) derivative of Ar/ Au is calculated on the basis of rij = (k7,)“6/al:/6 Ar,, x -(kT,,,)l16

.Aa,/6ai’6,

WI

[4bl

where k is a constant which incorporates the correlation time and other physical constants. This number can be set empirically if estimates of the correlation time are

GEOMETRY

0.200

0.100

479

FROM 2D NOE SPECTRA

0.050

0.020

0.010

0.005

0.002

0.001

INTENSITY

FIG. 2. Expected errors in calculated distances. The a priori error for any distance calculated by MARDIGRAS as a function of the cross-peak intensity. The curves are for random errors in the intensities of +O.OOl(-.-),*0.003(---).and+_0.005(-).

available using either ratios of Ti and T2 relaxation rates (11) or ratios of ROESY and 2D NOE cross-relaxation rates ( 12). Alternatively, k can be determined from the cross-peak rates determined for constrained distances. Then, for any distance rb determined via MARDIGRAS, its range Ar,, is calculated using Eq. [ 4b] and the larger of (a) the error calculated for its associated intensity or (b) the minimum error dictated by the noise level for the experimental intensities. Figure 2 shows the dispersion expected for various noise levels. The influence which actual spin geometry has on the distance calculations has been discussed elsewhere ( 13). The conclusion in that paper was that certain geometries of spin systems will preclude accurate distance determinations, even when a complete relaxation matrix approach is taken. A typical argument might go as follows. When spin diffusion dominates the intensities of weak cross peaks, there is no way that the value of the direct relaxation rate can be determined unless geometrical considerations are taken into account. The worst-case scenario occurs when two spins are in-

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fluenced by a third “spectator” spin that lies directly between them. Consider the three-spin system. Let the AB and BC distances both be 2.0 A, and set the AC distance to 4.0 A. Can the AC distance be reliably calculated? It is easy to demonstrate that any choice of a long distance for AC, with a small cross-relaxation rate, will have almost no effect on the AC cross-peak intensity. Therefore, the AC peak itself gives virtually no information about the AC distance. This is actually part of the problem with the ISPA approach. But the fundamental assumption of MARDIGRAS is complete rduxation matrix unulysis. There is no one-to-one correspondence between a single cross-peak intensity and the corresponding cross-relaxation rate or distance. In fact. MARDIGRAS obtains the AC cross-relaxation rate from the perturbation spin C causes in the AB cross peak (and, by symmetry, spin A causes in the BC peak). When CORMA-generated intensities corresponding to the in-line A-B-C three-spin system are analyzed by MARDIGRAS, the result is always the same. regardless of starting structure, and depends only on the actual intensities used. For intensities calculated with no noise and an assumed correlation time of 2 ns and 100 ms mixing time, the distances generated by MARDIGRAS are AB = BC = 2.01 A and AC = 4.02 A. For intensities calculated with noise at the 50.003 level, the distances generated by MARDIGRAS are AB = BC = 2.01 + 0.01 A, and AC = 3.5 +- 0.2-4.3 t 0.2 A. It is evident from these results that the error bounds supplied by the calculation are somewhat optimistic. However, the calculated values do approach the true distance. They are much closer than can be reliably assigned from even a cautious ISPA analysis. Additional tests with the spectator proton B positioned 1.65, 1.75, and 1.85 A away from A and in line with C gave similar results. MARDIGRAS always finds the correct result: the accuracy ofthe distances depends mostly on the errors in intensities. The geometry of the target spin system, or the model system, does not substantially limit the ability to calculate reasonable distances. This does not mean that MARDIGRAS can determine all distances with high accuracy. As stated above, only distances for which a corresponding cross peak is observed will ever be considered for final structure generation. RESULTS

Tests of MARDIGRAS were performed on the protein bovine pancreatic trypsin inhibitor (BPTI) using hypothetical intensities calculated by the program CORMA (5) for the structure Spti (14) deposited in the Protein Data Bank (1.5, 16). We use hypothetical data since we need to know the true structure and molecular motion exactly in order to measure the consequences of any random or systematic errors in the experimental intensities on distance calculation and ultimately structure generation. We can calculate, using CORMA, a theoretical 2D NOE spectrum for a known structure using an arbitrary motional model. In the present case that mode1 is an isotropically reorienting rigid molecule. We can also add random noise at any level desired and can consider any number of peaks to be overlapping or otherwise unmeasurable. This allows us to assessthe method in its ability to handle realistic spectral limitations. Protons were repositioned with a locally written program to idealize their geometry with respect to the heavy atom coordinates. Even with the protons replaced, there

GEOMETRY

FROM 2D NOE SPECTRA

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FIG. 3. Results of MARDIGRAS on BPTI. Conditions: rC = I .8 ns, 7, = 100 ms; intensities calculated by CORMA with noise at f0.003 level. Initial structure was taken from the true BPTI coordinates in which the coordinates were each shifted randomly by up to + I .5 A. The RMS error in initial HH distances was I. 19 A. (A) Plot of the initial H-H distances vs the H-H distances in the X-ray structure from which the intensities were generated. (B) Distances as in A after one cycle of MARDIGRAS. (C) Final distances after convergence (six cycles). The RMS error in the final H-H distances was 0.33 A for approximately 1100 distances.

remains an improbably short contact of I .38 A between atoms HA of residue 1 and HDl of residue 2. This adversely effects the performance of the program MARDIGRAS which assumes that no H-H distance can be less than 1.5 A (see below). The coordinates generated in this manner were used as the “ideal structure” despite this minor fault. One initial model for the MARDIGRAS analysis was a randomized structure with proton coordinates shifted by an RMSD of 2.6 A relative to the ideal structure. This starting structure tends to have severe distance errors for intraresidue distances, but on the whole was approximately correct in terms of the long-range structure. MARDIGRAS yielded short (~5 A) distances within a few percent of their ideal values. Long distances reflect the roundoff error of the weakest intensities and the random distribution of the input model (Fig. 3A).

482

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An important consideration in designing a distance calculation routine such as MARDIGRAS is the dependence of the algorithm on the starting structure. This is akin to the local minimum problem in minimization techniques which can preclude finding the optimal distance. To test the dependence on starting structure we also used an extended chain structure based on the BPTI primary sequence. This structure has the short-range intraresidue distances approximately correct, but the long-range distances are in gross error. The distances calculated by MARDIGRAS are dramatically better than those of the initial model (Figs. 4A and 4B) and not significantly different from the distances generated from the randomized structure. Additionally, we wanted to test the performance of the algorithm using different mixing times. Since the signal-to-noise ratio is better when the mixing time is increased, a logical question to address is whether the distance estimates from MARDIGRAS are improved with longer mixing times. MARDIGRAS was run using intensities for 200 and 400 ms mixing times (Figs. 4C and 4D). The results are mixed. The 200 ms calculation is slightly better than the 100 ms calculation in terms of the general dispersion of distances, but yields fewer distances greater than 4 A. At 400 ms the results are considerably worse. The multispin cross-relaxation effect commonly referred to as spin diffusion is beginning to impact the results in the guise of lost information for the shortest-distance intensities (which are no longer building up in intensity). This is an important point. While spin diffusion is taken into account by MARDIGRAS, the program cannot compensate for limitations in the data. Specifically, when the effects of spin diffusion are so pronounced that most of the cross peaks are already tending toward a common value, there is insufficient information remaining in their intensities to reliably determine the distances. The data should never be collected with such long mixing times that the average intensity of cross peaks in the 3-3.5 A range is decreasing. Instead, it is best to obtain data with mixing times which yield the most information globally (8). Finally, there remain the questions of whether the distance ranges calculated by MARDIGRAS are within expectations and whether the calculated distances are within the expected error from the true distance. To test the assumptions used in calculating the ranges, we ran MARDIGRAS on a fragment of BPTI ( 10 residues), using calculated intensities at 200 ms with random noise at the kO.003 level. The errors between the calculated and the true distances are shown in Fig. 5a. Most of the calculated distances are found to lie within the range defined by the kO.003 level. In comparison, Fig. 5b shows the analogous plot for distances estimated from the single exponential approximation to the initial intensity buildup for mixing times 50, 100, 200, and 400 ms. Figure 5c shows the distances calculated according to the relative intensities at 200 ms alone (assuming that 200 ms is short enough to give a reasonable estimate of the initial rate). The cross-relaxation rates for Fig. 5b were estimated according to the assumption U,(t)

= ai,

+ [aim - aij(f)le-k’J

[51

where ati(f) is the maximum value reached for cross peak ai,. (Intensities a,( T, > f ) are ignored, since the initial rate assumption is no longer valid for these points.) From Eq. [ 5 ] it can be shown that the initial rate is given by

GEOMETRY

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FIG. 4. Results of MARDIGRAS on BPTI using an extended chain conformation for the initial structure. Intensities are as in Fig. 3. (A) The initial H-H distances vs the distances in the X-ray structure from which the intensities were generated. RMS error in 3-4 A H-H distances: 1.64 A; RMS error in 4-5 A HH distances: 2.65 A; RMS error in all 1100 H-H distances: 1.30 A. (B) Final results for intensities at 7, = 100 ms. RMS error in 3-4 A H-H distances: 0.35 A; RMS error in 4-5 A H-H distances; 0.78 A; final RMS error: 0.32 A. (C) Final results for intensities at 5,,, = 200 ms. RMS error in 3-4 A H-H distances: 0.3 1 A; RMS error in 4-5 A H-H distances: 0.48 A; final RMS error: 0.20 A. (D) Final results for intensities at 7, = 400 ms. RMS error in 3-4 A H-H distances: 0.42 A; RMS error in 4-5 A H-H distances: 0.92 A; final RMS error: 0.30 A.

484

BORGIAS

AND JAMES

0.050

0.020

0.010

0.005

0.002

INTENSITY

FIG. 5. Comparison of errors in distance calculations by MARDIGRAS and ISPA. The data correspond to a test case on the first 100 protons of BPTI. Intensities were calculated by CORMA with noise at the kO.003 level. In each panel the error between the true distance and the calculated distance is plotted versus the intensity of the cross peak. The theoretical error ranges from Fig. 2 are superimposed over each figure. (a) Results from MARDIGRAS. The intensities analyzed were from a single 200 ms set. Note that only a few points extend beyond the AO.003 boundaries. (b) Results from ISPA using a single exponential fit to the initial rates. Note the systematic error and the large number of points that exceed the +0.003 range. (c) Results assuming that the intensities at 200 ms are reasonable estimators of the initial rates.

R;+A dadt

= k[a&f) - a(O

[61

1-O

with k calculated as the simple average,

k = i ln[a&“) - aii(O>l- ln[aijU) - aij(T,)l tYl=l

Tm

n I/ ’

171

where m ranges over the mixing times prior to the one from which a,(f) is obtained. The distances from the single exponential approximation are clearly better than those from the relative intensity calculation, but still are worse than the MARDIGRAS estimates, and a significant number of calculated distances lie beyond the limits defined by the noise level of the intensities. The systematic underestimation of distances

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FIG.

5-Continued 485

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BORGIAS

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using this approach has been discussed previously (8) and is an expected consequence of neglecting the local environment in the isolated spin pair approximation. PROBLEMS

AND

PITFALLS

One problem which seems to be a recurrent theme with the back-calculation of the rates from the intensity matrix is the generation of negative eigenvalues. Typically, one or two of the smallest eigenvalues are found to be negative, which inhibits calculating the logarithm of Xi. The occurrence of negative eigenvalues for the mixing coefficient matrix in the BPTI example used here could be traced to MARDIGRAS forcing the 1.38 A distance for IHA-2HD 1 in the test model up to 1.5 A during the forward calculation of rates and intensities. This results in an unresolvable conflict between the calculated and “experimental” intensities. Allowing the minimum distance to reach 1.3 A eliminated this problem in this particular case. Alternatively, some success has been found in interpolating the value for In Xi from the trend in the largest calculable values. Another alternative is to increase all the eigenvalues by an amount sufficient to make the most negative eigenvalue positive. The net result of this operation is to increase all the diagonal intensities by the same amount. When the eigenvalue is only slightly negative, the shift has little consequence. CONCLUSIONS

MARDIGRAS is capable of calculating distances from NOE intensities while taking into account spin diffusion and the effects of relaxation in a network of spins. It uses a model structure to generate a matrix of calculated intensities for aiding the distance calculation, but is relatively insensitive to the choice of model. Also, the method is relatively fast. Depending on the number of cycles performed, the time for calculating the distances from a set of 2D NOE intensities for a small protein (BPTI) is only about 3-4 h on a Sun 3 / 160 with Floating Point Accelerator, or about 35-60 s on a Cray Y-MP. These times show that this method, while substantially more involved than ISPA, is workable in essentially any computing environment and represents a relatively small investment in computer time in relation to the complete structure determination process. A “user-friendly” version of MARDIGRAS with suitable documentation is currently being developed and should be available soon. Use of the complete relaxation matrix methodology utilized by MARDIGRAS permits longer mixing times to be employed, with consequently larger intensities for weak cross peaks and the possibility of measuring longer distances. But, as described in this paper, spin-diffusion effects limit the extension to longer mixing times and larger cross-peak intensities. In an earlier paper (8) we described a program to refine a structure based on the 2D NOE intensities. That program, dubbed COMATOSE (complete matrix analysis torsion optimized structure), was capable of achieving similar results in terms of the fit between final distances and target distances. Its limitations were ( 1) it required the evaluation of numerical derivatives, (2) being a least-squares minimization routine it is prone to the problem of finding a local rather than the global minimum, and (3) it yielded a structure which still needed to be polished up using molecular mechanics or restrained molecular dynamics. The first limitation results in making the computa-

GEOMETRY

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tional effort considerably greater than required for MARDIGRAS. The acronym COMATOSE is well suited to the state it creates on the computer when trying to analyze a large molecule. The local minimum problem is pervasive in refinement techniques, but does not really enter into the MARDIGRAS solution (although the structure generation step using DG or r-MD will still be prone to any inherent local minima problems of those techniques); the use of very different starting structures had little impact on the final result obtained. The necessity of further structure refinement even after COMATOSE makes the net result very similar to that obtained by MARDIGRAS: what is obtained are good estimates for all the distances for which an NOE cross peak can be measured. Despite our own investment in the program COMATOSE, we feel that its ultimate applicability is limited. MARDIGRAS provides a much faster means to obtain distances. ACKNOWLEDGMENTS We thank Mr. John Thomason for assistance in generating the extended chain structure for BPTI. This work was supported by National Institutes of Health Grants GM 39247, CA 27343, and RR 01695. We gratefully acknowledge the use of the University of California, San Francisco Computer Graphics Laboratory (Director: Dr. R. Langridge, supported by NIH Grant RR 0 108 1). Some of the calculations and code development were performed at the Pittsburgh Supercomputer Center (Grant CHE880090P). REFERENCES 1. M. BILLETER, A. D. KLINE, W. BRAUN, R. HUBER, AND K. W~THRICH, J. Mol. Biol. 206,677 ( 1989). 2. P. J. M. FOLKERS, G. M. CLORE, P. C. DRISCOLL, J. DODT, S. KOHLER, AND A. M. GRONENBORN, Biochemistry28,2601 ( 1989). 3. A. BAX, Annu. Rev. Biochem. 58,223 (1989). 4. J. W. KEEPERSANDT. L. JAMES, J. Mugs. Reson. 57,404 ( 1984). 5. B. A. BORGIAS, P. D. THOMAS, AND T. L. JAMES, “COMPLETE RELAXATION MATRIX ANALYSIS (CORMA),” University of California, San Francisco, 1987, 1989. 6. E. T. OLEJNICZAK, R. T. GAMPE, JR., AND S. W. FESIK, J. n/lagn. Reson. 67,28 ( 1986). 7. P. A. MIRAU, J. Mugn. Reson. 80,439 ( 1988). 8. B. A. BORGIAS ANDT. L. JAMES, J. Magn. Reson. 79,493 ( 1988). 9. R. B~ELENS, T. M. G. KONING, AND R. KAPTEIN, J. Mol. Sfruct. 173,299 ( 1988). IO. R. B~ELENS, T. M. G. KONING, G. A. VAN DERMAREL, J. H. VAN BOOM, AND R. KAPTEIN, J. Mugn. Resort. 82,290 ( 1989). I I. E. SUZUKI, N. PATTABIRAMAN, G. ZON, AND T. L. JAMES, Biochemistry 25,6854 ( 1986). 12. B. T. FARMER II, S. MACURA, AND L. R. BROWN, J. Magn. Reson. 80, I ( 1988). 13. S. B. LANDY AND B. D. NAGASWARA RAO, J. Magn. Resort. 83,29 ( 1989). 14. A. WLODAWER, J. WALTER, R. HUBER, AND L. SJOLIN, J. Mol. Biol. 180,307 ( 1984). 15. F. C. BERNSTEIN,T. F. KOETZLE, G. J. B. WILLIAMS, E. F. MEYER, JR., M. D. BRICE, J. R. RODGERS, 0. KENNARD, T. SHIMANOUCHI, AND M. TASUMI, J. Mol. Biol. 112,535 ( 1977). 16. E. E. ABOLA, F. C. BERNSTEIN, S. H. BRYANT, T. F. KOETZLE, AND J. WENG, in “Crystallographic Databases-Information Content, Software Systems, Scientific Applications” (F. H. Allen, G. Bergerhoff, and R. Seivers, Eds.), pp. 107- 132, Data Commission ofthe International Union ofcrystallography, Bonn/Cambridge/Chester, 1987.