Wang-Landau Sampling Research Articles

Microcanonical Phase Diagram of the BEG and Ising Models

The density of states of long-range Blume—Emery—Griffiths (BEG) and short-range Ising models are obtained by using Wang—Landau sampling with adaptive windows in energy and magnetization space. With accurate density of states, we are able to calculate the microcanonical specific heat of fixed magnetization introduced by Kastner et al. in the regions of positive and negative temperature. The microcanonical phase diagram of the Ising model shows a continuous phase transition at a negative temperature in energy and magnetization plane. However the phase diagram of the long-range model constructed by peaks of the microcanonical specific heat looks obviously different from the Ising chart.

Optimized Wang-Landau sampling of lattice polymers: Ground state search and folding thermodynamics of HP model proteins

Coarse-grained (lattice-) models have a long tradition in aiding efforts to decipher the physical or biological complexity of proteins. Despite the simplicity of these models, however, numerical simulations are often computationally very demanding and the quest for efficient algorithms is as old as the models themselves. Expanding on our previous work [T. Wüst and D. P. Landau, Phys. Rev. Lett. 102, 178101 (2009)], we present a complete picture of a Monte Carlo method based on Wang-Landau sampling in combination with efficient trial moves (pull, bond-rebridging, and pivot moves) which is particularly suited to the study of models such as the hydrophobic-polar (HP) lattice model of protein folding. With this generic and fully blind Monte Carlo procedure, all currently known putative ground states for the most difficult benchmark HP sequences could be found. For most sequences we could also determine the entire energy density of states and, together with suitably designed structural observables, explore the thermodynamics and intricate folding behavior in the virtually inaccessible low-temperature regime. We analyze the differences between random and protein-like heteropolymers for sequence lengths up to 500 residues. Our approach is powerful both in terms of robustness and speed, yet flexible and simple enough for the study of many related problems in protein folding.

COMPUTER SIMULATIONS OF A POLYMER WITH EXACT SOLUTION

In this work we apply a refined Wang–Landau simulation to a simple polymer model which has an exact solution both in the microcanonical and the canonical formalisms. We investigate the behavior of the microcanonical and canonical averages during the Wang–Landau simulation. The simulations were carried out using conventional Wang–Landau sampling (WLS) and the 1/t scheme. Our results show that updating the density of states only after every N monomer moves leads to a much better precision. During the simulations the canonical averages such as the location of the maximum of the specific heat calculated from independent runs tend asymptotically to values around the correct value obtained from the exact calculations of the density of states and remain unchanged for some final modification factor. Since this f final is found for the model analyzed, one has a criterion to stop the simulations. We compare our results with the exact value and with those of the 1/t scheme.

Evaluation of the grand-canonical partition function using expanded Wang-Landau simulations. I. Thermodynamic properties in the bulk and at the liquid-vapor phase boundary

The Wang-Landau sampling is a powerful method that allows for a direct determination of the density of states. However, applications to the calculation of the thermodynamic properties of realistic fluids have been limited so far. By combining the Wang-Landau method with expanded grand-canonical simulations, we obtain a high-accuracy estimate for the grand-canonical partition function for atomic and molecular fluids. Then, using the formalism of statistical thermodynamics, we are able to calculate the thermodynamic properties of these systems, for a wide range of conditions spanning the single-phase regions as well as the vapor-liquid phase boundary. Excellent agreement with prior simulation work and with the available experimental data is obtained for argon and CO(2), thereby establishing the accuracy of the method for the calculation of thermodynamic properties such as free energies and entropies.

Wang-Landau sampling: Improving accuracy

In this work, we investigate the behavior of the microcanonical and canonical averages of the two-dimensional Ising model during the Wang-Landau simulation. The simulations were carried out using conventional Wang-Landau sampling and the 1/t scheme. Our findings reveal that the microcanonical average should not be accumulated during the initial modification factors f, and they outline a criterion to find this limit, which we define as f(micro). We show that updating the density of states only after every L(2) spin-flip trials leads to a much better precision. We present a mechanism to determine for the given model up to what final modification factor f(final) the simulations should be carried out. Altogether these small adjustments lead to an improved procedure for simulations with much more reliable results. We compare our results with 1/t simulations. We also present an application of the procedure to a self-avoiding homopolymer.

Massively parallel Wang–Landau sampling on multiple GPUs

Abstract Wang–Landau sampling is implemented on the Graphics Processing Unit (GPU) with the Compute Unified Device Architecture (CUDA). Performances on three different GPU cards, including the new generation Fermi architecture card, are compared with that on a Central Processing Unit (CPU). The parameters for massively parallel Wang–Landau sampling are tuned in order to achieve fast convergence. For simulations of the water cluster systems, we obtain an average of over 50 times speedup for a given workload.

Replica Exchange Statistical Temperature Molecular Dynamics Simulations of Peptide Dimerization

We present a new computer simulation algorithm called Replica Exchange Statistical Temperature Molecular Dynamics that combines the multicanonical sampling technique of Statistical Temperature Molecular Dynamics with temperature replica exchange. This algorithm is related to Wang-Landau sampling but uses a dynamical update of the density of states to achieve flat energy sampling within a replica-dependent temperature range. This algorithm is used to investigate the thermodynamics of dimerization of two polypeptide chains. Three two-peptide systems are investigated: two hydrophobic peptides, two hydrophilic peptides, and one of each. Each monomer is modeled using a coarse-grained peptide model that has an α-helix as the lowest energy configuration. For each dimer system, interesting folding behavior is observed. It is found that at low temperatures, both peptides are helical and the lowest-energy configuration maximizes inter-peptide contact; at high temperatures, both peptides are random coils; and at intermediate temperatures, one peptide is folded and the other unfolded. Formation of the peptide dimer causes one peptide to fold at a higher temperature than an isolated monomer and the other to fold at a lower temperature than an isolated monomer. Dimerization causes one peptide to become more stable and the other peptide to become less stable. It is also shown that at intermediate temperatures, neither peptide adopts a random coil configuration: the helical peptide induces a conformational change in the unfolded peptide. The Potential Energy Surface (PES) is also determined for each dimer and the effect of hydrophobic/hydrophilic nature of the peptide on the PES is discussed.

Evaporation-condensation transition of the two-dimensional Potts model in the microcanonical ensemble

The evaporation-condensation transition of the Potts model on a square lattice is numerically investigated by the Wang-Landau sampling method. An intrinsically system-size-dependent discrete transition between supersaturation state and phase-separation state is observed in the microcanonical ensemble by changing constrained internal energy. We calculate the microcanonical temperature, as a derivative of microcanonical entropy, and condensation ratio, and perform a finite-size scaling of them to indicate the clear tendency of numerical data to converge to the infinite-size limit predicted by phenomenological theory for the isotherm lattice gas model.

Phase transition of a single star polymer: A Wang-Landau sampling study

Star polymers, as an important class of nonlinear macromolecules, process special thermodynamic properties for the existence of a common connecting point. The thermodynamic transitions of a single star polymer are systematically studied with the bond fluctuation model using Wang-Landau sampling techniques. A new analysis method employing the shape factor is proposed to locate the coil-globule (CG) and liquid-crystal (LC) transitions, which shows a higher efficiency and accuracy than the canonical specific heat function. The LC transition temperature is found to obey the identical scaling law as the linear polymers. However, the CG transition temperature shifts towards the LC transition with the increasing of the arm number. The reason is that for the star polymer a lower temperature is needed for the attractive force to overcome the excluded volume effect of the polymer chain because of its high arm density. This work clearly proves the structural distinction of the linear and star polymers can only affect the CG transition while has no influence on the LC transition.

Method for sampling compact configurations for semistiff polymers

The sampling of compact configurations is crucial when investigating structural properties of semistiff polymers, like proteins and DNA, using Monte Carlo methods. A sampling scheme for a continuous model based on configuration biasing is introduced, tested, and compared with conventional methods. The proposed configuration biased Monte Carlo method, used together with the Wang-Landau sampling scheme, enables us to obtain any thermodynamic property within the statistical ensemble in use. Using the proposed method, it is possible to collect statistical data of interest for a wide range of compactions (from stretched up to several toroid loops) in a single computer experiment. A second-order-like stretched-toroid phase transition is observed for a semistiff polymer, and the critical temperature is estimated.

Two‐Dimensional Wang–Landau Algorithm for Osmotic Pressure Calculations in a Polyelectrolyte–Membrane System

AbstractThe Monte Carlo method based on two‐dimensional entropic sampling within the Wang–Landau (WL) algorithm is applied to simulation of a continuous model of a polyelectrolyte between membrane surfaces. Membranes are presented by parallel plane surfaces holding either fixed or mobile dipoles (representing lipid headgroups). A strongly charged polyion accompanied by neutralizing counterions is placed between the membranes. Periodic boundary conditions are imposed along X‐ and Y‐axes. The volume of the main cell is varied during the simulation by shifting one of the surfaces along Z‐axis. Within two‐dimensional WL sampling algorithm we obtain joint density of states as a function of energy and volume in a single run. In order to increase efficiency of our calculations we introduce a number of modifications to the original WL‐approach. Various properties of the system over wide temperature and volume or pressure ranges, i.e., conformational energy, heat capacity, and free energy, are obtained from the two‐dimensional density of states by simple integration. The osmotic pressure is calculated as a derivative of Helmholtz free energy. Alternatively, properties of the system, including average volume, can be obtained under condition of NPT ensemble. It is shown that the both approaches produce coinciding Posm(V) isotherms. In all considered cases we observe repulsive effective interaction between the membrane surfaces, and repulsion is stronger for the surfaces with fixed dipoles. magnified image

Structural properties and thermodynamics of water clusters: A Wang–Landau study

The temperature dependence of structural properties and thermodynamic behavior of water clusters has been studied using Wang-Landau sampling. Four potential models, simple point charge/extended (SPC/E), transferable intermolecular potential 3 point (TIP3P), transferable intermolecular potential 4 point (TIP4P), and Gaussian charge polarizable (GCP), are compared for ground states and properties at finite temperatures. Although the hydrogen bond energy and the distance of the nearest-neighbor oxygen pair are significantly different for TIP4P and GCP models, they approach to similar ground state structures and melting transition temperatures in cluster sizes we considered. Comparing with TIP3P, SPC/E model provides properties closer to that of TIP4P and GCP.

Monte Carlo simulations of the HP model (the “Ising model” of protein folding)

Using Wang–Landau sampling with suitable Monte Carlo trial moves (pull moves and bond-rebridging moves combined) we have determined the density of states and thermodynamic properties for a short sequence of the HP protein model. For free chains these proteins are known to first undergo a collapse “transition” to a globule state followed by a second “transition” into a native state. When placed in the proximity of an attractive surface, there is a competition between surface adsorption and folding that leads to an intriguing sequence of “transitions”. These transitions depend upon the relative interaction strengths and are largely inaccessible to “standard” Monte Carlo methods.

Static and dynamical aspects of the metastable states of first order transition systems

We numerically study the metastable states of the 2d Potts model. Both of equilibrium and relaxation properties are investigated focusing on the finite size effect. The former is investigated by finding the free energy extremal point by the Wang-Landau sampling and the latter is done by observing the Metropolis dynamics after sudden heating. It is explicitly shown that with increasing system size the equilibrium spinodal temperature approaches the bistable temperature in a power-law and the size-dependence of the nucleation dynamics agrees with it. In addition, we perform finite size scaling of the free energy landscape at the bistable point.

Monte Carlo analysis of the critical properties of the two-dimensional randomly bond-diluted Ising model via Wang–Landau algorithm

The influence of random bond-dilution on the critical properties of the two-dimensional Ising model on a square lattice with periodic boundary conditions is presented using the Monte Carlo process with Wang–Landau sampling. The lattice linear size is L = 20 – 140 and the concentration of the diluted bonds spans a wide range from weak to strong dilution, namely, q = 0.05 , 0.1 , 0.2 , 0.3 , 0.4 ; the respective percolation limit for the square lattice is q c P E R C = 0.5 . Its pure version ( q = 0 ) has a second-order phase transition with vanishing specific heat critical exponent, an example of inapplicability of the Harris criterion. The main effort is focused on the temperature dependence of the specific heat and magnetic susceptibility to estimate the respective maximum values and subsequent pseudocritical temperatures for extracting the relative critical exponents. We have also looked at the probability distribution of the susceptibility, pseudocritical temperature and specific heat for assessing self-averaging. The study is carried out in the appropriately restricted but dominant energy subspaces. By applying the finite-size scaling analysis, the critical exponents are estimated; the specific heat exponent vanishes, the correlation-length exponent ν is equal to one and the critical exponents’ ratio ( γ / ν ) retains its pure-Ising-model value, thus supporting the strong universality hypothesis.

Critical behavior of hard-core lattice gases: Wang–Landau sampling with adaptive windows

Critical properties of lattice gases with nearest-neighbor exclusion are investigated via adaptive-window Wang–Landau sampling (WLS) on the square and simple cubic lattices, for which the model is known to exhibit an Ising-like phase transition. We study the particle density, order parameter, compressibility, Binder cumulant and susceptibility, in efforts to test WLS, which has been used quite widely in recent years, in the context of lattice gases. Of considerable interest is whether it is possible to estimate critical exponents reliably using WLS with adaptive windows. We find that the method yields results in fair agreement with exact values (in two dimensions) and numerical estimates (in three dimensions).

Effects of counterion size on the attraction between similarly charged surfaces

Interaction between similarly charged surfaces can be attractive at high electrostatic coupling constants Ξ = l(B)Z(2)/μ(GC), where l(B) is the Bjerrum length, μ(GC) the Gouy-Chapman length, and Z the valency of counterions. While this effect has been studied previously in detail, as a function of surface charge density and valency of the pointlike counterions, much less is known about the effect of counterion size. We apply the Wang-Landau sampling Monte Carlo (MC) simulation method to compute the free energy F as a function of the scaled distance between the plates D̃=D/μ(GC) for a range of Ξ and scaled counterion radii R̃=R/μ(GC). We find that for large Ξ and small ion radius, there is a global equilibrium distance D̃=D̃(eq)=2(1+R̃), correctly giving the expected value at the point counterion limit. With increasing R̃ the global minimum in F(D̃) changes to a metastable state and finally this minimum vanishes when R̃ reaches a critical value, which depends on Ξ. We present a state diagram indicating approximate boundaries between these three regimes. The Wang-Landau MC method, as it is applied here, offers a possibility to study a wide spectrum of extended problems, which cannot be treated by the use of contact value theorem.

Number fluctuations of the condensing agent affects the coil–toroid coexistence forsemi-stiff polyelectrolytes

A semi-stiff charged polymer with counterions and salt was investigated using aMonte Carlo method with the Wang–Landau sampling scheme. Simulations show thecoexistence of thermodynamically stable structures—prolonged and compact (toroidal-like).The transition between the two states is accompanied by concentration fluctuations for thecondensing agent. Compact structures are accompanied by more multivalent counterionsthan prolonged ones.

Thermodynamics of RNA structures by Wang–Landau sampling

Motivation: Thermodynamics-based dynamic programming RNA secondary structure algorithms have been of immense importance in molecular biology, where applications range from the detection of novel selenoproteins using expressed sequence tag (EST) data, to the determination of microRNA genes and their targets. Dynamic programming algorithms have been developed to compute the minimum free energy secondary structure and partition function of a given RNA sequence, the minimum free-energy and partition function for the hybridization of two RNA molecules, etc. However, the applicability of dynamic programming methods depends on disallowing certain types of interactions (pseudoknots, zig-zags, etc.), as their inclusion renders structure prediction an nondeterministic polynomial time (NP)-complete problem. Nevertheless, such interactions have been observed in X-ray structures.Results: A non-Boltzmannian Monte Carlo algorithm was designed by Wang and Landau to estimate the density of states for complex systems, such as the Ising model, that exhibit a phase transition. In this article, we apply the Wang-Landau (WL) method to compute the density of states for secondary structures of a given RNA sequence, and for hybridizations of two RNA sequences. Our method is shown to be much faster than existent software, such as RNAsubopt. From density of states, we compute the partition function over all secondary structures and over all pseudoknot-free hybridizations. The advantage of the WL method is that by adding a function to evaluate the free energy of arbitary pseudoknotted structures and of arbitrary hybridizations, we can estimate thermodynamic parameters for situations known to be NP-complete. This extension to pseudoknots will be made in the sequel to this article; in contrast, the current article describes the WL algorithm applied to pseudoknot-free secondary structures and hybridizations.Availability: The WL RNA hybridization web server is under construction at http://bioinformatics.bc.edu/clotelab/.Contact: clote@bc.edu

Comment on “Six-state clock model on the square lattice: Fisher zero approach with Wang-Landau sampling”

Hwang in [Phys. Rev. E 80, 042103 (2009)] suggested that the two transitions of the six-state clock model on the square lattice are not of the Kosterlitz-Thouless type. Here we show from simulations that at the upper transition, the helicity modulus does make a discontinuous jump to zero. This gives strong evidence for a Kosterlitz-Thouless transition.