Flexibility of bovine pancreatic trypsin inhibitor

390

Biochimica el Biophysica Acla, 536 (1978) 390--405

© Elsevier/North-Holland Biomedical Press

BBA 38016

FLEXIBILITY OF BOVINE PANCREATIC TRYPSIN INHIBITOR

TATSUO OOI, KEN NISHIKAWA, MOTOttISA OOBATAKE and HAROLD A. SCHERAGA Institute ['or Chemical Research, Kyoto University, Uji, Kyoto-fu (Japan) and Deparlmenl o f Chemistry, Cornell University, lthaca, N. K 1.I853 (lAS.A.)

(Received May 2nd, 1978)

Summary The native c o n f o r m a t i o n of a protein may be expressed in terms of the dihedral angles, ¢'s and ~b's for the backbone, and X's for the side chains, for a given g e o m e t r y (bond lengths and bond angles). We have developed a m e t h o d to obtain the dihedral angles for a low-energy structure of a protein, starting with the X-ray structure; it is applied here to examine the degree of flexibility o f bovine pancreatic trypsin inhibitor. Minimization of the total energy of the inhibitor {including n o n b o n d e d , electrostatic, torsional, hydrogen bonding, and disulfide loop energies) yields a c o n f o r m a t i o n having a total energy o f - - 2 2 1 kcal/mol and a r o o t mean square deviation between all atoms o f the computed and experimental structures of 0.63 A. The optimal c o n f o r m a t i o n is not unique, however, there being at least two o t h e r c o n f o r m a t i o n s of low-energy {--222 and --220 kcal/mol), which resemble the experimental one (root mean square deviations of 0.66 and 0.64 A, respectively). These three conform at i ons are located in different positions in ¢, ~ space, i.e., with a total deviation of 81 ° , 100 ° and 55 ° from each ot her (with a r o o t mean square deviation of several degrees per dihedral angle from each other). The n o n b o n d e d energies of the backbones, calculated along lines in ¢, ~ space connecting these three conformations, are all negative, w i t hout any intervening energy barriers (on an energy c o n t o u r map in the ¢, ~ plane). Side chains were attached at several representative positions in this plane, and the total energy was minimized by varying the X's. The energies were of a p p r o x i m a t e l y the same magnitude as the previous ones, indicating that the c o n f o r m a t i o n of low energy is flexible to some e x t e n t in a restricted region of ¢, ~ space. Interestingly, the difference AC/+1 in ¢i+1 for the (i + l) th residue from one c o n f o r m a t i o n to anot her is approximately the same as - - ~ i for the ith residue; i.e., the plane of the peptide group between the ith and (i + 1) th residues re-orient w i t hout significant changes in the positions of the ot her atoms. The flexibility of the orientations of the planes o f the peptide groups is probably coupled in a cooperative manner to the flexibility o f the positions of the backbone and side-chain atoms.

391 Introduction During the course of development of procedures to adjust atomic coordinates of proteins to yield structures of low energy, with geometry {i.e., bond lengths and bond angles) similar to that of small peptides, it was found that a range of backbone conformations could be accommodated within a small range of energies; i.e., the native protein is flexible to some degree. In this paper, we described the range of this flexibility for bovine pancreatic trypsin inhibitor. A variety of methods have been formulated [1--7] to improve the threedimensional structures of proteins and thereby reduce the errors in the atomic positions deduced by X-ray diffraction analysis. These methods may be classified into two categories according to the variables u s e d - either the cartesian coordinates of the atoms or the dihedral angles (with fixed geometry and with the peptide groups usually taken in the planar trans conformation). In both methods, interatomic potentials are employed to calculate the conformational energy but, when the atomic coordinates are used, additional data are required (viz., force constants for varying the geometry). The principle of both methods is the same, i.e., variation of the conformation to minimize both the energy and the deviation of the structure from the experimental one. The main problems encountered in applying such procedures are the multiplicity of minima in the many-dimensional energy surface and the attainment of a conformation that resembles the experimental one. The multiple-minimum problem becomes more difficult as the number of variables increases, and an efficient search of the conformational space is required to reach the global minimum. In this paper, in which we demonstrate the nature of the flexibility of a native protein, we present a new approach to the problem of adjusting protein conformation, using fixed geometry. The reasons for using fixed geometry have been discussed extensively elsewhere [7]. The method is applied here to bovine pancreatic trypsin inhibitor, a relatively small protein (58 residues) whose X-ray structure has been refined to a considerable degree [8]; the coordinates of this structure are designated as X-ray-A. Procedure The basic idea of the procedure introduced here is that two structures can be compared by considering the relative distances between given pairs of atoms in both structures, thereby eliminating the need to transform one into the coordinate system of the other. A virtual-bond representation is used to describe the positions of the Ca atoms of the backbone, and a d u m m y atom attached to the Ca atom is used to define the orientation of the plane of the peptide group between two consecutive Ca atoms. The deviation of a calculated structure from the experimental one, and the energy of the computed one, are minimized to obtain a low-energy structure that resembles the native one.

Bond lengths and bond angles While a 'standard' set of fixed bond lengths and bond angles {which differ for the various residues of a polypeptide chain) is available [9], for simplicity

392

in programming in these initial calculations we adopted a uniform set for the backbones of all residues. Thus, the bond lengths and bond angles used [including T(NC~C ') = 111.5 °] were those of Ooi et al. [10]; however, some calculations were performed by using other uniform values of 7(NCC~C') in the range of 110 ° to 112.5 ° . The bond lengths and bond angles of the side chains were those of Momany et al. [9]. Using the matrix methods described by Ooi et al. [10], it was possible to generate any conformation of the backbone and side chain, using this set of fixed bond lengths and bond angles {with peptide groups taken in the planar trans conformation), for any set of backbone dihedral angles ¢i's and ~bi's, and side-chain dihedral angles (×i ]} 's.

Representation o f conformation While any conformation of a polypeptide chain can be represented in terms of either the cartesian coordinates of its constituent atoms or the backbone and side-chain dihedral angles of each residue, it was simpler (for the method used here) to use a virtual-bond representation, where the virtual bond is the vector between two consecutive Ca atoms. Since the backbone geometry has been taken to be fixed and uniform, all virtual bonds of the chain have the same length. It is possible to convert the description of the backbone of the chain in terms of ¢i's and ~i's to one in terms of the 'dihedral' angles for rotation about virtual bonds and the ' b o n d ' angles between successive virtual bonds [11]. In this way, the ends of the virtual bonds would locate the Ca atoms of the chain, and the 'dihedral' angles and ' b o n d ' angles would specify the orientation of the peptide planes between each pair of Ca atoms. However, in using a virtual-bond representation of the backbone, we have preferred to use an alternative representation of the orientation of the peptide planes because it enabled us to compare any two conformation of the chain in a very simple manner. As shown in Fig. 1, the orientation of the plane of the peptide group around the

Qcy

i•

V

Ci+l

\

/( I

I

), ! I

Fig. 1. R e p r e s e n t a t i o n of p o l y p e p t i d e c h a i n as a s e q u e n c e of p l a n a r g r o u p s . C v and Cv+l are d u m m y a t o m s c o n n e c t e d to C~' a n d C~+I, r e s p e c t i v e l y , with the v e c t o r f r o m C~ to C v being p e r p e n d i c u l a r to the plane o f t h e p e p t i d e g r o u p f o l l o w i n g C~.

393

vector connecting C~/ to Cai+l is specified by the direction of the vector (perpendicular to the plane following Ca/) from C% to the d u m m y atom CVi. The distance between C"/ and CVl depends on the weight assigned to the orientation of the plane (in matching the computed to the experimental structure) since, the larger this distance is, the more sensitive is the fit to the orientation. tiere, the distance is taken {fixed) as 5 X (see Discussion). Thus, the coordinates of the Ca and Cv atoms of each residue completely describe the backbone conformation; i.e., if the coordinates are given, we know the location of the N, C' and O atoms (and hence the orientation of the plane) from which we can obtain the ~b's and ~'s (and vice versa). Thus, in the procedure used in this paper, we always generate a chain in terms of ~b's and ~'s, and then express the conformation in terms of Ca's, CV's. Comparison o f two conformations o f a p o l y p e p t i d e chain With the representation of a conformation of a polypeptide chain specified as in Fig. 1, it is very easy to compare two different conformations, in particular a calculated conformation with an experimental one. When comparing a calculated conformation with an experimental one, both are usually expressed in different coordinate systems, and it would be necessary to transform one conformation to the coordinate system of the other. When the calculated conformation differs significantly from the experimental one, it is difficult to assign an o p t i m u m degree of coincidence of the two coordinate systems. Thus, we chose the alternative method of comparing the relative distances between the i TM and jth residues, because such a comparison (of the backbone conformation) can be made, independent of the coordinate system. In this procedure, it does not matter that the virtual bonds are of uniform length in the calculated conformation, but of variable length in the experimental one. If r~l.? is the calculated distance between the Ca atoms of residues i and ], and r~.° is the corresponding experimental value, and if rfi and r~° are the analogous distances between the d u m m y atoms, then the quantity f is a measure of the correspondence between the calculated and experimental conformations, where f : p + [~ = E (r~ ---,jr~°)2 + E (r,~ -- r~?) 2 i "j

(1)

i>j

In the final stage of the computation, where the calculated conformation is close to the experimental one, the usual comparison in terms of deviations of atomic coordinates is made to illustrate the correspondence between the two conformations. For this purpose, the two conformations are superimposed by transforming the coordinate system of the computed conformation to that of the experimental one. Also, after attachment of the side chains, the conformations are compared by superposition because, at this stage, the calculated and experimental conformations are very close to each other. While it may seem that the above m e t h o d is somewhat restrictive, in focusing so much attention on the C~ atoms, it should be pointed out that the positions of the Ca atoms are generally known (experimentally) more accurately than those of the other atoms of the chain (except for sulfur atoms). Thus, in essence, greater weight in the fitting is given to the more-accurately located atoms.

394

Generation of the chain The values of r~-° and r[j° for the experimental structure were computed directly from the Cartesian coordinates. The d u m m y atoms, C~', were placed at 5 A from C~, as described above, in a mean direction obtained by averaging the orientations of the vectors perpendicular to the (C~i, C'i, O~), (Cai, C'~, Ni), (Ca/, Ni, Cai+l) and (cai, C'i, Cai+l) planes. The values of r~ and r~ for the theoretical structure were computed after the chain was generated with the assumed (fixed) geometry by varying the values of the ~b's and ~'s (starting with the experimental values of these dihedral angles) to give a close fit to the experimental conformation, as described below. Once the backbone of the calculated conformation was established, the side chains were generated, using the geometry of Momany et al. [9] and varying the side-chain dihedral angles (also starting with the experimental values).

Minimization of f In order to adjust the calculated conformation to approach the experimental one as closely as possible, the value of f must be minimized with respect to all ¢i's and ~;'s. In addition, as discussed in the next section, the energy must also be minimized. In the minimization of f, it is only the backbone that is considered, i.e., the side chains are not yet included at this stage of the procedure. In order to avoid being trapped in local minima during the minimization of f, this procedure must be carried out step by step. Thus, instead of minimizing f initially, we begin by minimizing a function fa in a subspace of 2a residues rather than in the space of all N residues of the chain, where

ro =

E) li--jl
_

70

(r,jv _ risv 0 2)

(2)

li-j[
The function f~ was minimized with respect to all ~bi's and ffi's for values of a that were increased, step by step, beginning with a = 3 and proceeding to a = (N/2)--I in increments of 5 (see Discussion). The Davidon minimizer [12] was used for this procedure. The function to be minimized contains two terms, one arising from the relative distances between the C a atoms, and the other from the relative distances between the d u m m y atoms (i.e., the orientations of the planes of the peptide groups). Since the f~ and f~ terms are not independent of each other, they cannot be minimized separately. If the relative contribution of the f¢ term were great, the chain would undergo large movements and force f into incorrect local minima. This can be prevented by weighting the f~ term appropriately, first of all by choosing the distance between Ca; and CV~ properly (i.e., as 5 h ; see above), secondly, b y minimizing f~, initially, instead of f and, thirdly, by restricting the range of values of the dihedral angles during the minimization of f, so that remote parts of the chain do not move significantly. This can be accomplished by splitting the minimization of fa into two parts: (A) f~ is minimized with respect to all ¢i's and ~]i'S in the interval li --jl
395 is rotated about the C~i-C ' bond by an a m o u n t A~i and about the N-Cai+I bond by the same a m o u n t AOi+l (but in the opposite direction), then this rotation is accomplished with little movement of C~ and C~+1 or of the rest of the chain. This rotation of the peptide group does not alter the rest of the chain significantly because the C~-C ' and N-C~+I bonds are nearly parallel to the virtual bond connecting C~ to C~i+l. Because of the constraint in step B, we reduce the number of variables from 2N to N, thereby reducing the number of local minima.

Energy minimization The mininaization of f usually leads to atomic overlaps between backbone atoms which can be relieved in a separate calculation by minimization [12] of the conformational energy of the backbone with respect to the ~bi's and ffi's. In the early stages of the calculation where the energies are large and positive because of atomic overlaps, only the positive energies of a Lennard-Jones 6--12 potential [13] for colliding pairs are included in the computations; i.e., the negative energies are set equal to zero. For simplicity, a united-atom approximation [13] is used for the C~H group of the backbone, no side chains being included at this stage. After the positive energy is minimized, then the side chains are attached and the negative energies of the Lennard-Jones potential, and other energy terms, are included for further minimization. While more accurate potentials exist [9], the availability of a computer program with an earlier set of potentials [10], which was considered accurate enough for the purpose at hand, led to the adoption of the earlier set. The n o n b o n d e d energy between N or O and H atoms was not calculated in order to avoid unreasonably high energies in hydrogen-bonded configurations. Instead, a hydrogen-bond energy was included in the electrostatic interaction between polar atoms [10]. Partial charges, estimated from bond dipole moments, were used to calculate the (Coulombic) electrostatic interactions between polar atoms [10], with a dielectric constant of 4. Torsional energies of the form (E0/2) cos ( 1 - nx) were calculated with E0 = 2.8 kcal and n = 3 for rotation about side-chain C-C bonds; the other torsional barriers were those of the previous paper [10]. A potential of the form E s s ( l - - 10)2 was used to close disulfide loops, where Ess = 300 kcal/A 2 and 10 = 2.00 A. A united-atom approximation [13], for nonpolar atoms, was used in the calculation of the total energy.

Minimization algorithm The following sequence of computations is carried out in order to obtain a calculated conformation (of standard geometry) that fits the experimental structure as closely as possible and, at the same time, has a low conformational energy. (1) The value of f is minimized by minimizing fa (for the backbone) with increasing values of a. Both procedures A and B are used. The resulting calculated conformation might be very close to the native one but usually would have several severe atomic overlaps at this stage. (2) The atomic overlaps (involving only backbone atoms, including C~ atoms) are relieved by minimizing the positive energy of the conformation

396 obtained in step 1 with respect to only the Oi's and ~i's. In this energy minimization, there is also an alternation between procedures A and B, i.e., with and without the constraint that A~i = --~¢i+ 1. (3) The relief of the atomic overlaps in step 2 leads to an increase in f. Therefore, f is again minimized (using procedures A and B) with respect to all ¢~'s and ~i's, but the changes in conformation are usually small at this stage because the conformation is now close to the experimental one. Then steps 2 and 3 are repeated until a conformation of minimum f and minimum atomic overlap is obtained. Since both f and the extent of atomic overlaps are minimized, the following criteria are used as a compromise between the two minimization procedures: the value of f* should be less than 300 X 2 (corresponding to a root mean square, deviation {(ri~--rp)°)2} j': of about 0 . 4 A ) and the overlap energy should not exceed 5 kcal/mol. This criterion kept the computed structure close to the X-ray one with little atomic overlap. (4) The resulting backbone conformation is then held fixed, and the side chains are attached by varying the dihedral angles of each side chain separately to mimize the sum of the squares of the deviations between the calculated and experimental coordinates of the side-chain atoms. For this purpose the coordinate system of the calculated structure was transformed to that of the experimental one; i.e., the calculated backbone conformation was superimposed on the experimental one. (5) With the backbone still maintained fixed, the atomic overlaps involving side-chain atoms are relieved by minimizing the positive energy, by the same procedure used in step 2 for the backbone atoms. (6) Then the total nonbonded energy of the whole molecule is minimized with respect to all backbone and side-chain dihedral angles [12]. The alternation between steps 2 and 3 avoids the necessity of using a fitting potential [5,7 ] during the energy minimization. The program was written in Fortran S, and the final computations were performed on a FACOM 230-48 computer at the Institute for Chemical Research, K y o t o University. Results

Initially, the value of fo was minimized with a = 3, starting with values of the ~bi's and ~i's computed from the X-ray structure. Since only local structures were adjusted at this stage, the value of f for the whole molecule was not low; i.e., the generated conformation did not resemble the native one. As a was increased, the value of f for each minimum value of fo decreased significantly until a = 15, after which the value of f reached a minimum value, as shown in Fig. 2. The values of the ¢i's and ~i's changed similarly; these changes are expressed (in Fig. 2) as the square root of the sum of the squares of all A¢'s and ±~'s, the latter being the 'distance' moved from the initial values in conformation space. Both procedures A and B were used in each iteration. The final value of f was 268 A: (or 118 A 2 for f~ and 150 A 2 for f~). The set of ¢i's and ~i's at the global minimum of f is designated as A1. The conformation generated by the dihedral angles A1 has several overlaps among the backbone atoms. Its calculated nonbonded energy is given in

397 0 5

5 i

[0 4

I0 15 20 28 28 28 28

0

o

hO3 'o

o,~ ~ S

"'o-..o_._ o

o---o---o.

10 2

.~'~'-'-'-'-'-'-'-L

50°~

•

v

tO I

£3 I

L

I

[ J J I 5 No. of I t e r a t i o n I

I

[ I0

L

I

L

l

~o

F i g . 2. B e h a v i o r o f f ( o p e n c i r c l e s ) a n d [ V ( . ' , 0 2 + / ~ 2 ) ] 1 / 2 , e x p r e s s e d as ' d i s t a n c e ' m o v e d values in conformation space, (filled circles) in successive minimizations (iterations) of i n e a c h i t e r a t i o n is s h o w n o n t h e t o p s c a l e .

from

fa" T h e

the value

initial of a

Table I. The major overlaps occur between atoms near the C-terminus, with residues 52--56 accounting for 256 kcal/mol and residues 53--55 for 67 kcal/mol. These atomic overlaps were relieved in step 2, by minimizing the positive energy. However, this relief of atomic overlaps led to a significant deviation of the calculated conformation from the experimental one; i.e., the value of f increased to > 2 0 0 0 £2. Hence, in step 3, the value of f was reminimized (but with the use of only procedure A, for the reason given below}, and decreased from 2 1 9 4 to 983 A 2 (272 £2 for f~ and 711 A 2 for f~, i.e., greater deviations of CV's from the experimental values than for Ca's). Most of the

TABLE

I

CHARACTERISTICS Conformation

OF BACKBONE

f(fa

CONFORMATION

fv) , (A2)

A1

26R (118, 150)

A2 A3

983 (272,711) 1440 (237, 1203)

AT MINIMA

E n e r g y ** ( k c a l / m o l )

213 (410, --197) 256 (52--56) *** 67 (53--55) *** --115 (65, --180) --127 (60, --187)

, fa and fv are measures of the correspondence of the C a and C v atoms, respectively. ** Nonbonded energy for the backbone atoms. The positive and negative contributions to the nonbonded energy are listed separately in parentheses. * * * A l a r g e p o s i t i v e e n e r g y o f 2 5 6 k c a l / m o l is f o u n d b e t w e e n r e s i d u e s 5 2 a n d 5 6 . S i m i l a r l y , 6 7 k c a l / m o l is c o n t r i b u t e d b y r e s i d u e s 5 3 a n d 5 5 .

398 increase in f in step 2 had arisen from a change in the conformation near the Cterminus, where the severe overlaps had occurred. The triangle plot of the difference, ~rii = (r~ -- r~°), in Fig. 3 illustrates the large deviations near the C-terminus. The use of only procedure A in step 3 led to a local minimum, where the conformation differed from that at the earlier (lower) minimum value of f. If procedure B had also been used in step 3, the original conformation (global minimum of /) with atomic overlaps would have been reached again. The set of Oi's and ~i's obtained at the end of step 3 is designated as A2. Steps 2 and 3 were repeated (using procedures A and B in step 2, but only A in step 3) over and over again until the energy and the value of f were minimized to satisfy the criteria stated above. The set of ¢i's and ~i's thus obtained is designated as A3. Table I summarizes the results for A1, A2, and A3. Side chains were then attached to the (fixed) backbone conformation generated by A3 (step 4), using the values of {Xij}'s computed from the X-ray data. Since the backbone conformation differs from the X-ray structure by a root mean square deviation of 0.6 A, the calculated coordinates of the sidechain atoms generated in this manner did not agree with those of the X-ray structure. Therefore, the values of the {X~j}'s were first varied, all at once, to make the calculated positions of the side chains fit the X-ray ones as closely as possible (step 4). The root mean square deviation (of the calculated coordinates from the experimental ones) of the whole molecule was 0.62 A (0.65 A for the side-chain atoms, and 0.61 A for the backbone atoms). As indicated above (see also Fig. 3), large deviations (1 to 2 A) were observed for residues 56, 57 and 58. If these residues are excluded, the root mean square deviation for all atoms is reduced to 0.4 A. Although this conformation agrees well with the X-ray one, it contains overlaps among side-chain atoms. Therefore, the positive energy was minimized by first varying only the {Xi;}'s, then the ¢~'s and ~i's, and alternatively the {XJ} 's-plus-¢i's and ~ ' s until there was no further change in energy within 0.1 kcal/mol (step 5). At this stage, the presence of the side chains had not pro-

10-

-~

20~

z 30 ~ [M 40~

, I0

D~n, 2O Residue

, 50

, 40

50

Number

,~rii = ( r ~ - - r~iiO), ~1 1 A < ,"~rij < 2 A ; map for the difference Fig. 3. Triangle t h e b l a n k s p a c e s , t h e v a l u e s o f Arid w e r e a l l s m a l l e r t h a n t h e s e l i m i t s .

~, - - 1 A. > ",rij >

- 2 A. In

399 duced a significant change in the conformation of the backbone that had been obtained with A3. Then the total nonbonded energy (not including electrostatic or other energies) was minimized by changing the (×i j}'s and ¢i's and ~i's alternately (step 6). The decrease in energy is shown in Fig. 4. The set of q5i's and ~/'s obtained at the end of this minimization is designated as A4. Once this conformation (backbone and side chains) was obtained, the polar hydrogen atoms were attached, using the procedure of Momany et al. [9], and the total energy [nonbonded, electrostatic (including hydrogen bonding), torsional, and disulfide loop] was minimized with respect to the (X/}'s, keeping the q~i's and ~i's fixed. The final total energy of the structure (now designated as A4-1) was --221 kcal/mol (see Table II), only three residues had a positive energy (arising from the torsional energy term) (see Fig. 5), and the three disulfide bonds were formed correctly (with a loop energy of less than 0.5 kcal/mol). As indicated by the deviations in the last column of Table II, the final conformation (A4-1) is very close to the experimental one (except for the three residues at the C-terminus). The deviations, residue by residue, are shown in Fig. 6. The experimental X-ray coordinates of bovine pancreatic trypsin inhibitor were computed from 1.5 A-resolution data. In the structure, the bond angles T(NC~C ') vary from 95 ° to 124 ° with a mean value of 111.8 ° and a standard deviation of 5.8 °, and the dihedral angles (co) for rotations about the peptide bonds vary from 163 ° to --163 ° with a mean value of 179.3 ° and a standard deviation of 7.1 ° [8]. In a sense, there is a certain arbitrariness in having selected backbone conformation A3 to which to attach the side chains. If, instead, the side chains were attached to A2 (which differs only by a root mean square deviation of 0.34 A, from A3), and then steps 4, 5 and 6 were repeated, the conformation A2-1

2000I

-150

-160 oE

E

I000

-170

.~ c UA

-180

7 -~o~g Z

-200

0

I A I L J

5

I

,

I

I

IT

I IO

No.

of

Iteration

F i g . 4. D e c r e a s e in p o s i t i v e e n e r g y ( l e f t s i d e ) a n d in t o t a l n o n b o n d e d energy (right side). The arrow indic a t e s t h e s t a g e at w h i c h X 2 o f L e u - 2 9 , w h i c h w a s t r a p p e d in a l o c a l m i n i m u m , was varied to relieve an atomic overlap.

400 TABLE

II

TOTAL ENERGY INHIBITOR

* OF

THE

CALCULATED

STRUCTURES

OF

BOVINE

PANCREATIC

Conformation

Eq, o t

ENB

EElcc

ETo r

Root mean square deviations ~ * (A)

A4-1 A2-1 B-1

--221 --222 --250

--245 --248 --241

-49 --50 --52

73 76 73

0.60, 0.67, 0.63 0.66, 0.66, 0.66 0.62, 0.66, 0.64

• In kcal/mol. • * T h e first f i g u r e in t h e t r i p l e t p e r t a i n s tn t h e b a c k b o n e t h e t h i r d t o all a t o m s .

TRYPSIN

atoms, the second to the side-chain atoms, and

(Table II) was obtained. Alternatively, the whole procedure (used to obtain A4-1) was repeated, but starting with an earlier set of X-ray coordinates, designated X-ray-B (personal communication from W. Steigemann in 1974); the resulting low-energy conformation is designated as B-1 (Table II). Since we could obtain three slightly different conformations, all having energies within 2 kcal/mol of each other (i.e., three possibly different local minima), it is of interest to examine their conformational properties. First, the total distance (defined in the legend of Fig. 2) from A4-1 to A2-1 in (¢, ~)space is 81 °, that from A4-1 to B-1 is 110 °, and that from A2-1 to B-1 is 55 °,

I0

E

Backbone

±

0

u

o~ I

-I0

I

I

I

I

2.49 .~

Total

¢r

J III

OA

1.0

0.5,

-5C

I

I0

I

20

I

50 40 Residue Number

I

50

F i g . 5. E n e r g y p e r r e s i d u e o f b o v i n e p a n c r e a t i c mum total energy.

60

0

trypsin inhibitor

I0

20

50 4.0 Residue Number

for conformation

F i g . 6. R o o t m e a n s q u a r e d e v i a t i o n f o r e a c h r e s i d u e o f A 4 - 1 ; b a c k b o n e

50

A4-1, having a mini-

( t o p ) , all a t o m s ( b o t t o m ) .

401 with a root mean square deviation of several degrees per dihedral angle, and the root mean square deviations of the coordinates of these calculated structures from the experimental one (given in Table II) are similar. These values indicate that the conformations are very similar, b u t that the positions of the dihedral angles in conformational space differ considerably. Second, the energies of the b a c k b o n e (without the presence of side chains) were calculated along the lines (in conformational space) connecting every pair of the three conformations A4-1, A2-1, and B-1. Strikingly, the backbone energy changed monotonically along these lines, with no intervening energy barriers. Fig, 7 shows a backbone energy contour diagram in the region of conformational energy space of A4-1, A2-1, and B-l; this diagram was obtained by calculating the n o n b o n d e d energies of the backbones of 140 conformations inside and outside the triangle formed by these three backbone conformations. This result indicates that, as far as the backbone is concerned, the native conformation occupies a region, rather than a single position, of q~, V-space; i.e., there are no atomic overlaps in the region, and the root-mean-square deviation of the positions of the atoms (in the triangle region) from those of the experimental structure is a b o u t 0.6 A. The question then arises as to whether side chains can be attached to various backbones in this region, without introducing atomic overlaps. Therefore, the side chains were attached using the values of the X'S at the energy minima given in Table II, and the total energy was c o m p u t e d at several points (I-VI of Fig. 7) in the region. For all six points examined, the total energies, obtained b y energy minimization with respect to {Xij}'s, with the backbone conformations held fixed, were of similar magnitude (see Table III); i.e., the side -II0

-,20

I

I

I

,

I

bistonce(degrees)

Fig. 7. B a c k b o n e e n e r g y c o n t o u r d i a g r a m in t h e v i c i n i t y o f t h e t h r e e c o n f o r m a t i o n s A4-1, A 2 - 1 , a n d B-1. T h e p o i n t s I - - V I r e p r e s e n t b a c k b o n e e n e r g i e s f o r w h i c h side c h a i n s w e r e a d d e d , a n d t h e n t h e t o t a l e n e r g y w a s c o m p u t e d ( T a b l e I I l ) . T h e d i s t a n c e scale in t h i s m u l t i - d i m e n s i o n a l s p a c e c o r r e s p o n d s to t h e d e f i n i t i o n o f d i s t a n c e g i v e n in t h e l e g e n d to Fig. 2.

402 T A B L E Ill T O T A L E N E R G I E S a A T S E V E R A L P O I N T S IN T H E P L A N E OF B A C K B O N E C O N F O R M A T I O N A L E N E R G I E S C O R R E S P O N D I N G T O A 4 - 1 , A2-1 , A N D B-1

ETot

b

Conformation

I II Ill IV V VI a b c d

c

--229 --227 --218 --209 --196 --179

EN B

EElec

ETo r

Deviations d (A)

--255 --24~ --247 --240 --228 --211

--50 --52 --48 --48 --47 --45

75 73 76 79 79 77

0.62,0.66,0.64 0,59,0.63,0.61 0.71,0.86,0.79 0,68,0.76,0.72 0.68,0.72,0.70 0.69,0.77,0.73

In k c a l / m o l . The locations of these conformations are i n d i c a t e d in Fig. 7. T h e v a l u e s o f E T o t are t h e o n e s o b t a i n e d a f t e r m i n i m i z i n g t h e t o t a l e n e r g y w i t h r e s p e c t t o t h e (~(/J}'s. T h e first f i g u r e in e a c h t r i p l e t p e r t a i n s t o t h e b a c k b o n e a t o m s , t h e s e c o n d t o t h e s i d e - c h a i n a t o m s , a n d t h e t h i r d t o all a t o m s .

chains could be attached to the backbone conformations in this region, without introducting any atomic overlaps. Thus, the native conformation can move about in conformational space, although the movement is limited to a portion of this space. If one changes conformation from A2-1 to A4-1, the changes in dihedral angles, ~0i÷1 and - - ~ i , for each residue, shown in Fig. 8, are involved. It is apparent from this result that there is a strong correlation between A¢~÷~ and - - ~ i . This correlation is represented as a plot of A¢i÷ 1 vs. A~i for all residues in Fig. 9. The data lie on a straight line, passing through the origin and bisecting the second and fourth quadrants. As described in procedure B, this implies that, in going from conformation A2-1 to A4-1, the plane of each peptide group rotates (A¢~+~ = - - ~ ) leaving the rest of the molecule unaltered to any significant extent. This rotation of the peptide plane is illustrated by the

20 if

i

<

, I

J

i j

";'11¢,

j

÷

%. %\

-20 0

\

-20 I0

20 Residue

30

40

50

60

-2 '0

-i '0 Ag' i

Number

Fig. 8. M o v e m e n t , :~0i+1 ( s o l i d l i n e s ) a n d - - ' x ~ i ( d a s h e d Fig. 9. P l o t o f / , ~ b i + 1 vs. :x~ i.

0

lines), from conformation

I0

(degrees)

A 2 - 1 t o A4-1.

i 2O

403

Fig. 10. S t e r e o d i a g r a m s s h o w i n g the d i f f e r e n t o r i e n t a t i o n s of the p l a n e of t h e Deptide g r o u p b e t w e e n Gln-31 a n d T h r - 3 2 in c o n f o r m a t i o n A2-1 ( t o p ) a n d A4-1 ( b o t t o m ) .

stereo diagrams of Fig. 10 for the peptide group between residues Gln-31 and Thr-32. Similar results were obtained when comparing A4-1 and B-l, and A2-1 and B-l, B-1 having been derived from a different initial set of X-ray coordinates. These results indicate that there is flexibility in the native conformation, which allows the atoms to move (within a restricted region) even though the conformation was generated with fixed geometry. Even though T(NC~C ') was maintained fixed at the uniform value of 111.5 ° in these computations, similar results were obtained for other fixed uniform values in the range T(NCaC ') = 110°--112.5 °. This, too, suggests that the native molecule possesses a degree of flexibility. Discussion Before we developed the present method, we tried several others for adjusting atomic coordinates of proteins to yield structures of low energy. For example, when using the atomic coordinates themselves as the variables, and trying to fit the theoretical bond lengths and bond angles, a difficulty was encountered in that, if the coordinates were adjusted to fit the set of bond lengths, then the bond angles in some portions of the chain deviated significantly from a given set of theoretical bond angles (and vice versa). That is, it was not possible to adjust bond lengths and bond angles simultaneously because of the t e n d e n c y to be trapped in local minima. Also, depending upon how either was weighted, there was an oscillation between good fits to the bond lengths and good fits to the bond angles. This is one of the reasons why the geometry was maintained fixed and the value of f was chosen as a criterion

404 with which to judge a generated backbone conformation ill comparison with the experimental one. Also, as indicated in the Introduction, the reasons for using fixed geometry have been discussed extensively elsewhere [ 7 ]. Another reason for using the quantity f is that the relative distance r u does not depend on the coordinate system. When o n l y / ~ (rather than f or f,) was minimized various sets of ~i's and ~i's were obtained, and f did not decrease significantly by minimizing fa with increasing a [2]. The introduction of the d u m m y atoms at positions C v, however, restricted the range of ~i's and ~i's to conform to the experimental conformation. This is indicated by the large decrease in f until a = 15; in other words, the polypeptide chain folds into the native conformation only by minimizing ~ + ~ within a = 15. The choice of the distance from Ca to C V (i.e., the weighting factor) as 5 A was based on an examination of the results obtained by varying this distance from 1 to 10 A. When the distance is small, f is approximately twice as large as /~ (i.e., the d u m m y atoms are close to the Ca atoms, and f~ ~ F ) , and therefore the orientations of the planes of the peptide groups could not be adjusted. When the distance is large, the orientation of these planes is overweighted, and the fit of the generated positions of the C~'s to the experimental ones is poor. The value of 5 A appeared to be the best choice to optimize these effects. While the details of the computed conformation u n d o u b t e d l y depend on the geometry adopted for the calculations, the most important result(viz., that the native conformation is a flexible structure with few atomic overlaps) is independent of geometry with the (reasonable) range of r(NC~C ') tested. From the point of view of obtaining a refined structure (the subject of future work), use will be made of recently formulated geometry and energy parameters [9]. It is worth noting that the conformation of lowest total energy (when the side chains are attached to the backbone) is not obtained from the backbone conformation of lowest energy, but from one with a somewhat higher energy (see Fig. 7 and Tables IL-III). In order to generate a backbone chain that conforms to the native conformation, the dihedral angles, 6i's and ~i's, must be specified with an accuracy of ~0.01 ° since, as also observed elsewhere [14], small variations of dihedral angles lead to significant relative movements of residues that are far apart. Therefore, it is essential (in order to obtain a conformation of minimum f) to start with a small value of a, and adjust local conformations along the chain, and then increase the value of a progressively. Otherwise, the values of the 6~'s and ~i's move in a direction to minimize (r~ - - r ~ °) and (r~. - - r v°) of distant residue pairs at the expense of worse fits for nearby residue pairs, thereby leading f to incorrect local minima. This need to confine the initial adjustments of the 6i's and ~i's to small values of a was also observed by Swenson et al. [7]. It may seem strange that the ¢;'s and fai's are variable to a significant extent, from one low-energy conformation to another (see Fig. 8), in spite of the need for determining these values very accurately. One of the reasons for the flexibility is that the planes of the peptide groups can orient to some extent without significant movement of the C~ atoms (procedure B). However, as can be seen in Fig. 9, the points do not lie exactly on the line because any given peptide plane cannot be rotated independently; i.e., these deviations from linearity imply that there is a fine adjustment of these orientations along the chain, and

405

the a m o u n t of each movement is determined so as to give the experimental conformation. These movements are restricted to a certain region of ¢, ~-space, a part of which is shown in Fig. 7. Because the atoms inside a protein are packed compactly, the tilt of a peptide plane from its original position would result in atomic overlaps. Therefore, the potentially overlapping atoms have to move to make such re-orientation possible; i.e., flexibility of the orientations of the planes of the peptide groups must be coupled, in a cooperative manner, to flexibility in the positions of side chain and backbone atoms. These results, indicating some flexibility in the native conformation, are compatible with those of McCammon et al. [15], who applied molecular dynamics to a study of bovine pancreatic trypsin inhibitor. Since the present computations were performed on an isolated molecule, this flexibility should be realizable in solution. In the crystal, however, from which the experimental coordinates were obtained, the movements of the atoms in each molecule would be somewhat restricted by interactions with the surrounding molceules. Therefore, the molecule would be expected to be less flexible in the crystal. It would thus be worthwhile to examine the vibrational spectra of the inhibitor, e.g., by Raman spectroscopy, in the crystal and in solution.

Acknowledgements This work was supported in part by a research grant from the Ministry of Education of Japan (143031) and by research grants from the National Science F o u n d a t i o n (PCM75-08691), and from the National Institute of General Medical Sciences of the National Institutes of Health, U.S. Public Health Service (GM-14312). References 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

L e v i t t , M. a n d L i f s o n , S. ( 1 9 6 9 ) J. Mol. Biol. 4 6 , 2 6 9 - - 2 7 9 N i s h i k a w a , K. a n d O o i , T. ( 1 9 7 2 ) J. P h y s . S o c . J a p a n 32, 1 3 3 8 - - 1 3 4 7 W a r m e , P . K . , G ~ , N., a n d S c h e r a g a , H . A . ( 1 9 7 2 ) J. C o m p u t . P h y s . , 9, 3 0 3 - - 3 1 7 W a r m e , P.K. a n d S c h e r a g a , H . A . ( 1 9 7 3 ) J° C o m p u t . P h y s . 12, 4 9 - - 6 4 W a r m e , P.K. a n d S c h e r a g a , H . A . ( 1 9 7 4 ) B i o c h e m i s t r y 13, 7 5 7 - - 7 6 7 H e r m a n s , J., J r , , a n d M c Q u e e n , J r . , J.E. ( 1 9 7 4 ) A c t s C r y s t a l l o g r . A 3 0 , 7 3 0 - - 7 3 9 S w e n s o n , M.K., B u r g e s s , A.W. a n d S c h e r a g a , H.A. ( 1 9 7 8 ) in F r o n t i e r s in P h y s i c o - C h e m i c a l B i o l o g y P a r i s ( 1 9 7 7 ) ( P u l l m a n , B., ed.), A c a d e m i c Press, N e w Y o r k , in p r e s s D e i s e n h o f e r , J. a n d S t e i g e m a n n , W. ( 1 9 7 5 ) A c t a C r y s t a l l o g r . , B 3 1 , 2 3 8 - - 2 5 0 M o m a n y , F . A . , M c G u i r e , R . F . , B u r g e s s , A.W. a n d S c h e r a g a , H . A . ( 1 9 7 5 ) J. P h y s . C h e m . , 79, 2 3 6 1 - 2381 O o i , T., S c o t t , R . A . , V a n d e r k o o i , G. a n d S c h e r a g a , H.A. ( 1 9 6 7 ) J. C h e m . P h y s . 4 6 , 4 4 1 0 - - 4 4 2 6 N i s h i k a w a , K., M o m a n y , F . A . a n d S c h e r a g a , H . A . ( 1 9 7 4 ) M a c r o m o l e c u l e s 7, 7 9 7 - - 8 0 6 D a v i d o n , W.C. ( 1 9 5 9 ) U.S. A E C R e s e a r c h a n d D e v e l o p m e n t R e p o r t , A N L - 5 9 9 0 O o b a t a k e , M. a n d O o i , T. ( 1 9 7 7 ) J. T h e o r . Biol. 6 7 , 5 6 7 - - 5 8 4 . B u r g e s s , A.W. a n d S c h e r a g a , H . A . ( 1 9 7 5 ) P r o c . Natl. A c a d . Sci. U.S. 72, 1 2 2 1 - - 1 2 2 5 M c C a m m o n , J . A . , Gelin, B.R. a n d K a r p l u s , M. ( 1 9 7 7 ) N a t u r e , 2 6 7 , 5 8 5 - - 5 9 0