Comparison of line search minimization algorithms for exploring topography of multidimensional potential energy surfaces: Mg+Arn case

The most robust numerical algorithms for unconstrained optimization that involve a line search are tested in the problem of locating stable structures and transition states of atomic microclusters. Specifically, the popular quenching technique is compared with conjugate gradient and variable metric algorithms in the Mg⁺Ar_n clusters. It is found that the variable metric method BFGS combined with an approximate line minimization routine is the most efficient, and it shows global convergence properties. This technique is applied to find a few hundred stationary points of Mg⁺Ar₁₂ and to locate isomerization paths between the two most stable icosahedral structures found for Mg⁺Ar₁₂. The latter correspond to a solvated and a nonsolvated ion, respectively. © 1997 John Wiley & Sons, Inc. J Comput Chem 18 :1011–1022, 1997     

A parallel tabu search for conformational energy optimization of oligopeptides

We have developed and implemented a tabu search heuristic (TS) to determine the best energy minimum for oligopeptides. Our test molecule was Met‐enkephalin, a pentapetide that over the years has been used as a validation model for many global optimizers. The test potential energy function was ECEPP/3. Our tabu search implementation is based on assigning integer values to the variables to be optimized, and in facilitating the diversification and intensification of the search. The final output from the TS is treated with a local optimizer, and our best result competes both in quality and CPU time with those reported in the literature. The results indicate that TS is an efficient algorithm for conformational searches. We present a parallel TS version along with experimental results that show that this algorithm allows significant increases in speed. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 147–156, 2000     

A new Lamarckian genetic algorithm for flexible ligand‐receptor docking

We present a Lamarckian genetic algorithm (LGA) variant for flexible ligand‐receptor docking which allows to handle a large number of degrees of freedom. Our hybrid method combines a multi‐deme LGA with a recently published gradient‐based method for local optimization of molecular complexes. We compared the performance of our new hybrid method to two non gradient‐based search heuristics on the Astex diverse set for flexible ligand‐receptor docking. Our results show that the novel approach is clearly superior to other LGAs employing a stochastic optimization method. The new algorithm features a shorter run time and gives substantially better results, especially with increasing complexity of the ligands. Thus, it may be used to dock ligands with many rotatable bonds with high efficiency. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2010     

The minimized dead-end elimination criterion and its application to protein redesign in a hybrid scoring and search algorithm for computing partition functions over molecular ensembles

One of the main challenges for protein redesign is the efficient evaluation of a combinatorial number of candidate structures. The modeling of protein flexibility, typically by using a rotamer library of commonly-observed low-energy side-chain conformations, further increases the complexity of the redesign problem. A dominant algorithm for protein redesign is dead-end elimination (DEE), which prunes the majority of candidate conformations by eliminating rigid rotamers that provably are not part of the global minimum energy conformation (GMEC). The identified GMEC consists of rigid rotamers (i.e., rotamers that have not been energy-minimized) and is thus referred to as the rigid-GMEC. As a postprocessing step, the conformations that survive DEE may be energy-minimized. When energy minimization is performed after pruning with DEE, the combined protein design process becomes heuristic, and is no longer provably accurate: a conformation that is pruned using rigid-rotamer energies may subsequently minimize to a lower energy than the rigid-GMEC. That is, the rigid-GMEC and the conformation with the lowest energy among all energy-minimized conformations (the minimized-GMEC) are likely to be different. While the traditional DEE algorithm succeeds in not pruning rotamers that are part of the rigid-GMEC, it makes no guarantees regarding the identification of the minimized-GMEC. In this paper we derive a novel, provable, and efficient DEE-like algorithm, called minimized-DEE (MinDEE), that guarantees that rotamers belonging to the minimized-GMEC will not be pruned, while still pruning a combinatorial number of conformations. We show that MinDEE is useful not only in identifying the minimized-GMEC, but also as a filter in an ensemble-based scoring and search algorithm for protein redesign that exploits energy-minimized conformations. We compare our results both to our previous computational predictions of protein designs and to biological activity assays of predicted protein mutants. Our provable and efficient minimized-DEE algorithm is applicable in protein redesign, protein-ligand binding prediction, and computer-aided drug design.     

A comparison of heuristic search algorithms for molecular docking

This paper describes the implementation and comparison of four heuristic search algorithms (genetic algorithm, evolutionary programming, simulated annealing and tabu search) and a random search procedure for flexible molecular docking. To our knowledge, this is the first application of the tabu search algorithm in this area. The algorithms are compared using a recently described fast molecular recognition potential function and a diverse set of five protein–ligand systems. Statistical analysis of the results indicates that overall the genetic algorithm performs best in terms of the median energy of the solutions located. However, tabu search shows a better performance in terms of locating solutions close to the crystallographic ligand conformation. These results suggest that a hybrid search algorithm may give superior results to any of the algorithms alone.     

A new method for the gradient‐based optimization of molecular complexes

We present a novel method for the local optimization of molecular complexes. This new approach is especially suited for usage in molecular docking. In molecular modeling, molecules are often described employing a compact representation to reduce the number of degrees of freedom. This compact representation is realized by fixing bond lengths and angles while permitting changes in translation, orientation, and selected dihedral angles. Gradient‐based energy minimization of molecular complexes using this representation suffers from well‐known singularities arising during the optimization process. We suggest an approach new in the field of structure optimization that allows to employ gradient‐based optimization algorithms for such a compact representation. We propose to use exponential mapping to define the molecular orientation which facilitates calculating the orientational gradient. To avoid singularities of this parametrization, the local minimization algorithm is modified to change efficiently the orientational parameters while preserving the molecular orientation, i.e. we perform well‐defined jumps on the objective function. Our approach is applicable to continuous, but not necessarily differentiable objective functions. We evaluated our new method by optimizing several ligands with an increasing number of internal degrees of freedom in the presence of large receptors. In comparison to the method of Solis and Wets in the challenging case of a non‐differentiable scoring function, our proposed method leads to substantially improved results in all test cases, i.e. we obtain better scores in fewer steps for all complexes. © 2008 Wiley Periodicals, Inc. J Comput Chem, 2009     

Conformational search by potential energy annealing: Algorithm and application to cyclosporin A

Summary A major problem in modelling (biological) macromolecules is the search for low-energy conformations. The complexity of a conformational search problem increases exponentially with the number of degrees of freedom which means that a systematic search can only be performed for very small structures. Here we introduce a new method (PEACS) which has a far better performance than conventional search methods.To show the advantages of PEACS we applied it to the refinement of Cyclosporin A and compared the results with normal molecular dynamics (MD) refinement. The structures obtained with PEACS were lower in energy and agreed with the NMR parameters much better than those obtained with MD. From the results it is further clear that PEACS samples a much larger part of the available conformational space than MD does.     

Heuristic-based tabu search algorithm for folding two-dimensional AB off-lattice model proteins

The protein structure prediction problem is a classical NP hard problem in bioinformatics. The lack of an effective global optimization method is the key obstacle in solving this problem. As one of the global optimization algorithms, tabu search (TS) algorithm has been successfully applied in many optimization problems. We define the new neighborhood conformation, tabu object and acceptance criteria of current conformation based on the original TS algorithm and put forward an improved TS algorithm. By integrating the heuristic initialization mechanism, the heuristic conformation updating mechanism, and the gradient method into the improved TS algorithm, a heuristic-based tabu search (HTS) algorithm is presented for predicting the two-dimensional (2D) protein folding structure in AB off-lattice model which consists of hydrophobic (A) and hydrophilic (B) monomers. The tabu search minimization leads to the basins of local minima, near which a local search mechanism is then proposed to further search for lower-energy conformations. To test the performance of the proposed algorithm, experiments are performed on four Fibonacci sequences and two real protein sequences. The experimental results show that the proposed algorithm has found the lowest-energy conformations so far for three shorter Fibonacci sequences and renewed the results for the longest one, as well as two real protein sequences, demonstrating that the HTS algorithm is quite promising in finding the ground states for AB off-lattice model proteins.     

A genetic algorithm for flexible molecular overlay and pharmacophore elucidation

Summary A genetic algorithm (GA) has been developed for the superimposition of sets of flexible molecules. Molecules are represented by a chromosome that encodes angles of rotation about flexible bonds and mappings between hydrogen-bond donor proton, acceptor lone pair and ring centre features in pairs of molecules. The molecule with the smallest number of features in the data set is used as a template, onto which the remaining molecules are fitted with the objective of maximising structural equivalences. The fitness function of the GA is a weighted combination of: (i) the number and the similarity of the features that have been overlaid in this way; (ii) the volume integral of the overlay; and (iii) the van der Waals energy of the molecular conformations defined by the torsion angles encoded in the chromosomes. The algorithm has been applied to a number of pharmacophore elucidation problems, i.e., angiotensin II receptor antagonists, Leu-enkephalin and a hybrid morphine molecule, 5-HT_1D agonists, benzodiazepine receptor ligands, 5-HT₃ antagonists, dopamine D₂ antagonists, dopamine reuptake blockers and FKBP12 ligands. The resulting pharmacophores are generated rapidly and are in good agreement with those derived from alternative means.     

A critical evaluation of several global optimization algorithms for the purpose of molecular docking

Several global optimization algorithms were applied to the problem of molecular docking: random walk and Metropolis Monte Carlo Simulated Annealing as references, and Stochastic Approximation with Smoothing (SAS), and Terminal Repeller Unconstrained Subenergy Tunneling (TRUST) as new methodologies. Of particular interest is whether any of these algorithms could be used to dock a database of typical small molecules in a reasonable amount of time. To address this question, each algorithm was used to dock four small molecules presenting a wide range of sizes, degrees of flexibility, and types of interactions. Of the algorithms tested, only stochastic approximation with smoothing appeared to be sufficiently fast and reliable to be useful for database searches. This algorithm can reliably dock relatively small and fairly rigid molecules in a few seconds, and larger and more flexible molecules in a few minutes. The remaining algorithms tested were able to reliably dock the small and fairly rigid molecules, but showed little or no reliability when docking large or flexible molecules. In addition, to decrease the error in the typical grid-based energy evaluations a new form of interpolation, logarithmic interpolation, is proposed. This interpolation scheme is shown to both quantitatively reduce the numerical error and practically to improve the docking results. © 1999 John Wiley & Sons, Inc. J Comput Chem 20: 1740–1751, 1999     

A computationally efficient alternative to the Buckingham potential for molecular mechanics calculations

This paper describes a (6-8) variant of the Lennard-Jones (6-12) potential,for computing the energy of non-bonded interactions in molecular mechanicscalculations, which combines the overall precision of the Buckingham (6-exp)potential with the computational efficiency of the standard Lennard-Jones(6-12) potential. There is also a note on the radius of convergence of thefull matrix Newton–Raphson optimization procedure.     

The use of quantum molecular calculations to guide a genetic algorithm: a way to search for new chemistry

The process of gene-based molecular evolution has been simulated in silico by using massively parallel density functional theory quantum calculations, coupled with a genetic algorithm, to test for fitness with respect to a target chemical reaction in populations of genetically encoded molecules. The goal of this study was the identification of transition-metal complexes capable of mediating a known reaction, namely the cleavage of N(2) to give the metal nitride. Each complex within the search space was uniquely specified by a nanogene consisting of an eight-digit number. Propagation of an individual nanogene into successive generations was determined by the fitness of its phenotypic molecule to perform the target reaction and new generations were created by recombination and mutation of surviving nanogenes. In its simplest implementation, the quantum-directed genetic algorithm (QDGA) quickly located a local minimum on the evolutionary fitness hypersurface, but proved incapable of progressing towards the global minimum. A strategy for progressing beyond local minima consistent with the Darwinian paradigm by the use of environmental variations coupled with mass extinctions was therefore developed. This allowed for the identification of nitriding complexes that are very closely related to known examples from the chemical literature. Examples of mutations that appear to be beneficial at the genetic level but prove to be harmful at the phenotypic level are described. As well as revealing fundamental aspects of molecular evolution, QDGA appears to be a powerful tool for the identification of lead compounds capable of carrying out a target chemical reaction.     

GRID: A high‐resolution protein structure refinement algorithm

The energy‐based refinement of protein structures generated by fold prediction algorithms to atomic‐level accuracy remains a major challenge in structural biology. Energy‐based refinement is mainly dependent on two components: (1) sufficiently accurate force fields, and (2) efficient conformational space search algorithms. Focusing on the latter, we developed a high‐resolution refinement algorithm called GRID. It takes a three‐dimensional protein structure as input and, using an all‐atom force field, attempts to improve the energy of the structure by systematically perturbing backbone dihedrals and side‐chain rotamer conformations. We compare GRID to Backrub, a stochastic algorithm that has been shown to predict a significant fraction of the conformational changes that occur with point mutations. We applied GRID and Backrub to 10 high‐resolution (≤ 2.8 Å) crystal structures from the Protein Data Bank and measured the energy improvements obtained and the computation times required to achieve them. GRID resulted in energy improvements that were significantly better than those attained by Backrub while expending about the same amount of computational resources. GRID resulted in relaxed structures that had slightly higher backbone RMSDs compared to Backrub relative to the starting crystal structures. The average RMSD was 0.25 ± 0.02 Å for GRID versus 0.14 ± 0.04 Å for Backrub. These relatively minor deviations indicate that both algorithms generate structures that retain their original topologies, as expected given the nature of the algorithms. © 2012 Wiley Periodicals, Inc.     

A physically meaningful method for the comparison of potential energy functions

In the study of the conformational behavior of complex systems, such as proteins, several related statistical measures are commonly used to compare two different potential energy functions. Among them, the Pearson's correlation coefficient r has no units and allows only semiquantitative statements to be made. Those that do have units of energy and whose value may be compared to a physically relevant scale, such as the root-mean-square deviation (RMSD), the mean error of the energies (ER), the standard deviation of the error (SDER) or the mean absolute error (AER), overestimate the distance between potentials. Moreover, their precise statistical meaning is far from clear. In this article, a new measure of the distance between potential energy functions is defined that overcomes the aforementioned difficulties. In addition, its precise physical meaning is discussed, the important issue of its additivity is investigated, and some possible applications are proposed. Finally, two of these applications are illustrated with practical examples: the study of the van der Waals energy, as implemented in CHARMM, in the Trp-Cage protein (PDB code 1L2Y) and the comparison of different levels of the theory in the ab initio study of the Ramachandran map of the model peptide HCO-L-Ala-NH2.     

A genetic algorithm for the structural optimization of Morse clusters

This article describes the application of a genetic algorithm for the structural optimization of 19–50-atom clusters bound by medium-range and short-range Morse pair potentials. The GA is found to be efficient and reliable for finding the geometries corresponding to the previously published global minima [Doye JPK, Wales DJ (1997) J Chem Soc Faraday Trans 93: 4233]. Using the genetic algorithm, only a relatively small number of energy evaluations and minimizations are required to find the global minima. By contrast, a simple random search algorithm often cannot find the global minima of the larger clusters, even after many thousands of searches. Received: 27 October 1999 / Accepted: 7 December 1999 / Published online: 19 April 2000     

A hybrid genetic algorithm for estimating the equilibrium potential of an ion-selective electrode

Non-linear equations can be used to model the measured potential of ion-selective electrodes (ISEs) as a function of time. This can be done by using non-linear least squares regression to fit parameters of non-linear equations to an ISE response curve. In iterative non-linear least squares regression (which can be considered as local optimisers), the determination of starting parameter estimates that yield convergence to the global optimum can be difficult. Starting values away from the global optimum can lead to either abortive divergence or convergence to a local optimum. To address this issue, a global optimisation technique was used to find initial parameter estimates near the global optimum for subsequent further refinement to the absolute optimum. A genetic algorithm has been applied to two non-linear equations relating the measured potential from selected ISEs to time. The parameter estimates found from the genetic algorithm were used as starting values for non-linear least squares regression, and subsequent refinement to the absolute optimum. This approach was successfully used for both expressions with measured data from three different ISEs; namely, calcium, chloride and lead ISEs.     

PRO_LIGAND: An approach to de novo molecular design. 3. A genetic algorithm for structure refinement

Summary Recently, the development of computer programs which permit the de novo design of molecular structures satisfying a set of steric and chemical constraints has become a burgeoning area of research and many operational systems have been reported in the literature. Experience with PRO_LIGAND—the de novo design methodology embodied in our in-house molecular design and simulation system PRO-METHEUS—has suggested that the addition of a genetic algorithm (GA) structure refinement procedure can add value to an already useful tool. Starting with the set of designed molecules as an initial population, the GA can combine features from both high- and low-scoring structures and, over a number of generations, produce individuals of better score than any of the starting structures. This paper describes how we have implemented such a procedure and demonstrates its efficacy in improving two sets of molecules generated by different de novo design projects.     

New out-of-plane angle and bond angle internal coordinates and related potential energy functions for molecular mechanics and dynamics simulations

With currently used definitions of out-of-plane angle and bond angle internal coordinates, Cartesian derivatives have singularities, at ±π/2 in the former case and π in the latter. If either of these occur during molecular mechanics or dynamics simulations, the forces are not well defined. To avoid such difficulties, we provide new out-of-plane and bond angle coordinates and associated potential energy functions that inherently avoid these singularities. The application of these coordinates is illustrated by ab initio calculations on ammonia, water, and carbon dioxide. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 1067–1084, 1999     

A sparse algorithm for the evaluation of the local energy in quantum Monte Carlo

A new algorithm is presented for the sparse representation and evaluation of Slater determinants in the quantum Monte Carlo (QMC) method. The approach, combined with the use of localized orbitals in a Slater-type orbital basis set, significantly extends the size molecule that can be treated with the QMC method. Application of the algorithm to systems containing up to 390 electrons confirms that the cost of evaluating the Slater determinant scales linearly with system size.     

Adding energy minimization strategy to peptide‐design algorithm enables better search for RNA‐binding peptides: Redesigned λ N peptide binds boxB RNA

Our previously developed peptide‐design algorithm was improved by adding an energy minimization strategy which allows the amino acid sidechains to move in a broad configuration space during sequence evolution. In this work, the new algorithm was used to generate a library of 21‐mer peptides which could substitute for λ N peptide in binding to boxB RNA. Six potential peptides were obtained from the algorithm, all of which exhibited good binding capability with boxB RNA. Atomistic molecular dynamics simulations were then conducted to examine the ability of the λ N peptide and three best evolved peptides, viz. Pept01, Pept26, and Pept28, to bind to boxB RNA. Simulation results demonstrated that our evolved peptides are better at binding to boxB RNA than the λ N peptide. Sequence searches using the old (without energy minimization strategy) and new (with energy minimization strategy) algorithms confirm that the new algorithm is more effective at finding good RNA‐binding peptides than the old algorithm. © 2016 Wiley Periodicals, Inc.