Overcoming entropic barrier with coupled sampling at dual resolutions

An enhanced sampling method is proposed for ab initio protein folding simulations. The new method couples a high-resolution model for accuracy and a low-resolution model for efficiency. It aims to overcome the entropic barrier found in the exponentially large protein conformational space when a high-resolution model, such as an all-atom molecular mechanics force field, is used. The proposed method is designed to satisfy the detailed balance condition so that the Boltzmann distribution can be generated in all sampling trajectories in both high and low resolutions. The method was tested on model analytical energy functions and ab initio folding simulations of a beta-hairpin peptide. It was found to be more efficient than replica-exchange method that is used as its building block. Analysis with the analytical energy functions shows that the number of energy calculations required to find global minima and to converge mean potential energies is much fewer with the new method. Ergodic measure shows that the new method explores the conformational space more rapidly. We also studied imperfect low-resolution energy models and found that the introduction of errors in low-resolution models does decrease its sampling efficiency. However, a reasonable increase in efficiency is still observed when the global minima of the low-resolution models are in the vicinity of the global minimum basin of the high-resolution model. Finally, our ab initio folding simulation of the tested peptide shows that the new method is able to fold the peptide in a very short simulation time. The structural distribution generated by the new method at the equilibrium portion of the trajectory resembles that in the equilibrium simulation starting from the crystal structure.     

Monte Carlo temperature basin paving with effective fragment potential: an efficient and fast method for finding low-energy structures of water clusters (H2O)20 and (H2O)25

Determining low-energy structures of large water clusters is a challenge for any optimization algorithm. In this work, we have developed a new Monte Carlo (MC)-based method, temperature basin paving (TBP), which is related to the well-known basin hopping method. In the TBP method, the Boltzmann weight factor used in MC methods is dynamically modified based on the history of the simulation. The states that are visited more are given a lower probability by increasing their temperatures and vice versa. This allows faster escapes from the states frequently visited in the simulation. We have used the TBP method to find a large number of low-energy minima of water clusters of size 20 and 25. We have found structures energetically same to the global minimum structures known for these two clusters. We have compared the efficiency of this method to the basin-hopping method and found that it can locate the minima faster. Statistical efficiency of the new method has been investigated by running a large number of trajectories. The new method can locate low-energy structures of both the clusters faster than some of the reported algorithms for water clusters and can switch between high energy and low-energy structures multiple times in a simulation illustrating its efficiency. The large number of minima obtained from the simulations is used to get both general and specific features of the minima. The distribution of minima for these two clusters based on the similarity of their oxygen frames shows that the (H(2)O)(20) can have different variety of structures, but for (H(2)O)(25), low-energy structures are mostly cagelike. Several (H(2)O)(25) structures are found with similar energy but with different cage architectures. Noncage structures of (H(2)O)(25) are also found but they are 6-7 kcal/mol higher in energy from the global minimum. The TBP method is likely to play an important role for exploring the complex energy landscape of large molecules.     

A systematic study of minima in alanine dipeptide

The alanine dipeptide is a standard system to model dihedral angles in proteins. It is shown that obtaining the Ramachandran plot accurately is a hard problem because of many local minima; depending on the details of geometry optimizations, different Ramachandran plots can be obtained. To locate all energy minima, starting from geometries from MD simulations, 250,000 geometry optimizations were performed at the level of RHF/6-31G*, followed by re-optimizations of the located 827 minima at the level of MP2/6–311++G**, yielding 30 unique minima, most of which were not previously reported in literature. Both in vacuo and solvated structures are discussed. The minima are systematically categorized based on four backbone dihedral angles. The Gibbs energies are evaluated and the structural factors determining the relative stabilities of conformers are discussed. © 2018 Wiley Periodicals, Inc.     

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.     

Surveying a potential energy surface by eigenvector-following

We have developed a method to search potential energy surfaces which avoids some of the difficulties associated with trapping in local minima. Steps are directly taken between minima using eigenvector-following. Exploration of this space by low temperature Metropolis Monte Carlo is a useful global optimisation tool. This method successfully finds the lowest energy icosahedral minima of Lennard- Jones clusters from random starting configurations, but cannot find the global minimum in a reasonable time for difficult cases such as the 38-atom Lennard-Jones cluster where the face-centred-cubic truncated octahedron is lowest in energy. However, by performing searches at higher temperatures, we have found a pathway between the truncated octahedron and the lowest energy icosahedral minima. Such a pathway may be illustrative of some of the structural transformations that are observed for supported metal clusters by electron microscopy.     

Minima hopping: an efficient search method for the global minimum of the potential energy surface of complex molecular systems

A method is presented that can find the global minimum of very complex condensed matter systems. It is based on the simple principle of exploring the configurational space as fast as possible and of avoiding revisiting known parts of this space. Even though it is not a genetic algorithm, it is not based on thermodynamics. The efficiency of the method depends strongly on the type of moves that are used to hop into new local minima. Moves that find low-barrier escape-paths out of the present minimum generally lead into low energy minima.     

Global energy minimization by rotational energy embedding

Given a sufficiently good empirical potential function for the internal energy of molecules, prediction of the preferred conformations is nearly impossible for large molecules because of the enormous number of local energy minima. Energy embedding has been a promising method for locating extremely good local minima, if not always the global minimum. The algorithm starts by locating a very good local minimum when the molecule is in a high-dimensional Euclidean space, and then it gradually projects down to three dimensions while allowing the molecule to relax its energy throughout the process. Now we present a variation on the method, called rotational energy embedding, where the descent into three dimensions is carried out by a sequence of internal rotations that are the multidimensional generalization of varying torsion angles in three dimensions. The new method avoids certain kinds of difficulties experienced by ordinary energy embedding and enables us to locate conformations very near the native for avian pancreatic polypeptide and apamin, given only their amino acid sequences and a suitable potential function.     

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.     

Global optimization of binary Lennard-Jones clusters using three perturbation operators

Global optimization of binary Lennard-Jones clusters is a challenging problem in computational chemistry. The difficulty lies in not only that there are enormous local minima on the potential energy surface but also that we must determine both the coordinate position and the atom type for each atom and thus have to deal with both continuous and combinatorial optimization. This paper presents a heuristic algorithm (denoted by 3OP) which makes extensive use of three perturbation operators. With these operators, the proposed 3OP algorithm can efficiently move from a poor local minimum to another better local minimum and detect the global minimum through a sequence of local minima with decreasing energy. The proposed 3OP algorithm has been evaluated on a set of 96 × 6 instances with up to 100 atoms. We have found most putative global minima listed in the Cambridge Cluster Database as well as discovering 12 new global minima missed in previous research.     

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.     

Ab Initio prediction of possible crystal structures for general organic molecules

The performance of a new crystal packing procedure for the ab initio prediction of possible molecular crystal structures is presented. The method is based upon only molecular information, i.e., no unit cell parameters are assumed to be known. The search for the global crystal energy minimum and all local minima inside an energy window is derived from Monte Carlo simulated annealing methods and has been applied to various organic molecules containing heteroatoms and polar groups. A systematic evaluation of the search method and of the quality of the potential energy function has been established. It is demonstrated that the packing of general organic molecules is possible even with standard force fields like CHARMM provided that the charges defining the electrostatic interactions are based upon physical models rather than transferable empirical parameters. Concepts of crystal packing that were based till now upon assumptions and speculations could be proved or disproved by solving directly the extended global optimization problem related to crystal packing. Crystal structures of molecules as complex as those treated in this article have not been, till now, predicted by a computational approach. In one case, a disagreement between the predicted and experimental structure was evident and, based upon the computations, we suspect that the published structure is the wrong one. © 1992 by John Wiley & Sons, Inc.     

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.     

Equilibrium density of states and thermodynamic properties of a model glass former

This paper presents an analysis of the thermodynamics of a model glass former. We have performed equilibrium sampling of a popular binary Lennard-Jones model, employing parallel tempering Monte Carlo to cover the crystalline, amorphous, and liquid regions of configuration space. Disconnectivity graphs are used to visualize the potential energy landscape in the vicinity of a crystalline geometry and in an amorphous region of configuration space. The crystalline global minimum is separated from the bulk of the minima by a large potential energy gap, leading to broken ergodicity in conventional simulations. Our sampling reveals crystalline global minima that are lower in potential energy than some of the previous candidates. We present equilibrium thermodynamic properties based on parallel tempering simulations, including heat capacities and free energy profiles, which depend explicitly on the crystal structure. We also report equilibrium melting temperatures.     

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.     

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.     

A new hybrid algorithm for finding the lowest minima of potential surfaces: approach and application to peptides

A new algorithm is presented for finding the global minimum, and other low-lying minima, of a potential energy surface (PES) of biological molecules. The algorithm synergetically combines three well-known global optimization methods: the diffusion equation method (DEM), which involves smoothing the PES; a simulated annealing (SA) algorithm; and evolutionary programming (EP), whose population-oriented approach allows for a parallel search over different regions of the PES. Tests on five peptides having between 6 and 9 residues show that the code implementing the new combined algorithm is efficient and is found to outperform the constituent methods, DEM and SA. Results of the algorithm, in the gas phase and with the GBSA implicit solvent model, are compared with crystallographic data for the test peptides; good accord is found in all cases. Also, for all but one of the examples, our hybrid algorithm finds a minimum deeper than those obtained by a very extensive scan. TINKERs implementation of the OPLS-AA force field is employed for the structure prediction. The results show that the new algorithm is a powerful structure predictor, when a reliable potential function is available. Our implementation of the algorithm is time-efficient, and requires only modest computational resources. Work is underway on applications of the new algorithm to structural prediction of proteins and other biological macro-molecules.     

Conformational distribution of heptaalanine: Analysis using a new Monte Carlo chain growth method

A distribution of conformations of heptaalanine is obtained using a new Monte Carlo (MC) method that grows the chain atom by atom. Resulting configurations are energy minimized and a detailed analysis is performed of the minimum-energy conformers using a method of classification that partitions ?ψ space. The MC-generated configurations are compared with those generated from high-temperature molecular dynamics (MD) simulations. It is found that the new method generates a wide distribution of low-energy conformers at least 10 times more quickly than the MD. An analysis of the generated energy minima demonstrates that they can be divided into clusters in the space defined by the five pairs of ?—ψ angles of the inner residues. The space occupied by the minima populating each cluster is restricted. © 1992 by John Wiley & Sons, Inc.     

Force field development for poly(dimethylsilylenemethylene) with the aid of ab initio calculations

The molecular geometries, conformational energies, and zero-point energies of di(trimethylsilylene)methylene have been determined from high-level quantum chemistry calculations. The results are further used in the parametrization of a classical potential energy function suitable for performing simulations of the corresponding polymer, namely, poly(dimethylsilylenemethylene). Di(trimethylsilylene)methylene geometrical parameter optimizations for a proper location of the global minimum and other local minima, constrained at certain dihedral and bond angles, were performed at both the B3LYP/6-311G and MP2(full)/6-311G levels of theory. The global minimum configuration is slightly displaced from a perfectly staggered geometry, approximately by 16.0 degrees, at both levels of theory. Molecular mechanics and Monte Carlo calculations for isolated polymer chains together with molecular dynamics runs for the modeled dimer provide very good results in terms of conformational and thermodynamic properties.     

Computer optimization of small atomic clusters

We have successfully identified stable configurations of both rare-gas and NaCl clusters with a new optimization procedure. An initial cluster configuration is prepared in a so-called shoot-and-stay process. Its total energy is then minimized with respect to the atomic coordinates. To prevent the system from being locked in local minima, the step size of each move is chosen as the width of the energy well at a higher level. As the system evolves, the global minimum is contained in the volume bounded by the decreasing value of step sizes. We have also carried out the optimization of NaCl clusters by the simulated annealing technique, for comparison. The results show that for such heterogeneous systems, the latter method cannot always find the global minimum, because of large energy gaps between different catchment regions in phase space.