Protein Folding Simulation With Genetic Algorithmand Supersecondary Structure ConstraintsYan Cui,1 Run Sheng Chen,1* and Wing Hung Wong2*1Laboratory of Protein Engineering, Institute of Biophysics, The Chinese Academy of Sciences, Beijing, China2Program in Statistics, University of California, Los Angeles

ABSTRACT We describe an algorithm tocompute native structures of proteins fromtheir primary sequences. The novel aspects ofthis method are: 1) The hydrophobic potentialwas set to be proportional to the nonpolarsolvent accessible surface. To make computa-tion feasible, we developed a new algorithm tocompute the solvent accessible surface areasrapidly. 2) The supersecondary structures ofeach protein were predicted and used as re-straints during the conformation searchingprocesses. This algorithm was applied to fiveproteins. The overall fold of these proteins canbe computed from their sequences, with devia-tions from crystal structures of 1.48–4.48 Å forCa atoms. Proteins 31:247–257, 1998.r 1998 Wiley-Liss, Inc.

Key words: protein structure prediction; su-persecondary structure; genetic al-gorithm; solvent accessible sur-face area; hydrophobic potential

INTRODUCTION

The success of protein structure prediction de-pends on our knowledge of protein structure, interac-tions, and folding mechanism. At present, there areonly a few reliable conclusions: 1) Native structuresof proteins are compact and have well-packed coreswhich are highly enriched in hydrophobic resi-dues.1,2 2) Hydrophobic interaction is the drivingforce for protein folding.3,4 Native structures of pro-teins have minimal solvent-exposed nonpolar sur-face areas.5 3) Globular proteins are organized as astructural hierarchy,6,7 i.e., secondary structure, su-persecondary structure, tertiary structure, and qua-ternary structure. 4) The proteins employ foldingpathways to avoid extensively searching the wholeconformation space. They fold by hierarchic conden-sation.7 The folding pathway is suggested to besecondary structures, supersecondary structure, do-mains, and ultimately whole protein monomers.8–10

We developed a protein structure prediction algo-rithm based on this crude knowledge. First, thesupersecondary structures were predicted with anartificial neural network method.11 Then we searchedfor low-energy structures in the conformation space

under the constraints suggested by the supersecond-ary structures. The energy function is very simpleand has only two terms—a hydrophobic interactionterm, and a van der Waals interaction term. We useda genetic algorithm to search the conformation space.According to our design of this model, hydrophobicinteractions drive the peptide chain to fold; van derWaals forces are used to reject the incorrect compactstructures during the hydrophobic collapse. Only thestructures in which there is minimal conflict be-tween the hydrophobic interactions and the van derWaals interactions can survive and become domi-nant during the competition and selection process ofthe genetic algorithm. This algorithm was applied tofive proteins. The overall fold of these proteins werecomputed from their sequences by this algorithm,with deviations from crystal structures of 1.48–4.48Å for Ca atoms (Table I).

There have been several important advances incomputer algorithms intended to predict native 3-dimensional structures of globular proteins fromtheir amino acid sequences using simple energyfunctions.12–14 The novel aspects of our method are:1) We set the hydrophobic potential to be propor-tional to the nonpolar solvent-accessible surface area(NSASA). Although this is a reasonable way ofincluding the hydrophobic effects, it was not used inprevious protein structure prediction algorithms.12–14

The computation of solvent-accessible surface areawas very time-consuming. It would be prohibitivelyslow to incorporate it into protein structure predic-tion schemes which need to sample a large number ofstructures. We developed a new algorithm to calcu-late the solvent-accessible surface area rapidly. Withthis method we can compute the NSASA of everysampled structure in an acceptable time. 2) Thesupersecondary structures were predicted and usedto derive soft constraints for the conformation searchprocess. The identification of such structures repre-

Contract grant sponsor: Chinese National Scientific Founda-tion; Contract grant number: 39392900; Contract grant spon-sors: UCLA, the Institute of Mathematical Sciences at theChinese University of Hong Kong.

*Correspondence to: Run Sheng Chen, Institute of Biophys-ics, The Chinese Academy of Sciences, Beijing 100101, P.R.China; or Wing Hung Wong, Program in Statistics, UCLA, 8142Math Sciences Building, Los Angeles, CA 90095.

Received 1 August 1997; Accepted 13 November 1997

PROTEINS: Structure, Function, and Genetics 31:247–257 (1998)

r 1998 WILEY-LISS, INC.

sents important progress along the folding pathway.The conformation space is greatly reduced with theseconstraints.

METHODSReview of SupersecondaryStructure Prediction

Supersecondary structure is defined as the combi-nation of two secondary structural elements with ashort connecting peptide between one to five resi-dues in length. A short connecting peptide can have alarge number of conformations. They play an impor-tant role in defining protein structures. A connectingpeptide usually changes the trend of the proteinbackbones so as to form an antiparallel turn, avertical corner, a twist, or just a slight bend in apeptide chain.11 The conformations of the residues inthe short connecting peptides are classified into fivemajor types, namely, a, b, e, 1, or t,16 each repre-sented by a region on the f-c map, respectively (seeFig. 1b in Sun and Jiang16). Supersecondary struc-tures are classified according to their componentsecondary structural elements, the length of theconnecting peptide, and the type of residues in theconnecting peptide. In a survey of 240 proteins,16 itwas found that there are 34 types of supersecondarystructures which occur more than 5 times. Of these34 types there are 11 types of supersecondary struc-tures which occur more than 25 times. These 11types of supersecondary structures are called fre-quently occurring supersecondary structures. The34 types of supersecondary structures occurred alto-gether 766 times, among which the 11 frequentlyoccurring supersecondary structures occurred 568times. This result shows that about 75% of theshort-connecting peptides which occurred more than5 times belong to the 11 types of frequently occurringsupersecondary structures. Sun et al.11 developed anartificial neural network method to predict the 11

frequently occurring supersecondary structure:H-b-H, H-t-H, H-bb-H, H-ll-E, E-aa-E, E-ea-E,H-lbb-H, H-lba-E, E-aal-E, E-aaal-E, and H-l-E,where ‘‘H’’ and ‘‘E’’ represent a-helix and b-strand,respectively. Each of these corresponds to a well-defined 3-dimensional motif (see Fig. 4 in Sun andJiang16). The method of Sun et al. was used forsupersecondary structure prediction. The predictedsupersecondary structure will not be rigidly imposedon the conformation. Rather, it will serve to definesuitable constraints (Table II) on affected torsionangles. In this way, the size of the conformationspace is greatly reduced. However, under these con-straints the conformation is still highly flexible andthe structure can take on various shapes that arevastly different from the native shape.

Peptide Chain Representation

Amino acids are represented at the united-atom level.Bond lengths and bond angles are always fixed at theirideal values (according to Biosym’s residue library). Allthe peptide bond dihedral angles are fixed in the trans(v 5 180°) conformation. The degrees of freedom in thisreduced representation are the backbone and sidechaintorsion angles f, c, and x (some residues have morethan one sidechain torsion angle).

Potential Energy Function

Our potential function has two terms: a hydropho-bic interaction and a van der Waals interaction term,

Etotal 5 EHH 1 Evdw.

We define polar and nonpolar united atoms bytheir heavy atoms: carbon and sulphur are nonpolar;nitrogen and oxygen are polar. The hydrophobicpotential is proportional to the solvent-accessiblesurface of nonpolar atoms,

EHH 5 Ch · NSASA

where Ch is a constant and is set to 0.031 andNSASA (in units of Å2) is the nonpolar solventaccessible surface area.

We use a cut-off of 8 Å for van der Waals interactions,

Evdw 5 CV · o fvdw 1 rij

Ri 1 Rj2

where Cv is a constant that is set to 0.1, rij is thedistance between atom i and atom j, Ri and Rj are thevan der Waals radii of atom i and atom j, and thesummation is over all pairs of atoms with rij , 8Å.The function fvdw is a van der Waals potential with atapering-off at short distances (Fig. 3):

fvdw(r ) 5 51

r12 22

r6 (r . 0.8)

C (r # 0.8).

TABLE I. Summary of the Computed Proteins

Protein Ehp Evdw Ehb Etotal DME (Å)

1ROPCrystal 79.0 230.3 0.0 48.7 0.0Ga 76.9 231.7 0.0 45.2 1.481UTGCrystal 104.5 228.3 0.0 76.2 0.0Ga 88.6 237.1 0.0 51.5 3.471CRN*Crystal 63.2 6.8 215.0 41.4 0.0Ga 64.6 25.6 240.0 50.2 2.731R69Crystal 64.2 243.1 0.0 21.1 0.0Ga 59.3 220.3 0.0 39.0 4.481CTFCrystal 69.1 224.1 245.0 0.0 0.0Ga 84.6 41.4 270.0 56.0 4.00

*The native disulphide bond constraints were not used in thesimulation.

248 Y. CUI ET AL.

This tapered van der Waals potential will notreject a structure with low hydrophobic energy foronly a few steric conflicts.

A backbone-to-backbone hydrogen bonding term isadded for b-sheet structures. The distance betweenN···O pair should be no more than 3.5 Å and theout-of-plane dihedral angle between the oxygen andthe peptide plane of the nitrogen (C–N–Ca) shouldnot exceed 40°. If these criteria are satisfied, anH-bond energy of 25.0 units is assigned.

Two-Level Lattice Method forSolvent-Accessible Surface Area Calculation

The hydrophobic effect is considered a principalforce in the formation of the native protein struc-tures.3,4 A reasonable way of including the hydropho-bic effect is to set the hydrophobic potential asproportional to the solvent-accessible surface area(SASA) of nonpolar atoms. Many algorithms forcalculation of the SASA and molecule surface havebeen developed during the past 25 years.17–40 Severalfast numerical algorithms have been described inrecent years.38,39 However, calculation of the SASAhas not been used in the folding simulation of wholeprotein molecules, in which a large number of SASAneed to be calculated. In this work, we used a newalgorithm to calculate the nonpolar solvent-acces-sible surface areas of about 300,000 conformationsfor each protein. Our method is based on the cubealgorithm.23–27 The whole molecule was put in arectangular box. We introduced two levels of cubiclattice in the box, one was the coarse lattice (edgelength 0.5 Å), the other was the fine lattice (edgelength 0.1 Å), by which each cube in the coarse latticewas divided into 125 sub-cubes. (The cube in the finelattice is called a ‘‘sub-cube,’’ which differs from thecube in the coarse lattice.) Each of the 125 sub-cubeswas assigned a number from 0 to 124. Water mol-ecules were imitated by balls with a radius of 1.4 Å.Before calculating the SASA, we built a library of the

cubic decomposition of the accessible surfaces andinner parts of each kind of atoms. We selected thecenter of a particular cube to be the origin. Then weput a ‘‘hydrated’’ sphere with a radius equal to thesum of the van der Waals radii of the atom and awater molecule around the atom at the center of oneof the sub-cubes in this cube. If the center of a cubewas covered by the sphere, it was marked by ‘‘V’’(V-cube). Then every V-cube was checked. If a V-cubewas on the surface, this meant that at least one of itssix neighboring cubes was not a V-cube, then it wasmarked by ‘‘S’’ (S-cube). If a V-cube was not on thesurface, it was marked by ‘‘I’’ (I-cube). Thus, we havetwo kinds of V-cubes, i.e., S-cubes, which were on thesurface of the ‘‘hydrated’’ sphere, and I-cubes, whichwere at the inner part of the ‘‘hydrated’’ sphere. Thetotality of S-cubes was an approximation of thesurface of the ‘‘hydrated’’ sphere. The totality ofI-cubes was the cubic decomposition of the inner partof the ‘‘hydrated’’ sphere. The positions (lattice coor-dinates relative to the origin) of the S-cubes andI-cubes were recorded in the library. In such a way,the cubic decomposition of the ‘‘hydrated’’ sphere(include the surface and the inner part) whose centerwas at each of the 125 sub-cubes in the selected cubewas recorded in the library according to the order ofthe sub-cube number.

With this library, the SASA can be calculatedrapidly:

a) A protein molecule was put into the two-levellattice. For each atom, we determined which cubeand sub-cube it was in. In other words, theCartesian coordinates were transferred to latticecoordinates.

b) If an atom was in the cube at (lx, ly, lz), whichwere the lattice coordinates of the center of thecube, and in the sub-cube whose number is n,then we look for the record of the cubic decomposi-tion of the ‘‘hydrated’’ sphere surface of this atom.The record was for the cube at the origin, so wetranslated them to (lx, ly, lz). In such a way, weput the cubic approximation of the ‘‘hydrated’’sphere surface of every atom in the lattice. Thecubes that were occupied by the ‘‘hydrated’’ spheresurface were marked by ‘‘S.’’

c) In the same way, we put the cubic approximationof the inner part of each atom on the lattice. Thesecubes were marked by ‘‘I.’’ So, the S-cubes whichwere covered by the inner part of other ‘‘hydrated’’spheres were remarked ‘‘I.’’

d) Every S-cube was checked to ensure that it wasreally on the surface of the ‘‘hydrated’’ proteinmolecule.

e) We counted the number of S-cubes. The numberwas proportional to SASA. Similarly, the totalnumber of S-cubes belonging to nonpolar atomswould be proportional to NSASA.

Fig. 1. The modified van der Waals potential energy function.

249PROTEIN FOLDING SIMULATION

The advantage of this method is that it does notrequire any calculation of distances between points.Besides table look-up, it only needs to perform about6N floating point multiplications in transferring theCartesian coordinates to lattice coordinates, whereN is the number of atoms. With this method we cancalculate the SASA of N-terminal domain of the 434repressor (1R69, 484 nonhydrogen atoms) to within

1.2% accuracy in an average CPU time of 0.115 s on aSGI PowerChallege R10000 processor at 194Mhz.

Conformational Space Searching Algorithmand Supersecondary Structure Constraints

We use genetic algorithm (GA)42,43 to search thepeptide chain conformational space for low-energystructures. In recent years there have been many

Fig. 2. Comparison of the predicted supersecondary structures (predicted) and the X-ray-elucidated supersecondary structure (native) of (a) 1ROP, (b) 1UTG, (c) 1CRN, (d) 1R69, and (e)1CTF.

250 Y. CUI ET AL.

studies of the use of GAs for protein structureprediction and other related structure optimizationproblems.12,44–49 The basic idea of the genetic algo-rithm is to give better chances of survival andreproduction to the good individuals of the popula-tion (in our case, the low-energy structures). In thisway the good genes (structural factors) will accumu-late and combine gradually to dominate the wholepopulation. In the GA used in this work, a chromo-some consists of all the free variables in our peptidechain representation, i.e., it encodes the set of f, c,and x. Most residues have one, two, or three sidechaintorsion angles, so the length of the chromosome isabout 4N, where N is the number of residues. Thereare many versions of GA in its applications to proteinfolding simulation. Our GA procedure is:

The initial population

The initial population size was 500. These struc-tures were built by randomly selecting the backboneand sidechain torsion angles in the constrainedregions. The backbone torsion angles f and c of aresidue were sampled uniformly in a certain regionwith a fineness of 1°. This region was defined by theposition of the residue in the predicted super-secondary structures (Table II). The backbone tor-sion angles of the residues in the short connectingpeptide of any super-secondary structure were con-strained to lie in the corresponding regions on thef-c map (Fig. 1). The backbone dihedral angles ofthe residues in the a-helix and b-strand were re-strained to within 610° of their ideal value, whichwas set to (265°, 240°) and (2120°, 120°) respec-tively. In this way, predicted supersecondary struc-tures were used to greatly reduce the allowablevariation of the affected torsion angles. For the otherresidues not predicted to lie in any supersecondarystructure, their backbone torsion angles are sampledrandomly on the left half (f , 0) of the f-c map (for

glycine, it was the whole f-c map). For the sidechaintorsion angles, the constraints were based on thesidechain rotamer library.41 The mean value of thesidechain torsion angle in the rotamer library wasselected according to its occurring ratio. After amean value was selected, an integer value is selectedrandomly from the interval [mean value 2 standarddeviation, mean value 1 standard deviation]. These500 structures were the parent individuals of thefirst generation.

Fitness criterion

The potential energy of the 500 parent individualswere computed. Potential energy was used as theobjective function. Then we mapped the objectivefunction onto a fitness scale. If there are a fewextraordinary individuals in the early stage of theGA process, they will take over a significant propor-tion of the population after several generations. Thisis a leading cause of premature convergence. On theother hand, if there is still significant diversitywithin the population in the later stages of the GAprocess, then the population average fitness may beclose to the population best fitness and the bestmembers may not become dominant in the popula-tion. In this case, the GA process becomes a randomwalk.43 To prevent premature convergence and ran-dom walk, we used a generation-dependent fitnessscaling.

Fitnessgn,i 5 1 1 Cgn

Egn,max 2 Egn,i

Egn,max 2Egn,min

Cgn 5 C0 1 incr · gn

where Fitnessgn,i is the fitness of the ith individual ingnth generation, Egn,max, Egn,min, and Egn,i is highest,lowest, and the ith individual’s potential energy inthe gnth generation, C0 is a constant that is set to be0.02, incr is increment of the ratio of fitness of thebest individual (with lowest energy) to the worstindividual (with highest energy in each generation).We set incr 5 0.0016 in our computations.

In each generation, the fitness of the worst indi-vidual was always set to 1, the best individual was1 1 C0 1 incr · gn. This scaling strategy is a variantof the fitness scaling methods that focus on the ratioof the fitness of the best individual and the averagefitness. Premature convergence and random walkcan be prevented by this scaling strategy.

The crossover operation

Pairs of individuals were selected randomly forcrossover operation. The probability for an indi-vidual to be selected was fi/Sf, where fi was thefitness of the individual and Sf was the summation ofthe fitness of all the individuals in the population.

TABLE II. Corresponding Regions of theSupersecondary Structure Constraints†

Supersecondarystructures f c

a-helix 275° , 255° 250° , 230°b-strand 2130° , 2110° 110° , 130°a 2150° , 230° 2100° , 50°b 2230° , 230° 100° , 200°e 30° , 130° 130° , 260°l 30° , 150° 260° , 90°t 2160° , 250° 50° , 100°undefined* 2180° , 0°** 2180° , 180°

†The rectangles were used as substitutes for the irregularregions of a, b, e, l, and t on the f-c map. The most populatedarea of each region was included in the corresponding rect-angle.*The residues that did not belong to any predicted supersecond-ary structures were classified as undefined.**For glycine this should be 2180° , 180°.

251PROTEIN FOLDING SIMULATION

The chromosome looked like:

f1c1x11x120f2c2x21x22x230 f3c3x310

········0 fncnxn1xn20 ········0fNcNxN1xN2,

where n was the residue number, xn1 and xn2 was thex1 and x2 of the nth residues, 0 is a possible crossoversite. In order to keep the correlation between f and cin the super-secondary structures, and the correla-tion of x1, x2, and x3 in the sidechain rotamer library,the crossover site is not allowed to occur between thef, c, and x of the same residue. A selected pair ofchromosomes would undergo a fixed number (chosento be 1 in this study) of crossovers at randomlychosen allowable sites.

The mutation operation

Two kinds of mutation operators were used. Thefirst mutation operator may change the conforma-tion dramatically. When this operator acted on apeptide chain, all the values of the backbone andsidechain torsion angles of a randomly chosen resi-due were reselected from their corresponding con-strained regions. We made a copy of the 500 parentindividuals and modified this copied population M1

times, each time by applying the operator to arandomly selected individual from this population.An individual can be selected more than one time, sothere may be changes in torsion angles in more thanone residue in a chromosome. The second mutationoperator is for a more local search of conformationalspace.12 It will perturb some residues’ torsion angles

(f, c, and x) by a random angle between 25° and 5°. Thenumber of perturbed residues of each individual is M2.This operator was also applied to every parent indi-vidual so that in total 500 offspring were produced.

Again, for the purposes of preventing prematureconvergence and random walk, we made M1 and M2

decrease as the search proceeds:

M1 5 1 1 P · exp(2gn/Neff )

M2 5 1 1N4

· exp(2gn/Neff )

where P is set to 500, N is the number of residues inthe protein, gn is the generation, and Neff is aconstant set to 150.

Selection

Now the population consists of 500 parent conforma-tions, 500 crossed offspring, and 1,000 mutated off-spring. The total population is 2,000. The potentialenergy of these 2,000 conformations were computed andonly the 500 lowest-energy conformations were selectedinto the next generation as parent conformations.

Convergence

At least 100 generations of GA were performed foreach protein. After 100 generations, the GA processwill stop only if the decrease of the lowest energy inthe population is less than 1 unit during the last 20generations. On average, about 150 generations ofGA were performed for each protein.

Fig. 3. Structure comparison between the crystal structure (native) and the computed structure(predicted) of repressor of primer.

252 Y. CUI ET AL.

RESULTSRepressor of Primer (1ROP)

Repressor of primer is a 4-helix bundle proteinthat is composed of two identical monomers. Eachmonomer has 56 residues and forms a a–turn–astructure, which does not belong to the 11 frequentlyoccurring supersecondary structures. The predictedsecondary structures (Fig. 2a) were used as con-straints. After computing the conformation using ouralgorithm, we calculated its distance matrix error(DME) to the crystal structure. The computed struc-ture matches the crystal structure with a DME of1.48 Å (Fig. 3).

Uteroglobin (1UTG)

Uteroglobin is a 4-helix protein that has 70 resi-dues. The predicted supersecondary structures area-bb-a-lbb-a-bb-a (Fig. 2b). With these supersecond-ary structure constraints we computed the 3-dimensional structure. The computed structurematches the crystal structure with a DME of 3.47 Å(Fig. 4).

Crambin (1CRN)

Crambin is a 46-residue protein with two a-helixand a pair of b-strands. It has three disulphidebonds. We did not use the disulphide bond con-straints. The predicted supersecondary structuresare b-loop-a-lbb-a-l-b-loop-a (Fig. 2c). The computedstructure matches the crystal structure with a DMEof 2.73 Å (Fig. 5).

N-Terminal Domain of the 434Repressor (1R69)

The crystal structure of the N-terminal domain ofthe 434 repressor has 63 residues and is composed offive helices. The predicted supersecondary struc-tures are a-lbb-a-lbb-a-loop-a-lbb-a (Fig. 2d). Thecomputed structure matches the crystal structurewith a DME of 4.48 Å (Fig. 6).

C-Terminal Domain of the L7(SLASH)*L12 50S Ribosomal Protein (1CTF)

This protein has 68 residues. It has six secondarystructures—three a-helix and three b-strands. Thisprotein is the most complex example in this study.The predicted supersecondary structures are b-a-lbb-a-b-a-l-b (Fig. 2e). The computed structure matchesthe crystal structure with a DME of 4.00 Å (Fig. 7).

DISCUSSION

The predicted super-secondary structures and thenative supersecondary structures of these five pro-teins are shown in Fig. 2. In these five proteins, thereare 21 secondary structures and 16 short connectingpeptides. Ten short connecting peptides were identi-fied to be in one of the 11 frequently occurringsupersecondary structures. Most of the supersecond-ary structures are correctly predicted. For these fiveproteins the correctness ratio is 90.1%. Although theaccuracy is high, in some instances the predictedstructures do not align precisely with those observedin the crystal structures. If the backbone torsionangle (f, c) of a few consecutive residues were

Fig. 4. Structure comparison between the crystal structure (native) and the computed structure(predicted) of Uteroglobin.

253PROTEIN FOLDING SIMULATION

restrained in wrong regions, the peptide chain mayhave a wrong trend at this segment. If the segmentwas at the central part of the peptide chain, theoverall fold may be misdetermined. In this study,such a fatal mistake has not occurred in the supersec-ondary structure prediction. The most serious struc-ture distortion caused by the errors of the supersec-ondary structure prediction was in the crambin. Atthe C-terminal of the peptide chain, an incorrectlypredicted a-helix (from residue 41 to 45) was im-posed on the peptide chain as a constraint (Fig. 4).

As a result, a wrong structure was formed in thisterminal (Fig. 7).

In Figure 2 one can find that in some casessupersecondary structure was not correctly pre-dicted at only one or two residues, while the neighbor-ing residues were all restrained in the correct re-gions. In these cases, the residue the peptide chainwill turn to a wrong direction at this point. But if thenative-like structures are favored by the potential,the nearby residues will move to compensate for thismistake. As a result, a native-like profile can still be

Fig. 5. Structure comparison between the crystal structure (native) and the computed structure(predicted) of repressor of Crambin.

Fig. 6. Structure comparison between the crystal structure (native) and the computed structure(predicted) of N-terminal domain of the 434 repressor.

254 Y. CUI ET AL.

formed. For example, the computed structure of lutg(Fig. 4) was native-like, while at two central residues(15 and 28) the supersecondary structures werepredicted incorrectly. This is possible because of theflexibility of the model. In this model, the peptidechains are more flexible than those in the fixedsecondary structure model12 where the affected tor-sion angles are fixed at their ideal values (withoutany degree of freedom). In contrast, the torsionangles in our model can rotate in some regions,which is determined by the predicted supersecond-ary structures and the sidechain rotamer library.

With the supersecondary structure constraints theconformation space of the peptide chain is greatlyreduced, but the DME of the peptide chain can stillbe large (over 15 Å). For example, in the simulationof the 434 repressor we observed many conforma-tions with a DME exceeding 15 Å, especially in theearly generations. This indicates that the overallfold of a protein molecule cannot be well definedonly by the supersecondary structures. Genetic algo-rithm was used to search the reduced conformationspace for low-energy structures. Four of the fivecomputed structures are similar to the correspond-ing X-ray elucidated structures. The DME of repres-sor of primer, uteroglobin, crambin, and C-terminaldomain of L7(SLASH)*L12 50 S ribosomal proteinare all smaller than or equal to 4.0 Å. For theN-terminal domain of the 434 repressor, there arethree supersecondary structures. One of the connect-ing peptides, a six-residue loop, cannot be recognizedas belonging to any of the 11 kinds of supersecondarystructures. The peptide chain is divided into twofragments which are connected by the loop region.The computed structure of each of these two frag-

ments is similar to their corresponding parts in thecrystal structure, but the relative position of the twofragments is incorrectly determined by the loopregion.

An ideal potential should give higher values tonon-native conformations than the native conforma-tion. Recent studies50–52 indicated that some poten-tials can distinguish between correct and certainincorrect structures with a high degree of success.However, in the protein folding simulation there isan astronomical number of candidates in the confor-mation space. If a potential can identify 99.99%non-native structures, while it gives 0.01% of themlower energy than the native structure, then thereare still uncountably many minima with lower en-ergy than that of the native structure on the energylandscape. In this situation, our hope of finding anative-like structure lies in the possibility that mostof the low-energy structures are ‘‘near’’ the nativestructure to form a cluster of native-like structures.This appears to be the case for the repressor ofprimer and uteroglobin, where the energy of thecomputed structures is lower than that of theirnative structures. The other three proteins are morecomplex; the energy of the computed structures arehigher than that of their native structures. This ismainly caused by the van der Waals term and itindicates that a more efficient local search method isneeded.

In recent years, important progress has been madein computing the 3-dimensional structures of pro-teins from their sequences using simple energyfunction. The attractive aspect of this method is thata simple model should be much easier to improvethan highly parameterized ones, and the prediction

Fig. 7. Structure comparison between the crystal structure (native) and the computed structure(predicted) of C-terminal domain of the ribosomal protein L7/*L12.

255PROTEIN FOLDING SIMULATION

results of these simple models is arguably compa-rable to those of the more complex models.14 Encour-aging results have been reported by Sun et al.12 Theydeveloped a model that predicted reasonably wellthe known tertiary folds of 7 out of 10 small proteins.Their method used experimental secondary struc-tures, in which the backbone dihedral angles (f, c)are fixed at the ideal values. Three of these tenproteins are also considered in this study. They arerepressor of primer, crambin, and N-terminal do-main of the 434 repressor. The DME reported in Sunet al. are 1.65 Å, 4.87 Å, and 5.55 Å, respectively. Inour results, the DME of these proteins are 1.48 Å,2.73 Å, and 4.48 Å, respectively. This suggests thatsupersecondary structure constraints and better mod-eling of the hydrophobic interaction are of consider-able utility in protein structure computation.

CONCLUSION

One important step toward building a tertiarystructure is to identify how secondary structures asbuilding blocks arrange themselves in space. Goodsupersecondary structure prediction methods canprovide important information in the prediction ofprotein tertiary structure.

The structure of a protein is determined by thecompetition and cooperation of all of the interac-tions, especially hydrophobic interaction and van derWaals interaction. Correctly including the hydropho-bic interaction is extremely important. Although thenature of hydrophobic interaction is not completelyunderstood, it is suggested that protein–solvent inter-action depends on the solvent-accessible surfacearea of the protein molecule.53,54

The goal of this study is to suggest a way tocapture these two main features in our currentunderstanding of protein structure, interaction, andfolding mechanism. The results show that somesmall protein structures can be determined by amodel that carefully adduces these points.

ACKNOWLEDGMENTS

We thank the National Laboratory of Scientificand Engineering Computing, Institute of Computa-tional Mathematics & Scientific and EngineeringComputing, Chinese Academy of Sciences, and theComputer Network Information Center, ChineseAcademy of Sciences, for providing free CPU time.We are grateful to Professor Zhirong Sun for his helpin the prediction of the supersecondary structures ofthe five proteins used in this study.

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