Multicanonical jump walk annealing: An efficient method for geometric optimization

Enhanced sampling algorithms

In biomolecular systems (especially all-atom models) with many degrees of freedom such as proteins and nucleic acids, there exist an astronomically large number of local-minimum-energy states. Conventional simulations in the canonical ensemble are of little use, because they tend to get trapped in states of these energy local minima. Enhanced conformational sampling techniques are thus in great demand. A simulation in generalized ensemble performs a random walk in potential energy space and can overcome this difficulty. From only one simulation run, one can obtain canonical-ensemble averages of physical quantities as functions of temperature by the single-histogram and/or multiple-histogram reweighting techniques. In this article we review uses of the generalized-ensemble algorithms in biomolecular systems. Three well-known methods, namely, multicanonical algorithm, simulated tempering, and replica-exchange method, are described first. Both Monte Carlo and molecular dynamics versions of the algorithms are given. We then present various extensions of these three generalized-ensemble algorithms. The effectiveness of the methods is tested with short peptide and protein systems.

Complex network classification using partially self-avoiding deterministic walks

Complex networks have attracted increasing interest from various fields of science. It has been demonstrated that each complex network model presents specific topological structures which characterize its connectivity and dynamics. Complex network classification relies on the use of representative measurements that describe topological structures. Although there are a large number of measurements, most of them are correlated. To overcome this limitation, this paper presents a new measurement for complex network classification based on partially self-avoiding walks. We validate the measurement on a data set composed by 40000 complex networks of four well-known models. Our results indicate that the proposed measurement improves correct classification of networks compared to the traditional ones.

Global optimization of ionic Mg(n)F(2n) (n=1-30) clusters

The global optimization basin-hopping (BH) method has been used to locate the global minima (GM) of Mg(n)F(2n) (n=1-30) clusters using a Born-Mayer-type potential. Some of the GM were particularly difficult to find, requiring more than 1.5 x 10(4) BH steps. We have found that both the binding energy per MgF2 unit and the effective volume of the GM isomers increase almost linearly with n, and that cluster symmetry decreases with cluster size. The data derived from the BH runs reveal a growing density of local minima just above the GM as n increases. Despite this, the attraction basin around each GM is relatively large, since after all their atomic coordinates are randomly displaced by values as high as 2.0 bohrs, the perturbed structures, upon reoptimization, relax back to the GM in more than 50% of the cases (except for n=10 and 11). The relative stabilities derived from energy second differences suggest that n=8,10,13,15, and 20 are probably the magic numbers for these systems. Mass spectrum experiments would be very useful to clarify this issue.

Asynchronous multicanonical basin hopping method and its application to cobalt nanoclusters

The multicanonical basin hopping (MUBH) method, which uses a multicanonical weight in the basin hopping (BH) Monte Carlo method, was found to be very efficient for global optimization of large-scale systems such as Lennard-Jones clusters containing more than 150 atoms. We have implemented an asynchronous parallel version of the MUBH method using the message passing interface (MPI) to take advantage of the full usage of multiprocessors in either a homogeneous or heterogeneous computational environment. Based on the intrinsic properties of the Monte Carlo method, this MPI implementation used the task parallelism to minimize interthread data communication. For a Co nanocluster consisting of N atoms, we have applied the asynchronous multicanonical basin hopping (AMUBH) method (for 181 < N < or = 200), together with BH (for 2 < or = N < 150) and MUBH (for 150 < or = N < or = 180), to search for the molecular configuration of the global energy minimum. AMUBH becomes the only practical computational scheme for locating the energy minimum within realistic computational time for a relatively large cluster.

Local-structural diversity and protein folding: Application to all-β off-lattice protein models

Global measures of structural diversity within a distribution of biopolymers, such as the radius of gyration and percent native contacts, have proven useful in the analysis of simulation data for protein folding. In this paper we describe a statistical-based methodology to quantify the local structural variability of a distribution of biopolymers, applied to 46- and 69-"residue" off-lattice, three-color model proteins. Each folds into beta-barrel structures. First we perform a principal component analysis of all interbead distance variables for a large number of independent, converged Boltzmann-distributed samples of conformations collected at each of a wide range of temperatures. Next, the principal component vectors are subjected to orthogonal (varimax) rotation. The results are displayed on so-called "squared-loading" plots. These provide a quantitative measure of the contribution to the sample variance of the position of each residue relative to the others. Dominant structural elements, those having the largest structural diversity within the sampled distribution, are responsible for peaks and shoulders observed in the specific heat versus temperature curves, generated using the weighted histogram analysis method. The loading plots indicate that the local-structural diversity of these systems changes gradually with temperature through the folding transition but radically changes near the collapse transition temperature. The analysis of the structural overlap order statistic suggests that the 46-mer thermodynamic folding transition involves the native state and at least three other nearly native intermediates. In the case of the 46-mer protein model, data are generated at sufficiently low temperatures that squared-loading plots, coupled with cluster analysis, provide a local and energetic description of its glassy state.

Generalized-ensemble algorithms: enhanced sampling techniques for Monte Carlo and molecular dynamics simulations

In complex systems with many degrees of freedom such as spin glass and biomolecular systems, conventional simulations in canonical ensemble suffer from the quasi-ergodicity problem. A simulation in generalized ensemble performs a random walk in potential energy space and overcomes this difficulty. From only one simulation run, one can obtain canonical ensemble averages of physical quantities as functions of temperature by the single-histogram and/or multiple-histogram reweighting techniques. In this article we review the generalized ensemble algorithms. Three well-known methods, namely, multicanonical algorithm (MUCA), simulated tempering (ST), and replica-exchange method (REM), are described first. Both Monte Carlo (MC) and molecular dynamics (MD) versions of the algorithms are given. We then present five new generalized-ensemble algorithms which are extensions of the above methods.

Multicanonical molecular dynamics algorithm employing an adaptive force-biased iteration scheme

We present one effective multicanonical molecular dynamics (MCMD) algorithm accelerating the convergence of rough energy landscapes simulations via an adaptive force-biased iteration scheme. Our method utilizes several short MCMD simulations with dynamically updated weights and combines them to estimate the density of states via multiple histogram technique. The key step of our algorithm is the adaptive refinement for the derivative of multicanonical weight, which allows the system to enlarge the sampling energy range maintaining the statistical accuracy. The performance of our method has been validated for atomic Lennard-Jones clusters.

Parameter space minimization methods: Applications to Lennard-Jones–dipole-dipole clusters

The morphology of the uniform Lennard-Jones-dipole-dipole cluster with 13 centers (LJDD)13 is investigated over a relatively wide range of values of the dipole moment. We introduce and compare several necessary modifications of the basin-hopping algorithm for global optimization to improve its efficiency. We develop a general algorithm for T=0 Brownian dynamics in curved spaces, and a graph theoretical approach necessary for the elimination of dissociated states. We find that the (LJDD)13 cluster has icosahedral symmetry for small to moderate values of the dipole moment. As the dipole moment increases, however, its morphology shifts to an hexagonal antiprism, and eventually to a ring.

Dynamical origin of uniform sampling in multicanonical ensemble

The stochastic model describing the sampling process in multicanonical ensemble has been derived by considering the sampling process as an overdamped Brownian motion on the free energy surface. The essential dynamics of the multicanonical sampling has been characterized by a Langevin equation in a piecewise multivalleyed free energy landscape, modulated by a temperature-dependent curvature. Based on the stochastic model we showed that the multicanonical weight can be determined by interpolating maximum probability energy points of the canonical samplings at different temperatures.