R E V I E W : C H E M I S T R Y

Global Optimization of Clusters,Crystals, and Biomolecules

David J. Wales1* and Harold A. Scheraga2

Finding the optimal solution to a complex optimization problem is of great impor-tance in many fields, ranging from protein structure prediction to the design ofmicroprocessor circuitry. Some recent progress in finding the global minima ofpotential energy functions is described, focusing on applications of the simple“basin-hopping” approach to atomic and molecular clusters and more complicatedhypersurface deformation techniques for crystals and biomolecules. These methodshave produced promising results and should enable larger and more complex systemsto be treated in the future.

T he global optimization problem is asubject of intense current interest. Ap-plications of obvious economic impor-

tance include traveling salesman–type prob-lems and the design of microprocessor cir-cuitry. In the domain of atoms and molecules,discovering the lowest-energy isomer or crys-tal structure for a system with a given com-position is frequently a goal. For example, itseems likely that the native structure of aprotein is often related to the global minimumof its potential energy surface (PES). Hence,considerable research efforts are being madeto predict the three-dimensional structure of aprotein solely from its amino acid sequenceby computer simulation. Also, experimentalmicrocalorimetry for free sodium clusters (1)has revealed highly irregular thermodynamicproperties as a function of size. To explainthese results, the global potential energy min-imum, which is expected to be the favoredstructure for the low-temperature experiments,must first be identified. As these two exam-ples show, developing methods for treating adiverse range of systems spanning the fieldsof chemistry, biochemistry, and materials sci-ence is important.

In treating any nontrivial global optimiza-tion problem, the principal difficulty arisesfrom the number of minima on the PES,which usually increases exponentially withthe size of the system. An example is thecluster of 55 atoms interacting by a Lennard-Jones (LJ) potential, LJ55 (Fig. 1B), wherethe number of minima (excluding permuta-tional isomers) is at least 1010. Nevertheless,in this specific case the global minimum is

easily located because of the form of thepotential energy landscape (2). For systemsof different sizes, locating the global mini-mum can be much harder and may requiresophisticated search routines. Similarly, arandom conformational search for the globalminimum of a typical protein would take anunfeasibly long time. However, it is nowgenerally agreed that the search is not ran-dom, but instead the PES is biased toward theglobal minimum, which results in efficientrelaxation (2, 3). Hence, the amino acid se-quences of naturally occurring proteins arepresumed to have evolved to fold rapidly intoa unique native structure. This does not mean,however, that the corresponding computa-tional search becomes trivial. Although thereis some evidence that it may be possible tosimulate protein folding by brute force mo-lecular dynamics (4), much faster computerswill be needed before such tasks can be rou-tinely undertaken.

Numerous approaches to solving the glob-al optimization problem have been suggested(5–13). One difficulty with the global optimi-zation literature is that it is spread over jour-nals in many different fields; our citationsreflect our knowledge of applications to prob-lems in molecular science. Another problemis that detailed comparisons of different ap-proaches are rare, partly because gatheringreliable statistics for nontrivial problems istime-consuming. In this article we focus on afew particular strategies that we have foundto work best for problems involving clusters,

crystals, and biomolecules. We focus on thePES, rather than the free energy, to avoid theadditional complications arising from finitetemperature. We also emphasize that, forglobal optimization to succeed in predictingthe properties of real systems, it is necessaryto have a sufficiently accurate potential ener-gy function. An efficient way to incorporatemore realistic potentials has been suggestedby Hartke (8).

The LJ model of inert gas clusters hasbeen investigated intensively and provides auseful testing ground for putative global op-timization algorithms (14). In terms of thetopology of the energy landscape and thestructure of stationary points, this potentialalso provides a useful model of noble gasclusters (15). Furthermore, some of the non-icosahedral global minima first discoveredfor this potential have recently been identi-fied in nickel and gold clusters (16). A briefoverview of global optimization strategies istherefore first presented in the context of LJclusters (17). The prediction of crystal struc-tures is also a problem of current interest (18)and is discussed subsequently. Finally, wedescribe several strategies for tackling theprotein folding problem (6).

ClustersMost global minima for LJ clusters contain-ing fewer than 100 atoms are based on ico-sahedral packing (Fig. 1B). The exceptions,LJ38 and LJ75–77, serve as particularly inter-esting test cases, because the correspondingenergy landscapes consist of two families ofstructures (2, 19). At these sizes, the lowest-energy minimum based on icosahedral pack-ing acts as a trap and is widely separated fromthe true global minimum (Fig. 1, A and C).Actually, it is quite easy to find the globalminima of these clusters by seeding the start-ing geometry with a core of the appropriatemorphology. Indeed, most of the lowestknown minima up to LJ147 were first ob-tained by Northby by counting nearest-neigh-

1University Chemical Laboratories, Lensfield Road,Cambridge, CB2 1EW, UK. 2Baker Laboratory ofChemistry and Chemical Biology, Cornell University,Ithaca, NY 14853-1301, USA.

*To whom correspondence should be addressed. E-mail: dw34@cus.cam.ac.uk

Fig. 1. Global minimaof the LJ potential for(A) 38 atoms (truncatedoctahedron), (B) 55 at-oms (Mackay icosahe-dron), and (C) 75 atoms(Marks decahedron).

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bor interactions for icosahedral packingschemes (20). However, our focus here is onunbiased methods that may be transferable toother systems.

Simulated annealing (21) probably pro-vided the first generally applicable tech-nique for global optimization. In this ap-proach the state of the system is followedby simulation as the temperature is de-creased slowly from a high value, in thehope that it will eventually come to rest atthe global potential energy minimum. Anew global minimum was located for LJ24

with this approach (22), but otherwise sim-ulated annealing does not appear to havebeen very successful for LJ clusters, evenin more sophisticated forms (10, 23). Theproblem is that the free-energy global min-imum can change at a temperature wherethe energy barriers are too high for the systemto escape from a local minimum.

An alternative approach is based upon“hypersurface deformation” where the func-tional form of the potential energy is delib-erately altered. Some transformations smooththe surface and reduce the number of min-ima, thereby making the global optimiza-tion problem easier (10, 24 –28). However,the global minimum of the smoothed sur-face must then be mapped back to the realsurface, and this reversing procedure is thekey problem associated with such ap-proaches. It is now known that the globalminimum can change rather dramaticallyunder some of the smoothing procedures(18, 29). Hence, it is necessary to couplesmoothing with an efficient local searchprocedure, which can be applied in map-ping minima back from the deformed hy-persurface to the original one (18, 30). Toimprove efficiency, more than one mini-mum of the smoothed surface must betracked backwards (18, 30).

One hypersurface deformation ap-proach, the distance scaling method (DSM,see the section on biomolecules), has beenapplied with a reversing procedure that in-volves both structural perturbations andshort molecular dynamics (MD) simula-tions (27 ). A short molecular dynamics runwas carried out for each minimum beingtracked back at each step of the reversingprocedure, and another molecular dynamicsrun was carried out for the resulting mini-mum on the undeformed surface. The de-formation lowered barriers between mini-ma, allowing the molecular dynamics pro-cess to explore configurational space moreefficiently. Coupled in this way to molec-ular dynamics, the DSM was used to locatethe nonicosahedral global minimum forLJ38; however, it failed in some other cases(27 ). Using local search and self-consistentmapping of deformed and undeformed min-ima, a deformation-based method has now

been used to locate global minima for LJclusters of sizes up to 100 atoms, except forthose containing 75 to 77 atoms.

A third approach uses genetic algo-rithms (31) that mimic the evolutionaryprocess by evolving a “genetic code” basedon the structure and using the concepts offitness, mutation, and crossover. A varietyof different implementations have beenproposed (32). In discrete genetic algo-rithms, the conformational space is subjectto a binary encoding corresponding to the“genotype” (33). However, it is also possi-ble to work directly in physical coordinatespace, corresponding to the “phenotype”(34), and such a “continuous” genetic algo-rithm located new global minima for LJ88

and LJ98 (35 ). The latter study also in-cludes an implicit “catchment basin” trans-formation of the energy landscape, whichappears to be common to several of themore successful global optimization algo-rithms. This transformation can be separat-ed from consideration of how the resultingsurface is actually searched and is de-scribed in the following section.

Basin-HoppingWe now outline the particular hypersurfacedeformation that appears to be a constituentof several studies in which new global min-ima have been found for LJ clusters(35–39). This “basin-hopping” approach hasproved to be useful for a range of atomicand molecular clusters as well as biomol-ecules, and is easy to implement (40). Thefunctional form is explained in the Appen-dix, part A, and in Fig. 2. In this transfor-mation, the potential energy for every pointin the catchment basin of each local mini-mum becomes the energy of that minimum.These catchment basins partition all of con-figuration space, and so the potential ener-gy can vary only in discrete steps when thegeometry moves from one basin to another.The transformation must be combined witha search strategy, and Monte Carlo sam-pling provides two possibilities if the struc-ture is reset to that of the current localminimum, or allowed to vary continuously.We refer to search techniques coupled tothe catchment basin transformation as ba-sin-hopping. The “Monte Carlo plus energyminimization” (MCM) procedure of Li andScheraga (41) corresponds to coordinateresetting and is generally found to be themore effective approach (19, 39, 42). Stepsare proposed by perturbing the latest set ofcoordinates and carrying out a minimiza-tion from the resulting geometry. A step isaccepted if the energy of the new minimum,Enew, is lower than the starting point, Eold.If Enew is greater than Eold, then the step isaccepted if exp[(Eold 2 Enew)/kT] is greaterthan a random number drawn from the

interval [0,1]. The temperature, T, becomesan adjustable parameter, but is not used forannealing.

In an application of the MCM procedureto LJ clusters, all the lowest known minimawere located up to 110 atoms, and globalminima for LJ69, LJ75–78, LJ103–104, andLJ107 were located for the first time inunbiased searches (39). We have found thatan unbiased genetic algorithm can also findall the lowest LJ minima up to 110 atoms,if each member of the population is mini-mized after every step (43). Use of theminimization step means that the samecatchment basin landscape is searched.This enabled Deaven and Ho (34 ) to findnew global minima for LJ88 and LJ98. Sim-ilarly, the “two-level simulated annealing”approach enabled Xue to find new globalminima for LJ65 and LJ66; this methodbasically corresponds to basin-hoppingwithout resetting the coordinates to those ofthe current minimum (36 ). The “exponen-tial tunneling” approach of Barron et al.was also used to search the catchment basinsurface and to locate the truncated octahe-dron for LJ38 (37 ) (Fig. 1A). The sameauthors also located some of the new globalminima reported in (39) by a seeded search,which again involves local minimization ofstructures (38). Wolf and Landman suc-cessfully located the LJ global minima upto LJ110 using local minimization and aseeded genetic algorithm (44 ).

To explain why the catchment basintransformation is useful for global optimi-zation, it is helpful to examine the thermo-dynamics of the deformed landscape. ForLJ clusters with nonicosahedral global min-ima, Doye and Wales found that the catch-ment basin transformation broadens the oc-cupation probabilities of the different mor-phologies (19). On the original surface theoverlap between these distributions issmall, and so the probability of escape fromthe large basin of icosahedral structures israther low. Other groups have experiment-ed with different sampling distributionswithin the framework of simulated anneal-ing, with the same intention of broadeningtransition regions (45 ).

Some timings for the basic MCM pro-cedure applied to LJ clusters are given inFig. 3 and Table 1 (40). There are two freeparameters: the temperature was fixed, andthe maximum step allowed in the perturba-tion of coordinates that precedes each min-imization was adjusted dynamically to givea particular acceptance ratio. The meannumber of basin-hopping steps and cputime required to find the global minimum ata fixed temperature, are shown in Fig. 3 forLJn up to n 5 74. The results are averagesover 100 different random starting points ineach case. In fact the optimal temperature

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increases somewhat with size, and so Fig. 3does not correspond to the best possibleparameterization for each cluster. Somemore detailed statistics are presented inTable 1 for selected sizes. These resultsillustrate how difficult it is to find thetruncated octahedron for LJ38, and howeasy it is to find the Mackay icosahedronfor LJ55 (Fig. 1B).

It will certainly be possible to improveupon these results by exploring the trans-formed landscape more efficiently, or per-haps by a quite different approach. One pos-sibility is to find pathways by means of truetransition states (46). Another is to use mo-lecular dynamics to simulate the system’sevolution between minimizations. The opti-mal algorithm in any given case is almostcertainly system-dependent and may involvea combination of different methods. Unfortu-nately, testing the efficiency of different al-gorithms for nontrivial problems, such asLJ38 and LJ75, is rather time-consuming.

CrystalsCrystal engineering is an important branchof materials science whose aim is to designsolids with particular properties (47 ). This

effort would be greatly assisted if it werepossible to predict crystal structures fromthe intermolecular potential alone. Unfor-tunately, many compounds of interest arepolymorphic, exhibiting alternative pack-ing modes. This phenomenon causes par-ticular problems for the pharmaceutical in-dustry, where polymorphic “impurities”can lead to undesirable physical properties,which, for example, led to litigation surround-ing the drug Zantac (48). Polymorphismmay also provide a stringent test of empir-ical intermolecular potentials.

The application of global optimizationtechniques to crystal structure prediction isat an early stage of development. Somestudies have attempted to generate plausi-ble starting points for energy minimizationusing common coordination geometries,most probable crystal symmetries, close-packing arguments, and statistical correla-tions (49). Monte Carlo-simulated anneal-ing has also been considered (50) and mo-lecular dynamics techniques have enabledsolvent and kinetic effects to be simulated(51). Williams (52) has approached thisglobal optimization problem by MonteCarlo sampling to predict the crystal struc-

ture of benzene without prior assumption ofthe space group.

Deformation procedures, initially adopt-ed for treating biomolecules, have nowbeen applied to predict crystal structureswithout making use of ancillary informa-tion such as the space group (18). In thiswork the diffusion equation method (DEM)and DSM (Appendix, part B were used topredict the crystal structures of hexasulfurand benzene, which were treated as rigidmolecules (18). After fixing only the mo-lecular geometry and the interaction poten-tial, the unit cell dimensions, space groups,and the number of molecules in the unit cellwere all computed, and the experimentalcrystal structures were successfully locat-ed. For benzene, the calculation succeededeven when the number of molecules in theunit cell was set to twice the experimentalvalue, which made the global optimizationproblem considerably harder.

Genetic algorithms have also been appliedto analyze powder diffraction data, whichdoes not require an intermolecular potentialat all. For example, Kariuki et al. correctlypredicted the previously unknown crystalstructure of ortho-thymotic acid (53). Thisapproach could prove very useful when sin-gle crystals of sufficient size or quality forroutine structure solution are impossible toobtain.

BiomoleculesPredicting the native structure of a proteinfrom its amino acid sequence alone is anarea of intense current research. The poten-tial savings of experimental time and effortalone have stimulated a number of ap-proaches: sequence-homology employs theknown structures of sequences that are sim-ilar to the one in question (54 ), whereas

D

B

C

A

Fig. 2. Illustration ofthe E(X) energy land-scape transformationin two dimensions. (A)Original surface. (B)Transformed surface.Each local minimumof E(X) corresponds toa plateau or catchmentbasin for E(X). The sur-faces are colored con-sistently according tothe energy. (D) Viewof the transformed sur-face from above. (C)Cut through the com-bined E(X) and E(X)surfaces for the redboxed region shown inall the other panels.The catchment basinscan have complicatedboundaries because ofthe finite step size usedin the minimizations.

Fig. 3. Mean time required to find the globalminimum with the MCM procedure for LJn upto n 5 74 (abscissa); the averages are over100 random starting points in each case. Thecpu time (seconds) in the lower (red) curvecorresponds to a 250 MHz Sun Ultra II pro-cessor, whereas the number of basin-hoppingsteps in the (black) upper plot is dimension-less (ordinate).

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threading uses energy (or energy-like)functions to compare the sequence withstructural motifs from a database of knownstructures (55 ). Some previous methodshave combined the global optimization of apotential energy function with constraintsbased on secondary-structure predictionand multiple-sequence alignments, andthese have also been quite successful (56 ).Here we focus on three global optimizationprocedures, DEM (25), DSM (27, 57 ) andconformational space annealing (CSA) (58)(see Appendix, part C ), which have recent-ly produced encouraging results withoutthe use of any sequence or structure anal-ogies and databases (59).

We have applied potential function defor-mation to this global optimization problem,with careful mapping between deformed andundeformed minima using local search tech-niques. The underlying principle is to locatelarge regions of conformational space con-taining low-energy minima by coupling themto some of the greatly reduced number ofminima on the highly deformed surface. TheDSM and the DEM have been implementedto carry out the deformation, each being ap-plied to a different part of the potential func-tion (see Appendix, part B).

The DSM has been applied to united-residue polyalanine chains with a length of upto 100 residues and to staphylococcal proteinA (SPA) (57). It has successfully locatedlow-energy structures of polyalanine chains,predicting that the most stable structure in theabsence of solvent is a straight a-helix for upto 70 residues. For 70–80 residues the moststable form is bent in the middle of the a-he-lix and, from 80 residues upward, the moststable structure is a three-helix bundle. ForSPA, a minimum very close to the experi-mental structure has been located. These re-sults show that hypersurface deformation canbe applied to the conformational analysis ofglobular proteins.

The greatest advantage of the CSAmethod (see Appendix, part C ) is that itfinds distinct families of low-energy con-formations. It was successfully applied toall-atom polypeptide chains for the pen-tapeptide enkephalin (58) and to the 20-residue membrane-bound portion of melit-tin (60). An application using a united-residue representation (61) was also suc-cessful; the method located very lowenergy structures for the fragment of SPAconsisting of residues 10 through 55 andfor apo calbindin (ACB), with a root-mean-square deviation of 2.1 Å and 3.9 Å, re-spectively, from the a-carbon trace of theexperimental structure (62). CSA is also acore part of a recently developed hierarchi-cal approach to protein-structure prediction(59) first fully used on Critical Assessmentof Techniques for Protein Structure Predic-

tion (CASP3) (63) proteins. Figure 4 illus-trates the quality of one of the blind pre-dictions (59).

ConclusionsWe have provided an overview of some se-lected recent developments in the field ofglobal optimization as applied to clusters,crystals, and biomolecules. For atomic andmolecular clusters the basin-hopping ap-proach coupled to search strategies based onMonte Carlo sampling or genetic algorithmsseems to work well. Unbiased algorithms canoften treat systems with at least 100 atoms ormolecules reliably, and we expect biased orseeded approaches to be useful for signifi-cantly larger systems.

The basin-hopping approach has demon-

strated transferability between atomic andmolecular clusters and has also yielded usefulresults for biomolecules (11). However, morecomplicated methods such as conformationalspace annealing have been shown to performbetter for SPA, ACB, the periplasmic proteinHDEA, and a bipartite transcriptional activa-tor MarA (59).

For biomolecules, development of betterpotential functions is clearly a priority. Im-proved potential functions must better de-scribe structures that exist in nature, wherethe global minimum is expected to be sepa-rated by an energy gap from higher energystructures. The quality of the potential is alsoa serious issue in structure prediction forcrystals.

With further improvements in both algo-

Fig. 4. Superposition of the crystal (red) and predicted (yellow) structures of the CASP3 periplasmicprotein HDEA. The Ca atoms of the fragment included between residues Asp25 (D25) and Ile85 (I85)were superposed. The root-mean-square deviation is 4.2 Å. Helices 3, 4, and 5 are indicated as H-3,H-4, and H-5, respectively (59).

Table 1. Mean time and number of basin-hopping steps taken to find the global minimum for selectedLJn clusters with the MCM procedure. The standard deviation is similar to the mean time in each case. Thestatistics were obtained for 1000 random starting points for each size with a fixed acceptance ratio of 0.5and temperature T* (reduced units). T* is the size-dependent optimal temperature that gives the shortestmean time for the given acceptance ratio; it was determined by varying T in steps of 0.1 and gatheringstatistics for samples of 100 random starting points. The cpu times are for a 250 MHz Sun Ultra IIprocessor. rel., relative.

n T*Time

(s)Time(rel.)

StepsSteps(rel.)

30 1.4 8 1 132 1.038 1.1 216 27 2068 16.440 1.4 18 2 152 1.250 1.1 26 3 187 1.555 1.8 22 3 126 1.060 1.4 42 5 198 1.670 1.6 157 20 522 4.180 1.6 243 30 635 5.090 1.6 982 123 1957 15.5100 1.7 1971 246 3335 26.5110 1.7 4268 534 6288 50.0

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rithms and potential energy functions, weexpect to see solutions of previously intrac-table global optimization problems in manydifferent fields (64).

Note added in proof: Using a basin-hop-ping approach, R. H. Leary has found a newnonicosahedral global minimum of Ta sym-metry for LJ98 (14).

AppendixA. A transformed energy landscape. The following trans-

formation of the energy landscape does not change therelative energies of any minima: E(X) 5 min{E(X)}, whereX represents the three-dimensional vector of nuclearcoordinates and “min” signifies that an energy minimiza-tion is carried out starting from X. The transformedenergy, E(X), at any point, X, becomes the energy of thestructure obtained by minimization. Each local minimumis, therefore, surrounded by a catchment basin of constantenergy consisting of all the neighboring geometries fromwhich that particular minimum is obtained. The overallenergy landscape becomes a set of plateaus, one for eachcatchment basin (Fig. 2), but the energies of the localminima are unaffected by the transformation. Aside fromremoving all the transition state regions from the surface,the catchment basin transformation also accelerates thedynamics, because the system can pass between basins allalong their boundaries. Atoms can even pass through eachother without encountering prohibitive energy barriers.

B. The diffusion equation and distance scaling methods.The DEM achieves a smoothing of a potential functionf(X) by transforming it into the function F(X, t), which isthe solution of the diffusion equation with f(X) as thestarting condition and t (time) the deformation parame-ter. The DSM, which is applicable to pairwise interactions,transforms the distance between the centers of interac-tions according to the formula r 5 (r 1 r0t)/(1 1 t),where r0 is the equilibrium distance for a pairwise inter-action. The whole procedure consists of macro-iterationsin which the parameter t controls the deformation chang-es between two extreme values, tmax and tmin (t 5 0corresponds to the original energy surface).

C. Conformational space annealing. The CSA method(58, 60) searches the whole conformational space in its earlystages and then narrows the search to smaller regions withlow energy as the distance cut-off, Dcut, which defines thesimilarity of two conformations, is reduced. As for geneticalgorithms (33), CSA starts with a preassigned number ofrandomly generated and subsequently energy-minimizedconformations. This pool of conformations is called thebank. At the beginning, the bank is a sparse representation ofthe entire conformational space. A number of dissimilarconformations are then selected from the bank, excludingthose that have already been used; they are called seeds.Each seed conformation is modified by changing from onevariable to one-third of the total number of variables per-taining to a contiguous portion of the chain; the new vari-ables are selected from one of the remaining bank confor-mations rather than being picked at random. Each confor-mation is energy minimized to give a trial conformation.

For each trial conformation, a, the closest conformation Afrom the bank (in terms of distance DaA) is determined. IfDaA , Dcut (Dcut being the current cut-off criterion), a isconsidered similar to A; in this case a replaces A in the bankif it is also lower in energy. If a is not similar to A, but itsenergy is lower than that of the highest-energy conforma-tion in the bank, B, a replaces B. If neither of the aboveconditions holds, a is rejected. The narrowing of the searchregions is accomplished by setting Dcut to a large valueinitially (usually one-half of the average pair distance in thebank) and gradually diminishing it as the search progresses.Special attention is paid to selecting seeds that are far fromeach other. One round of the procedure is completed whenthere is no seed to select, that is, all conformations from thebank have already been used. The round is repeated apredetermined number of times.

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64. D.J.W. is grateful to J. Doye for his comments on thismanuscript and to the Royal Society and the Engi-neering and Physical Sciences Research Council forfinancial support. H.A.S. is grateful to J. Pillardy forhelp in writing parts of this manuscript and to theNIH and the NSF for financial support.

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