Normal Mode Refinement: Crystallographic Refinement of Protein Dynamic Structure Applied to Human Lysozyme A K l N O R l KIDERA,' K O J I INAKA,' MASAAKI MATSUSHIMA,' and N O B U H I R O GO'** 'Protein Engineering Research Institute, 6-2-3 Furuedai, Suita, Osaka 565; and *Department of Chemistry, Faculty of Science, Kyoto University, Kyoto 606, Japan
SYNOPSIS
A new method of dynamic structure refinement of protein x-ray crystallography, normal mode refinement, is developed. In this method the Debye-Waller factor is expanded in terms of the low-frequency normal modes and external normal modes, whose amplitudes and couplings are optimized in the process of crystallographic refinement. By this method, internal and external contributions to the atomic fluctuations can be separated. Also, anisotropic atomic fluctuations and their interatomic correlations can be determined experimentally even with a relatively small number of adjustable parameters. The method is applied to the analysis of experimental data of human lysozyme to reveal its dynamic structure.
INTRODUCTION The x-ray crystallography provides a wealth of information about the dynamic as well as static protein structure. However, the isotropic B-factor model for the expression of the Debye-Waller factor in the conventional method of refinement has two fundamental problems. At first, the model is based on an assumption that the atomic fluctuations are (1)isotropic and ( 2 ) independent of each other. At second, many factors other than the real internal fluctuations in a protein molecule, such as static disorder and rigid body external motions, contribute to the B factor. To solve these problems we propose a new method of refinement of protein dynamic structure-normal mode refinement.
FORMULATION If the distribution of atomic positions around their mean is gaussian, the complete description of the distribution, including anisotropy and interatomic correlations, is given by a matrix U , whose elements * To whom correspondence should be addressed. Biopolymers, Vol. 32, 315-319 (1992) 0 1992 John Wiley & Sons, Inc.
CCC 0006-3525/92/040315-05$04.00
are ( ArikArjl),where i and j designate atom numbers, and k and 1 are x , y , z or 1,2, 3, Arik is the deviation of the kth component of atom i from its mean; and the angle brackets indicate the thermal equilibrium average. If the assumption of the gaussian distribution holds exactly, the matrix U can be calculated by the normal mode as
u = axat
(1)
where 9 is a matrix whose columns are eigenvectors of the normal modes and X is a diagonal matrix whose diagonal elements are amplitudes of fluctuations that are inversely proportional to the eigenvalues, with t meaning transpose. Both 9 and X are calculated theoretically from the hessian matrix at a minimum of the potential energy surface corresponding to the native state of a protein. It has been shown4r5that U can be well approximated by external and a relatively small number of low-frequency internal normal modes. In this sense these modes define an important subspace of the conformational space. We write a part of a and X consisting of the important normal modes as CP' and XI, respectively, (I for important) and the rest as aUand Xu ( U for unimportant). Therefore @ = (a', 9 ' ) . The important subspace is defined by a' and U can be well approximated by @'XI( Let the 315
316
KIDERA ET AL.
dimension of the subspaces @ I be M ' . For the external modes, the TLS model of Schomaker and Trueblood' can be used. Because there are multiple minima in the native state region of the conformational space, the normal mode analysis can be carried out at any one of them. It has been known that, even though each individual normal mode depends appreciably on the minimum at which it is calculated, the subspace corresponding to the low-frequency modes depends very little on which minimum. Therefore, most of the molecular motions contributing to U are expected to be taking place in the subspace @ I , even though it is constructed at one particular minimum with the assumption of the harmonicity. The following formulation is developed, taking this expectation as the basic assumption. The real distribution of atomic positions may not be gaussian due to the anharmonicities of the potential surface in the native state region. Yet treatments based on the assumption of the gaussian distribution still have practical value. Now, according to the basic assumption stated above, U' = instead of for the purely harmonic expression, should be a good approximation to U, where is a certain M' X M' matrix to be determined experimentally. Then, the Debye-Waller factor for atomj is given by
The number of adjustable parameters in Eq. ( 2 ) is M' (MI 1) / 2 20. The number 20 comes from the external terms, which is not 6 X ( 6 1) / 2 = 21, because the trace of the part of for the screw motion is arbitrary? We propose to use Eq. ( 2 ) as an expression for the Debye-Waller factor in the later stage of crystallographic refinement of human lysozyme. In the initial stage, the conventional isotropic B-factor refinement is to be done. Then, by using the structure thus determined, the normal mode analysis was carried out to calculate normal mode eigenvectors, which was further followed by the normal mode refinement based on Eq. ( 2 ) . Intensities at Bragg angles do not contain information concerning interatomic correlations. They only contain information pertaining to block diagonal elements in matrix U , each corresponding to a 3 X 3 intraatomic distribution tensor. Because there are six independent elements in each of them, the intensities at the Bragg angles contain information pertaining to 6N independent elements in
+
+
xE
+
@'z(+')t,
3
1.0
8
0.8
23
0.6
s a
0.4
B
0.2 40
20
60
80
100
120
Residue number
where q k is a component of vector q = 27reth, with X 3 matrix that converts the Cartesian coordinates to the fractional coordinates and h being a reciprocal lattice vector; 4 j k m is a component oi the mth normal mode; urnnis an element of the internal part of the matrix djEkrnis an element 01 the mth external normal mode; and,:a is an element of the external part of the matrix Refinement of protein x-ray crystallography can be carried out by taking Eq. ( 2 ) as a model of conformational dynamics. This model was proposed independently by Kidera and Go, and by Diamond.' Because the external normal mode eigenvectors &E are orthogonal to the internal normal modes, all thc factors including crystal defects orthogonal to thc internal modes should contribute as external modes
8 being a 3
c;
x.
0 45
\
miernal .
Figure 1. ( A ) Root mean square fluctuation (r;)*'* of nonhydrogen atoms averaged within residuej. Four curves are for the following fluctuations: upper thick curve is apparent (i.e., external internal); broken curve is external; lower thick curve is internal; thin curve is theoretically calculated by the normal mode analysis a t a minimum corresponding to the native state. The last two curves agree with each other even quantitatively except for a few short segments. They occur only at residues where strong intermolecular contacts exist in the crystal. ( B ) The amplitudes of the apparent, external, and internal fluctuations averaged over all nonhydrogen atoms are shown schematically.
+
NORMAL MODE REFINEMENT
+
R
i?
2.0
+
U . If 6N > M' ( M I 1) / 2 20, such information would be enough to determine the adjustable parameters in Eq. ( 2 ) . Once these parameters are determined, the intra- and interatomic distribution tensor U is given as a good approximation by ( 9I ) t. This means that not only the anisotropic intraatomic distribution but also interatomic correlations are determined from the diffraction intensities. Note that the theoretical normal mode analysis is used only to define the important conformational subspace @ I . The power of this new method of refinement has been tested theoretically by using simulated diffraction data.5s7 Diamond applied a similar method to the analysis of experimental data of bovine pancreatic trypsin inhibitor by taking only ten diagonal elements as adjustable parameters for the internal fluctuations.' Our analysis indicates that this is too small for U' to be a good approximation.
@'z
1.6
20
40
60
80
100
120
Residue number
Figure 2. Anisotropy of the atomic distribution defined by the axial ratio [ 2X1/ ( X2 X,) ] "* averaged within each residue, where XI, Xz, and X3 are the variances along the first, second, and third principal axes of distribution of each atom. The three curves are for the following fluctuations: thick curve is apparent, broken curve is external, and thin curve is internal. The external fluctuations are found to be nearly isotropic, while the internal fluctuations are highly anisotropic.
+
317
Figure 3. Stereo ORTEP" drawings of thermal ellipsoids of (2"'s for the apparent ( A ) and the internal ( B ) fluctuations.
318
KIDERA E T AL.
EXPERIMENTAL The method is applied to human lysozyme to reveal its dynamic structure. A crystal of human lysozyme was grown in a solution containing 20 mg/mL protein, 2.5M NaC1, and 30 m M sodium phosphate buffer at pH 6.0 in a chamber controlled a t 13.0"C. The size of the rod-shaped crystal was 0.4 X 0.4 X 0.8 mm3. Diffraction intensities were collected with an imaging plate. The space group was P212121. Refinement was carried out with the data of 1.8 A resolution. In the initial stage the conven-
tional isotropic B-factor refinement was done by PROLSQ.9 In this stage the R factor was 15.67%. In the stage of the normal mode refinement of this molecule, in which the number N of nonhydrogen atoms is 1029, we have treated 100 diagonal elements of E corresponding to the 100 lowest frequency normal modes, 903 off-diagonal elements of E corresponding to the 43 lowest frequency normal modes, and the 20 parameters in ZEas adjustable parameters. Because the total number, 1023, of the adjustable parameters in the Debye-Waller factor is much less than 6 X 1029, the values of the adjustable
20 40
60
80 100 120 20
40 60 80 100 120 Residue number
Figure 4. Covariances in the atomic fluctuations (Ar, Ar,) between a pair of C"'s i and j : ( X ) positive covariance > 0.05 A'; (0)negative covariance < -0.05 A'. The loci of highly negative covariances a - ( correspond to the correlation between two segments of human lysozyme illustrated in the figure of the main-chain trace. Those of p, 7 , and 6 suggest the existence of the hinge-bending motions. Triangular areas of highly positive covariances correspond to two loops of A and B.
NORMAL MODE REFINEMENT
319
parameters can be determined from the intensities at the Bragg angles. In this stage the sum of conformational energy and the crystallographic residual was minimized. An R factor of 15.21% was attained in this stage.
We are grateful to Dr. S. Hayward for a careful reading of the manuscript. Computation was done at the Protein Engineering Research Institute. This work has been supported at Kyoto University by grants from MESC and HFSP to NG.
RESULTS
REFERENCES
Results of the experimental study are summarized in Figures 1-4. They indicate the following: (1)Internal and external fluctuations can be determined separately. ( 2 ) Root mean square atomic fluctuations consist of two parts-highly anisotropic internal fluctuations and almost isotropic external fluctuations. The former is smaller than the latter. ( 3 ) The internal fluctuations are found to agree even quantitatively with those predicted by the theoretical normal mode analysis. (4)Correlations of fluctuations are detected between the two lobes forming the active site cleft, which move simultaneously in opposite directions, i.e., are undergoing the hingebending motion. The new method of refinement presented here is expected to enlarge the applicability of the protein x-ray crystallography to detect functionally important soft collective fluctuations in proteins.
1. Go, N., Noguti, T. & Nishikawa, T. (1983) Proc. Natl. Acad. Sci. USA 80,3696-3700. 2. Brooks, B. & Karplus, M. (1983) Proc. Natl. Acad. Sci. USA 80,6571-6575. 3. Levitt, M., Sander, C. & Stern, P. S. (1983) Int. J . Quant. Chem. Quant. Biol. Symp. 10, 181-199. 4. Go, N. (1990) Biophys. Chem. 35,105-112. 5. Kidera, A. & Go, N., submitted. 6. Schomaker, V. & Trueblood, K. N. (1968) Acta Crystallogr. B 24, 63-76. 7. Kidera, A. & Go, N. (1990) Proc. Natl. Acad. Sci. USA 87, 3718-3722. 8. Diamond, R. (1990) Acta Crystallogr. A 46,425-435. 9. Hendrickson, W. A. (1985) Methods Enzymol. 115, 252-270. 10. Johonson, C. K. (1976) Report ORNL-5138, Oak Ridge National Laboratory, TN.
Received June 10, 1991 Accepted July 21, 1991
SYNOPSIS
A new method of dynamic structure refinement of protein x-ray crystallography, normal mode refinement, is developed. In this method the Debye-Waller factor is expanded in terms of the low-frequency normal modes and external normal modes, whose amplitudes and couplings are optimized in the process of crystallographic refinement. By this method, internal and external contributions to the atomic fluctuations can be separated. Also, anisotropic atomic fluctuations and their interatomic correlations can be determined experimentally even with a relatively small number of adjustable parameters. The method is applied to the analysis of experimental data of human lysozyme to reveal its dynamic structure.
INTRODUCTION The x-ray crystallography provides a wealth of information about the dynamic as well as static protein structure. However, the isotropic B-factor model for the expression of the Debye-Waller factor in the conventional method of refinement has two fundamental problems. At first, the model is based on an assumption that the atomic fluctuations are (1)isotropic and ( 2 ) independent of each other. At second, many factors other than the real internal fluctuations in a protein molecule, such as static disorder and rigid body external motions, contribute to the B factor. To solve these problems we propose a new method of refinement of protein dynamic structure-normal mode refinement.
FORMULATION If the distribution of atomic positions around their mean is gaussian, the complete description of the distribution, including anisotropy and interatomic correlations, is given by a matrix U , whose elements * To whom correspondence should be addressed. Biopolymers, Vol. 32, 315-319 (1992) 0 1992 John Wiley & Sons, Inc.
CCC 0006-3525/92/040315-05$04.00
are ( ArikArjl),where i and j designate atom numbers, and k and 1 are x , y , z or 1,2, 3, Arik is the deviation of the kth component of atom i from its mean; and the angle brackets indicate the thermal equilibrium average. If the assumption of the gaussian distribution holds exactly, the matrix U can be calculated by the normal mode as
u = axat
(1)
where 9 is a matrix whose columns are eigenvectors of the normal modes and X is a diagonal matrix whose diagonal elements are amplitudes of fluctuations that are inversely proportional to the eigenvalues, with t meaning transpose. Both 9 and X are calculated theoretically from the hessian matrix at a minimum of the potential energy surface corresponding to the native state of a protein. It has been shown4r5that U can be well approximated by external and a relatively small number of low-frequency internal normal modes. In this sense these modes define an important subspace of the conformational space. We write a part of a and X consisting of the important normal modes as CP' and XI, respectively, (I for important) and the rest as aUand Xu ( U for unimportant). Therefore @ = (a', 9 ' ) . The important subspace is defined by a' and U can be well approximated by @'XI( Let the 315
316
KIDERA ET AL.
dimension of the subspaces @ I be M ' . For the external modes, the TLS model of Schomaker and Trueblood' can be used. Because there are multiple minima in the native state region of the conformational space, the normal mode analysis can be carried out at any one of them. It has been known that, even though each individual normal mode depends appreciably on the minimum at which it is calculated, the subspace corresponding to the low-frequency modes depends very little on which minimum. Therefore, most of the molecular motions contributing to U are expected to be taking place in the subspace @ I , even though it is constructed at one particular minimum with the assumption of the harmonicity. The following formulation is developed, taking this expectation as the basic assumption. The real distribution of atomic positions may not be gaussian due to the anharmonicities of the potential surface in the native state region. Yet treatments based on the assumption of the gaussian distribution still have practical value. Now, according to the basic assumption stated above, U' = instead of for the purely harmonic expression, should be a good approximation to U, where is a certain M' X M' matrix to be determined experimentally. Then, the Debye-Waller factor for atomj is given by
The number of adjustable parameters in Eq. ( 2 ) is M' (MI 1) / 2 20. The number 20 comes from the external terms, which is not 6 X ( 6 1) / 2 = 21, because the trace of the part of for the screw motion is arbitrary? We propose to use Eq. ( 2 ) as an expression for the Debye-Waller factor in the later stage of crystallographic refinement of human lysozyme. In the initial stage, the conventional isotropic B-factor refinement is to be done. Then, by using the structure thus determined, the normal mode analysis was carried out to calculate normal mode eigenvectors, which was further followed by the normal mode refinement based on Eq. ( 2 ) . Intensities at Bragg angles do not contain information concerning interatomic correlations. They only contain information pertaining to block diagonal elements in matrix U , each corresponding to a 3 X 3 intraatomic distribution tensor. Because there are six independent elements in each of them, the intensities at the Bragg angles contain information pertaining to 6N independent elements in
+
+
xE
+
@'z(+')t,
3
1.0
8
0.8
23
0.6
s a
0.4
B
0.2 40
20
60
80
100
120
Residue number
where q k is a component of vector q = 27reth, with X 3 matrix that converts the Cartesian coordinates to the fractional coordinates and h being a reciprocal lattice vector; 4 j k m is a component oi the mth normal mode; urnnis an element of the internal part of the matrix djEkrnis an element 01 the mth external normal mode; and,:a is an element of the external part of the matrix Refinement of protein x-ray crystallography can be carried out by taking Eq. ( 2 ) as a model of conformational dynamics. This model was proposed independently by Kidera and Go, and by Diamond.' Because the external normal mode eigenvectors &E are orthogonal to the internal normal modes, all thc factors including crystal defects orthogonal to thc internal modes should contribute as external modes
8 being a 3
c;
x.
0 45
\
miernal .
Figure 1. ( A ) Root mean square fluctuation (r;)*'* of nonhydrogen atoms averaged within residuej. Four curves are for the following fluctuations: upper thick curve is apparent (i.e., external internal); broken curve is external; lower thick curve is internal; thin curve is theoretically calculated by the normal mode analysis a t a minimum corresponding to the native state. The last two curves agree with each other even quantitatively except for a few short segments. They occur only at residues where strong intermolecular contacts exist in the crystal. ( B ) The amplitudes of the apparent, external, and internal fluctuations averaged over all nonhydrogen atoms are shown schematically.
+
NORMAL MODE REFINEMENT
+
R
i?
2.0
+
U . If 6N > M' ( M I 1) / 2 20, such information would be enough to determine the adjustable parameters in Eq. ( 2 ) . Once these parameters are determined, the intra- and interatomic distribution tensor U is given as a good approximation by ( 9I ) t. This means that not only the anisotropic intraatomic distribution but also interatomic correlations are determined from the diffraction intensities. Note that the theoretical normal mode analysis is used only to define the important conformational subspace @ I . The power of this new method of refinement has been tested theoretically by using simulated diffraction data.5s7 Diamond applied a similar method to the analysis of experimental data of bovine pancreatic trypsin inhibitor by taking only ten diagonal elements as adjustable parameters for the internal fluctuations.' Our analysis indicates that this is too small for U' to be a good approximation.
@'z
1.6
20
40
60
80
100
120
Residue number
Figure 2. Anisotropy of the atomic distribution defined by the axial ratio [ 2X1/ ( X2 X,) ] "* averaged within each residue, where XI, Xz, and X3 are the variances along the first, second, and third principal axes of distribution of each atom. The three curves are for the following fluctuations: thick curve is apparent, broken curve is external, and thin curve is internal. The external fluctuations are found to be nearly isotropic, while the internal fluctuations are highly anisotropic.
+
317
Figure 3. Stereo ORTEP" drawings of thermal ellipsoids of (2"'s for the apparent ( A ) and the internal ( B ) fluctuations.
318
KIDERA E T AL.
EXPERIMENTAL The method is applied to human lysozyme to reveal its dynamic structure. A crystal of human lysozyme was grown in a solution containing 20 mg/mL protein, 2.5M NaC1, and 30 m M sodium phosphate buffer at pH 6.0 in a chamber controlled a t 13.0"C. The size of the rod-shaped crystal was 0.4 X 0.4 X 0.8 mm3. Diffraction intensities were collected with an imaging plate. The space group was P212121. Refinement was carried out with the data of 1.8 A resolution. In the initial stage the conven-
tional isotropic B-factor refinement was done by PROLSQ.9 In this stage the R factor was 15.67%. In the stage of the normal mode refinement of this molecule, in which the number N of nonhydrogen atoms is 1029, we have treated 100 diagonal elements of E corresponding to the 100 lowest frequency normal modes, 903 off-diagonal elements of E corresponding to the 43 lowest frequency normal modes, and the 20 parameters in ZEas adjustable parameters. Because the total number, 1023, of the adjustable parameters in the Debye-Waller factor is much less than 6 X 1029, the values of the adjustable
20 40
60
80 100 120 20
40 60 80 100 120 Residue number
Figure 4. Covariances in the atomic fluctuations (Ar, Ar,) between a pair of C"'s i and j : ( X ) positive covariance > 0.05 A'; (0)negative covariance < -0.05 A'. The loci of highly negative covariances a - ( correspond to the correlation between two segments of human lysozyme illustrated in the figure of the main-chain trace. Those of p, 7 , and 6 suggest the existence of the hinge-bending motions. Triangular areas of highly positive covariances correspond to two loops of A and B.
NORMAL MODE REFINEMENT
319
parameters can be determined from the intensities at the Bragg angles. In this stage the sum of conformational energy and the crystallographic residual was minimized. An R factor of 15.21% was attained in this stage.
We are grateful to Dr. S. Hayward for a careful reading of the manuscript. Computation was done at the Protein Engineering Research Institute. This work has been supported at Kyoto University by grants from MESC and HFSP to NG.
RESULTS
REFERENCES
Results of the experimental study are summarized in Figures 1-4. They indicate the following: (1)Internal and external fluctuations can be determined separately. ( 2 ) Root mean square atomic fluctuations consist of two parts-highly anisotropic internal fluctuations and almost isotropic external fluctuations. The former is smaller than the latter. ( 3 ) The internal fluctuations are found to agree even quantitatively with those predicted by the theoretical normal mode analysis. (4)Correlations of fluctuations are detected between the two lobes forming the active site cleft, which move simultaneously in opposite directions, i.e., are undergoing the hingebending motion. The new method of refinement presented here is expected to enlarge the applicability of the protein x-ray crystallography to detect functionally important soft collective fluctuations in proteins.
1. Go, N., Noguti, T. & Nishikawa, T. (1983) Proc. Natl. Acad. Sci. USA 80,3696-3700. 2. Brooks, B. & Karplus, M. (1983) Proc. Natl. Acad. Sci. USA 80,6571-6575. 3. Levitt, M., Sander, C. & Stern, P. S. (1983) Int. J . Quant. Chem. Quant. Biol. Symp. 10, 181-199. 4. Go, N. (1990) Biophys. Chem. 35,105-112. 5. Kidera, A. & Go, N., submitted. 6. Schomaker, V. & Trueblood, K. N. (1968) Acta Crystallogr. B 24, 63-76. 7. Kidera, A. & Go, N. (1990) Proc. Natl. Acad. Sci. USA 87, 3718-3722. 8. Diamond, R. (1990) Acta Crystallogr. A 46,425-435. 9. Hendrickson, W. A. (1985) Methods Enzymol. 115, 252-270. 10. Johonson, C. K. (1976) Report ORNL-5138, Oak Ridge National Laboratory, TN.
Received June 10, 1991 Accepted July 21, 1991
