 Methodology article
 Open Access
 Published:
Cyndi: a multiobjective evolution algorithm based method for bioactive molecular conformational generation
BMC Bioinformaticsvolume 10, Article number: 101 (2009)
Abstract
Background
Conformation generation is a ubiquitous problem in molecule modelling. Many applications require sampling the broad molecular conformational space or perceiving the bioactive conformers to ensure success. Numerous in silico methods have been proposed in an attempt to resolve the problem, ranging from deterministic to nondeterministic and systemic to stochastic ones. In this work, we described an efficient conformation sampling method named Cyndi, which is based on multiobjective evolution algorithm.
Results
The conformational perturbation is subjected to evolutionary operation on the genome encoded with dihedral torsions. Various objectives are designated to render the generated Pareto optimal conformers to be energyfavoured as well as evenly scattered across the conformational space. An optional objective concerning the degree of molecular extension is added to achieve geometrically extended or compact conformations which have been observed to impact the molecular bioactivity (J Comput Aided Mol Des 2002, 16: 105–112). Testing the performance of Cyndi against a test set consisting of 329 small molecules reveals an average minimum RMSD of 0.864 Å to corresponding bioactive conformations, indicating Cyndi is highly competitive against other conformation generation methods. Meanwhile, the highspeed performance (0.49 ± 0.18 seconds per molecule) renders Cyndi to be a practical toolkit for conformational database preparation and facilitates subsequent pharmacophore mapping or rigid docking. The copy of precompiled executable of Cyndi and the test set molecules in mol2 format are accessible in Additional file 1.
Conclusion
On the basis of MOEA algorithm, we present a new, highly efficient conformation generation method, Cyndi, and report the results of validation and performance studies comparing with other four methods. The results reveal that Cyndi is capable of generating geometrically diverse conformers and outperforms other four multiple conformer generators in the case of reproducing the bioactive conformations against 329 structures. The speed advantage indicates Cyndi is a powerful alternative method for extensive conformational sampling and largescale conformer database preparation.
Background
One of the imperative aspects in drug design and development is to perceive corresponding bioactive conformations which determine the physical and biological properties of drugs [1]. Conformation generation is the kernel in computeraided drug design (CADD) methods such as molecular docking [2–4], pharmacophore construction and matching [5, 6], 3D database searching [7–9], 3DQSAR [10–12], and molecular similarity/dissimilarity analysis [13], to name a few. The ability to account for conformational flexibility is highly valued by these methods as it presumes that small molecules have to adopt energyreasonable conformations in respect of different environments. However, according to Boltzmann Law, the properties observed for "one molecule" are actually the conformerensemble averages [14]. The conformers with high energies contribute little to the ensembleaverage properties quantitatively and consequently have to be discarded during the conformation generation process. To select those lowenergy conformers, a bruteforce method can be applied to enumerate a set of conformations to describe the reallife distribution of the molecular conformational ensemble across the energy surface. Unfortunately, thorough conformational sampling may lead to combinatorial explosion problem even if the molecules are decomposed into fragments first and recombined into new conformers using predefined torsion library [15]. Therefore, a practical conformational ensemble should guarantee the conformers are energy reasonable and can span available conformational space evenly.
Recent studies on crystal structures of ligandprotein complexes revealed that the bioactive molecules tend to adopt more extended conformations than compact ones [16] and may be several kcal/mol higher in energy than their respective global energy minima [17]. Although our knowledge about the pharmacologically allowed conformational space is still limited, one of the criteria for accessing conformer generation tools remains to be to what extent the experimental determined conformations can be reproduced as quickly as possible since it's not applicable to cover the whole conformational space in short time. Researchers are referred to the works by Bostrom who evaluated the capability of reproducing the bioactive conformations of several stateofart conformation generation programs [18–20]. When it comes to conformational analysis in which multiple lowenergy conformations are required, the generated conformers need to be geometrically distinct in case that some "hot spots" of the conformational space are over sampled, which cannot reflect the molecular flexibility because duplicated conformers failed to provide new information about the system. From this point of view, conformation generation may be formulated as a multiobjective optimization process in which the optima are not dominated by sole criteria exclusively. Moreover, besides of potential energy and geometrical diversity restraints, other sophisticated or ruleofthumb criteria such as pharmacophore and binding pocket mapping can be implemented to sample more biased conformers fulfilling these objectives.
As a nondeterministic optimization method, genetic algorithm (GA) has been broadly applied in molecular docking, pharmacophore construction, and conformation generation [21–31]. Most traditional GA implementations of conformation generation perturb the dihedral torsions of rotatable bonds (sometimes plus flipping the ring conformations by modifying the geometry of sp^{3} hybridized atoms at the junctions of two (or more) fused aliphatic rings) using single objective function, which can be either potential energy of single individual conformers or some sophisticated designed metric of conformational space completeness/saturation in conjunction with the population each individual belongs to. For a detailed review of application of single objective GA in computeraided drug design see the references [32, 33]. However, in practical application, an ideal conformer ensemble must meet both energy and diversity criteria simultaneously (even though they conflict to each other sometimes), and it's imperative to consider all the objectives to find a set of equally valid optima (namely Paretooptimal solutions) to maintain the balance between these restraints. Consequently multiobjective genetic algorithm (MOGA) or multiobjective evolution algorithm (MOEA) is an alternative appropriate method to solve such tradeoff problems.
MOGA or MOEA method has been successfully applied to diverse areas in CADD, e.g. molecular docking [34], pharmacophore generation [35, 36], combinational library building [37], and QSAR analysis [38]. During the preparation of this manuscript, Vainio used the MOGA method to generate conformer ensembles for drugsize organic molecules [39]. In his work, two uncorrelated objective functions, van de Waals (VDW) energy and dihedral torsion energy, were calculated individually using the MMFF94 force field. A niche filter on the basis of calculating the root mean square deviation (RMSD) between conformers as well as a userdefinable energy cutoff value were used to reduce the size of the generated ensemble in the post processing step. Tested against the CCDC/Astex test set containing 311 entries with known bioactive conformations, the authors declared their method successfully produced lowenergy conformers that are geometrically distinct from each other, which was the result their method designed to obtain.
In this work, a MOEAbased conformation generation method, named Cyndi, is presented with four specially designed objective functions. The conformers are encoded into GA individuals with the dihedral torsions of rotatable bonds, VDW and torsional energy terms are selected as two distinctive objectives to separate the generated conformers in energy space using Tripos force field [40]; moreover, the geometric dissimilarity (GD) of the individuals is characterized by comparing the RMSD value between each individual and the corresponding initial conformer. As an optional objective, the molecular gyration radius is employed to sample the conformations with various geometric extension degrees. As a proofofconcept test, a combined test set composed of 329 small molecules, whose bioactive conformers are available from Protein Data Bank (PDB) [41], is benchmarked against our method. The ability of Cyndi in reproducing bioactive conformations as well as generating geometrically diverse conformational ensembles is well assessed and the results demonstrate that Cyndi provides an effective means to sample the conformational space and find the set of conformers with expected energy stability and geometric diversity.
Results and discussion
Flowchart
The design and application of the genomes and objectives are based on οMOEA method developed by Deb et al. [42]. The general flowchart of Cyndi is elucidated in Figure 1 and summarized below:

a.
Read the input molecules, and then initialize the population of N individuals by perturbing the input 3D conformer with randomly generated dihedral angles. Afterwards, step through the population to pick up the nondominant individuals into archives.

b.
Pick up the parents using tournament selection to generate new child with crossover and mutation operations, calculate the fitness for the new child and update the archive if the new child dominates any individuals in the archive. Repeat the generation procedure until the termination condition is achieved.

c.
Discard the geometrical redundant conformers with the RMSD filter.

d.
Optimize the remaining conformers with energy minimization and discard the conformers outside the energy cutoff, then discard the geometrical redundant conformers again.

e.
Output the final generated conformers.
Profiles of conformer ensembles generated by MOEA method
With traditional singleobjective approach, the solutions identified through several independent runs may be local minimum in a singleobjective space. Hence, compromise between energy and geometrical feasibility has to be explicitly considered to select the solutions from the singleobjective conformational space [37]. Cyndi surmounted this problem by offering a set of pareto solutions from which the users can select a conformer bearing favoured VDW energy but low GD value or vice versa. In this way the Pareto frontier actually covers the different region of the solution space and suppresses the "randomness" inherited with the EA algorithm. Different conformations are generated and possess rationality with respect to at least one objective.
For the case with 2objective functions, the Pareto optimal solutions in the final archive should distribute across a 2D Pareto frontier curve and nonoverlap with each other. For the case with 3objective functions, the Pareto front curve transformed into a 3D surface whose profile is hard to predict. Therefore, to visualize the distribution of the solutions on the Pareto front, the exemplary scatter points as well as surface landscape for the ligand of 1mcr (PDB ID) is plotted in the 3D space composed of 3 objective values of each solution in the final archive (see Figures 2a and 2b). To clarify the result, the 3D hypersurface are mapped onto the plane formed by the other two objectives in contour mode that using continuous deepening colors to represent increasing GD values (see Figure 2c). Then corresponding solutions are also mapped on the same plane as scattered points whose sizes varied with corresponding RMSD values. One can see the solutions in the final archive are separated quite well in the threeobjective space and only few points overlapped on the solution surface (marked with red ovals as outliers). These solutions are the representatives of the feasible solution space.
Computational Performance of Cyndi
The ability for conformation generation of Cyndi was tested against a data set containing 329 drugsized molecules whose bioactive conformations were extracted from the PDB database (see Methods Section). Figure 3 presents the distribution of the rotatable bonds of the test set, the PDB IDs and rotatable bonds numbers corresponding to each molecule are listed in Table A1 (see Additional file 2). Most of the molecules posses less than 11 rotatable bonds. To access the computational cost of Cyndi, the histogram of processing times is presented in Figure 4. For the overwhelming majority of the molecules, the conformation generation time distributes from 0.2 seconds to 0.6 seconds and the average computation time is about 0.49 ± 0.18 seconds per molecule. The computational cost of post minimization was not considered during the evaluation because energy minimization is the major time limit step for most stochastic conformational analysis algorithms and intensively depends on the applied local optimization algorithms, adopted force field, and specified minimization iterations. Alternatively the conformer ensembles generated with Cyndi can also be energy minimized by external softwares implemented with alternative force fields, therefore the energy optimization is discarded from computational time evaluation. As a matter of fact, the average computational time of Cyndi amounts to 23.68 seconds and never exceeds 50 seconds for each molecular conformer ensemble in an independent run, which is still acceptable and superior to that of Balloon or Catalyst Best (at least 100 seconds per ligand conformer ensemble averagely). The result presented here proves that Cyndi is extremely fast and capable of generating conformers in nearly constant time disregarding the molecular flexibility, and therefore, can be utilized to prepare conformational database for largescale molecular database used in virtual screening.
Cyndi is originally designed as an "upstream" conformation generator for ligandbased virtual screening in the pipeline of computeraided drug design. The numbers of generated conformers have to be limited due to the fact that "downstream" calculations are probably highly computationally expensive. Theoretically the sizes of the conformational ensembles may increase significantly for highly flexible molecules. The number of conformers obtained after post processing are plotted against the number of rotatable bonds of each molecules (Figure 5). Interestingly, the numbers of generated conformers don't increase linearly with the numbers of rotatable bonds as expected, instead, a skewed distribution with a peak value of 7 rotatable bonds was observed (as plotted in grey line). For highly flexible molecules with more than 20 rotatable bonds, the RMSD filter scaled on number of rotatable bonds tends to drive the geometric similarity criterion more stringent by increasing RMSD tolerance. Therefore the conformational space in the archive is considered to be more "crowded" and more conformers were discarded for the highly flexible molecules. This treatment was a tradeoff between elimination of conformational "combinatorial explosion" and enumeration of excessive geometrically diverse conformers. Cyndi employs fixed population size, and if one hopes to sample the conformational space for the flexible molecules more thoroughly, multiple runs or increased population size is recommended. Cyndi combines the Pareto optimal conformers from independent runs and outputs final conformation ensemble after energy as well as geometric filter.
Assesing the ability of reproducing bioactive conformations of Cyndi
To evaluate Cyndi's capability of reproducing the bioactive crystal structures, the distribution of minimum RMSD values between bestfit conformers and crystal structures is shown in Figure 6 by plotting against the number of rotatable bonds of corresponding molecules. The minimum RMSD values are linearly correlated to the numbers of rotatable bonds of ligands with R^{2} = 0.68, which is in accordance to the expectation that the predictive inaccuracy increases with the molecular flexibility. The average minimum RMSD value of total 329 conformational ensembles to their crystal bound conformers is 0.864 ± 0.687 Å for 3objective case (0.831 ± 0.601 Å for 4objective case), and in addition, nearly 67.2% of the minimum RMSD values are below 1.0 Å (93.6% below 2.0 Å). As a comparison, both the results of Cyndi with and without the optional objective (gyration radius) are compared with Balloon, and three conformation generation methods provided in Catalyst. Figure 7 shows the histogram of the cumulative distribution of the 329 molecules whose minimum RMSD to crystal conformers is within specific cutoff thresholds. It's apparent that the results of Cyndi are better or equal to those of the 4 methods considering that they share similar percentage of conformers within 2 Å threshold to the bioactive ones. With regard to the apparent potency of bioactive conformations reproduction (35.2% of the minimum RMSD of generated conformations are within 0.5 Å intervals from the bioactive ones) as well as speedup advantage, Cyndi outperforms other methods both in accuracy and efficiency. Table A1 (see Additional file 2) lists more detailed comparisons of Cyndi and other methods for each molecule. The result of initial single 3D conformers generated by ChemAxon's Standardizer module is also tabulated in Table A1 (see Additional file 2) as reference.
Comparing with 3objective case, additional objective of gyration radius slightly bettered the result (lowering the average minimum RMSD from 0687 Å to 0.607 Å) and the number of minimum RMSD within 0–0.5 Å cutoff to the bioactive conformers increased from 107 to 116. Essentially, considering the definition of gyration radius and essence of VDW energy, there would be certain overlapping between these two objectives on penalizing the steric bumped conformations, which has been observed (data not shown). This is part of the reason that gyration radius was formulated as an optional objective. Including gyration radius may improve the qualities of generated conformations for some flexible molecules, but it is not always the case. Actually the Pareto solutions in 4objective space do not necessarily bear larger gyration radius because MOEA algorithm never optimizes towards the extremum of any single objective. Figure 8 depicts the alignments of bestfit conformers for two flexible molecules (PDB entries 1if8 and 1sme) generated with and without gyration radius objective. For 1if8, the bioactive structure adopts a relatively extended conformer with a gyration radius of 5.250 Å; comparing with the conformer generated in 3objective case, the minimum RMSD (to the bioactive one) of conformer generated by Cyndi with 4objective is reduced from 1.186 Å to 0.501 Å with gyration radius increasing from 4.830 Å to 5.233 Å (see Figure 8a); while for 1sme, the minimum RMSD increases from 1.754 Å to 2.569 Å with gyration radius increasing from 6.093 Å to 6.274 Å for 3objective and 4objective cases respectively (see Figure 8b). In fact, additional objective separates the solutions across more refined hyper grids on the Pareto frontiers so as to sample the conformational space more thoroughly. For linear molecules, including gyration radius may boost the procedure of conformational sampling if the bioactive conformers adopt more extended conformation; however, maximizing gyration radius in Cyndi may blur the unbiased sampling procedure for highly branched molecules (as 1sme in Figure 8b) which may harbor multiple reasonable spatial extensions. Practically, because of the inherit irreproducibility flaw of evolution algorithm, one is advised to repeat several independent runs from diverse multiple random conformations both with and without gyration radius objective to sample the conformational space more thoroughly, as Table A2 (see Additional file 3) shows, the results from 3 independent runs of Cyndi outperformed that of single run to some extent.
Assessment geometrical diversity of generated conformer ensembles
The RMSD filter was applied to remove the conformational redundancy. To quantify the geometrical variation, agglomerative hierarchical clustering analysis was performed on the populated distance matrix via calculating the RMSD values between each pair of conformers. The resulted clustering dendrograms as well as corresponding reordered similarity heatmaps for four selected conformer ensembles are shown in Figure 9. In principle, if the conformers in the ensemble are totally geometrically different from each other, explicit nonoverlapping diagonal blocks should emerge from the similarity matrix. The ligand of 1apt (isovaleryl (iva)valvallystaoet) is typically a peptide mimic with 3 peptide bonds and a total of 18 rotatable bonds. Since the stiffness induced by the peptide bonds (energyconstrained close to 0 and 180 degrees), the conformer ensemble is generally clustered into two major groups indicated by two large ondiagonal blocks, the members in each cluster are at least within 3.0 Å RMSD threshold against each other (see Figure 9a). Each of the large clusters is divided into much smaller independent ones if we lower the dissimilarity threshold to 0.5 or 1.0 Å, indicating the generated conformers are geometrically dissimilar to each other at this level. Similar results have also been observed in the case of all other three molecules because they all bear one or more peptide bonds. Comparing with 1apt, more distinguishable smallsize ondiagonal blocks are exemplified by ligand of 1tmn (Ncarboxymethyl dipeptide, 14 rotatable bonds) and that of 1epo (morphenlechfnme, 20 rotatable bonds) in Figures 9b and 9d. The conformers are geometrically distinguishable with each other below the RMSD threshold of 2.0 Å, illustrating the generated conformers are highly distantly spread in conformational space. As for a less flexible case, the ligand of 8gch (glyalatrp tripeptide with 9 rotatable bonds) in Figure 9c exhibits a more blurred profile of clustering heatmap below the RMSD threshold of 2.0 Å. Allowing for the more compact conformational space formed by relatively small number of rotatable bonds, the RMSD threshold for dissimilarity should be lowered by reducing the scaling factor in GD filter.
Theoretically, the generated conformers should maximize diversity by distinguishing geometrically with each other completely. Practically, this goal can not be achieved because for a molecule bearing middle level flexibility (like 1epo), the energyfavoured accessible areas are limited with respect to broadly sampled conformational space. Herein many conformers have to be discarded which failed to pass the lowenergy filter despite their geometrical dissimilarity. The RMSD threshold in the RMSD filter increases with molecular flexibility, and for those molecules with moderate flexibility (with number of rotatable bonds up to 15), the threshold is about 0.5 Å with the scaling parameter set to 0.1. From the observation of the clustering heatmaps, most of the conformers are geometrically different from each other at the threshold level of 0.5 Å considering the corresponding ondiagonal blocks are smallsize and nonoverlapped (filled by deepest red colour). When increasing the threshold, the areas of ondiagonal blocks become larger and the corresponding borders turn out fuzzier. Cyndi employs a scalable RMSD threshold instead and achieves a good compromise to generate limited number (usually no more than 100 averagely) of diverse conformers.
It may be argued that the generated conformers can be biased geometrically to mimic the starting conformation because GD objective reflects essentially minimized RMSD between the individuals and initial conformation. Actually, different from single objective GA in which the population evolves towards the extremum in the single objective space exclusively, the optimization essential of Cyndi is to generate a nondominant solution set in the hyperspace of multiple objectives because no single or a few objectives can exclusively harness the optima towards the extrema in single or parts of objective space. For the Pareto optimal conformers in the final archive, most of them are geometrically distant from the initial conformers because they are favourable in other objective spaces such as VDW and torsion energy. Namely the objectives divide the objective hyperspace into grids with the size of corresponding ο values and the Pareto solutions scatters across those nondominant grids consisting of Pareto frontier.
Currently, Cyndi only supports generating conformer ensemble from an initial 3D conformation, and moreover, since Cyndi merely perturbs the dihedral torsions in optimization procedure, it is not compulsive but highly recommended starting from the conformer converted by the 2Dto3D conversion toolkits or the one energyminimized to local minimum with appropriate force field to ensure the generated conformers bear reasonable bond lengths and bond angles. The work of integrating the 1Dto3D and 2Dto3D features to perfect Cyndi in handling the problem of conformation generation is under progress.
Conclusion
A new conformation generation method Cyndi was designed based on a fast and robust MOEA theory. Using multiple objectives controlling energy accessibility as well as geometric diversity, Cyndi is capable of searching the conformational space in nearly constant time and sampling the Pareto frontier on which both energy and diversity features are favoured. Three independent objectives (VDW energy, torsion energy and geometrical dissimilarity) are employed and tested against 329 ligands with available crystal determined conformations, the resulted minimum RMSD to bioactive conformers amounts to 0.864 Å in average. The objectives can be implemented easily so it's very simple to customize Cyndi to generate different conformation ensembles which are pareto optimals scattering in the hyperspace formed by different objectives.
Methods
Overview of Multiobjective Evolution Algorithm
In real life, most optimization problems involve multiple objectives (possibly in conflict), which should be satisfied simultaneously. Generally, optimization with single objective often results in "optimal" solutions against only one objective but unacceptable with respect to the other ones. To solve this tradeoff problem, multiple instead of single reasonable solutions should be reserved, incorporating multiple objectives optimized simultaneously. A solution is said to be Pareto optimal if it is not dominated by any other solutions in the solution space, which cannot be improved with respect to any objective without worsening at least one of other objectives. The set of all feasible nondominated solutions is referred to as the nondominated or Pareto optimal solutions and the Pareto frontier can be mapped out by ranking the fitness of the Pareto optimal solutions in the search space, see references [43–45] for more details.
Through the operation on the population of individuals with inherent parallel mechanism, GA or EA is suitable to solve multiobjective optimization problems. The crossover operator may exploit structures of good solutions with respect to different objectives to create new nondominated solutions in unexplored parts of Pareto front. There are many MOGA and MOEA methods tailored for different practical problems, and generally speaking they are merely different with respect to fitness assignment procedure, elitism, or diversification approaches [46].
We adopt οMOEA method, a robust algorithm based on the concept of οdomination developed by Deb et al. [47, 48] as our conformational searching engine. In οMOEA, two populations coevolve, one is the general evolving population P(t) identical to those used in singleobjective GA, the other is the "archive" population A(t) serving as the collection of Pareto solutions generated in the t th generation. A(0) is the οnondominated solution set of P(0) after initialization. The crossover operator is applied to two individuals randomly selected from P(t) and A(t), respectively, and each new child is compared with the individuals in the two populations by the οdomination method to check if it would be accepted or not. Each individual in the archive is defined as identification array B as following:
where is the minimum probable value of the j th objective function and ο_{ j }is the userdefinable tolerance between two objective values. Identification array divides the whole j th objective function space into the hyperboxes with the size of ο_{ j }. If the identification array of the new child dominates any archive individual, then the dominated individual is substituted by the new child; otherwise the new child is discarded. If none of the two conditions is satisfied, the new child is a οnondominated solution. Following criteria are used to handle the new child:

1.
When the new child and one archive individual share the same identification array B, namely they are in the same box, the new child would only be accepted either if it dominates the archive individual or it is nearer to B.

2.
When the new child doesn't share one identification array B with any of archive individuals, the new child would be accepted.
In this way, the distribution of Pareto optimal solution set is attained under the condition that each box of the Pareto frontier can only be occupied by one solution. The number of Pareto optimal solutions can be restrained by scaling the ο value.
Similarly, the selection procedure in the population P(t) is also determined by epsilon domination. If the new child dominates one or more individuals in P(t), the new child would substitute one of such individuals randomly, or else it should be discarded. If they are nondominated by each other, one of them would be substituted by the other randomly to fix the size of the population.
Details of MOEA in Cyndi
2.1 Genome Encoding
Similar to the encoding scheme used in traditional single objective GA operated on molecules, namely the molecules are divided into a set of rigid fragments linked by several rotatable single bonds, Cyndi encodes the genome into a vector of real numbers within the interval of [0, 2π), which represents the torsion angle each rotatable bond to be rotated during the conformation generation. The phenotypes, namely the conformers, are easily translated through rotating the single bond by the values coded in each gene. There have been some reports declaring that reducing the continuous torsion space into discrete one (using torsion grid or biasing the torsion value towards the favourable ranges discovered through mining torsion space) may boost convergence [15], however, this assumption may cause missing some conformational significant areas due to the limited accessible information, for this reason, we still manipulate the genomes in continuous torsion space.
2.2 Optimization Objective Functions
Two terms of potential energy calculated with Tripos force field, VDW and torsion energy, are taken as parts of the objectives for separating the solutions in energy space, during which both bond lengths and bond angles are frozen. The torsion term is directly subjected to the geometric variations during the evolution, consequently it is selected as one of the objectives; the second objective is VDW energy in that it increases exponentially with intensive steric bumps so as to penalize the "ugly" conformers. The format of VDW term is a conventional LenardJones 126 function defined in Tripos force field and no cutoff is employed regarding the small size of most drugsize molecules. Since all the energy calculations are performed without considering solvent effects, electrostatic term is also discarded to avoid abnormal compact conformations induced by intramolecular interaction such as internal hydrogen bonds and salt bridges.
The profile of energy landscape is often described as a zigzagged hypersurface. Two geometrically similar conformers may locate quite distant in energy hypersurface but converge to one minimum after energy optimization. Therefore a mechanism is necessary to promote geometric diversity in the population of evolving conformers; the geometric dissimilarity (GD) between each individual and the input conformation is set as the third objective, to control the geometrical distribution of the Pareto optimal solutions. The GD calculation is formulated as eq. (2):
where x_{indiv, i}and x_{input, i}are the positions of the i th heavy atom in each individual and the input conformer. Here only the positions of the heavy atoms are considered in GD calculation.
Previous study indicated most of the bioactive conformers tend to adopt relatively extended geometries [16], in light of this important observation, Cyndi provides an optional objective to direct the evolution towards generating conformers with different geometrical extension degree. Similar to the work by Izrailev et al., the objective that quantifies the degree of extension is formulated by calculating the gyration radius for each molecule, as defined in eq.(3):
where x_{ i }and m_{ i }are the position and mass, respectively, of the i th atom; is the geometrical centre of the molecule. This objective may improve the conformational diversity if we have no idea about whether the bioactive conformer is compact or extended.
The multiobjective optimization model of Cyndi consists of a set of n parameters (design variables), a set of l objective functions, and a set of m constraints. Objective functions and constraints are functions of the design variables. It can be formulated mathematically as follows:
where x is the design vector x= {T_{b 1}, ⋯, T_{ bn }}^{T}, in which T_{b 1}, ⋯, T_{ bn }are the torsion angles of the n th rotatable bonds. Accordingly, the constraints for the design variables (g ( x )s) can be represented as
0 ≤ T_{b 1}, ⋯, T_{ bn }< 2π
y is the objective vector, which consists of VDW term f_{1}(x), torsion score f_{2}(x), and the diversity of geometry f_{3}(x) or f_{4}(x), respectively. Note that the maximal problem for GD or gyration radius of each conformation is converted to minimal problem through reciprocal to ensure the most conformational distribution in geometric space. The maximal X is denoted as the decision space, and Y is denoted as the objective space.
The objective functions of Cyndi are easy to extend and customize without modifying kernel of the algorithm. By integrating other objectives as constraints, such as the interaction energy of ligand with the binding receptor or pharmacophore model matching, Cyndi would evolve to facilitate structure or ligand based virtual screening easily.
2.3 Archive Post Processing
The final archive contains the optimal solutions on Pareto frontier which have been separated by the objectives and are nondominant to each other. However, the conformers in the archive may be energyunfavourable or geometrically crowded because they may be suboptimal in the single objective case. Therefore, the final set of conformations is the pruned result of archive members by userdefinable energy and RMSD filters.
The conformational energy and geometry are optimized using the conjugated gradient (CG) method with Tripos force field, and all energy terms excluding electrostatic term and torsion term lest inner electrostatic interaction and torsion angle perturbation undermine generated conformation. After energy minimization, following energy cutoff is applied to discard those conformers beyond the energy window:
E_{ cutoff }= E_{ w }+ k × N_{ rot }
where E_{ w }is a userdefinable energy cutoff (default is 20 kcal/mol), N_{ rot }is the number of rotatable bonds of current molecule and k is the scaling parameter (userdefinable and the default value is 0.5). Similar to Balloon's approach, such energy cutoff takes into account of the effect imposed by molecular flexibility and reflects the observed dependence between the local minima and number of rotatable bonds of bioactive conformation [49]. Only those conformers within the energy threshold are labelled as energyfavoured and survived to subsequent RMSD filter.
The conformers survived from the energy filter are submitted to a RMSD filter to remove geometrical duplicates. We adopted the filter scheme defined in Catalyst [50]. Conformers are compared with each other in terms of RMSD and any conformer within the RMSD tolerance (vide infra) of another conformer is discarded:
where c is a userdefinable scaling parameter, and N_{ rot }, again, is the number of rotatable bonds. The involvement of molecular flexibility can efficiently reduce the geometrical redundancy for highly flexible molecules.
Testing of Cyndi
The ability for conformation generation of Cyndi was tested against a data set containing 329 drugsized molecules whose bioactive conformations were extracted from the PDB database (see Additional file 1). This is a combined set of part of CCDC/Astex subset [51, 52] used in Balloon's validation and the subset used by Izrailev et.al. [53, 54]. To avoid the influences on the generation of conformers from existing crystal structures, single 3D conformer was regenerated for each molecule from 2D structures using the Standardizer module of ChemAxon package [55].
Cyndi test run was performed on each of the molecules in the test set with 200 populations and 200 generations. The probabilities for crossover and mutation operation were set to 0.85 and 0.1, respectively. The relatively larger mutation probability was designated to sample the conformational space more broadly by mutating the existing conformers more frequently. The epsilon values for the four objectives (VDW energy, torsion energy, GD value, and gyration radius) were set as 20 kcal/mol, 5 kcal/mol, 0.2 Å and 0.1 Å, respectively. The maximum iteration for post processing CG minimization was set to 100, and the convergence criterion based on gradient RMS was set to 0.1 kcal·mol^{1}·Å^{1}. No initial optimization against the input conformers was applied and the input conformers were discarded from the final conformer ensembles. Other parameters were set to the default values mentioned previously.
To make a comparison, the Pentium4 processor optimized version of Balloon (version 0.6.6.4641) [56] as well as three conformer generation methods (Catalyst Best, Catalyst Fast and CEASAR) in DS2.0 [57–60] were run against the same test. All default parameters were applied and the maximum number of generated conformers was limited to 100. The initial single 3D conformers generated by Standardizer of ChemAxon were also retained for further comparison (see Table A1 in Additional file 2).
To make an impartial comparison, the minimum RMSD values between the crystal bound conformers and the conformers generated by all methods were calculated with a thirdparty program, Vrms 1.0, which is an RMSD calculator removing artificial differences caused by symmetry [61]. In this study, a conformation is considered to be successfully reproduced if the RMSD value is less than 0.5 Å, as compared to the original Xray structure. Balloon, Cyndi, Vrms and DS 2.0 were all run on a workstation with an Intel Pentium Dual Core Processor (2.66 GHz).
Hierarchical clustering was applied to analyze the conformational diversity of each conformer ensemble generated by Cyndi. Vrms calculated RMSD values between each pairs of conformers were used to populate a distance matrix, which was then analyzed by applying complete linkage agglomerative hierarchical clustering. The clustering results were visualized by dendrograms and reordered heatmaps of the distance matrix.
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Acknowledgements
This work was supported by the National Natural Science Foundation of China (grants 20803022, 30672539 and 20721003), the Shanghai Committee of Science and Technology (Grant 07dz22004), the 863 HiTech Program of China (Grant 2007AA02Z304), the International Collaboration Project of China (Grant 2007DFB30370) and the National Basic Research Program of China (Grants 2009CB918501 and 2009CB918502). H.L. was also supported by Knowledge Innovation Program of the Chinese Academy of Sciences (Grant SIMM0709QN09). The authors thank Professor Deb for providing MOEA source code, Dr. Vainio for providing Balloon, Dr. Michael J. Potter from VeraChem for Vrms. We also thank Dr. Jiabo Li and Dr. Deqiang Zhang from Accelrys for their helpful discussions and suggestions.
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XL designed and validated Cyndi method and also contributed to analysis and data interpretation. FB contributed to the design of method. SO contributed to method validation. HL conceived the idea of the MOEA in conformation generation and codrafted the manuscript with XL. HL, XW and HJ participated in Cyndi's design, provided direction for its development and revised the subsequent drafts of this manuscript. All authors read and approved the final manuscript.
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Keywords
 Pareto Front
 Root Mean Square Deviation
 Pareto Optimal Solution
 Conformational Space
 Pareto Frontier