Cluster optimization simplified by interaction modification

Single-root networks for describing the potential energy surface of Lennard-Jones clusters

Potential energy surface (PES) holds the key in understanding a number of atomic clusters or molecular phenomena. However, due to the high dimension and incredible complexity of PES, only indirect methods can be used to characterize a PES of a given system in general. In this paper, a branched dynamic lattice searching method was developed to travel the PES, which was described in detail by a single-root network (SRN). The advantage of SRN is that it reflects the topological relation between different conformations and highlights the size of each structure energy trap. On the basis of SRN, to demonstrate how to transform one conformation to another, the transition path that connects two local minima in the PES was constructed. Herein, we take Lennard-Jones (LJ) clusters at the sizes of 38, 55, and 75 as examples. It is found that the PES of these three clusters have many local funnels and each local funnel represents one morphology. If a morphology is located more frequently, it will lie in a larger local funnel. Besides, certain steps of the transition path were generated successfully, such as changing from icosahedral to truncated octahedral of the LJ₃₈-cluster. Though we do not exhibit all the parts of the PES or all transition paths, this method indeed works well in the local area and can be used more widely.

Ab initio random structure searching

It is essential to know the arrangement of the atoms in a material in order to compute and understand its properties. Searching for stable structures of materials using first-principles electronic structure methods, such as density-functional-theory (DFT), is a rapidly growing field. Here we describe our simple, elegant and powerful approach to searching for structures with DFT, which we call ab initio random structure searching (AIRSS). Applications to discovering the structures of solids, point defects, surfaces, and clusters are reviewed. New results for iron clusters on graphene, silicon clusters, polymeric nitrogen, hydrogen-rich lithium hydrides, and boron are presented.

Balancing simulation accuracy and efficiency with the Amber united atom force field

We have analyzed the quality of a recently proposed Amber united-atom model and its overall efficiency in ab initio folding and thermodynamic sampling of two stable beta-hairpins. It is found that the mean backbone structures are quite consistent between the simulations in the united-atom and its corresponding all-atom models in Amber. More importantly, the simulated beta turns are also consistent between the two models. Finally, the chemical shifts on H alpha are highly consistent between simulations in the two models, although the simulated chemical shifts are lower than experiment, indicating less structured peptides, probably due to the omission of the hydrophobic term in the simulations. More interestingly, the stabilities of both beta-hairpins at room temperature are similar to those derived from the NMR measurement, whether the united-atom or the all-atom model is used. Detailed analysis shows high percentages of backbone torsion angles within the beta region and high percentages of native contacts. Given the reasonable quality of the united-atom model with respect to experimental data, we have further studied the simulation efficiency of the united-atom model over the all-atom model. Our data shows that the united-atom model is a factor of 6-8 faster than the all-atom model as measured with the ab initio first pass folding time for the two tested beta-hairpins. Detailed structural analysis shows that all ab initio folded trajectories enter the native basin, whether the united-atom model or the all-atom model is used. Finally, we have also studied the simulation efficiency of the united-atom model as measured in terms of how fast thermodynamic convergence can be achieved. It is apparent that the united-atom simulations reach convergence faster than the all-atom simulations with respect to both mean potential energies and mean native contacts. These findings show that the efficiency of the united-atom model is clearly beyond the per-step dynamics simulation of about 2 over the all-atom model. Thus, reasonable reduction of a protein model can be achieved with improved sampling efficiency while still preserving a high level of accuracy for applications in both ab initio folding and thermodynamic sampling. This study motivates us to develop more simplified protein models with sufficient consistency with the all-atom models for enhanced conformational sampling.

The ruggedness of protein-protein energy landscape and the cutoff for 1/r(n) potentials

MOTIVATION

Computational studies of the energetics of protein association are important for revealing the underlying fundamental principles and for designing better tools to model protein complexes. The interaction cutoff contribution to the ruggedness of protein-protein energy landscape is studied in terms of relative energy fluctuations for 1/r(n) potentials based on a simplistic model of a protein complex. This artificial ruggedness exists for short cutoffs and gradually disappears with the cutoff increase.

RESULTS

The critical values of the cutoff were calculated for each of 11 popular power-type potentials with n=0/9, 12 and for two thresholds of 5% and 10%. The artificial ruggedness decreases to tolerable thresholds for cutoffs larger than the critical ones. The results showed that for both thresholds the critical cutoff is a non-monotonic function of the potential power n. The functions reach the maximum at n=3/4 and then decrease with the increase of the potential power. The difference between two cutoffs for 5% and 10% artificial ruggedness becomes negligible for potentials decreasing faster than 1/r(12). The analytical results obtained for the simple model of protein complexes agree with the analysis of artificial ruggedness in a dataset of 62 protein-protein complexes, with different parameterizations of soft Lennard-Jones potential and two types of protein representations: all-atom and coarse-grained. The results suggest that cutoffs larger than the critical ones can be recommended for protein-protein potentials.

Chasing funnels on protein-protein energy landscapes at different resolutions

Studies of intermolecular energy landscapes are important for understanding protein association and adequate modeling of protein interactions. Landscape representation at different resolutions can be used for the refinement of docking predictions and detection of macro characteristics, like the binding funnel. A representative set of protein-protein complexes was used to systematically map the intermolecular landscape by grid-based docking. The change of the resolution was achieved by varying the range of the potential, according to the variable resolution GRAMM methodology. A formalism was developed to consistently parameterize the potential and describe essential characteristics of the landscape. The results of gradual landscape smoothing, from high to low resolution, indicate that i), the number of energy basins, the landscape ruggedness, and the slope decrease accordingly; ii), the number of near-native matches, defined as those inside the funnel, increases until the trend breaks down at critical resolution; the rate of the increase and the critical resolution are specific to the type of a complex (enzyme inhibitor, antigen-antibody, and other), reflect known underlying recognition factors, and correlate with earlier determined estimates of the funnel size; iii), the native/nonnative energy gap, a major characteristic of the energy minima hierarchy, remains constant; and iv), the putative funnel (defined as the deepest basin) has the largest average depth-related ruggedness and slope, at all resolutions. The results facilitate better understanding of the binding landscapes and suggest directions for implementation in practical docking protocols.

Interaction cutoff effect on ruggedness of protein-protein energy landscape

The concept of the energy landscape is important for better understanding of protein-protein interactions and for designing adequate docking procedures. The intermolecular landscape has a rugged terrain that impedes search procedures. Its inherent ruggedness is related to the conformational characteristics of the molecules and to the form of the potential function--more rugged for short-range potentials and less rugged for "soft," typically long-range potentials. Our study determined that the landscape ruggedness is further substantially exacerbated by truncation of the potentials. This additional ruggedness appears below certain critical interaction ranges that depend on the form of the potential. The theoretical model describing the cutoff effect on the landscape ruggedness is confirmed by the energy calculation on a dataset of protein-protein complexes. The negative effect of the potentials cutoff is well known. However, revealing its physical basis in terms of the energy landscape is important for better understanding of intermolecular interactions.

Preferential attachment during the evolution of a potential energy landscape

It has previously been shown that the network of connected minima on a potential energy landscape is scale-free, and that this reflects a power-law distribution for the areas of the basins of attraction surrounding the minima. Here, the aim is to understand more about the physical origins of these puzzling properties by examining how the potential energy landscape of a 13-atom cluster evolves with the range of the potential. In particular, on decreasing the range of the potential the number of stationary points increases and thus the landscape becomes rougher and the network gets larger. Thus, the evolution of the potential energy landscape can be followed from one with just a single minimum to a complex landscape with many minima and a scale-free pattern of connections. It is found that during this growth process, new edges in the network of connected minima preferentially attach to more highly connected minima, thus leading to the scale-free character. Furthermore, minima that appear when the range of the potential is shorter and the network is larger have smaller basins of attraction. As there are many of these smaller basins because the network grows exponentially, the observed growth process thus also gives rise to a power-law distribution for the hyperareas of the basins.

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.

Global minimum structure searches via particle swarm optimization

Novel implementation of the evolutionary approach known as particle swarm optimization (PSO) capable of finding the global minimum of the potential energy surface of atomic assemblies is reported. This is the first time the PSO technique has been used to perform global optimization of minimum structure search for chemical systems. Significant improvements have been introduced to the original PSO algorithm to increase its efficiency and reliability and adapt it to chemical systems. The developed software has successfully found the lowest-energy structures of the LJ(26) Lennard-Jones cluster, anionic silicon hydride Si(2)H(5) (-), and triply hydrated hydroxide ion OH(-) (H(2)O)(3). It requires relatively small population sizes and demonstrates fast convergence. Efficiency of PSO has been compared with simulated annealing, and the gradient embedded genetic algorithm.

Characterizing the network topology of the energy landscapes of atomic clusters

By dividing potential energy landscapes into basins of attractions surrounding minima and linking those basins that are connected by transition state valleys, a network description of energy landscapes naturally arises. These networks are characterized in detail for a series of small Lennard-Jones clusters and show behavior characteristic of small-world and scale-free networks. However, unlike many such networks, this topology cannot reflect the rules governing the dynamics of network growth, because they are static spatial networks. Instead, the heterogeneity in the networks stems from differences in the potential energy of the minima, and hence the hyperareas of their associated basins of attraction. The low-energy minima with large basins of attraction act as hubs in the network. Comparisons to randomized networks with the same degree distribution reveals structuring in the networks that reflects their spatial embedding.

Gravitational smoothing as a global optimization strategy

An optimization scheme for atomic cluster structures, based on exaggerating the importance of the gravitational force, is introduced. Results are presented for calculations on Lennard-Jones clusters of 13, 38, and 55 atoms, and the 13-atom Morse cluster.

Global minimization of an off-lattice potential energy function using a chaperone-based refolding method

A global energy minimization method based on what is known about the mechanisms of the GroEL/GroES chaperonin system is applied to two 22-mers of an off-lattice protein model whose native states are beta-hairpins and which have structural similarity to short peptides known to interact strongly with the GroEL substrate binding domain. These model substrates have been used by other workers to test the effectiveness of a number of global minimization techniques, and are regarded as providing a significant challenge. The minimization method developed here is progressively elaborated from an initial simple form that targets exposed hydrophobic regions for unfolding to include a refolding phase that encourages the later recompactification of partly unfolded substrate; this refolding phase is seen to be crucial in the successful application of the method. The optimal handling of hydrophilic monomers within the model is also systematically explored, and it is seen that the best interpretation of their role is one that allows the chaperonin model to operate in "proofreading" mode whereby misfolded substrates are recognized by their surface exposure of a large proportion of hydrophobic monomers. The final version of the model allows native-like structures to be found quickly, on average for the two 22-mer substrates after 6 or 7 chaperone contacts. These results compare very favorably with those that have been obtained elsewhere using generic global minimization methods such as those based on thermal annealing. The paper concludes with a discussion of the place of the technique within the general category of hypersurface deformation methods for global minimization, and with suggestions as to how the chaperone-based method developed here could be elaborated so as to be effective on longer substrate chains that give rise to more complex tertiary structures in their native states.

Structural relaxation in atomic clusters: master equation dynamics

The role of the potential energy landscape in determining the relaxation dynamics of model clusters is studied using a master equation. Two types of energy landscape are examined: a single funnel, as exemplified by 13-atom Morse clusters, and the double funnel landscape of the 38-atom Lennard-Jones cluster. Interwell rate constants are calculated using Rice-Ramsperger-Kassel-Marcus theory within the harmonic approximation, but anharmonic model partition functions are also considered. Decreasing the range of the potential in the Morse clusters is shown to hinder relaxation toward the global minimum, and this effect is related to the concomitant changes in the energy landscape. The relaxation modes that emerge from the master equation are interpreted and analyzed to extract interfunnel rate constants for the Lennard-Jones cluster. Since this system is too large for a complete characterization of the energy landscape, the conditions under which the master equation can be applied to a limited database are explored. Connections are made to relaxation processes in proteins and structural glasses.

Global optimization of clusters, crystals, and biomolecules

Finding the optimal solution to a complex optimization problem is of great importance in many fields, ranging from protein structure prediction to the design of microprocessor circuitry. Some recent progress in finding the global minima of potential energy functions is described, focusing on applications of the simple "basin-hopping" approach to atomic and molecular clusters and more complicated hypersurface deformation techniques for crystals and biomolecules. These methods have produced promising results and should enable larger and more complex systems to be treated in the future.

Novel methods of sampling phase space in the simulation of biological systems

New advances in molecular dynamics and Monte Carlo simulations have led to impressive increases in the speed of sampling phase space for complex biological systems. These methods have been combined with new fast algorithms for computing long range electrostatic interactions for new polarizable force fields. In addition, new methods for sampling low energy molecular conformations allow the rapid determination of thermodynamically dominant regions on the potential-energy surface. Accurate measures of the rate of phase-space sampling should allow both the optimization and the comparison of methods for a particular problem of interest.

COMPUTATIONAL STUDIES OF CLUSTERS:Methods and Results

▪ Abstract  The experimental and computational study of clusters has been an active field of research for over a decade. This review provides an overview of some of the methods that have been developed to study clusters and some of the results that have been obtained. Included are computational approaches to explore the potential energy surface for clusters, methods to extract thermodynamic properties from the potential surface information and approaches to insure simulation studies are performed in an ergodic fashion. The methods have proved to be useful in studying the structural transition from clusters to bulk phases, phase changes in small clusters and the importance of quantum effects. The review ends with a discussion of problems in cluster chemistry and physics that are of interest for future investigations.