J. Phys. Chem. 1993,97,260-266
260
Mean Field Theory as a Tool for Intramolecular Conformational Optimization. 2. Tests on the Homopolypeptides Decaglycine and Icosalanine K. A. Olszewski,+~* L. Piela,+** and H. A. Scberaga'v* Quantum Chemistry Laboratory, Department of Chemistry, Warsaw University, Pasteura 1, 02-093 Warsaw, Poland, and Baker Laboratory of Chemistry, Cornell University, Ithaca, New York 14853-1301 Received: September 18,I992
Tests of the self-consistent multitorsional field (SCMTF) method for global optimization of the conformational energy of decaglycine (10-residue polyglycine) and icosalanine (20-residue poly-L-alanine) show that it is applicable for global minimization of larger oligopeptides than considered heretofore. The method is based on the fact that the maximum of the square of the ground-state wave function is very often close to the global minimum of the potential energy. Using a mean field approximation, a set of coupled Schrainger equations is solved iteratively in a dihedral-angle space, each equation describing the changes of a single dihedral angle in the averaged field of the others. To obtain the effective averaged potential energy for each dihedral angle, a Monte Carlo averaging of the ECEPP/2 (empirical conformational energy program for peptides) potential is carried out over all other dihedral angles. Some minor changes of the algorithm have been introduced in order to avoid oscillations of the mean field solutions of the Schriidinger equation. The method was found to converge, when locating the global minimum of the potential energy of the two molecules being tested and to be quite insensitive to the initial choice of the starting conformation. Very rapid convergence was noted even when the initial conformation was far from the global minimum. In the close neighborhood of the global minimum, the convergence was slightly slower.
1. Introduction We previously published a self-consistent multitorsional-field (SCMTF) method for global optimization of the energy of polypeptides and tested it on terminally blocked alanine and Metenkaphalin.1 In this paper, we extend the method to larger molecules, the homopolypeptides decaglycine ( 10 residues) and icosalanine (20 residues). The basic idea of the SCMTF method, which was designed to solve the multiple-minimaproblem in the conformationalanalysis of polypeptides and proteins,2is that the ground-state solution of the SchrMinger equation gives information directly about the location of the global minimum, even for a potential with a very complex structure of local extrema. Since it is not possible to solve the many-body SchrBdingerequation exactly, we have made use of a mean field approximation. The possibility of using the SchrBdinger equation to surmount the multiple-minima problem has been pointed out by many a~thors,~-s although their algorithms could not be used directly in many-dimensional problems. On the other hand, the mean field approach in its classical version has been applied to molecular problems by Finkelstein and Reva6 and by Roitberg and ElberS7 Their approach does lead to some smoothingofthe potential energy surface but still leaves numerous minima on it, whereas the quantum mechanical treatment gives global information about the approximate location of the lowest minimum, when using a ground-state wave function. Bond lengths and bond angles are maintained fixed throughout the whole procedure, as in the ECEPP (empirical conformational energy program for peptides) algorithm.8-IO Internal rotations (variations of dihedral angles) in the molecule under study are described by using an appropriate Hamiltonian operator. When applying the mean field approximationin order to solve the related Schrddinger equation, one has to solve a set of coupled onedimensional equations,' the variable in each equation being a Warsaw University. Cornell University. * To whom correspondence should be addressed.
+
single dihedral angle. The procedure is repeated iteratively until self-consistency is achieved for every dihedral angle in the molecule. Homopolypeptideshave been the subject of numerous studies, e.g., global minimization of decaglycine has been examined by Meirovitch et al.I1 and by Ripoll et a1.12 A 19-residue poly-^alanine chain served as a test molecule for the SCEFI3 and EDMC14methods. Many shorter fragmentsof poly-L-alanine,ISJ6 as well as icosalanine, were studied by means of molecular dynamics, e.g., in an analysis of the helix-coil transition,17 in a calculation of the free energy of the interaction of a helix dipole with charged terminal residues,]*and in an analysis of collective motion in an a-helix.l9 In this paper, we have introduced some minor changes in the SCMTFalgorithml toavoid oscillationsof themean field solutions of the SchrBdinger equation. Two additional sets of vectors &Id and Parthat defineconformationsare proposed in order to provide a more extensive test of convergence of the single dihedral angle probability density distributions. These vectors, as well as the ones introduced earlier' (Pin,Bmaxand P ) ,are also used, to force the convergence by including them in the set of conformations generated with the related probability density distributions.
2. Self-consistent Multitorsionel Method In this section, we sumarize the basic steps of the SCMTF method together with some recent improvements, introduced in the original algorithm.' The SchrBdingerequation for the motion of the nuclei is given by
H* = E+
(1)
where a circumflex over any symbol denotes the operator of the corresponding quantity. In the Cartesian coordinate representation, the Hamiltonian operator k is defined by
h2 &=-c-An
+
fj
n=12mn where An is the Laplacian operator for the nth nucleus, and vis 0 1993 American Chemical Society
The Journal of Physical Chemistry, Vol. 97, No. 1, 1993 261
Intramolecular Conformational Optimization. 2
TABLE I: SCMTF First Macroiteration for Decaelwine Startine from the ALLC Conformation' _ _ _ ~ ~~
~
iteration
GLY I
1
C C
2 3 4 a
A
A
GLYz
GLY3
GLYi
GLYs
GLY6
GLY7
GLYa
GLY9
GLYio
C A A A
C
C
C
A
A A A
A A A
C C
C
C
C
C
A A A
A A A
A A A
A
These conformational regions pertain only to
PId. The
A A
A A
A A
starting conformation is considered as a zeroth iteration.
TABLE 11: SCMTF First Macroiteration for haglycine Starting from the ALLD Conformation' iteration
GLY I C
GLY2
GLYj
GLYd
GLYs
GLY6
GLY7
GLYa
GLYq
GLYio
D
D
D
D
D
D
D
2 3
C
C
C
C
A
A A A A
A A A A A
C A A
5 6 7 8
A A A A A A
C A A A A A A
D C
A
C A
4
C A A A A A A
D C
1
0
A
A A A A A A
A A A
A A
A A A A
A
D
H
C
A A A A
C C A
GLYlo
A
These conformational regions pertain only to PId.The starting conformation is considered as a zeroth iteration.
TABLE 111: SCMTF First Macroiteration for Decaglycine Starting from the ALLF Conformation' iteration
GLY I
GLY2
GLY,
GLY4
GLYs
GLY6
GLY7
GLYs
GLY9
1
C
F
F D
F D
F
D
F D
F
C
F D
F
2 3 4-5 6 7 8
F D
D
D
D
A A A A A
A A A A A
A A A
A A A A A
A A A A A
A A A A
A A A A A
A A A A A
A A A A A
C G* C A A
GLYo
GLYln
G
A A
A
These conformational regions pertain only to PId. The starting conformation is considered as a zeroth iteration.
TABLE I V SCMTF First Macroiteration for Decaglycine Starting from the ALLC Conformation' iteration
GLY I
GLY2
GLY3
GLY4
1 2 3
G
G
G
A* A* A*
A* A* A*
A* A* A*
4 a
GLYs
GLYh
GLY7
G
D
G
G
A* A* A*
G A* A*
G
A* A* A*
A* A* A*
A* A* A*
F*
C*
A* A*
G G
A*
GLYR
These conformational regions pertain only to Bold. The starting conformation is considered as a zeroth iteration.
TABLE V iteration
SCMTF First Macroiteration for Decaglycine Starting from the ALLE Conformation' GLY2
GLY3
GLY4
GLYs
GLY6
GLY7
GLYa
GLYq
GLYio
1
E
D
E
2 3
D D
G G
E D D D* D*
E D
E D
E E
D
D
C* C*
E D D F* F*
E D D
4
C C A
E D D
C* C*
A A A* A*
C* C* A A C
E D* D*
A A A A A A A A
C C C* C* C C
5 6 7-1 3 14 15-1 8 19-22 23-27 28 29
GLY I
A A A A
A A A A
A A A A A A
A A A
A A A A A A A A A A
C* C* A A A A
A A A A
A* A A A
D* D*
B*
C* C* A
A A A
A A A
E C C A A
A
D* C C C
D* A A
These conformational regions pertain only to &'Id. The starting conformation is considered as a zeroth iteration.
the potential energy operator. Bond lengths and bond angles are kept fixed so that the only remaining variables are the dihedral angles 8 = (el, ..., O N ) . To solve the Schrainger equation in dihedral-angle space (Le., the space of molecular conformations with fixed bond lengths and bond angles), we have to transform the Hamiltonian in eq 1 appropriately. Assuming that the Hamiltonian may be approximated by diagonal terms only,l the resulting operator in the 8 representation may be written as (3)
where I , is an averaged moment of inertia.' Assuming that the solution of the corresponding eigenproblem is approximated by
the Hartree-like product of the normalized one-angle functions 4&), i.e. N
we,,e2,...,eN)= n 4 i ( e i )
(4)
i= I
we obtain a set of N coupled one-dimensional equations,' viz.
hi@,? =
i = 1,
..., N
(5)
where 4f1stands for the eigenfunction corresponding to the kith excited state of the ith dihedral angle and k, = 0 is the ground state. The Hamiltonian for the single dihedral angle is given by
262
The Journal of Physical Chemistry, Vol. 97, No. 1, 1993
Olszewski et al.
f
L Figure 1. (a) Starting conformation for decaglycine in the ALLE first macroiteration. (b-g) Lowest energy conformation for the Znd, 4th, 9th, 14th, 19th, and 23rd iteration, respectively, in the ALLE first macroiteration for decaglycine. (h) Global minimum conformation of decaglycine.
The Journal of Physical Chemistry, Vol. 97, No. I, 1993 263
Intramolecular Conformational Optimization. 2
TABLE VI: SCMTF First Macroiteration for Decaglycine Starting from the ALLH Conformation' iteration
GLY i
GLYz
GLYi
GLYr
GLY5
GLYn
GLY7
GLYn
GLYo
A
H
D*
A A A A A A A A A A A
A A A A C A A A A A A A
GLYin
~~
1 2 3 4-1 5 16 17 18-19 20 21 2242 43
B
A
A
A
C
A A*
C
C C C
C* C* C*
C C C
B
A*
A A A A A A A A
C A*
C C C
44 0
A
C
F
C
C*
A
A
A* A*
C
E*
C A A
A A A
A A A A A A
A A A A A
H D* D*
A A A A A A A A A A
A A A A A A A A A
H A
C
D C C C C C
C A
A
These conformational regions pertain only to PId.The starting conformation is considered as a zeroth iteration.
TABLE VII: SCMTF First Macroiteration for Icosalanine Starting from the ALLC Conformation' iteration ALA1 ALA2 ALA,
I 2
C C A A
3 4 0
A A A A
ALA4 ALAS ALA6 ALA7 ALA8 ALA9 ALAIO ALAII ALA12 ALA13 ALA14 ALAis ALA16 ALA17 ALA18 ALA19 ALA20
C A A A
C A A A
C A A A
C A A A
C A A A
C A A A
C A A A
C A A A
C A A A
C A A A
C A A A
C A A A
C A A A
C A A A
C A A A
C A A A
C C A A
C C C C
These conformational regions pertain only to PId. The starting conformation is considered as a zeroth iteration.
TABLE VIII: SCMTF First Macroiteration for Icosalanine Starting from the ALLD Conformation' ~~
iteration ALA1 ALA2 ALA, D A A
I 2 3
D A A
ALA4 ALA,
C A A
D A A
~
ALA6 ALA7 ALA8 ALA9 ALAIO ALAII ALA12 ALA13 ALA14 ALA13 ALA16 ALA17 ALA18 ALA19 ALA20
D A A
D A A
D A A
D A A
D A A
D A A
D A A
D A A
D A A
D A A
D A A
D A A
D A A
D A A
D A A
D A A
These conformational regions pertain only to BOid. The starting conformation is considered as a zeroth iteration.
TABLE I X SCMTF First Macroiteration for Icosalanine Startinn from the ALLF Conformation' ~~~
~
~
iteration ALA1 ALA2 ALA, 1 2 3 4 5 6 16 17
C C C C C -
1 A A
F F C C A 5 A A
F F F D A A
A A A
~
~
~
~
_
_
_
_
~
ALA4 ALAS ALA6 ALA7 ALA8 ALA9 ALAIO ALA11 ALA12 ALA13 ALA14 ALA15 ALA16 ALA17 ALA18 ALA19 ALA20 F F F C A A A A
F F F C A A A A
F F F C A A
F F F C A A
A A
A A A
F F F C A A A A
F F F C A A A A
F F F C A A
F F F C A A
A A
F F F C A A
A A
F F F C A A
A A
F F F C A A
A A
A A
F F F C A A A A
F F F C A A A A
F F F C A A A A
F F F C A A A A
F F F A A A A A
F F F C C C A A
These conformational regions pertain only to BOid. The starting conformation is considered as a zeroth iteration.
cff(Oj)
where the effective potential depends on the mean field created by averaging over the other dihedral angles, e/(/ # i ) , according to the probability density distribution p p = I&*, defined as the square of the absolute value of the ground state solutions of eq 5 . To calculate a Monte Carlo procedure has been used, i.e.
cff,
where the summation extends over Mc locally minimized trial conformations in the (N - 1)-dimensional space of all dihedral angles 81 except Bi. The solution of eqs 5 gives a new set of 4: and, therefore, a new set of pp. The procedure is repeated iteratively until self-consistency of the p p distributions is reached. To provide for a more detailed exploration of the dihedralangle space, a larger number ( W , instead of Mc) of trial conformationswere generated, and their energies were minmized locally with the secant unconstrained minimization solver (SUMSL) algorithm.Z0 Then, only the Mc lowest energy conformations of this set were chosen to approximate The W trial conformations were generated by means of the von Neumann algorithm21 using pp as the probability density distribution. The values of M Eand hP were usually in the ranges
c".
of 1-5 and 15-75, respectively. The ECEPP/2 potentialg-10 was used for v(0). The differential equations ( 5 ) were solved by using the renormalized Numerov-Cooley ~ r o c e d u r e ~ with ~ J 3 a step size equal to 2 r / K g ( K g = 640). Then, the probability densities pp were constructed from the solutions 4; of eq 5 and used to generate another set of the W trial conformations, and so forth. Actually, we constructed a temperature dependent probability density distribution pi( by averaging over the ki excited states using a Boltzmann weighting factor that depends on thedifference between the energy of the kith excited state and the ground-state of each single dihedral angle (cf. ref 1). This makes it possible to extend the searches of the conformational space to conformations related to the excited states given by the SchrBdinger equation. Five different criteria (three of them have been defined in the previous work1) for convergence of the probability density distributions in the consecutive steps of the iterative procedure were used in this study. The criteria are related to the following sets of dihedral angles, respectively: (i) Omin, corresponding to the minimum value of the mean field potential, (ii) Omax, corresponding to the maximum value of the probability density, (iii) ea", the mean value of the probability density distribution pi(T;B), i = 1, ..., N , (iv) @id, which corresponds to the lowest energy conformation found previously, and (v) Par, the components of which correspond to the lowest value of Fff'found throughout the whole procedure, up to the most recent iteration. The Par set has been introduced to damposcillations of the mean field solutions
264
Olszewski et al.
The Journal of Physical Chemistry, Vol. 97, No. 1, 1993
S
b
C
a
d Figure 2. (a) ALLE starting conformation for the first macroiteration for icosalanine. (b, c) Lowest energy conformation from the 5th and 10th iteration, respectively, in the ALLE first macroiteration for icosalanine. (d) Global minimum conformation for icosalanine.
that appear when the conformation of the macromolecule is close to that of the global minimum. We often encountered a situation in which, in the subsequent iteration, almost all except a few dihedral angles had values close to those of the global minimum, whereas the values of the remaining dihedral angles tended to oscillate, e.g., when one dihedral angle came close to the global
minimum, the other moved in the opposite direction. To solve this problem, we added the above sets of dihedral-angles, @mi", emax, @av, @Id, and to the set of ML trial conformations. This turned out to accelerate convergence and, when oscillations were detected, we restarted the calculation by choosing P a r as the starting conformation. Another way to solve the oscillation
The Journal of Physical Chemistry, Vol. 97, No. 1, 1993 265
Intramolecular Conformational Optimization. 2
TABLE X
SCMTF First Macroiteration for Icosalanine Starting from the ALLE Conformation.
~~
~
iteration ALA1 ALA2 ALA] ALA4 ALAS ALA6 ALA7 ALA8 ALA9 ALAIO ALAII A L A u ALAII
E
I 2 5
4
6 7 a-9 1 26
27 0
E A F A
c 0
D
A A
E D
E G
E
E E
E E
E E
E E
A * C * C * A A A A D G B A A A A A * C A A A A A A* c A A A A A 2 5 C A ' A A A A B A A A A A A B A A A A A A
E E
E E
A
E
A A
A A A
A A A A A
A A
A A A
E E
A A A
A A
A A A
E E
E E
E E
E E
E E
E E
E E
A
A
A A A A A A A
A A A A A A A
A A A A A A A
A A A A A A A
A A A A A A A
A A A A A A A
A
A
A A A
ALA14 ALAIJ ALA16 ALA17 ALA18 ALA19 ALA20
E E A A A A
A A A A
A A
A A A
These conformational regions pertain only to BOid. The starting conformation is considered as a zeroth iteration.
problem is to increase the temperature temporarily. This allows the molecule to escape from the oscillatingstate and usually yields smoother (but slower) convergence. Each separate run with a new starting conformation determined by the previous set of successive iterations, but with different values of the parameters (M, Mc,T,etc.) is called a macroiteration. We terminated the procedure when the probability density had converged, Le., when every one of the sets of dihedral angles defined above converged to the same pattern of Zimmermann et al.24 This means that, for each residue, any pair of dihedral angles 4, (irrespective of whether they belong to Omin, Oav, Omax, &Id, or Par) corresponds to a Zimmermann et al. region of the 4, J-map, which is thesameat all timesafter theprobabilitydensity convergence is reached. If, instead of convergence, oscillations of the probability density distributions were observed, the next macroiteration would be started, using BYar as the starting conformation. This procedure is related to the one that is widely used in mean field calculations of electronicstructures of molecules (SCF), in which density matrices from preceding iterations are mixed. The convergence of the probability densities was not our only criterion that thecalculation wascomplete. We alsochecked whether the energy of the &Id conformation changed by less than 0.2 kcal/mol with respect to the previous iteration; only then were iterations terminated. Situations, in which the energy converged but the probability densities did not, were treated as if the solutions oscillated.
+
3. Results and Discussion
3.1. -glycine. The decaglycine molecule was constructed by adding acetyl and methylamidegroups to theN and C termini, respectively. All 30 backbone dihedral angles, including 4% Ps, and w's were allowed to vary. The starting conformations for global minimization were generated with a's fixed at 180'. Six global minimizations were carried out, starting from the following arbitrarily chosen regular conformations,described by the notation of Zimmermann et al.z4 ALLX means an XXX ...set of conformations for all residues in the sequence,where X is a Zimmermann et al. region of the 4, J. map:
+ = 80°
1.
ALLC, Le., 4 = -80' and
2.
ALLD, i.e., 4 = - 1 5 0 O and $ = 80'
3.
ALLE, Le., 4 = -150' and
+ = 150'
4.
ALLF, Le., 4 = -8OO and J, = 170'
5.
ALLG, i.e., 4 = -150' and J, = -50'
ALLH, Le., 4 = ' 0 and $ = ' 0 For each of the above starting conformations (except ALLG), we wereabletolocatetheglobal minimum (aright-handed a-helix with an energy of -2 1.8 kcal/mol) as reported previously by Ripoll et a1.12 In the case of ALLG, an equivalent left-handed a-helix (ALLA*) was obtained (with the same energy). In each case, the computations were started at a relatively high temperature T 5 30 000 K to allow a full search of conformational space to
6.
E
be made. In the cases of ALLC, ALLD, and ALLF, convergence was very rapid. The correct conformationalpattern, Le., ALLA, was achieved at the fourth, seventh and seventh iteration, respectively, according to all the criteria, Omin, Omax, Oav, &Id, Par (Tables 1-111 show only the Bald conformational regions in the first few iterations). It should be noted that, in the ALLC calculations, the ALLA pattern was located by the &Id criterion already in the third iteration. For the ALLG starting conformation, the convergence was also very rapid, but led to the A*A*A*A*A*A*A*A*A*G pattern in seven iterations (see Table IV) and achieved the global minimum ALLA* pattern in the next macroiteration. As with the ALLC calculations, this low-energy pattern was located by the &Id criterion already in the third iteration. For the ALLE starting conformation,convergence was slightly slower, leading to the correct a-helical pattern in 30 iterations (see Table V). In this case, the trajectory of stable states is shown in Figure la-h. In the case of the ALLH starting conformation, convergencewas the slowest,leading, nevertheless, to the global minimum pattern in 43 iterations (see Table VI). Since one iteration took approximately 2.5 min of CPU time on one processor of an IBM 3090-600Ecomputer, the correct a-helical pattern was obtained in the best case after 10 min and in the worst case after almost 2 h. After the probability density converged, one additional macroiteration was sufficient to reach the global minimum in approximately 1 h. The SCMTF method is much slower near the global minimum because of correlation effects that cannot be taken into account in the mean field approach; i.e., the mean field approach approximates the potential energy for a single dihedral angle by averaging over all other dihedral angles. This approximation breaks down when the dihedral angles are not independent of each other. For a method designed to accelerate the convergence, see the accompanying paper.25 The total time to locate the global minimum (as opposed to the global minimum pattern), therefore, varied from 1 to 3 h, which should be compared with 3.7-8.5 h, required by the EDMC procedure14to locate the global minimum12for the same molecule. 3.2. Icosalanine. In the case of icosalanine, acetyl and methylamide terminal groups were also used. We allowed only the dihedral angles 4 and to vary and kept the dihedral angles w and x fixed at 180' and 60°, respectively; therefore, a total of 40 dihedral angles were varied while the other 40 were fixed. We carried out four global minimizations,starting from conformations that are related to extended structures, Le., ALLC, ALLD, ALLE, ALLF. For the ALLC starting conformation, the first macroiteration converged after three iterations to the pattern A& (A AC) (see Table VII), and the energy converged after ten iterations to -40.0 kcal/mol. For the ALLD starting conformation, convergence of the probability densities was even faster, leading to the Azo (correct global minimum pattern) in the second iteration (Table VIII). Energy convergence was slightly slower, yielding a value of -44.7 kcal/mol in 20 iterations. On the other hand, in the ALLF case, although the calculations leading to the A20conformation was considered as having converged after only 17 iterations (Table IX, with all density distribution criteria fulfilled),energy convergencewas much better, giving -42.2 kcal/ mol in this iteration and -43.4 kcal/mol at the 22nd iteration. The initial macroiteration for the ALLE required a longer time,
+
...
266
The Journal of Physical Chemistry, Vol. 97, No. 1, 1993
because oscillations of the solutions appeared. Although Omin, and PId converged quickly (within 10 iterations) to the structure CA*Als(with energy-12.6), Omax and Oav did not. This macroiteration was restarted twice, each time starting from updated values of P a r before it finally converged to the pattern ABAI8 (-28.0 kcal/mol). (The B region corresponds to qj = -105O and = 81'). The ALLE starting conformation for the first macroiteration, the first conformation with the long a-helix, the most persistently occurring conformation, and the globalminimum conformation,are depicted in Figure 2 a 4 , respectively (see Table X for the conformational regions of eld). As in the case of decaglycine, restarting from the conformation obtained in the first macroiteration in all cases leads to the global minimum (-47.9 kcal/mol) in 1-2 h. OneiterationoftheSCMTF method took approximately 4-1 2 min; therefore, the global minimizationfor icosalanine took from 2 h (for ALLC and ALLD) to as much as 6-7 h (for ALLE) on one processor of the IBM 3090 computer. This may be compared with the 27 h (average) required by EDMC14on the same machine (but using four to six processors) to minimize a 19-residue poly-L-alanine chain. In ref 14, however, all dihedral angles were allowed to vary (twice as many as in the present work). Comparison with the EDMC method shows that the SCMTF method can be considered as a valuable tool for global minimization of the conformational energy of polypeptides and proteins, even larger than 20-residues, since, for medium-size molecules, the method still scales approximately linearly with the number ofvariables asdiscussed in ref 1 (not counting thecost of evaluating the energy). Although, for larger polypeptides and proteins, correlation effects can be so strong that the mean field approximation may breakdown, it is still possible to develop an extension of the method by including some important correlation effects, e.g., the total wave function may be taken as a product of functions that depend on pairs of 4 and $. Work on this is in progress in our laboratories. &a',
+
Acknowledgment. This work was supported by research grants from the National Institutes of Health (GM-14312), from the National Science Foundation (DMB 90-15815),and from the
Olszewski et al. National Research Council (KBN) of Poland. The computations werecarriedout at thecornell Supercomputer Facility, a resource of the Cornel1 Center for Theory and Simulation in Science and Engineering, which receives major funding from the National Science Foundation and the IBM Corp., with additional support from New York State and members of its Corporate Research Institute.
References and Notes (1) Olszewski, K. A.; Piela, L.; Scheraga, H. A. J . Phys. Chem. 1992, 96, 4672. (2) Gibson, K. D.;Scheraga, H. A. InSarma, M. H.,Sarma, R. H., Eds.; Siruciure and Expression From Proteins i o Ribosomes, Adenine Press: Guilderland, NY, 1988; Vol. 1, p 67. (3) Ebeling, W.; Engel, A,; Esser, B.; Feistel, R. J . Srar. Phys. 1984,37, 369. (4) Rujdn, P. 2.Phys. E-Condens. Marrer 1988,73, 391, (5) Somorjai, R. L. J . Phys. Chem. 1991, 95,4141. (6) Finkelshtein, A. V.; Reva, B. A. Biofirika 1990,35, 402. (7) . . Roitbern. A.; Elber. R. J . Chem. Phvs. 1991. 95.9277. (8) Momany, F. A.; McGuire, R. F.; Burgess, A. W.; Scheraga, H. A. J . Phys. Chem. 1975,79, 2361. (9) Nbmethy, G.; Pottle, M. S.;Scheraga, H. A. J . Phys. Chem. 1983, 87, 1883. (10) Sippl, M. J.; Nimethy, G.; Scheraga, H. A. J. Phys. Chem. 1984, 88, 6231. (1 1) Meirovitch, H.; Vasquez, M.; Scheraga, H. A. J . Chem. Phys. 1990, 92, 1248. (12) Ripoll, D. R.; Vdsquez, M. J.; Scheraga, H. A. Biopolymers 1991, 31, 319. (13) Piela, L.;Scheraga, H. A. Biopolymers 1987,26,S33. (14) Ripoll, D. R.; Scheraga, H. A. Biopolymers 1988,27, 1283. (15) Brooks, B. R. Chem. Scr. 1989,29A, 165. (16) Kitao, A,; GO, N . J . Compur. Chem. 1991,12, 359. (17) Daggett, V.; Kollman, P. A.; Kuntz, I. D. Biopolymers 1991,31, 1 1 15.
(18) Daggett, V.; Kollman, P. A.; Kuntz, I D. Chem. Scr. 1989,29A, 205. (19) Pleiss, J.; JBhnig, F. Biophys. J . 1991,59, 795. (20) Gay, D. M. ACM Trans. Math. Software 1983,9, 503. (21) von Neumann, J. Nor. Bur. Stand. Appl. Math. Ser. 1951,12,36. (22) Cooley, J. W. Marh. Compui. 1961,15, 363. (23) Johnson, B. R. J . Chem. Phys. 1977,67,4086. (24) Zimmermann, S. S.; Pottle, M. S.;Nbmethy, G.; Scheraga, H. A. Macromolecules 1977,10, 1 . (25) Olszewski, K. A.; Piela, L.; Scheraga, H. A. J . Phys. Chem.,following paper in this issue.
