Proc. Nati. Acad. Sci. USAVol. 88, pp. 11076-11080, December 1991Biophysics

A strategy for finding classes of minima on a hypersurface:Implications for approaches to the protein folding problem

(nonlnear optimizatdon/"antlion")

TERESA HEAD-GORDON, FRANK H. STILLINGER, AND JULIO ARRECIS*AT&T Bell Laboratories, Murray Hill, NJ 07974

Contributed by Frank H. Stillinger, September 13, 1991

ABSTRACT Locating the native structure of a given pro-tein is a task made difficult by the complexity of the potentialenergy hypersurface and by the huge number of local minimait contains. We have explored a strategy (the "antlion"method) for hypersurface modification that suppresses allminima but that of the native structure. Transferrable penaltyfunctions with general applicability for modifying a hypersur-face to retain the desired minimum are identified, and twoblocked oligopeptides (alanine dipeptide and tetrapeptide) areused for specific numerical illustration of the dramatic simpli-fication that ensues. In addition, an intermediary role forneural networks to manage some aspects of the antlion strategyapplied to large polypeptides and proteins is introduced.

Section 1. Introduction

The protein folding problem is one ofthe most significant andintriguing challenges in moleculat biophysics (1-3). Thenative forms that have been determined for naturally occur-ring proteins display a fascinating variety of three-dimensional structures exquisitely tailored tp biological func-tion. In addition, the experimentally observed' folding kinet-ics of naturally occurring proteins involves time scales on theorders of microseconds to minutes, rather than the millenniaexpected for random-walk searches among conformationalalternatives (4). General principles by which any linearsequence of amino acid residues encodes information aboutthe native structure, and the most efficient kinetic pathway tothis structure, still remain largely out of reach. The presentpaper is devoted to the exploration of a strategy that, wehope, will eventually illuminate these general principles.From the theoretical viewpoint, the protein folding prob-

lem comprises three components. The first involves speci-fying the free energy (potential of mean force) hypersurfacefor arbitrary configurations of a given polypeptide immersedin'the solvent' of interest. The second concerns the kineticpathway by which any non-native structure (in particular thatof the newly synthesized protein emerging from the ribo-some) manages to attain the native structure. The thirdamounts to nonlinear optimization on the free energy hyper-surface to identify the native structure and any feasiblealternative folding structures (low-lying relative free energyminima) and to show how they are determined by the aminoacid sequence and solvent. In relation to this final point, wenote that there is some question as to whether the nativestructure is always the global free energy minimum (5, 6) ora very long-lived metastable state (7). For the present weconcentrate on the last of these three components, assuming

that at least an approximation to the relevant free energyhypersurface is available.

Chemically realistic approximations to the conformationalbehavior ofpolypeptides inevitably entail hypersurfaces withenormous complexity. It is generally believed that fl, thenumber of distinct local minima, rises approximately expo-nentially with N, the number of residues:

In iQ aN. [1.1]

A rough range ofN for naturally occurring proteins is 100 to1000, while a probably lies in the range of 1 to 10. Searchingfor the native structure among such a large number ofcandidates is daunting to say the least.

Theoretical and computational strategies for solving thenative protein minimization problem have been quite varied.Some examples include brute force minimization (8), statis-tical mechanical models ranging from that ofZimm and Bragg(9) to Monte Carlo simulations of highly simplified latticemodels (10), and the application of neural network conceptsto prediction of protein secondary structure (11-14) andtertiary structure (15-18). The adaptation ofspin-glass theoryto associative memory Hamiltonians for proteins (15-17) andexplicit neural network training on distance matrices (18)offer promise for overcoming the deficiencies of traditionalneural network implementations (11-14), where only a max-imum of -67% reliability has been achieved for prediction ofsecondary structure.This paper reports results of an exploratory investigation

that was undertaken to determine the applicability of ageneral optimization strategy to the protein folding problem,the so-called "antlion" method (19, 20), This approach relieson the ability to deform the objective function hypersurfacein such a way that the basin surrounding the global minimum(or a metastable minimum) widens and dominates. It takes itsname from the family of subterranean insects that lie in waitat the bottom of victim-entrapping basins. In the presentcontext it is the job of the antlion method to replace thecomplicated protein hypersurface by one for which a = 0 inEq. 1.1. Any elementary minimization routine such as steep-est descent on the modified hypersurface would then auto-matically converge to the single remaining minimum type,which by construction should be identical to the-global (oreven a preselected metastable) minimum of the' startingproblem, or at least a close approximation thereto. The finalstep in the antlion method is to optimize on the undeformedhypersurface, using the converged structure derived from thesimplified' potential energy surface as an initial guess. Wenote that this same strategy is possible for classes of minima,where the complexity of the hypersurface is reduced to 0 <a << 1. Stillinger (21) and Piela et al. (22) have proposed theuse of a diffusion equation method for deforming hypersur-faces to retain only the global minimum. The antlion method

*Present address: Chemistry Department, Massachusetts Institute ofTechnology, Cambridge, MA 02139.

11076

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differs from the diffusion equation method in several re-spects; the most important difference is that the diffusionequation method has been explicitly demonstrated on one-and two-dimensional model systems only (22), whereas theantlion method presented here is applicable to an arbitrarilylarge number of dimensions of biological interest (23).To create a generally useful antlion strategy, at least part

of the hypersurface modification algorithm must containmathematical operations that are transferable between dif-ferent polypeptides, and in particular from oligomers tohigher molecular weight polypeptides. The success of trans-ferability may ultimately allow insight into the nature of thepathways by which 4 random coil accesses the correcttertiary-structure miinium, in addition to identifyingwhether a protein is kinetically or thermodynamically stabi-lized. In this spirit we have focused considerable attention onthe blocked alanine dipeptide (Fig. 1) and alanine tetrapeptide(Fig. 2). As discussed in more detail below, we foreseeultimately employing a neural network automation procedureto manage some aspects of the antlion modification.

Section 2 presents the details of the objective potentialfunction for both alanine dipeptide and alanine tetrapeptide,as well as the specific algorithms for searching conformationspace for minima. Section 3 reports energies and structuresfor the alanine dipeptide minima, identifies some transferablemodifications ofthe original potential energy, and shows howthe resulting modified potential is drastically simplified to onesurviving minimum. In the same section, we extend theseconsiderations to the alanine tetrapeptide, again demonstrat-ing the capacity for dramatic simplification to a single min-imum hypersufface. Section 4 summarizes our results andoutlines our best projection for future development of theantlion approach.

Section 2. Methods and Models

Potential Function. A reasonable approximation to thehypersurface of an arbitrary polypeptide is the followingwell-established empirical potential energy function:

bonds angles

Vo= kbi(bi - b o)2 + E kei(i- 0,0)2

immoper torsions+ Z ki(, _- Ti0)2 + z k1i[l + cos(niwi + S,)]

i i

M M+ E>X {C qiqj/rij + Ej[(Rj/rj)12 - 2(Rj>/rj)6]}. [2.1]

i<j

The first four terms provide the connectivity potential; thebond length, bond angle, and improper torsion deformationsare represented as harmonic potential functions with forceconstants kb, k9, and k, and equilibrium values bo, 60, and To,respectively. The torsional potential is represented as aFourier cosine expansion, where k. is the force constant, 8is the phase, and n is a multiplicity factor that allows forinclusion of higher harmonics. While the chirality of thea-carbon center of all amino acids except glycine dictates theuse of a general Fourier series, the cosine series will beadequate for the current study. The empirical parametersused and the specific torsions evaluated (only one dihedralterm is evaluated for rotation around a given bond) for bothalanine dipeptide and alanine tetrapeptide are presented inTable 1 (24).The remaining terms in Eq. 2.1 are nonbonded interac-

tions, which are modeled as pairwise coulomb electrostaticand Lennard-Jones interactions. The i, j sums are restrictedto pairs of atoms separated by three or more interveningbonds between the pair. The Lennard-Jones interaction

FIG. 1. Structure of the global-minimum conformer for theblocked alanine dipeptide.

parameters are evaluated using simple mixing rules of theindividual atomic parameters:

Eij =(60 )

Rij= (Ri, + RU)/2. [2.21

In addition, the electrostatic interactions are scaled by afactor C = 0.5 when the pair under consideration are sepa-rated by exactly three bonds; otherwise C = 1.0. In Table 2we list the Lennard-Jones parameters (24) used for bothalanine dipeptide and alanine tetrapeptide. In Table 3 weprovide the charges for alanine dipeptide (24), which differslightly from the charges for alanine tetrapeptide in the sametable, in order to ensure that charge neutrality is maintained.The set of the connectivity and nonbonded parameters inTables 1-3 for the di- and tetrapeptide will henceforth bereferred to as yielding the unmodified interaction, VO. Adescription of the corresponding sets for the modified inter-actions is left to Sections 3 and 4.

Characterization of Minima. Given an objective functionsuch as that in Eq. 2.1, we require a method for obtaining amajority, or if possible all, of the minima on the hypersurfaceit represents. For simplicity we employ a Monte Carlo

FIG. 2. Structure of the global-minimum conformer for theblocked alanine tetrapeptide.

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Table 1. Parameters of the intramolecular potential energyfunction for the alanine dipeptide and tetrapeptide

Bond type

CT-HACT-CC-0C-NN-HN-CTCT-CTAngle type

CT-C-0CT-C-NHA-CT-HAHA-CT-CC-N-HC-N-CTO-C-NN-CT-HAN-CT-CTN-CT-CH-N-CTCT-CT-HAImproper type

C-CT-N-0N-C-CT-HDihedral type

CT-C-N-CTHA-CT-C-NHA-CT-N-CN-CT-C-NN-CT-CT-HA

Force constantkb, kcal/(mol.A2)

340.0279.0640.0350.0465.0310.0268.0

ke, kcal/(mol-rad2)66.064.037.045.03-2.044.098.046.576.058.023.037.5

kf, kcal/(mol rad2)125.028.0

ko, kcal/mol9.52.20.30.71.6

Equilibnium valuebo, A1.0901.5151.2251.3351.0001.4601.515

00, deg

122.6113.9109.8109.5120.9117.6125.0109.2111.0112.8120.8109.5

To, deg

0.00.0

8, deg (n)180 (2)0(3)0 (3)

180 (2)0 (3)

CT corresponds to Ca, C , CTR, and CTL. HA corresponds to Ha,Hp, HTR, and HTL.

heating and quenching protocol, and subsequent minimiza-tion, to search exhaustively for minima of V0 and of all of themodified functions discussed below. In some cases we use aminimization procedure with starting structures expected tobe near stable stationary points.The heating phase ofany given Monte Carlo run consists of

specifying an initial configuration of the atoms of alaninedipeptide oralanine tetrapeptide and generating configurationsat a temperature of20,000K by using the Metropolis algorithm(25). A step size in Cartesian space of ±0.125 A for every atomat every step results in a 50% acceptance rate for the 500,000-step run. Configurations are sampled every 10,000 steps,resulting in 50 configurations, which are used as startingstructures for the quenching portion of the Monte Carlosearch.

Table 2. Parameters of the Lennard-Jones function for thealanine dipeptide and tetrapeptide

Atom type eii, kcal/mol R11, AHTL/HTR 0.0450 1.468CTL/CTR 0.0903 1.800C 0.1410 1.8700 0.2000 1.560N 0.0900 1.830H 0.0498 0.800Ha 0.0450 1.468Ca 0.0903 1.800COl 0.0903 1.800Hp 0.0450 1.468

Proc. Nat!. Acad. Sci. USA 88 (1991)

Table 3. Parameters of the electrostatic function for the alaninedipeptide and tetrapeptide

qu, e

Atom type Dipeptide Tetrapeptide

HTL/HTR 0.00)0 0.0000CTL/CTR 0.0W)0.0000C 0.5500 0.55(K0 -0.5500 -0.5500N -0.3500 -0.3500H 0.2500 0.2500Ha 0.1000 0.1000Co 0.0000 0.0000C. -0.2600 -0.2917Hp 0.1200 0.1083

The Monte Carlo quenching stage consists of generatingconfigurations at 10 K using a step size of 0.0005 A, againresulting in an acceptance rate of 50% for Metropolis sam-pling. The Monte Carlo quench is terminated after 30,000steps, and a BFGS minimization algorithm (26) is then usedto determine the closest stationary p6int.

Sc 3. H face Mod ation for MIA

Al Dipie. We provide an enumeration in Table 4of all L mindima found by the Monte Carlo/minimizationprotocol outlined in Section 2; we note that we have alsofound most of the mirfor images (D form) of the entries inTable 4. A large majority of the mimma correspond tostructures where a cis-trans isomerization of one or bothpeptide groups has occurred. The remaining four minima notrepresented in the preceding class are those which are foundby a search through the two-dimensional space defined by theinternal coordinate irsions 0 (C-N-Ca-C) And* (N-C-C-N).Fig. 3 displays an energy contour map showing these fourenergy minima, whtbh is generated by cotraining and Vand allowing relaxation of the remaining degrees of freedom(27). The lowest-energy structure of the fl 35 minimuacorresponds to the C7.q all-trans L conformer (Fig. 1), whichmay be described as a seven-membered ring closed by anintramolecular hydrogen bond, with the side-chain methylgroup equatorial to the plane of the ring.

Table 4.

*, deg-78.1163.7

-163.371.674.8

-157.9-169.7-53.870.1

-72.865.4152.3

-166.7152.571.6

-68.1-49.576.053.268.4

Enumeration of the alanine dipeptide minia*, deg

72.5-163.9149.9-65.9

-143.278.3

-55.3-50.524.2

145.236.6

-148.6-42.3-160.7-156.7-39.8-48.8167.3

-131.6-178.8

0)1

transtranstranstranstranstranstranstrans

cis

cistrans

cis

trans

ciscis

ciscis

transcis

cis

t2

transtranscistransciscistranscis

transcis

cis

Ciscis

trans

trans

cis

trans

cis

transtrans

E, kcal/mol-32.391-30.997-30.766-30.339-29.630-28.430-26.258-26.215-26.037-25.995-25.870-25.716-25.284-24.841-24.833- 24.170-23.615-23.375-20.799-20.744

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180 -

90

-901

-180

-180 -90 0 90 180

0FIG. 3. The 4, 4 surface derived from the unmodified Hamilto-

nian, V0 (Eq. 2.1) for alanine dipeptide. The and 4i variables are heldfixed at each grid point (10° spacing), and all other degrees offreedomare relaxed. The dashed lines denote contours of 0.5 kcal/mol andextend from the zero of energy (the C7,q conformer) to 7.0 kcal/mol.Solid contours are drawn every 1.0 kcal/mol thereafter.

The first type of modification of VO is to eliminate allminima where one or both peptide groups are in the cisconformation (where exceptions are to be made if the residueis a proline) (1). We note that the peptide torsion potentialused in Eq. 2.1 is specifically

V = 9.5[1 + cos(2w + ff)], [3.1]

which favors minima at both w = 0 and w = r. The obviousmodification of Eq. 3.1 to favor the trans form is to changethe multiplicity factor of 2 to 1, and to change the phase from,r to 0. To maintain the correct curvature at the minimum, weuse a force constant of 38 kcal/mol, so that

V' = 38.0[1 + cos(wO)]. [3.2]

In addition, we will always desire the L configuration of apolypeptide sequence. To maintain the desired chirality, weincorporate an improper dihedral function,

V"= 125.0(T - To)2, [3.3]

for the torsions C,-N-C-Cp (T0 = 33.00) and Ca-N-C-Ha (To =-330). While the V' and V" modifications are trivial and insome sense physically unimportant in relevant areas ofconfiguration space for biological molecules of interest, thisserves as an illustrative example for what is to follow. For thecase of alanine dipeptide, this modification permits us tovisualize transforming the energy surface in Fig. 3 to retainonly the C7eq conformer.We have considered a number of modifications to the

potential energy function representing the surface in Fig. 3 inorder to retain only the global energy minimum. Our criteriafor a successful modification are (i) that the penalty functionexplicitly or implicitly incorporate information about thetertiary structure of any peptide, (ii) that the functional formof the modification is transferable across any polypeptidesequence, and (iii) that a variety of conformations can bedistinguished in a given segment of polypeptide ranging fromthe random coil to secondary structure conformers such asthe a-helix and /8-sheet.The generic penalty function

V"' = kp[l - cos(o - 40)] + k,4[l - cos(qi - io)]

fulfills the above objectives. For the case of polypeptideswith minimal side chains such as glycine or alanine, the 4,

variables most directly define the tertiary structure; forpolypeptides with more complex side chains, the same typeof penalty function can be applied to the Xi dihedrals as well,so that the functional form is transferable to any sequence ofamino acid residues. Finally, for appropriately defined 40 and41o parameters, it allows discrimination among the pool ofrelevant conformers observed in large polypeptides andproteins. This function (using the parameters defined in Table5) indeed accomplishes the simplification of the surface inFig. 3 to a single minimum: the global, C7eq minimum asexhibited in Fig. 4.

Transferability to Alanine Tetrapeptide. The number ofunique minima for the alanine tetrapeptide case is quite large,even when cis peptides and D isomers are eliminated fromconsideration. However, the alanine tetrapeptide systemoffers some simplification for classifying these minima whenone considers the conformational space of the tetrapeptide tocomprise three sets of 4, dihedrals. The alanine tetrapep-tide shows the following minima in any given 4), 4i) space:C7eq, C7a,,, C5, a', aR, aL, and polyglycine II, and several"unusual" minima that occur infrequently relative to thepreceding seven. Thus a large majority of minima fall into aclassification where the three sets of 4, 4i variables can adoptany combination of the seven conformers C7eq (-75°, 750),C7a,, (750, -750), C5 (-1650, 1650), a' (-1650, -550), aR(-600, -450), aL (600, 600), and polyglycine II (-80°, 1500).Enumeration of these possibilities indicates that the numberof unique minima is approximately

Q-73+ M, [3.5]

where M is the small number of minima that do not fall intothe above seven conformer classification (M 25 in oursearch). In the case of stable minima for all possible combi-nations of the seven conformers (343), fQ 2 368 or a- 3(again, ignoring the possibility of cis and D conformers). Wenote that our search found most (but not all) of the 73 = 343simple possibilities, which can be attributed both to a lack ofstability ofa particular combination (aR, aL, aR, for example)and to the likelihood of incomplete sampling.With these Monte Carlo results in hand for the unmodified

tetrapeptide hypersurface, we then tested the transferabilityof the modification functions in Eqs. 3.2-3.4. As before, theintention was to produce a modified potential surface pos-sessing only a single minimum that corresponds closely to apreselected minimum of the complicated starting hypersur-face. We have successfully achieved this goal in a mannerthat demonstrates considerable latitude in the character ofthe single minimum that is permitted to survive modification.Specifically, successful use of the transferable functions(with appropriate 00, 0 choices) has been demonstrated inthe following independent cases: (i) retention of the globalminimum, [C7eq, C7a,, polyglycine II], (ii) retention of apreselected metastable minimum, [aR, aR, aR], and (iii)retention of any one of the class of minima [C7eq, C7ax, *1

where * denotes a "wild card" specification for the third 4,

4) pair (we have found this third 4), 4) pair on the unmodifiedsurface to be either polyglycine II, aR, C7eq, or C7ax).

Table 5. Parameters for the alanine dipeptide potential, V'0Dihedral type ko, ko, kcal/mol 40, 4io, deg n

C-N-C,-C 7.5 -75.0 1N-C0-C-N 7.5 75.0 1

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180

90

-90

180

-180 -90 0 90 180

FIG. 4. The +, .r surface derived from the modified Hamiltonian(Eq. 3.1-3.4) for alanine dipeptide. The and variables are heldfixed at each grid point (100 spacing), and all other degrees offreedomare relaxed. The dashed lines denote contours of 2.0 kcal/mol andextend from the zero of energy (the C7,q conformer) to 8.0 kcal/mol.Solid contours are drawn every 3.0 kcal/mol thereafter.

Section 4. Discussion and Conclusions

We have implemented a strategy for greatly simplifyingpeptide energy hypersurfaces in order to retain only oneconformationally distinct minimum. To the extent that thesurviving minimum corresponds closely to the desired min-imum, the original conformational optimization problem un-dergoes drastic simplification. This approach has been illus-trated by specific calculation for two small peptides, theblocked alanine dipeptide and tetrapeptide. For the former,20 local minima (40, with mirror-image configurations) col-lapse to a single minimum upon application of suitablepotential energy penalty functions. The functional forms ofthese penalty functions are immediately transferable to thetetrapeptide case (or indeed to larger polypeptides) andsucceed in suppressing a much larger number oflocal minimaon the starting hypersurface to favor, once again, a singlesurviving minimum. For both the dipeptide and the tetrapep-tide, we have been able to arrange for the surviving minimumto closely approximate the desired minimum of the originalsurface, thereby simplifying the corresponding conforma-tional search problem. These examples illustrate the promisefor the extension of the antlion method to larger polypeptidesand proteins, where determining the native structure fromamong the vast number of minima on the unmodified hyper-surface is an intractable proposition, and therefore representsthe case where simplification is highly desirable.

In the general context of protein conformational predic-tion, implementation of the antlion approach might at firstglance seem to require at the outset knowledge of thesecondary and tertiary structure sought. In particular, itwould seem that sets of angles 40, *0 have to be identified toconstruct the necessary penalty functions; the 40 and 40values used for short peptides would not necessarily transferto longer peptides (28, 29). We foresee a fundamental role forneural networks (trained on a suitable protein database) tomanage this aspect of the multidimensional optimizationproblem. However we hasten to stress that an importantdistinction exists between this intended role and that con-ventionally required of neural networks in the protein foldingarea. For the latter, the outputs of the network are the directstructure predictions, whether they concern secondary struc-

ture predictions (11-14) or residue contact-distance classifi-cation (18). The antlion method, however, would require only

network predictions for the 'o, Xb penalty parameters (andperhaps the corresponding force constants); subsequent min-imization first on the modified potential hypersurface andthen on the unmodified hypersurface serves as the tertiarypredictor. Local violations of the neural network angle pre-dictions become feasible, even likely, as the entire systemseeks and finds its optimal final structure. In this respect ourapproach accommodates the presence of locally frustratedinteractions in the interests of attaining a global minimumtertiary structure. It is the frequent occurrence of suchintrinsic frustrated interactions that, in our view, has thus farlimited the success rate of neural networks as direct predic-tors of secondary structure in proteins.

In summary, the simple-peptide-system results obtainedthus far provide strong encouragement that the antlion strat-egy can be adapted to larger peptides and proteins. Indeed,we have been able to apply the antlion/neural networkstrategy, outlined in this section, successfully to the naturallyoccurring 26-residue polypeptide mellitin; details will bepresented elsewhere (23).

J.A. is a recipient of an AT&T Cooperative Research Fellowship.

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