Deterministic global optimization of molecular structures using interval analysis

The search for the global minimum of a molecular potential energy surface is a challenging problem. The molecular structure corresponding to the global minimum is of particular importance because it usually dictates both the physical and chemical properties of the molecule. The existence of an extremely large number of local minima, the number of which may increase exponentially with the size of the molecule, makes this global minimization problem extremely difficult. A new strategy is described here for solving such global minimization problems deterministically. The methodology is based on interval analysis, and provides a mathematical and computational guarantee that the molecular structure with the global minimum potential energy will be found. The technique is demonstrated using two sets of example problems. The first set involves a relatively simple potential model, and problems with up to 40 atoms. The second set involves a more realistic potential energy function, representative of those in current use, and problems with up to 11 atoms.     

On the investigation of the low-energy conformational space of molecules

A new perspective on traditional energy minimization problems is provided by a connection between statistical thermodynamics and combinatorial optimization (finding the minimum of a function depending on many variables). The joint use of a new method for uncovering the global minimum of intramolecular potential energy functions, based on following the asymptotic behavior of a system of stochastic differential equations, and an iterative-improvement technique, whereby a search for relative minima is made by carrying out local quasi-Newton minimizations starting from many distinct points of the energy hypersurface, proved most effective for investigating the low-energy conformational space of molecules.     

Comparing search strategies for finding global optima on energy landscapes

We provide some tests of the convex global underestimator (CGU) algorithm, which aims to find global minima on funnel-shaped energy landscapes. We use two different potential functions—the reduced Lennard–Jones cluster potential, and the modified Sun protein folding potential, to compare the CGU algorithm with the simplest versions of the traditional trajectory-based search methods, simulated annealing (SA), and Monte Carlo (MC). For both potentials, the CGU reaches energies lower on the landscapes than both SA and MC, even when SA and MC are given the same number of starting points as in a full CGU run or when all methods are given the same amount of computer time. The CGU consistently finds the global minima of the Lennard–Jones potential for all cases with up to at least n=30 degrees of freedom. Finding the global or near-global minimum in the CGU method requires polynomial time [scaling between O(n³) and O(n⁴)], on average. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 1527–1532, 1999     

Protein structure prediction using a combination of sequence homology and global energy minimization I. Global energy minimization of surface loops

A procedure has been developed for global energy minimization of surface loops of proteins in the presence of a fixed core. The ECEPP potential function has been modified to allow more accurate representations of hydrogen bond interactions and intrinsic torsional energies. A computationally efficient representation of hydration free energy has been introduced. A local minimization procedure has been developed that uses a cutoff distance, minimization with respect to subsets of degrees of freedom, analytical second derivatives, and distance constraints between rigid segments to achieve efficiency in applications to surface loops. Efficient procedures have been developed for deforming segments of the initial backbone structure and for removing overlaps. Global energy minimization of a surface loop is accomplished by generating a sequence (or a trajectory) of local minima, the component steps of which are generated by searching collections of local minima obtained by deforming seven-residue segments of the surface loop. The search at each component step consists of the following calculations: (1) A large collection of backbone structures is generated by deforming a seven-residue segment of the initial backbone structure. (2) A collection of low-energy backbone structures is generated by applying local energy minimization to the resulting collection of backbone structures (interactions involving side chains that will be searched in this component step are not included in the energy). (3) One low-energy side-chain structure is generated for each of the resulting low-energy backbone structures. (4) A collection of low-energy local minima is generated by applying local energy minimization to the resulting collection of structures. (5) The local minimum with the lowest energy is retained as the next point of the trajectory. Applications of our global search procedure to surface segments of bovine pancreatic trypsin inhibitor (BPTI) and bovine trypsin suggest that component-step searches are reasonably complete. The computational efficiency of component-step searches is such that trajectories consisting of about 10 component steps are feasible using an FPS-5200 array processor. Our procedure for global energy minimization of surface loops is being used to identify and correct problems with the potential function and to calculate protein structure using a combination of sequence homology and global energy minimization.     

A strategy to find minimal energy nanocluster structures

An unbiased strategy to search for the global and local minimal energy structures of free standing nanoclusters is presented. Our objectives are twofold: to find a diverse set of low lying local minima, as well as the global minimum. To do so, we use massively the fast inertial relaxation engine algorithm as an efficient local minimizer. This procedure turns out to be quite efficient to reach the global minimum, and also most of the local minima. We test the method with the Lennard–Jones (LJ) potential, for which an abundant literature does exist, and obtain novel results, which include a new local minimum for LJ₁₃, 10 new local minima for LJ₁₄, and thousands of new local minima for . Insights on how to choose the initial configurations, analyzing the effectiveness of the method in reaching low‐energy structures, including the global minimum, are developed as a function of the number of atoms of the cluster. Also, a novel characterization of the potential energy surface, analyzing properties of the local minima basins, is provided. The procedure constitutes a promising tool to generate a diverse set of cluster conformations, both two‐ and three‐dimensional, that can be used as an input for refinement by means of ab initio methods. © 2013 Wiley Periodicals, Inc.     

Do intelligent configuration search techniques outperform random search for large molecules?

We compare three global configuration search methods on a scalable model problem to measure relative performance over a range of molecule sizes. Our model problem is a 2-D polymer composed of atoms connected by rigid rods in which all pairs of atoms interact via Lennard–Jones potentials. The global minimum energy can be calculated analytically. The search methods are all hybrids combining a global sampling algorithm with a local refinement technique. The sampling methods are simulated annealing (SA ), genetic algorithms (GA ), and random search. Each of these uses a conjugate gradient (CG ) routine to perform the local refinement. Both GA and SA perform progressively better relative to random search as the molecule size increases. We also test two other local refinement techniques in addition to CG , coupled to random search as the global method. These are simplex followed by CG and simplex followed by block-truncated Newton. For small problems, the addition of the intermediate simplex step improved the performance of the overall hybrid method. © 1992 John Wiley & Sons, Inc.     

Conformational search using a molecular dynamics–minimization procedure: Applications to clusters of coulombic charges,Lennard–Jones particles,and waters

A new conformational search method, molecular dynamics–minimization (MDM), is proposed, which combines a molecular dynamics sampling strategy with energy minimizations in the search for low-energy molecular structures. This new method is applied to the search for low energy configurations of clusters of coulombic charges on a unit sphere, Lennard–Jones clusters, and water clusters. The MDM method is shown to be efficient in finding the lowest energy configurations of these clusters. A closer comparison of MDM with alternative conformational search methods on Lennard–Jones clusters shows that, although MDM is not as efficient as the Monte Carlo–minimization method in locating the global energy minima, it is more efficient than the diffusion equation method and the method of minimization from randomly generated structures. Given the versatility of the molecular dynamics sampling strategy in comparison to Monte Carlo in treating molecular complexes or molecules in explicit solution, one anticipates that the MDM method could be profitably applied to conformational search problems where the number of degrees of freedom is much greater. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 60–70, 1998     

Gradient tabu search

This paper presents a modification of the tabu search called gradient tabu search (GTS). It uses analytical gradients for a fast minimization to the next local minimum and analytical diagonal elements of the Hessian to escape local minima. For an efficient blocking of already visited areas tabu regions and tabu directions are introduced into the tabu list (TL). Trials with various well-known test functions indicate that the GTS is a very promising approach to determine local and global minima of differentiable functions. Possible application areas could be optimization routines for force field parameters or conformational searches for large molecules.     

Conformational search of peptides and proteins: Monte Carlo minimization with an adaptive bias method applied to the heptapeptide deltorphin

The energy function of a protein consists of a tremendous number of minima. Locating the global energy minimum (GEM) structure, which corresponds approximately to the native structure, is a severe problem in global optimization. Recently we have proposed a conformational search technique based on the Monte Carlo minimization (MCM) method of Li and Scheraga, where trial dihedral angles are not selected at random within the range [-180 degrees,180 degrees ] (as with MCM) but with biased probabilities depending on the increased structure-energy correlations as the GEM is approached during the search. This method, called the Monte Carlo minimization with an adaptive bias (MCMAB), was applied initially to the pentapeptide Leu-enkephalin. Here we study its properties further by applying it to the larger peptide with bulky side chains, deltorphin (H-Tyr-D-Met-Phe-His-Leu-Met-Asp-NH(2)). We find that on average the number of energy minimizations required by MCMAB to locate the GEM for the first time is smaller by a factor of approximately three than the number required by MCM-in accord with results obtained for Leu-enkephalin.     

Many-body optimization using an ab initio monte carlo method

Advances in computing power have made it possible to study solvated molecules using ab initio quantum chemistry. Inclusion of discrete solvent molecules is required to determine geometric information about solute/solvent clusters. Monte Carlo methods are well suited to finding minima in many-body systems, and ab initio methods are applicable to the widest range of systems. A first principles Monte Carlo (FPMC) method was developed to find minima in many-body systems, and emphasis was placed on implementing moves that increase the likelihood of finding minimum energy structures. Partial optimization and molecular interchange moves aid in finding minima and overcome the incomplete sampling that is unavoidable when using ab initio methods. FPMC was validated by studying the boron trifluoride-water system, and then the method was used to examine the methyl carbenium ion in water to demonstrate its application to solvation problems.     

Electrostatic solvation free energy of amino acid side chain analogs: implications for the validity of electrostatic linear response in water

Electrostatic free energies of solvation for 15 neutral amino acid side chain analogs are computed. We compare three methods of varying computational complexity and accuracy for three force fields: free energy simulations, Poisson-Boltzmann (PB), and linear response approximation (LRA) using AMBER, CHARMM, and OPLS-AA force fields. We find that deviations from simulation start at low charges for solutes. The approximate PB and LRA produce an overestimation of electrostatic solvation free energies for most of molecules studied here. These deviations are remarkably systematic. The variations among force fields are almost as large as the variations found among methods. Our study confirms that success of the approximate methods for electrostatic solvation free energies comes from their ability to evaluate free energy differences accurately.     

Analysis of the genetic algorithm method of molecular conformation determination

Many important problems in chemistry require knowledge of the 3-D conformation of a molecule. A commonly used computational approach is to search for a variety of low-energy conformations. Here, we study the behavior of the genetic algorithm (GA) method as a global search technique for finding these low-energy conformations. Our test molecule is cyclic hexaglycine. The goal of this study is to determine how to best utilize GAs to find low-energy populations of conformations given a fixed amount of CPU time. Two measures are presented that help monitor the improvement in the GA populations and their loss of diversity. Different hybrid methods that combine coarse GA global search with local gradient minimization are evaluated. We present several specific recommendations about trade-offs when choosing GA parameters such as population size, number of generations, rate of interaction between subpopulations, and combinations of GA and gradient minimization. In particular, our results illustrate why approaches that emphasize convergence of the GA can actually decrease its effectiveness as a global conformation search method. © John Wiley & Sons, Inc.     

Powerful simulated-annealing algorithm locates global minimum of protein-folding potentials from multiple starting conformations

Protein-folding potentials, designed with the explicit goal that the global energy minimum correspond to crystallographically observed conformations of protein molecules, may offer great promise toward calculating native protein structures. Achieving this promise, however, depends on finding an effective means of dealing with the multiple-minimum problem inherent in such potentials. In this study, a protein-folding-potential test system has been developed that exhibits the properties of general protein-folding potentials yet has a unique well-defined global energy minimum corresponding to the crystallographically determined conformation of the test molecule. A simulated-annealing algorithm is developed that locates the global minimum of this potential in four of eight test runs from random starting conformations. Exploration of the energy-conformation surface of the potential indicates that it contains the numerous local minima typical of protein-folding potentials and that the global minimum is not easily located by conventional minimization procedures. When the annealing algorithm is applied to a previously developed actual folding potential to analyze the conformation of avian pancreatic polypeptide, a new conformer is located that is lower in energy than any conformer located in previous studies using a variety of minimization techniques.     

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.     

Optimising mobile phase composition, its flow-rate and column temperature in HPLC using taboo search

A chemometric methodology is proposed to study the separation of seven p-hydroxybenzoic esters in reversed phase liquid chromatography (RPLC). Fifteen experiments were found to be necessary to find a mathematical model which linked a novel chromatographic response function (CRF) with the column temperature, the water fraction in the mobile phase and its flow rate. The CRF optimum was determined using a new algorithm based on Glover's taboo search (TS). A flow-rate of 0.9 ml min(-1) with a water fraction of 0.64 in the ACN-water mixture and a column temperature of 10 degrees C gave the most efficient separation conditions. The usefulness of TS was compared with the pure random search (PRS) and simplex search (SS). As demonstrated by calculations, the algorithm avoids entrapment in local minima and continues the search to give a near-optimal final solution. Unlike other methods of global optimisation, this procedure is generally applicable, easy to implement, derivative free, conceptually simple and could be used in the future for much more complex optimisation problems.     

Gradient gravitational search: An efficient metaheuristic algorithm for global optimization

The adaptation of novel techniques developed in the field of computational chemistry to solve the concerned problems for large and flexible molecules is taking the center stage with regard to efficient algorithm, computational cost and accuracy. In this article, the gradient‐based gravitational search (GGS) algorithm, using analytical gradients for a fast minimization to the next local minimum has been reported. Its efficiency as metaheuristic approach has also been compared with Gradient Tabu Search and others like: Gravitational Search, Cuckoo Search, and Back Tracking Search algorithms for global optimization. Moreover, the GGS approach has also been applied to computational chemistry problems for finding the minimal value potential energy of two‐dimensional and three‐dimensional off‐lattice protein models. The simulation results reveal the relative stability and physical accuracy of protein models with efficient computational cost. © 2015 Wiley Periodicals, Inc.     

A reliable prediction of the global phase stability for liquid–liquid equilibrium through the simulated annealing algorithm: Application to NRTL and UNIQUAC equations

The simulated annealing algorithm is introduced to search the global optimal solutions for the multipeak phenomena which generally exist in the phase stability problems with continuous variables. The Gibbs free energy criterion was modeled by the NRTL and UNIQUAC activity coefficient equations. When previous approaches fail, it is usually because they locate local minima due to the nonconvex and nonlinear natures of the models used to predict phase equilibrium. In this paper, the preliminary results show that the global minimum of the tangent plane distance function (TPDF) can be obtained by using the simulated annealing algorithm. The effects of the initial and the final values of the control parameter, the decrement of the control parameter and the length of the Markov chains are analyzed. The optimal `cooling schedule' was obtained according to the calculation results of the phase stability problems for one ternary mixture. The liquid–liquid equilibrium compositions were calculated by the Newton–Raphson method on the basis of the global minimum of TPDF. The results of four examples show that the simulated annealing algorithm can effectively solve the global phase stability problem.     

Simulated annealing to locate various stationary points in semiempirical methods

In this work, an algorithm was developed to study the potential energy surfaces in the coordinate spaces of molecules by a nonlocal way, in contrast to classic energy minimizers as the BFGS or the DFP method. This algorithm, based on the specificities of semiempirical methods, mixes simulated annealing and local searches to reduce computation costs. By this technique, the global energy minimum can be localized. Moreover, local minima that are close in energy to the global minimum are also obtained. If the search is not only for minima but for all stationary points (minima, saddle points…), then the energy is replaced by the gradient norm, which reaches its minimum values at stationary points. The annealing process is stopped before having accurately reached the global minimum and generates a list of geometries whose energies (respectively, whose gradients) are optimized by local minimizers. This list of geometries is shortened from the nearly equivalent geometries by a dynamic single-clustering analysis. The energy/gradient local minimizers act on the clustered list to produce a set of minima/stationary points. A targeted search of these points and reduction of the costs are reached by the way of several penalty functions. They eliminate—without energy calculation—most of the points generated by random walks on the potential energy surface. These penalty functions (on the total moment of inertia or on interatomic distances) are specific to the class of problem studied. They account for the nonrupture of either specific chemical bonds or rings in cyclic molecules, they assure that molecular systems are kept bonded, and they avoid the collapsing of atoms. © 1992 John Wiley & Sons, Inc.     

A powerful truncated Newton method for potential energy minimization

With advances in computer architecture and software, Newton methods are becoming not only feasible for large-scale nonlinear optimization problems, but also reliable, fast and efficient. Truncated Newton methods, in particular, are emerging as a versatile subclass. In this article we present a truncated Newton algorithm specifically developed for potential energy minimization. The method is globally convergent with local quadratic convergence. Its key ingredients are: (1) approximation of the Newton direction far away from local minima, (2) solution of the Newton equation iteratively by the linear Conjugate Gradient method, and (3) preconditioning of the Newton equation by the analytic second-derivative components of the “local” chemical interactions: bond length, bond angle and torsional potentials. Relaxation of the required accuracy of the Newton search direction diverts the minimization search away from regions where the function is nonconvex and towards physically interesting regions. The preconditioning strategy significantly accelerates the iterative solution for the Newton search direction, and therefore reduces the computation time for each iteration. With algorithmic variations, the truncated Newton method can be formulated so that storage and computational requirements are comparable to those of the nonlinear Conjugate Gradient method. As the convergence rate of nonlinear Conjugate Gradient methods is linear and performance less predictable, the application of the truncated Newton code to potential energy functions is promising.     

Potential energy and free energy landscapes

Familiar concepts for small molecules may require reinterpretation for larger systems. For example, rearrangements between geometrical isomers are usually considered in terms of transitions between the corresponding local minima on the underlying potential energy surface, V. However, transitions between bulk phases such as solid and liquid, or between the denatured and native states of a protein, are normally addressed in terms of free energy minima. To reestablish a connection with the potential energy surface we must think in terms of representative samples of local minima of V, from which a free energy surface is projected by averaging over most of the coordinates. The present contribution outlines how this connection can be developed into a tool for quantitative calculations. In particular, stepping between the local minima of V provides powerful methods for locating the global potential energy minimum, and for calculating global thermodynamic properties. When the transition states that link local minima are also sampled we can exploit statistical rate theory to obtain insight into global dynamics and rare events. Visualizing the potential energy landscape helps to explain how the network of local minima and transition states determines properties such as heat capacity features, which signify transitions between free energy minima. The organization of the landscape also reveals how certain systems can reliably locate particular structures on the experimental time scale from among an exponentially large number of local minima. Such directed searches not only enable proteins to overcome Levinthal's paradox but may also underlie the formation of "magic numbers" in molecular beams, the self-assembly of macromolecular structures, and crystallization.