Molecular docking methods play an important role in the field of computeraided drug design. In the work, on the basis of the molecular docking program AutoDock, we present QLDock as a tool for flexible molecular docking. For the energy evaluation, the algorithm uses the binding free energy function that is provided by the AutoDock 4.2 tool. The new search algorithm combines the features of a quantumbehaved particle swarm optimization (QPSO) algorithm and local search method of Solis and Wets for solving the highly flexible proteinligand docking problem. We compute the interaction of 23 proteinligand complexes and compare the results with those of the QDock and AutoDock programs. The experimental results show that our approach leads to substantially lower docking energy and higher docking precision in comparison to Lamarckian genetic algorithm and QPSO algorithm alone. QPSOls algorithm was able to identify the correct binding mode of 74% of the complexes. In comparison, the accuracy of QPSO and LGA is 52% and 61%, respectively. This difference in performance rises with increasing complexity of the ligand. Thus, the novel algorithm QPSOls may be used to dock ligand with many rotatable bonds with high accuracy.
Computeraided tools have wide applications in various fields, such as biotechnology engineering, medical engineering, mathematical modeling, and electronic information [
Molecular docking has been developed with primarily two parts varying: scoring function and searching method [
During the last decade, many different tools have been used for the docking problem, such as DOCK [
In this study, we propose a novel search method called QPSOls for solving highly flexible docking problem, which is a hybrid of quantumbehaved particle swarm optimization (QPSO) and a local search method. In 2004, inspired by quantum mechanics and trajectory analysis of PSO [
The remaining part of the paper is arranged as follows. First in Section
Quantumbehaved particle swarm optimization (QPSO) was motivated by concepts from quantum mechanics and particle swarm optimization (PSO); it is a probabilistic optimization algorithm. In QPSO, the state of a particle is depicted by wavefunction
In [
Parameter
Hence, the particle’s position is updated according to the following equation:
This equation is iterative equation of position in quantumbehaved particle swarm optimization (QPSO) algorithm.
The procedure for implementing the QPSO is given by Algorithm
Initialize population: random
do
for
If
find out
for
if
else
endif
endfor
endfor
Until termination criterion is met.
The local search method of Solis and Wets was employed in this study. The method of Solis and Wets is a stochastic heuristic for continuous parameter spaces, which introduces a probabilistic element. Its primal purpose is the local optimization of functions that do not provide gradient information. Basically, the local optimization starts by exploring a random direction in search space and generally follows this direction with random movements as long as the objective function keeps improving. Continued improvements lead to an expansion of the random search steps, whereas continued failing narrows the search. AutoDock adopts a LGA which is a hybrid of GA and a variant of Solis and Wets local search, and the algorithm is illustrated in detail in [
The landscape of energy function contains many local minima for flexible docking, so the performance of using only individual global search or local search may be not satisfactory. By analogy to combining GA with LS, we combined QPSO with LS and applied a hybrid algorithm to solve high flexible docking problem by proper design to enhance advantages of both algorithms as well as reducing their disadvantages to solve specific optimization problem.
In the hybrid search, QPSO can globally explore promising regions with low energy; the local search modifies the position parameters of a particle without referring to other particles and also maintains the diversity among particles. The local search may be applied to the particle according to a predefined probability. Hence, the algorithm QPSOls is beneficial to maintain diversity of particles and in turn prevent premature convergence. The global search method of QPSO and the Solis and Wets local search technique can efficiently compensate with each other in the new QPSOls hybrid algorithm. Simulation results show also that QPSO with local search is more efficient than QPSO without local search. A general structure of the QPSOls hybrid is as shown in Algorithm
Initialize the population;
Random
Evaluate docking energy
do
for
Evaluate docking energy of
Endfor;
Apply the Solis and Wets local search algorithm;
For
update
endfor;
Until termination criterion is met;
Return the best position
Search algorithms are able to quickly generate large number of possible conformations. The purpose of a scoring function is to compare the “quality" of these possible solutions and then select best binding modes. The scoring functions in common use attempt to approximate the binding free energy for the ligand binding to the receptor; a low energy indicates stable system and thus a likely receptorligand binding interaction.
AutoDock 4.2 [
The force field includes six pairwise evaluations (
Each of the pairwise energetic terms is defined by the following energy:
The first term is a typical LennardJones 126 dispersion repulsion interaction. The second term is a directional 1210 hydrogen bond potential. The third term is a screened Coulombic electrostatic potential. The final term is a desolvation potential based on the volume of atoms (
The prepared protein and ligand structures were saved in the PDBQT file format with the AutoDock version 4.2. All the following procedures were performed using AutoDock Tools. First, the protein input files were obtained according to the AutoDock manual:
The 23 proteinligand complexes that are used for the computational experiments are listed in Table
Proteinligand complexes used for the experiments.
PDB  Ligand  Torsions  Dimension  PDB  Ligand  Torsions  Dimension 

1aaq  psi  20  27  1hvr  Xk2  10  17 
1apt  iva  19  26  1nnb  dan  9  16 
1epo  mor  17  24  1nsd  dan  9  16 
1apu  iva  16  23  3tmn  val  7  14 
1icn  ola  15  22  2mcp  pc  4  11 
4phv  vac  15  22  1abf  fca  4  11 
1hpv  478  14  21  1tnk  pra  4  11 
2ifb  plm  14  21  1tnj  pea  3  10 
1htf  g26  13  20  1gsp  sgp  3  10 
1phg  hem  12  19  1tng  amc  2  9 
1cdg  mal  12  19  1tnl  tpa  2  9 
1nsc  sia  10  17 




The docking experiments were performed on the same computer. All methods use the same protein coordinate and start with exactly the same initial location, conformation, and orientation of the ligand. Each docking experiment has been repeated 30 times. The initial population was set to 50 individuals; maximum number of energy evaluations was 2.5 × 10^{5}; maximum number of generations was 27,000. The local search was based on the Solis and Wets local search. The probability of performing local search on individual was 0.06; the iterations of Solis and Wets local search were 300. The other parameters provided by the default setting were the same as in AutoDock.
To compare the performance of the docking problem, we applied the same parameter settings for all algorithms. For the evaluation of each complex, we recorded the docking energy, root mean square deviation (RMSD), running time, and the total number of scoring function evaluations in all docking runs. We rewrote the AutoDock program using C++ under Linux. The docking tests run on an Intel Pentium dual core 3.0 GHz PC with 2 G memory. The OS is Red Hat Enterprise Linux.
To investigate the efficiency of the hybrid algorithm of QPSOls, 30 independent docking experiments were performed for QLDock, QDock, and AutoDock. Optimization algorithms are guided only by energy function; their search ability can be evaluated in terms of docked energy. Table
Comparison of docking energy and RMSD values of QLDock, QDock, and AutoDock.
PDB  QLDock  QDock  AutoDock  



RMSD 


RMSD 


RMSD  
1aaq  −13.18  −3.96  1.64  −7.88  −2.75  3.90  −5.98  −2.84  2.96 
1apt  −11.35  −2.43  1.73  −3.72  −2.21  1.38  −7.74  −0.79  1.92 
1epo  −10.67  −1.54  2.64  −6.27  −4.34  3.11  −5.97  −0.21  5.13 
1apu  −10.25  −2.53  2.11  −6.79  −2.66  1.89  −3.03  −3.19  1.54 
1icn  −10.57  −0.82  1.40  −7.78  −0.48  1.94  −5.47  −0.26  3.49 
4phv  −15.17  −2.86  1.37  −11.67  −1.65  2.88  −10.15  −1.88  2.17 
1hpv  −11.25  −3.53  1.60  −5.9  −2.48  2.16  −6.03  −2.87  2.51 
2ifb  −9.70  −0.55  1.06  −6.87  −0.41  1.61  −10.91  −0.49  1.40 
1htf  −9.4  −5.09  1.64  −10.25  −2.40  3.87  −4.87  −4.04  3.09 
1phg  −19.23  −3.53  0.62  −16.26  −1.94  2.30  −13.5  −3.57  0.91 
1cdg  −7.04  −2.82  3.67  −6.1  −2.28  3.63  −9.40  1.19  3.33 
1nsc  −2.54  −5.49  1.58  −4.5  −2.95  2.49  −1.04  −3.83  1.65 
1hvr  −14.79  −3.64  0.95  −8.18  −2.72  1.81  −10.7  −4.28  0.78 
1nnb  −3.68  −3.72  1.96  −3.66  −1.67  1.12  −3.82  −1.98  1.03 
1nsd  −6.14  −2.97  1.75  −3.68  −2.82  1.16  −4.24  −2.28  2.21 
3tmn  −7.69  −1.86  4.15  −2.40  −2.50  4.20  −6.52  −1.03  6.47 
2mcp  −4.08  −1.00  2.38  −3.17  −1.04  1.50  −2.39  −0.74  1.92 
1abf  −6.91  −0.96  0.78  −5.62  −0.89  0.36  −6.85  −1.20  0.61 
1tnk  −7.09  −0.09  0.63  −5.57  0.05  1.93  −7.38  −0.12  0.82 
1tnj  −6.51  −0.06  1.10  −6.10  −0.09  1.31  −6.34  −0.08  0.91 
1gsp  −6.99  −1.10  1.69  −3.52  −0.65  4.01  −7.35  −0.51  1.79 
1tng  −6.95  0.04  0.96  −5.85  0.01  0.98  −6.62  0.02  0.99 
1tnl  −6.90  −0.12  0.59  −4.14  −0.12  2.25  −6.95  −0.12  0.72 
In Figure
Convergence performance of the lowest docking energy for 1aaq, 1icn, 1hvr, and 1abf during the optimization process.
Moreover, QPSOls finds this ligand complex with lower binding energy after approximately 140,000 computing steps in comparison to 150,000 computing steps found by the LGA. It shows that QPSOls had better convergence performance than LGA, which also employs a local search strategy. The average execution time of QPSOls per independent run was 27.53 s, while that of LGA was 25.69 s. The execution time of QPSOls was slightly greater than that of LGA using the same number 250,000 of function evaluations.
For the complexes with less flexible ligands, the evolution of the lowest energy among the individuals through generations displays a similar behavior for three algorithms, and the algorithm converges very rapidly to lower energies. In most of the cases, the lowest energy is reached almost after 70000 computing steps and remains stable after that for the 1abf/fca complexes. Similar results are obtained also for other less flexible ligands.
For the complexes with more than 15 rotatable bonds, the algorithm QPSOls is able to find conformations with much lower binding energies than QPSO and LGA. In this situation, QPSO and LGA lose the diversity in the population quickly and converge in some local optima. Moreover, the results show that the good performance of QPSOls with respect to QPSO is due to the concept of local search during the optimization, which can help to escape local minima. Although the local search procedure of QPSOls algorithm is computationally more demanding, the chance to find the optimum is higher. Consequently, this algorithm should be well suited for docking of ligands with many rotatable bonds.
To assess the docking accuracy of three programs, the root mean square deviation (RMSD) value with respect to the reference structure was chosen as a measure for the quality of the prediction. RMSD can be calculated using the following formula:
Figure
Comparison of the RMSD values with respect to three various algorithms for all instances.
Comparing the performance of algorithm with respect to the flexibility of a ligand, we divided the investigated complexes into two classes depending on the number of rotatable bonds of the ligand: when the rotatable bond is more than five, ligand is regarded as highly flexible molecule. As listed in Table
In Figure
The docked ligand conformation with the lowest energies for 1aaq, 1hvr, and 1abf (from top to bottom). The native conformations (shown in green) and the predicted conformations (shown in magenta) are represented by sticks and balls. (a) QPSOls of QLDock. (b) QPSO of QDock. (c) LGA of AutoDock.
A QPSOls algorithm for the molecular docking problem was presented in this paper. We evaluated its performance by a series of experiments using 23 complexes. The proposed algorithm uses the energy evaluation function of the wellknown tool AutoDock 4.2. To ensure the preservation of diversities of the particles and prevent the convergence procedure from prematurity, a local search strategy has been implemented. Although the number of computing steps needed to reach convergence increases, the QPSOls algorithm clearly outperforms the QPSO algorithm and the default LGA in terms of docked energy, convergence performance, robustness, and accuracy of the highdimensional molecular docking problem. As demonstrated above, these results indicate also that successful identification of binding modes might be further improved by combining the results from multiple programs, and the QPSO based program seems to be excellent approach and alternative to solve the flexible docking problem in molecular docking.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors gratefully acknowledge financial support from “Qing Lan Project” of Jiangsu province and National Natural Science Foundation of China (60774079 and 61300149).