A new Lamarckian genetic algorithm for flexible ligand-receptor docking

A New Lamarckian Genetic Algorithm for FlexibleLigand-Receptor Docking

JAN FUHRMANN,∗ ALEXANDER RURAINSKI,∗ HANS-PETER LENHOF, DIRK NEUMANNCenter for Bioinformatics, Saarland University, 66123 Saarbrücken, Germany

Received 11 August 2009; Revised 20 October 2009; Accepted 5 November 2009DOI 10.1002/jcc.21478

Published online 15 January 2010 in Wiley InterScience (www.interscience.wiley.com).

Abstract: We present a Lamarckian genetic algorithm (LGA) variant for flexible ligand-receptor docking which allowsto handle a large number of degrees of freedom. Our hybrid method combines a multi-deme LGA with a recently publishedgradient-based method for local optimization of molecular complexes. We compared the performance of our new hybridmethod to two non gradient-based search heuristics on the Astex diverse set for flexible ligand-receptor docking. Ourresults show that the novel approach is clearly superior to other LGAs employing a stochastic optimization method. Thenew algorithm features a shorter run time and gives substantially better results, especially with increasing complexity ofthe ligands. Thus, it may be used to dock ligands with many rotatable bonds with high efficiency.

© 2010 Wiley Periodicals, Inc. J Comput Chem 31: 1911–1918, 2010

Key words: meta-heuristic; ligand-receptor docking; unconstrained minimization; gradient based minimization

Introduction

Ligand-receptor docking is a key task in rational drug design.Although there exist many different algorithms and programs, theproblem is far from being solved. In general, the complexity of thetask rises with increasing flexibility of the molecules.1

In ligand-receptor docking, the ligand’s position is usuallyencoded as a set of real-valued variables, while the bindingfree energy is approximated by a force-field or scoring functionknowledge-based potential energy or scoring function.2 As thereis no practical method that guarantees to find the global optimumof such an objective function in acceptable time, heuristic searchalgorithms are applied to sample the conformational space globally.However, the performance of search heuristics may be improvedconsiderably by local optimization of solutions. Basically, there aretwo distinct classes of local optimization methods: (1) approacheswhich need only function values (non gradient-based methods) and(2) approaches utilizing the function’s derivatives (gradient-basedmethods).

Internally, many docking algorithms and programs representa ligand typically by its translation and orientation and a set oftorsional angles only. While this allows for freezing nonrelevantdegrees of freedom (DoFs) (e.g., bond lengths and angles), thecomputation of the derivatives is either impossible or extremelydifficult. Hence, in practice local optimization is usually performedusing non gradient-based methods. A well-known example for thisapproach is the Lamarckian genetic algorithm (LGA) implementedin autodock.3

The local optimization methods of the second class benefit fromemploying derivatives of the objective function4 and are expectedto find better results faster, e.g., “deeper minima” while requiring

shorter time. Therefore, these approaches are preferred, wheneveruseful derivative information is available. According to our knowl-edge there are only a few programs available, which employ thisapproach. Examples are the recently developed programs LGA-Dock/EM-Dock,5 GlamDock,6 and AutoDock Vina.7 Accordingto our knowledge, GlamDock uses Cartesian coordinates for theinternal molecular representation and a steepest descent minimizer,which results in a large number of DoFs and may lead to knownproblems during optimization.4 Using other, more compact repre-sentations like, e.g., the internal coordinates modeling approach maylead to fewer DoF,8 but may suffer from the so-called gimbal lockphenomenon or other singularities.9

Recently, we have shown how to circumvent the issue of the gim-bal lock when using unit quaternions by exponential mapping andhow to calculate the derivatives for our compact representation.9

These derivatives may be used for the effective local optimiza-tion methods of the second class. Here, we propose the usage ofa multi-deme LGA employing gradient-based local optimizationfor flexible ligand-receptor docking. Although the basic idea of aLGA is not new in itself, to the best of our knowledge, the computa-tional ideas behind our approach have never been applied before inmolecular docking. In short, we combine a multi-deme genetic algo-rithm with a Lamarckian gradient-based genetic algorithm. To thisend, the ligand is represented using a minimal number of DoFs andour gradient-based minimization procedure avoids singularities thatarise during optimization by reparametrisation with high efficiency.

∗These authors contributed equally to this work.

Correspondence to: D. Neumann; e-mail: [email protected]

© 2010 Wiley Periodicals, Inc.

1912 Fuhrmann et al. • Vol. 31, No. 9 • Journal of Computational Chemistry

Using the Astex diverse set, we show that our approach leads tosubstantially higher docking precision and shorter running times incomparison to genetic algorithm using conventional non gradient-based local optimization. This difference in performance rises withincreasing complexity (higher flexibility) of the ligands. In sum-mary, the results indicate that our method is superior to other LGA,which employ the commonly used stochastic algorithm of Solis andWets10 for local optimization.

The Docking Problem as an Optimization Problem

In this section, we briefly formulate the docking problem as an opti-mization problem and sketch the situation before an optimizationalgorithm is applied.

Molecular Representation

Our representation of a molecule closely follows the explanationsin Fuhrmann et al.9 The afore-mentioned compact representation isimplemented as follows: Like other applications,3, 11, 12 we employa model of limited flexibility using translation, orientation, and a setof flexible bonds that connect rigid entities. Thus, we need three realvalues (tx , ty, tz) for the translation and n real values (φ1, .., φn) for nrotatable bonds. For representing the orientation of a body in space,unit quaternions have become a quasi standard. However, becausethe unit quaternion space is only a subset of R

4 with three DoFs,direct optimization of the four interdependent unit quaternion valuesis awkward.13 Therefore, we use exponential mapping14 to map apoint p = (p1, p2, p3) ∈ R

3 onto a unit quaternion q representingthe orientation in the usual way by

q ={

(0, 0, 0, 1) if p = (0, 0, 0)

(sin(0.5‖p‖) p‖p‖ , cos(0.5‖p‖)) otherwise

.

Thus, all possible orientations can be denoted by parameter vectorsp with

‖p‖ ≤ π .

Our local optimization method9 ensures to stay within this shell forthe orientational parameters. In summary, a molecule is representedby a parameter vector

x = (tx , ty, tz, p1, p2, p3, φ1, . . . , φn).

Translations in the directions x, y, and z as well as all torsionsare implemented to form a ring. Hence, for rotations, the distancebetween −180◦ and 180◦ is 0◦ rather than 360◦. This enables thesearch algorithm to rotate freely around flexible torsional angles.For translation, we used periodic boundary condition.

Scoring Function

It is possible to rapidly convert from the afore-mentioned compactrepresentation to the atomic representation. Thus, all continuous

energy or scoring functions which operate on the atomic coordinatesare applicable. The optimization problem is to find conformationswith very deep scores or energy values, in an ideal case global min-ima (if there is more than one) assuming that deeper values of thepicked energy function correspond to energetically more favorableconformations in nature.

In this work, we chose the Gehlhaar scoring function15 becauseof a less frustrated landscape compared to other scoring functionsand a sufficient differentiation of native binding poses from decoystructures.

The score

E = Etor + Epair ,

is composed of one bonded term for the torsional potential

Etor = A · (1 + cos(n · φ − φ0)),

with parameters A, n, and φ0, where only sp3 − sp3 and sp2 − sp3

bonds are considered, and one non-bonded term

Epair =∑i �=j

f (dij),

for a kind of van der Waals interaction, where f is an interval piece-wise linear function of the pairwise atom distance dij of atoms iand j, with each type having different parameters.15 This functionis inherently not continuously differentiable which is a prerequesitefor the application of gradient-based optimization methods, so, weadded a quadratic transition function at each junction of the originallinear segments as described in Fuhrmann et al.9 Furthermore, theoriginal Gehlhaar function provides a separate energy term for theinternal non bonded interaction of the ligand by assigning a penaltyof 104 if two ligand atoms that do not share a bond come closerthan 2.35 Å. This kind of energy calculation is entirely unsuited forthe computation of a gradient for it is highly non continuous. Tocircumvent this problem we use the same term for internal ligand–ligand interactions as for ligand–receptor interactions. For furtherdetails on the parameters and settings, the reader is kindly referredto Fuhrmann et al.9

In the following, we briefly describe how to calculate the gradient

g := ∂E

∂x= ∂(Epair + Etor)

∂x,

of E for the parameters x that enables us to apply our gradient-based local optimization method.9 The gradient of Etor affects onlythe torsional parameters φ1, . . . , φn

∂Etor

∂φ= A · ∂(1 + cos(n · φ − φ0))

∂φ

= −n · A · sin(n · φ − φ0).

The derivatives of the pairwise term Epair can be interpreted as aforce gi acting on each atom i of the ligand. Thus, we have

gi =∑j �=i

f ′(dij)vi − vj

dij,

Journal of Computational Chemistry DOI 10.1002/jcc

A new LGA for Ligand-Receptor Docking 1913

where vi and vj are the position vectors of atoms i and j, respec-tively. For a more detailed account on mapping these forces on ourparameters x, please refer to Fuhrmann et al.9 The derivatives withrespect to the translational parameters are9

∂Epair

∂tx= (1, 0, 0) ·

∑i

gi,

∂Epair

∂ty= (0, 1, 0) ·

∑i

gi,

∂Epair

∂tz= (0, 0, 1) ·

∑i

gi.

Furthermore, if atoms k and j are connected by a flexible bond andatom i is moved by rotating this bond, we obtain that the derivativeof Epair with respect to the corresponding torsional parameter φjk is

∂Epair

∂φjk=

∑i

((vk − vj) × (vi − vj))T gi.

For the orientational parameters p1, p2, p3 we showed that it holds

∂Epair

∂pj=

∑i

(∂R∂pj

Rvi

)T

gi, j = 1, 2, 3,

where R is the rotation matrix corresponding to the rotationrepresented by p1, p2, p3.

Implemented Search Heuristics

In this section, we describe briefly the algorithms and techniques towhich we compared our new method and give the parameter settingswe tested.

A genetic algorithm (GA)16 closely follows the principles ofDarwinian evolutionary theory, in particular natural selection andreproduction. It applies a set of genetic operations to a populationof solution candidates (individuals) of an optimization problem,iteratively producing better results.

The algorithm starts by creating an initial random population offixed size. Afterwards, the objective function is evaluated for eachindividual to calculate its fitness score. Individuals with the worstscores are discarded, while the remaining solution candidates maycreate progeny. Mutation modifies existing individuals, new individ-uals are produced by mating. However, mutations are not applied to anumber of top–ranked individuals, which is called elitism. When thepopulation has grown again to the fixed size, the algorithm iteratesuntil a convergence criterion is met. GAs are widely used for ligand-receptor docking.17–20 Variants of the GA are used in well-knowndocking suites like autodock3, 21 and fitted.22

Two general modifications are known to influence the behaviourof a GA. In the first variant, the so-called Lamarckian genetic algo-rithm23 (LGA), a dedicated local search procedure is applied toimprove the fitness of existing individuals. This variant has beenpopularized for ligand-receptor docking by autodock.3 In the

second variant, the so-called multi-deme GA24 (MDGA), two ormore populations evolve independently, while there is a limitedmigration between them, increasing the genetic diversity of a pop-ulation and in turn facilitating breaking out of local minima. Thisapproach has been employed successfully for the superimpositionof flexible molecules25 and flexible ligand-receptor docking in theprogram gold.11 Finally, combining both modifications leads to themulti-deme LGA (MDLGA), which forms the basis of our hybridmethod.

The Lamarckian Genetic Algorithm

To compare our novel approach to a GA similar to the one used in thewell-known docking suite autodock,3, 21 we implemented a LGAemploying the local search method of Solis and Wets.10 The methodof Solis and Wets10 is a stochastic heuristic for continuous parameterspaces, which, in contrast to deterministic methods, like the Pow-ell26 or the Simplex algorithm,27 introduces a probabilistic element.Its primal purpose is the local optimization of functions that do notprovide gradient information. Our implementation closely followsthe version of autodock 3.1, making adaptations where necessarydue to using the C++ environment of BALL.28 Basically, the localoptimization starts by exploring a random direction in search spaceand generally follows this direction with random movements as longas the objective function keeps improving. Continued improvementslead to an expansion of the random search steps, whereas continuedfailing narrows the search. The algorithm iterates until either a max-imum number of function evaluations is reached or convergence isestablished by the step size falling below a certain threshold value.

In each iteration, the fitness of all individuals was assessed andonly the top 50% of the individuals were allowed to create progenyto replenish the genetic pool, while the other 50% of the populationwere discarded. Selection of the survivors for mating was based onthe rank-order of the individuals. The probability for random muta-tion was the same for all individuals except for the fittest individualwhich was protected from mutation. Finally, a randomly chosenindividual, which was not subject to local optimization in previousiterations, was optimized using the method of Solis and Wets.

The Multi-Deme Lamarckian Genetic Algorithm

To ensure that our approach is not only superior because we use aMDGA variant in contrast to the above-mentioned standard variant,we also expanded the ordinary LGA to a MDLGA using the samelocal search method of Solis and Wets.10 As for the ordinary LGA,we adapted the population size for best results.

The MDLGA uses five island populations which evolve similarto the single populations in the LGA. In addition, starting after 20iterations we allow for migration to occur every four iterations. Forthe migration operations, the populations are rank-ordered accord-ing to the fitness of their best individuals. Migration is allowed onlyif the least fittest population has converged, that is if the differ-ence between the scores of the fittest and the least fittest individualsis below a threshold value of 0.1. Migration is then performed byreplacing the least fittest individual with the information of the fittestindividual from the fittest population.

The New Hybrid Method

Our new hybrid method follows the tradition of evolutionary algo-rithms for molecular docking purposes. It consists of a MDLGA and



Figure 1. Flowchart of our hybrid search heuristic. The algorithm isbased on a multi-deme Lamarckian genetic algorithm and employsa gradient-based optimization method for the local optimization ofindividuals.

our numerical gradient-based local optimzation approach, which iscalled by the MDLGA for (re)optimization of the calculated con-formations. Figure 1 shows the basic schedule of our hybrid searchheuristic.

Numerical gradient-based local optimization techniques proceedin an iterative fashion by walking downhill on the objective functionE. We employed our modified L-BFGS29, 30 quasi-Newton methodproposed in Fuhrmann et al.9 An iteration starts with the calculationof a descent direction d based on a limited amount of accumulatedknowledge about the function of interest collected in previous iter-ations, that is the positions and the gradients of the recent m steps.Note that the computation of d can be done very efficiently withessentially only n(4m + 1) multiplications, where n denotes thenumber of parameters. We used m = 5 and the adaptive scalingdescribed by Liu and Nocedal.31

Later on, our line search proposed in Rurainski et al.32 looks foran approximate local minimum from the current position x into thedirection d, i.e., it solves approximately

λ := arg minµ>0

E(x + µd).

We modified this algorithm by restricting the general line searchproblem to

‖xori + λdori‖ ≤ 3

2π ,

where xori and dori denote the orientational parameters of x and d,respectively. Our line search algorithm 32 can be easily modified toincorporate an upper bound on λ. This ensures that we always stayin a region far away from the singularity 2π where the orientationalderivatives are usable but possibly tending to this singularity. Thus,after the line search has finished, the orientational part xori of theresulting position

x := x + λd

is reparameterized if

‖xori‖ ≥ π

via replacing xori by the equivalent orientation14 (but with betterderivatives∗)

(1 − 2π

‖xori‖)

xori.

Finally, the current iteration finishes by returning the possiblyreparameterized x as the next position.

We implemented four different MDLGAs with gradient-basedoptimization: smallOne and largeOne optimize one individual periteration and population whereas smallAll and largeAll optimizeall individuals. Furthermore, smallOne and smallAll have smallerpopulations than largeOne and largeAll. The maximum numberof iterations for the local search was set to 50. Table 1 lists theparameters used in this work for all tested GA.

Comparison of Search Heuristics

To compare the performance of our new hybrid method to the nongradient-based search heuristics we chose the Astex33 diverse set forflexible ligand-receptor docking. This test set consists of 85 high-resolution structures of protein-ligand complexes with all ligandsfeaturing drug-like properties.

For each optimization method and for each ligand of the Astexdata set we performed 300 docking runs. For this purpose, wedefined a translation box with an edge length of 10 Å centered on thereported binding site. Before each docking run, the ligand’s positionand orientation was randomized. To compare the running time, weapplied the same stopping criterion for all methods: the algorithmstopped when the best score had not improved for a given numberof function evaluations. This maximum number of function evalu-ations per run was calculated as a linear function of the number ofrotatable bonds in the ligand being docked and was between 3000and 5000.

For the evaluation, we recorded the best score and the total num-ber of scoring function evaluations for each docking run. For eachcomplex, we also recorded the energetically most feasible binding

∗We choose π because the derivatives in this region are excellent, while allpossible orientations are still representable.



Table 1. Parameters for the Genetic Algorithms Implemented in this Work.

Search Number of Initial Population Mutation Individualsheuristic populations population size Survivors Elitism rate optimized

LGA 1 100 200 100 1 0.05 1MDLGA 5 20 40 20 1 0.05 1SmallOne 5 5 10 5 1 0.05 1SmallAll 5 5 10 5 1 0.05 5LargeOne 5 10 20 10 1 0.05 1LargeAll 5 10 20 10 1 0.05 5

LGA, Lamarckian genetic algorithm using the Solis and Wets minimizer; MDLGA, LGA using mul-tiple populations; smallOne/largeOne, MDLGA with small/large population size using gradient-basedlocal search optimizing a single individual only; smallAll/largeAll, same as smallOne/largeOne butoptimizing all individuals. Note that the values for multi-deme algorithms are given per population,e.g., the MDLGA features an overall number of individuals of 5 × 40 = 200.

pose found in all our docking runs (including docking results frompreliminary studies). We call this position the target binding pose inthe following and define a hit as a ligand position featuring an rootmean square deviation (RMSD) smaller than 2 Å to the correspond-ing target binding pose.† In addition, we divided the complexes intosets

Sl,u

:= {complex where: l ≤ number of rotatable bonds of ligand ≤ u}

containing ligands with a similar degree of difficulty. In this studywe used three sets S0,3, S4,7, S8,11.

To compare the search methods in terms of their ability to findthe presumed global optimum, we calculated the average number ofhits, the average mean score, and the average best score for eachalgorithm a and each set Sl,u. The average number of hits wascomputed by ∑

j∈Sl,uHa(j)

|Sl,u| ,

where Ha(j) is the number of hits for complex j divided by the totalnumber of docking runs kmax (in our case kmax = 300). The averagemean score was computed by

∑j∈Sl,u

∑kmaxk=1 Ea(j, k)

kmax|Sl,u| ,

where Ea(j, k) is the score for complex j and docking run k. Finally,the average best score was calculated by

∑j∈Sl,u

mink=1...kmax {Ea(j, k)}|Sl,u| .

†The target binding pose does not necessarily coincide with the native bind-ing pose. However, in preliminary studies we found that using the Gehlhaarscoring function the RMSD between target binding pose and native bindingpose is < 2 Å for 79% of the complexes in the Astex diverse set.

To assess the reliability of the results, we defined a saturationmeasure. If h is the number of hits, then we define saturation as thenumber of hits in the h top-ranked results divided by h. For exam-ple, if a meta-heuristic produced 10 hits in one docking experimentand out of the 10 top-ranked results, three were hits, the algorithmachieved a saturation of 0.3.

For a more in-depth comparison of the search heuristics, we alsocomputed the normalized number of hits Ha and the normalizedratio Ra between Ha and the number of function evaluations Fa foreach algorithm a as follows: Let T(i) be the set of complexes withi ∈ Z rotatable bonds of the ligand. Then Ha is calculated for eachset by

Ha(i) :=∑

j∈T(i) Ha(j)

maxα∈A

{∑j∈T(i) Hα(j)

} .

Let Ha(j) be the total number of hits and let Fa(j) be the total numberof function evaluations performed by algorithm a in all docking runswith complex j, then the ratio between Ha(j) and Fa(j) is given by

Ra(i) :=∑

j∈T(i) Ha(j)∑j∈T(i) Fa(j)

.

Finally, the normalized Ra is simply given by:

Ra(i) := Ra(i)

maxα∈A

{∑j∈T(i) Rα(j)

} .

The statistical significance of differences between the searchheuristics was assessed using the Mann–Whitney test as imple-mented in R, a system for statistical computation and graphics.34

Results

The results for the new gradient-based search heuristics and for thenon gradient-based heuristics are summarized in Table 2. We dividedthe test set into three classes, namely ligands of low (0–3 flexible



Table 2. Comparison of Meta-heuristics in Terms of Average Number ofHits, Average Mean and Best Score, Average Number of FunctionEvaluations as well as Saturation.

# Rot. bonds ∅ Hits ∅ Mean ∅ Best ∅ Function ∅

(# ligands) Name [%] score score eval. Saturation

LGA 21.8 −77.7 −99.7 8424 0.79MDLGA 46.6 −89.1 −99.8 14360 0.87

0–3 SmallOne 60.0 −92.6 −99.7 5657 0.93(n = 40) LargeOne 63.9 −92.7 −99.7 6178 0.89

SmallAll 78.7 −96.6 −99.6 6720 0.94LargeAll 84.8 −97.0 −99.6 7369 0.94

LGA 4.9 −84.8 −116.6 11792 0.61MDLGA 15.4 −98.4 −122.3 20275 0.55



LGA 0.1 −89.0 −124.7 15055 —MDLGA 3.8 −106.9 −147.7 26048 0.42



The results are partitioned for small, medium, and large number of flexi-ble torsional angles. Bold letters indicate best results for each column andcategory.

torsional angles, n = 40), medium (4–7 flexible torsional angles,n = 37) and high (more than seven torsional angles, n = 8)complexity.

In general, the gradient-based methods have a higher chance tofind the target binding pose than the non gradient-based methods,that is they produce more hits than LGA and MDLGA. This differ-ence is even more pronounced for ligands of high complexity forwhich the LGA fails almost completely. A similar trend is observedfor the average mean scores, where LGA and MDLGA give worseresults than the gradient based search heuristics. Statistical analysisof the differences in the number of hits confirmed this observa-tion. For ligands with < 8 rotatable bonds, the search algorithmsfall into four distinct “groups” (p < 0.05). LGA and MDLGAeach form a group of their own, while the other two groups containsmallOne/largeOne and smallAll/largeAll, respectively. However,for ligands with more than seven rotatable bonds (n = 8), thedistinction between these groups becomes blurred except for LGA,for which results still differs significantly from all other heuristics.

The average best scores for ligands with few rotatable bonds arecomparable for all methods and this finding seems to hold for lig-ands of medium complexity, too. For ligands of high complexity theperformance of the non gradient-based search methods deterioratesseverely. In contrast, the gradient-based methods, with the exceptionof largeOne, deliver consistent results.

Additionally, the gradient-based heuristics require fewer func-tion evaluations than the non gradient-based methods before meet-ing the stop citerion. However, this gap decreases as ligands getmore complex and the gradient-based methods perform more localsearch steps. Statistical analysis of the number of function evalua-tions for few rotatable bonds shows the same groupings as observed

for the number of hits (p < 0.05): (1) LGA, (2) MDLGA, (3)smallOne/largeOne, and (4) smallAll/largeAll. For ligands withmore than three rotatable bonds, it is not possible to classify LGAinto a single group anymore. However, the differences between thegroups MDLGA, smallOne/largeOne, and smallAll/largeAll are stillsignificant.

As for the saturation, the gradient-based methods optimizing allindividuals have a higher chance to rank hits higher than the othersearch heuristics. For the gradient-based methods optimizing just asingle individual per population and iteration, the one featuring asmaller population seems to produce slightly better results.

Figure 2a displays the chance to find the target-binding poserelative to the best performing search heuristic. Obviously, thegradient-based methods are always well above their competitors,and again, the difference becomes larger as the complexity of theligands increases. Nonetheless, the MDLGA produces a consider-ably higher number of hits than the LGA and seems to perform quitewell.

When evaluating the number of hits per number of functionevaluations, the difference between the gradient-based and the nongradient-based GA is even more pronounced, while the differences

Figure 2. Comparison of the various genetic algorithms with respectto (a) the normalized number of hits and (b) the normalized number ofhits per number of function evaluations. The best performing methodwas used for normalizing the values of all other methods.



between the search heuristics using the same approach for localoptimization are less marked (Fig. 2b). Here, LGA and MDLGAshow a similar profile. Thus, the higher number of hits found bythe MDLGA is counter-balanced by a higher number of functionevaluations and hence a longer run-time of the search algorithm.

Discussion

Our results strongly indicate that population-based search heuristicsbenefit from incorporating a gradient-based search method. Regard-less of the complexity of the ligand, the gradient-based methodsdeliver better results with fewer iterations. For ligands of high com-plexity, the performance of non gradient procedures breaks down,while gradient-based methods are still feasible.

The advantage of the new hybrid method can be exemplified bythe direct comparison between the MDLGA and smallOne, whichboth perform local optimization of a single individual. AlthoughMDLGA features a much larger total population of 200 individualsthan smallOne (50 individuals), the gradient-based search heuris-tic produces more hits while performing much fewer iterations(approximately 50%) than MDLGA — irrespective of the numberof rotatable bonds.

A simple stochastic calculation may help to underline theimprovement. The probability pr to find at least a single hit afterr docking runs is given by

pr = 1 −r−1∏i=0

rtotal − h − i

rtotal − i,

where rtotal is the total number of docking runs and h is the numberof hits. Thus, if we want to achieve a 99% chance to find the targetbinding pose at least once, we have to find min{r | pr ≥ 0.99}. Theenhancement of using the gradient-based approach may be shown byinserting the appropriate figures into the equation for pr . For examplefor ligands of high complexity, we have rtotal = #ligands × kmax =2400 and h = 90 for MDLGA, whereas h = 518 for largeAll. Weeasily see that we must perform at least 118 docking experimentswith MDLGA, but only 19 with largeAll. If we take into account,that largeAll also requires fewer function evaluations, we gain analmost 10-fold speedup.

The improvement in performance indicates that a higher numberof DoFs may be studied during docking. Thus, our method mayfacilitate docking of highly flexible ligands or to incorporate proteinflexibility in the docking algorithm (e.g., in cross docking).

We also tried to answer the question of how many individu-als should be locally optimized. Both heuristics that optimized allindividuals (smallAll, largeAll) performed slightly better than thoseperforming local search for only one individual per population anditeration (smallOne, largeOne). This finding is supported by the factthat smallOne performed better than largeOne: smallOne dedicatesa larger fraction of its function evaluations to the gradient-basedlocal search procedure than largeOne.

Despite the advances in molecular docking presented in thiswork, there is no guarantee that any however sophisticated searchalgorithm will identify the native binding pose correctly. The per-formance of finding a true hit depends directly on the quality ofthe scoring or energy function. Especially when using an unsuited

scoring function, the probability of finding a hit is small and evenif the native binding pose is reconstructed by chance, it is veryimprobable that this result is top-ranked.

Conclusion

By combining a MDGA with a gradient-based optimization method,we evolved a novel search heuristic for flexible ligand-receptordocking that is superior to its non gradient relative and allows forhandling a higher number of DoFs. While the GA is responsiblefor generating new starting positions for local optimization, thegradient-based optimization explores nearby local minima.

However, the good performance of our approach is based onthe general concept of joining a search heuristic for global sam-pling with an efficient gradient-based optimization method. Thus,the usage of our gradient-based optimization method is not confinedto the GA presented here, but it may be combined with other searchheuristics as well.3, 35–37

Although we cannot make a statement on the impact of gradient-based minimization in heuristic global optimization of a real-valuedobjective function in general, ligand-receptor docking seems to beamenable to our approach.

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