Sidechain Torsional Potentials and Motion of Amino Acids in Proteins:

Proc. Nat. Acad. Sci. USA Vol. 72, No. 6, pp. 2002-2006, June 1975 Sidechain Torsional Potentials and Motion of Amino Acids in Proteins: Bovine Pancreatic Trypsin Inhibitor (nuclear magnetic resonance of proteins/theory of protein structure/ energy functions applied to proteins/sidechain rotation rates/conformational potentials) BRUCE R. GELIN AND MARTIN KARPLUS Department of Chemistry, Harvard University, Cambridge, Massachusetts 02138 and Laboratoire de Chimie Th6orique*, Universit6 de Paris VII, 2, place Jussieu, Paris Ve, France Contributed by Martin Karplus, February 25, 1975 ABSTRACT Conformational potentials of sidechains in the bovine pancreatic trypsin inhibitor have been studied with an empirical energy function. Calculated minimumenergy positions are in excellent agreement with the x-ray structure for sidechains in the core or at the surface of the protein; as expected, angles for sidechains that are directed out into the solvent do not agree with the calculated values. The contributions to the potentials are analyzed and compared with the potentials for the free amino acid. Although there is a large restriction in the available conformational space due to nonbonded interactions, the minimum energy positions in the protein are close to those of the free amino acid; the significance of this result is discussed. To estimate the effective barriers for rotation of the aromatic rings (tyrosine and phenylalanine), calculations are done in which the protein is permitted to relax as a function of the ring orientation. The resulting barriers, which are much lower than the rigid rotation barriers, are used to evaluate the rotation rates; comparison is made with the available nuclear magnetic resonance data. An analysis of the structure and function of a protein requires, first, a knowledge of the interactions involved in producing its native conformation and, second, an understanding of the possible motions that lead to conformational changes. Consideration has to be given in such an analysis to both the polypeptide backbone and to the amino-acid sidechains. From the available x-ray and chemical data for proteins, it is clear that the molecular conformation is based on a sensitive balance of many factors. Substitution of a single residue can result in significant changes. Further, in oligomeric systems like hemoglobin, displacements in one part of the molecule can produce sizeable movements in another part. There have been a large number of studies directed toward elucidating the conformational elements of proteins. Most of these have been concerned with the polypeptide backbone. They have been based on statistical analyses of the available structural data for proteins (1) and on model calculations of the conformational energies of simple peptides (2, 3). In addition, empirical energy functions have been used to "refine" the structure of a protein by local energy minimization, starting with the x-ray structure (4, 5). In this paper, we concern ourselves with the behavior of amino-acid sidechains of the bovine pancreatic trypsin inhibitor (6, 7). This trypsin inhibitor is a small protein of 58 residues comprising 454 heavy (non-hydrogen) atoms. The molecule is large enough to represent a real protein system, * Eqtmipe de Recherche Associee, Centre National de Recherche Scientifique. and yet small enough to facilitate calculations. Furthermore, recent nuclear magnetic resonance studies of this trypsin inhibitor have provided information on the freedom of motion of certain sidechains (8-10). By means of empirical energy functions, we examine the nature of the potentials holding the sidechains in their equilibrium positions in the protein and compare them with those existing for the free amino acids. We also estimate the rotation barriers for the aromatic sidechains by a calculation in which the protein is allowed to relax as a function of the ring orientation and use the results for a preliminary analysis of the rate of ring rotation. Section I briefly outlines the method. The results and discussion are presented in Section II. I. METHOD The method used for evaluating the conformational energy of the protein is similar to that developed by Levitt and Lifson (5, 11). The empirical energy function is expressed as a sum of approximately separable contributions; they are the bond stretching terms, the bond angle bending terms, the torsional (dihedral) angle twisting terms, the hydrogen bonds, the nonbonded (van der Waals) interactions, and the electrostatic potential terms. As in refs. 5 and 11, the energy function is simplified by combining the hydrogen atoms with the heavy atoms to which they are bonded, so that the energy parameters correspond to "extended atoms." From the empirical energy function, the analytic gradient (the vector of derivatives with respect to the Cartesian coordinates of all of the extended atoms) can be obtained. This makes it possible to perform energy minimizations via the method of steepest descents in the complete conformational space of the protein. To evaluate sidechain rotational potentials for the dihedral angle X(AB-CI)) with the rest of the protein fixed rigidly in position, the only atoms displaced are 1) and those farther from the backbone. Consequently, only a small fraction of the energy function needs to be recalculated for each value of x and the rotational potential can be determined extremely rapidly. For the same reasons, it is easy to produce energy contour maps as a function of the sidechain dihedral angles, XI = X(NCa-CA-Y) and X2 = X(CaC6-A3yBa). However, such "rigid" rotations can lead to unrealistically high energies as a result of short nonbonded contacts that are easily relieved by small changes in the structure of the protein. To include relaxation effects, the dihedral angle of interest is rotated to a desired value and constrained there by a large potential, while 2002

Proc. Nat. Acad. Sci. USA 72 (1975) Sidechain Potentials of Amino Acids in Proteins 2003 m - a 80, 70 /\ti-'erg %, TYRlO0 I I I1 40 I X-ray~~'I I I I 2700 a 20 20-- 2 6 42 0~~~~~~~2-111 &0 60 120 180 240 300 360 X2 180 20 900 6 4 2 10 24 20 00 900 1800 2700 Xi 3600 co ẇ0 X2 1800 FIG. 1. Rigid rotation barriers for the aromatic ring (X2) of Tyrosine 10 and 21; both the x-ray and energy-refined geometry (ERG) results are shown. all other conformational variables are allowed to change in accord with the steepest-descent procedure (12, 13). Details of the formulation and the energy parameters will be given separately, 11. RESULTS AND DISCUSSION In this section, we describe first the comparison between the x-ray and energy-refined structures of the bovine pancreatic trypsin inhibitor, second the study of the rotational potential of the sidechains in the protein, and finally the possible aromatic ring motions of the tyrosine residues; preliminary results for the phenylalanine sidechains are also given. Overall Structure. An analysis of the x-ray structure (7) for the trypsin inhibitor showed that it is a very "good" structure from an energetic viewpoint; that is, there are no large repulsive nonbonded or hydrogen-bonded contacts, the torsional angles (0, ', Xi, w) generally have reasonable values, and the distribution of the bond angle T(N-Ca,-C') that was varie(l is rather narrow. In comparing the energy refined structure with the x-ray result, the largest changes were found in the backbone dihedral angles 0, V, and w; the angles 0 and ; changed by up to 30 and w by almost 200 in a few cases; the sidechain dihedral angles (Xi) generally changed less. The distribution of T(N- Ca-C') narrowed from the initial standard deviation o f 6 to approximately 3. The decrease in the width of the T(N-Ca- C') angle distribution relative to that obtained in the x-ray 00 900 1800 2700 3600 FIG. 2. (XlX2) maps for Tyr 21: (a) free dipeptide; (b) peptide in protein; the black dot corresponds to the (XlX2) values in the protein (0 = 253.230, i, = 146.770, = 170.930; see text for description). structure resulted from the greater flexibility of the energy minimization procedure, which permitted variation of the bond and dihedral angles and the bond lengths. In all cases very narrow distributions from the mean resulted in these structural parameters. The typical root-mean-square change in the coordinate of an atom that resulted from the energy refinement is about 0.1 A. This change is within the overall limits of error of the x-ray structure determination (14). In spite of the small changes in atom positions, the energy of the refined structure is reduced by about 300 kcal/mol (1 cal = 4.184 J) relative to the x-ray structure. This stabilization is due mostly to a large number of very small shifts in nonbonded contacts; approximately 3000 nonbonded atom pairs closer together than 8.0 A were included in this calculation. It was found that the values of the dihedral angles in the protein are such that there is; significant strain energy associated with them (141 kcal/mol for the x-ray structure and 90 kcal/mol for the energy refined structure); the bond angles are also slightly strained (44 kcal/mol in the x-rav structure and 49 kcal/mol in the energy refined structure). The nonbonded and hydrogen-bonded interactions stabilize the structure. Xi

2004 Chemistry: Gelin and Karplus Proc. Nat. Acad. Sci. USA 72 (1975) TABLE 1. Close contacts responsible for tyrosine aromatic ring (X2) rotation barriersa Distance,unrotated* Closest approach during rotation Energy- X-ray geometry Energy-refined geometry Flexible geometryb refined Atom pair X-ray geometry 1)ist. Energy Dist. Energy Dist. Energy C210-Cal2 3.39 3.36 2.64 12.3 2.51 24.4 3.03 1.5 C6210-N12 3.88 3.66 2.92 1.0 2.64 5.8 2.99 0.6 CQ210-N12 3.92 3.72 3.00 0.5 2.72 3.7 3.12 0.1 Ca121-N48 3.62 3.45 2.28 44.0 2.16 89.4 2.82 2.0 C,221-Cy232 3.54 3.54 2.45 44.5 2.32 88.9 2.99 2.6 C,121-N48 3.74 3.64 2.43 19.1 2.42 19.7 2.92 1.0 Ca121-C47 3.52 3.45 2.51 17.8 2.53 16.2 3.00 1.1 C,223-O55 3. 54 3.93 2.30 22.7 2.32 20.5 2.87 0.8 C6223-C,655 3.72 4.00 2.61 14.2 2.46 32.2 2.95 2.4 Ca123-N24 3.79 4.07 2.53 10.7 2.81 2.1 2.98 0.6 Ca135-N6244 3.50 3.55 2.19 94.1 2.24 72.7 2.82 2.9 C,235-N37 3.63 3.72 2.36 28.0 2.31 37.8 2.88 1.4 Ce235-Ca37 3.49 3.43 2.56 18.7 2.40 44.9 3.10 1.0 C6235-N37 3.69 3.72 2.45 17.0 2.37 27.1 2.81 2.1 a Distance (Dist.) in Angstroms; energies in kilocalories. b 150 steepest descent cycles (see text). However, since no attempt has been made to refer the energy to that corresponding to the denatured protein in aqueous solution or to explicitly include hydrophobic terms, the absolute values of certain energy contributions (e.g., hydrogen bonding) may not be meaningful (15). Sidechain Dihedral Angles. Of the 58 residues in the trypsin inhibitor, 36 residues have sidechains with dihedral angles that can be rotated; the 22 other residues are the 6 glycines, the 6 alanines (whose XI moves only hydrogen atoms), the 4 prolines, and the 6 cysteines, paired by 3 disulfide bonds. It is of considerable interest to determine whether the sidechains of these residues are at their calculated minima. To simplify the problem, we have rotated each sidechain dihedral angle individually, keeping all others and the rest of the protein fixed according to the rigid rotation procedure described in Section I. For the x-ray coordinate set, it was found that 56 out of the 97 bonds about which heavy atoms may undergo rotation have a global or local minimum in their rotation potentials within 100 of the x-ray values, and an additional 16 are within 300. The remaining 25 bonds are associated with the nine polar residues Glu 7, Lys 15, Arg 17, Lys 26, Arg 39, Lys 41, Arg 42, Lys 46, and Glu 49, all of whose sidechains are directed out of the protein either into solvent or toward an adjacent molecule in the crystal (6); Glu 7, Lys 15, and Lys 26 are not completely located by the x-ray density map (7). No terms representing solvent or adjacent-molecule interactions are included in the potential energy function. As has been pointed out (6), Asn 43 is the only internal polar residue and its position is in accord with the energy calculations. For five other polar residues (Arg 1, Asp 3, Arg 20, Asp 50, and Arg 53) the computed results are in a satisfactory agreement with the x-ray values; these residues lie near the surface (especially the short Asp sidechains) or have important electrostatic interactions with peptide groups in the core. The results of the sidechain dihedral angle variation for the energy-refined geometry are very similar to those for the x-ray coordinates. The calculated coordinates are in many cases even closer to the local potential minima; now 63 out of 97 dihedral angles are within 100 of the calculated minimum. As expected (see above), no improvement is found for the external polar sidechainis. In most cases, the barriers to rotation obtained after energy refinement are steeper anld more symmetrical than those calculated for the x-ray structure, primarily as a consequence of fine adjustments in nonbonded interactions (see below). Some studies were also made of sidechain contour maps involving the rotation of two angles at a time. Again generally good agreement was found between the calculated potential minimum and the actual position of the residue, the most striking exceptions involving the polar residues, as already (liscussed. Detailed Sthdy of Aromtatic Sidechains. The brief survey given above indicates that the present method can be of value for the study of sidechains whose primary interactions are with other parts of the protein. We now report more detailed results for the eight aromatic sidechains present in this trypsin inhibitor. The four phenylalanines (4, 22, 33, and 45) are fairly well buried in the interior of the protein. Of the four tyrosines, only Tyr 10 lies roughly flat along the molecular surface; Tyr 21, 23, and 35 have their rings buried, though the OH groups of 21 and 23 are close to the surface. Fig. 1 shows the barriers for rotation about the ring dihedral angle X2 obtained for the Tyr 10 and 21; the barriers for Tyr 23 and 35 are very similar in form to those for Tyr 21, though the heights are different (see Table 2). The x-ray and energyrefined geometry barriers are very similar, the latter being slightly higher, as expected. The location of the maximum of the barrier near 900 and its symmetric form (in all cases except for the Tvr 10) is also of interest. An important comparison can be made between the tor- the protein and that sional potential seen by a sidechain in seen by the same sidechain as part of a "dipeptide" free in solution. To illustrate the difference, we compare in Fig. 2a (free) and b (protein) the (X1,X2) potential energy map of Tvr 21; the backbone angles of the free dil)eptide are chosen to be the same as in the protein. The minimum energy conforma-

Proc. Nat. Acad. Sci. USA 72 (1975) Sidechain Potentials of Amino Acids in Proteins 2005 TABLE 2. Barrier heights and rate constant for tyrosine aromatic ring rotation Energy- X-ray refined Flexible geometry geometry geometry barriers barriers barriers Reorientation (kcal/ (kcal/ (kcal/ rate constant" Residue mol) mol) mol) (sec-1) Tyr 10 22 43 0 6.2 X 1012 Tyr 21 140 230 12 1.3 X 104 Tyr 23 63 75 7 5.3 X 107 Tyr 35 175 200 23 1.4 X 10-4 a Obtained with Eq. 2 as described in text. tions are found to be very similar in the two cases. However, the sidechain is much more rigidly fixed in position by its neighbors in the protein than it is by interactions with the backbone of the chain in solution. The same result is found for the sidechains of most of the amino acids that are located in the interior of the protein. To analyze the potential, we have determined the contributions to the tyrosine barriers around X2; Table 1 lists the major nonbonded contacts involved. We have plotted in Fig. 3 the dominant nonbonded interactions appearing in the Tyr 21 rotational barrier. Many of the interactions involve residues that are distant in sequence number so that the contacts arise from the specific folding of the polypeptide chain. For Tyr 21, for example, the two largest contributions come from contacts with the backbone nitrogen of Ala 48 and C72 of Thr 32 (see Fig. 3). These two atoms are located nearly symmetrically above and below the approximate center of the Tyr 21 ring. Atoms C0, and C,0 of Ala 48 are somewhat further from the ring, are asymmetrically placed, and make a smaller contribution. In addition, the carbonyl carbon of Ser 47 interacts significantly with the ring; C47 is asymmetrically placed on the other side of N48 from C.48 and C$48. Energy refinement brings all the atoms somewhat closer to the ring, in slightly more symmetric positions. The sum of the listed nonbonded contributions leads to a symmetric potential (see Fig. 3) that is very similar to the overall potential (Fig. 1), which includes the other interactions (see Section I). The above result, which is typical of the behavior found for the sidechains in the protein, raises an important question. Since the interactions of a residue with the other groups in the protein are large, it seems unlikely that the correspondence between the position of the sidechain in the protein and the free residue minimum is simply a consequence of the energetics involved. Instead, it may well have its origin in the requirements of the folding process. As the protein folds, the "free" sidechains are undergoing rotation and torsional oscillations, but spend most of their time in the neighborhood of the "free" minimum. Thus, it considerably simplifies the folding problem if the residue is incorporated into the native structure with its dihedral angles corresponding to the "free" values. Of course, once the protein has folded, these dihedral angle values are fixed considerably more rigidly than in the free peptide by the non-bonded interactions, as described above. The bulky and rigid sidechains of tyrosine and phenylalanine must by their presence in the interior of the protein exert considerable influence on the packing of atoms around 00 300 600 900 1200 1500 lf FIG. 3. Dominant Tyr 21 barrier nonbonded components for energy-refined coordinates: ( ) sum of the following six interactions: (---) C6121-N 48; (-- ) C,221-C,232; Q (-- C1l21-N 48; (---)Cj121-C 47; ( ) C7,21-C, 48; (---) CEl21-C$48. them. The existence of only a few strongly interacting nonbonded contacts between them and the rest of the protein argues that these may also be of importance in evolutionary development. A mutation that disturbed a close contact could be energetically very costly and the distortion of the structure required to compensate for this might prevent the proper functioning of the molecule. For the x-ray and energy-refined geometries, it is clear from Fig. 1 and Table 2 that rigid rotation of the aromatic ring gives rise to very large barriers. These barriers are so large that the protein is not rigid enough to maintain them. Instead, during the rotation of an aromatic ring, it can be expected that "relaxation" of the protein would occur and that the effective barriers would be much reduced. This is made particularly likely by the fact that the dominant contributions to the barriers come from a few nonbonded contacts (Table 1). Since such interactions are short range (r'l2 dependence), a small displacement that costs little in energy can lead to a large reduction in the effective barrier. To evaluate the importance of the relaxation, "flexible" geometry barriers were determined (see Section I). At the minimum ( = 00) and maximum energy orientation of the aromatic ring, 150 steepest-descent cycles of energy minimization were performed starting with the energy-refined geometry coordinates; this number of cycles appears to be sufficient for convergence of the energy difference between the two orientations (i.e., for the barrier height), although both structures are still slowly decreasing in energy. In addition, more limited energy minimizations (50 cycles) were performed for other values of X2 to clarify the nature of the atom motions involved in reducing the static barrier. Table 1 shows the changes in the close constant distances and energies and Table 2 lists the barriers; Tyr 35 stands out as having the highest barrier. The Tyr 21 barrier, whose contributions in the absence of relaxation are shown in Fig. 3, provides a good examlple of the mechanism by which steric repulsions may be lessened as a result of shifts in atom positions. For the 50-cycle minimization, the barrier is about 15 keal/mol of which 11 keal/mol is nonbonded in origin; the bond angle strain is about 5 kcal/mol and all other contributions amount to -1 kcal/mol. The nonbonded part of the barrier has thus been reduced by about 215 kcal/mol at the cost of 5 keal/mol in bond angle strain.

2006 Chemistry: Gelin and Karplus The Tyr 21 ring moves off the axis of the C#-CQ bond, as shown by changes in the angles Cg-C7-Ca, (-5.40) and Cf-C--C52 (8.70). This displacement, combined with a change in Xi of Tyr 21 of up to 100, leads to a ring motion that tends to increase the distance between C5221 and N48 and between C,221 and C,232. Simultaneously, C.232 moves away by a 9 opening of its angle CaCt-C.C2, and a -170 change in its dihedral angle N-Ca-CO-Cz2. Atom N48 is shifted by deformation of the backbone via an increase in 130 of the peptide torsional angle W47-48 and a decrease by about- 130 of the torsional angle, /48. In the 150-cycle minimization, the listed distortions increase very slightly and there is a tendency toward smaller, longer-range changes that continue the energy decrease. The high Tyr 35 barrier is composed of 13 kcal/mol nonbonded interactions, 8 kcal/mol bond-angle bending, and 2 kcal/mol from all other terms. Three very close contacts remain (C235-N37, 2.811 i; Ca135-N6244, 2.817 A; and CQ235- N37, 2.876 A); see Table 1. The Asn 44 sidechain angle CO- C'-N62 has closed by almost 190, representing an energy cost of about 3.5 kcal/mol for the potential functions used. There are also smaller deformations of the backbone near N37 in the direction toward the C-terminus; that is, 37 increases by 160, 4'7 decreases by 130, and C037-8 increases by 7. The various deformations are becoming so large as to overbalance possible decreases in the nonbonded energy. The reason for this is that the region of the protein surrounding Tyr 35 forms a rather rigid, tight cage about the ring (6, 7). Roughly parallel with the ring rotation axis is a section of the twisted antiparallel,8-sheet (Ala 16 through Asn 24 antiparallel to Leu 29 through Gly 36); Cys 38 is linked by a disulfide bond to Cys 14, somewhat extending the,-network; the Asn 44 sidechain forms a strong hydrogen bond with the sidechain of Arg 20 on the opposite side of the,3-sheet, thus completing the enclosure of Tyr 35. With the barrier results (Table 2), it is possible to estimate the rate constant for ring rotation by making a model for the dynamics involved. The simplest assumption is to regard the rotation as a unimolecular process that can be treated by transition state theory, with the rate constant K equal to K = Tqlt e-bairt h Q. where Qt and Q. are the partition functions for the transition state and initial state, respectively, and EA is the activation energy. For the present case, it is reasonable to assume Q: - Q, in the first approximation, so that K _ -e-ea4irt h The results obtained from Eq. 2 with the flexible geometry barriers are listed in the last column in Table 2. It can be seen that all of the aromatic residues except Tyr 35 are expected to [1] [2] Proc. Nat. Acad. Sci. USA 72 (1976) appear as freely rotating on the time scale of about 200 seccorresponding to the nuclear magnetic resonance measurements, in agreement with the analysis of Snyder et al. (10). The phenylalanine results are similar, in that the rigid barriers of 60-250 kcal/mol are reduced to values on the order of those found for the tyrosines. Among the phenylalanines, 4 and 45 are calculated to have the lowest barriers and 22 and 33 have significantly higher values. The latter pair are in the interior of the protein near a well-localized water molecule; refined calculations including the oriented water molecules have been made to obtain more definitive barrier values. The agreement between the calculations and experiment found in the present work is encouraging. However, it must be cautioned that the energy function used is highly approximate and that the rate constants given in Table 2 are only estimates. The large difference between the rigid and flexible barriers is an important result that is certainly valid. Further, it is evident that from rigid barrier heights, little can be deduced concerning the flexible barrier values; the latter depend not only on where the interacting groups are located but also on how easily they can get out of each other's way. We thank J. Deisenhofer, R. Huber, S. Karplus, G. H. Snyder, and B. D. Sykes for many discussions. We thank J. Deisenhofer and W. Steigemann for providing the x-ray coordinates. The calculation of the tyrosine sidechain potential was suggested by S. Karplus. This research was supported in part by grants from the National Science Foundation (USA) and the National Institutes of Health (USA). B.R.G. was supported in part by a grant from Roussel Velaf. 1. Chou, R. Y. & Fasman, G. D. (1974) Biochemistry 13, 211-245. 2. Lewis, P. N., Momany, F. A. & Scheraga, H. A. (1974) Is. J. Chem. 11, 121-152. 3. Pullman, B. & Pullman, A. (1974) Advan. Protein Chem. 28, 347-526. 4. Warme, P. K. & Scheraga, H. A. (1974) Biochemistry 13, 757-767. a. Levitt, M. (1974) J. Mol. Biol. 82, 393420. 6. Huber, R., Kukla, D., Ruhlman, A. & Steigemann, W. (1970) in Proceedings of the International Research Conference on Proteinase Inhibitors, Munich (Walter de Gruyter, Berlin), pp. 56-64. 7. Deisenhofer, J. 0. & Steigemann, W. (1975) Acta Cryst. B31, 238-250. 8. Karplus, S., Snyder, G. H. & Sykes, B. D. (1973) Biochemistry 12, 1323-1329. 9. Masson, A. & Wuthrich, K. (1973) FEBS Lett. 31, 114-118. 10. Snyder, G. H., Rowan, R., Karplus, S. & Sykes, B. D. (1975) Biochemistry, in press. 11. Levitt, M. & Lifson, S. (1969) J. Mol. Biol. 46, 269-279. 12. Warshel, A. & Karplus, M. (1972) J. Amer. Chem. Soc. 94, 5612-5625. 13. Gelin, B. R. & Karplus, M. (1975) J. Amer. Chem. Soc., in press. 14. Huber, R., Kukla, D., Bode, W., Schwager, P., Bartels, K., Deisenhofer, J. 0. & Steigemann, W. (1974) J. Mol. Biol. 89, 73-101. 15. Chothia, C. H. (1974) Nature 248, 338-339.