A comparison of heuristic search algorithms for molecular docking

A comparison of heuristic searchalgorithms for molecular dockingBy David Westhead, David Clark and ChristopherMurrayPublished in: Journal of Computer-AidedMolecular Design, 11 (1997) 209-228Seminar talk 17.10.2005Kerstin Kunz (K.Kunz@bioinf.uni-sb.de)AbstractThis work describes the implementation andcomparison of four different heuristic searchalgorithms and a random search procedure w. r. t.their performance on five test cases for moleculardocking.IntroductionDocking tries to find the energetically most feasiblethree-dimensional arrangement of two molecules inclose contact with each other. Molecular docking isof great interest in the development ofpharmaceutical agents. The selective recognition ofthe drug molecule by its appropriate target receptoris crucial for a save and effective action of a drugwithin the body. To discover the binding geometryand affinity of two molecules without expensiveand time consuming experimental techniques likesynthesis, co-crystallization and assay, couldsignificantly speed up the early stages of drugdiscovery. There exist several algorithmicsolutions. Whereas the rigid body dockingconsiders only translational and rotational degreesof freedom of the ligand, this method isn’tappropriate to receptor-ligand complexes withconformational changes during the binding process.A more sophisticated solution is provided byflexible docking algorithms, which allow an internalconformational flexibility of the ligand. But withincreasing degrees of freedom, the size of thesearch space grows exponentially. Hence, fast andeffective search algorithms are needed! So far, theonly acceptable solutions are given by heuristicsearch algorithms. A heuristic is an optimizationtechnique that sometimes will work, but not always.It might help us to find solutions that are good, butnot necessarily optimal. The basic idea of heuristicsearch is that, rather than trying all possible searchpaths, you try and focus on paths that seem to takeyou nearer to the goal.Problem representationTo restrict the search space, there must be somesimplifications w. r. t. the degrees of freedom.Neither receptor nor ligand can be considered fullyflexible. Regarding the ligand, there are onlyrotations about rotatable bonds allowed. Thereceptor has some limited flexibility at the activesite, where it is restricted to a box within which theligand is moved w. r. t. translation, orientation andinternal flexibility. The docking variables representthe relative position of ligand and receptor as wellas their internal conformations that are representedby an internal coordinate tree. They are attached tothe ligand. At the root of the tree is a triplet ofvirtual atoms that serves as reference for bondlengths, valence and torsion angles. The variablesare stored as a string of real numbers, on which thealterations are applied to change the ligandsorientation. There must be a measure to evaluate thequality of the docked complexes. As docking iscoupled with a minimization problem, that meansthat the ligand is always trying to be in a state oflowest energy, an energy function is applied toscore the results. The idea is that minimum valuesshould correspond to the preferred binding mode ofthe ligand and that there is a correlation between thevalues of the energy function and the bindingaffinity of the ligand.Search algorithmsEvolutionary programming and genetic algorithmsimitate the natural process of evolution to solve aproblem. Both are similar, they start with apopulation of individuals which properties arerepresented by a string of variables. This string ismodified until no better solutions are found or nosignificant change is observed. The difference isthat evolutionary programming models only thebehavioral linkage between parents and theiroffspring, while genetic algorithms apply alsorecombination between different parents to createnew offspring. The disadvantage of thesealgorithms is the risk to get trapped in localminima.Simulated annealing is a method of solvingminimization problems with a very large number offree parameters, typically with an objective functionthat can be evaluated quickly. It imitates thegrowing of a crystal. To grow a crystal, one beginsby heating the raw materials to a liquid state. Thismolten material is then slowly cooled, until thecrystal structure is frozen in. If the temperature isdecreased too quickly, defects in the crystalstructure develop. A slow temperature decreaseallows these defects to “work themselves out”forming a much better crystal. This process is

called annealing. A perfect crystal contains theminimum of energy of all the final possibilities.Simulated annealing works just like a local searchby examining exchanged solutions and alwaystaking the better solution. The moves are generatedrandomly and so they may cause the objectivefunction to increase, to decrease or to remainunchanged. By analogy with the physical process,the temperature is initially high and so does theprobability of accepting a move increasing theobjective function and barriers can be overcomeeasily. The temperature is gradually decreased asthe search progresses. So the system is cooledslowly and in the end, the probability of accepting amove that increases the objective function becomesvanishing small. In general the temperature islowered in accordance with an annealing schedule.Applying the Metropolis criterion, decreasingmoves are always accepted and increasing movesare accepted with probability e −ΔE T .Tabu search was formally known in operationsresearch and combinatorial optimization problems.€Here, tabu search is applied to the docking problemfor the first time. While other algorithms likeGenetic algorithm risk to get trapped in localminima, preventing perpetually visiting alreadyknown energetically favorable conformationsprevents this in tabu search. This is realized byapplying a “tabu list” to store previously foundsolutions. Only one current solution is maintainedduring the course of a search. An initial solution ischosen at the start of the run, evaluated and addedto the tabu list. Then a user-defined number ofmoves are generated. A move describes a mutationlike procedure on the docking variables. Thescoring is done by the energy function. The movesare examined in rank order. A move is consideredtabu if it generates a solution not sufficientlydifferent from the solutions in the tabu list. Thehighest-ranking move is always accepted if theenergy of the current solution is lower than theenergy of the best solution so far. Otherwise thebest non-tabu move is chosen. The algorithmterminates if neither of these criteria can be met.New current solutions are added to the tabu listsimply at the end until the list is full, afterwardsreplacing already existing solutions. Each but thenext iteration of the search procedure starts fromthe solutions found in the previous iteration. If nosignificant change to the best solution occurs duringa defined number of iterations, the algorithmrestarts from a random position. Else the algorithmterminates with the best solution found.As part of the basis of comparison the authorsimplemented a random search procedure as control.It was expected that this would perform poorer thanthe more sophisticated search algorithms. Becauseof the randomization of start variables it wasimportant to evaluate a statistical performance overa sufficiently large number of independent trials.An important factor is the characteristics of theenergy distribution of the results containing theaverage energy of a solution and the width ofdistribution around this average.The ideal case is a single energy minimum wherethe algorithm produces median energies close to thevalue of this minimum and a narrow distribution ofresults around this value. As the real world is not sosimple and often there is more than one minimum,another criterion is introduced, the rms distance ofthe docking result to the known crystal structure.This helps to classify competing minima if there ismore than one possible solution. Solutions that liewithin a distance of 1.5 Angstrom of thecrystallographic ligand conformation are counted inthe success rate. The test cases describe relevantproblems in computer aided molecular design. Theyare of varying difficulty and reflect different aspectsof molecular recognition.ResultsDHFR-MTX is a standard test case for docking andwas also used in the parameterization of the energyfunction. What it makes an easy case is that there isa single deep minimum near the crystal structure. Itwas shown that success rate and median energywere not correlated. Here, the GA was trapped inlocal minima farther than 1.5 Angstrom from thecrystal structure.Neuraminidase-DANA was chosen because of theelectrostatic and hydrogen bonds considered todominate recognition. The success rates were muchlower than in the first example. Reason for thelower success rate is a more complicated energysurface, because there is possibly more than oneminimum. TS found two dominant clusters of lowenergysolutions and proved to be more effective atglobal search. It was also noticed that there weredeficiencies in the energy function which had astrong bias towards steric fit.HIV-1 protease and XK263 make lipophilicinteractions that is particularly important for goodbinding regarding the steric fit. Here, SA performedbest at median energy and success rate, so there wasa correlation observed for the first time. Thealgorithms showed no big differences in medianenergy, but the semi-interquartile range suggested astatistical difference. This is due to a greater spreadof good solutions around the crystal conformationbecause of the large molecule with less directionalnature of binding in the active site. E. g., theorientation of the ring systems has greater effect onthe calculated rms than on the value of the energyfunction.The next two examples have both strong lipophilicand electrostatic interaction in different bindingpockets. Thus the thrombin examples provide agood test of the ability of the docking potential

function to differentiate between the two types ofinteraction.Thrombin-NAPAP produced the lowest rate ofsuccessful solutions. But in terms of the authors’criterion for success, this example is an exception.The lowest energy is not related to the lowest rms.There are three clusters of solutions found. The firstis close to the crystallographic minimum. Thesecond cluster is farther from the crystal structurebut favored by the scoring function. Here are lowerenergy values than for the first cluster found thatmislead TS more often from the correct solutionthan the other algorithms. The third cluster reflectsagain a weakness of the energy function byfavoring steric fit over salt bridges.Thrombin-argatroban showed some correlationbetween median energy and success rate, thoughthe success rates are relatively low compared to theother examples. A reason can be competing minimaon the energy surface. GA, EP and SA show atendency to get trapped into a deep local minimumwhile TS can escape and do global searching, whichmay be due to its random restart procedure. As thisexample turned out to be a difficult case, becausethere wasn’t a single low-energy minimum easily tolocate, the authors decided to rerun the dockingprocedure with an increasing number of rotatablebonds. The first five rather terminal bonds showedno great effect with the exception of RS that wasseriously compromised. This indicates that thenumber of rotatable bonds is an indicator for thesize of the search space but not for its difficulty.Introducing bond 6 and 7 where bigger parts of themolecule were rotated makes evident that thedifficulty depends on the presence and the characterof competing low energy minima.Summaryoriginal crystallographic conformation. That refersagain to the deficiencies of the energy function thatis biased towards steric fitting and gives nosufficient weight to salt bridges or contains sometoo large energy terms. Here, the docking wasperformed with well-known structures. For thefuture when the procedures will be applied fordocking ligands with unknown bound conformationit is crucial to have a reliable energy function. Sothere is an investigation of various docking energyfunctions needed to learn about their ability topredict binding mode and affinity correctly. It alsoturned out that not the number but the situation ofthe rotatable bonds is crucial. A rotatable bond in ahinge position will make more difficulties than onein a terminal position. This leads to competingminima that make a docking case difficult.A critical point is, that the authors used the lowestenergy as scoring criterion. The realistic positionof a ligand isn’t necessarily to find at the globalenergy minimum, e. g. if this is in the core of theprotein. Generally, it is difficult to define a criterionby that docking results can be evaluated correctly.OutlookPossible future directions are:1. An improved GA without getting trapped in localminima, e. g. by introducing a greater geneticdiversity.2. TS could be adapted to a more local searchbehavior in the end of the search run, e. g. byscaling down the tabu threshold or concomitantlyreducing the length of the tabu list during thedocking run.3. A combination of heuristic search algorithmscould join global and local search qualities.4. As tabu search turned out to be very effective, itcould be applied in other search and optimizationprocedures.It was obvious that random search performed verypoorly because of the size of the search space. Allheuristic methods were effective and gavesatisfactory performance. GA is a very effectivelocal search algorithm but has tendencies to gettrapped in low-energy local minima. TS samplesthe global minimum more frequently, but localregions less deeply. The energy function has a biginfluence on the results.DiscussionTS and GA performed best within different aspects.While TS headed the table at docking success ratethat is joined with the rms distance to the “correctanswer” known from the original complex, GA wasbest w. r. t. the median energy of a solution. It mustbe said that in these examples, the lowest energiesemerging from successful docking runs aregenerally much better than that obtained for the