Determining the density of states for classical statistical models: a random walk algorithm to produce a flat histogram

Monte Carlo algorithm for free energy calculation

We propose a Monte Carlo algorithm for the free energy calculation based on configuration space sampling. An upward or downward temperature scan can be used to produce F(T). We implement this algorithm for the Ising model on a square lattice and triangular lattice. Comparison with the exact free energy shows an excellent agreement. We analyze the properties of this algorithm and compare it with the Wang-Landau algorithm, which samples in energy space. This method is applicable to general classical statistical models. The possibility of extending it to quantum systems is discussed.

Theoretical studies on the kinetics of hydrogen abstraction reactions of H and CH3 radicals from CH3OCH3 and some of their H/D isotopologues

The hydrogen abstraction reactions by H and CH3 radicals from CH3OCH3 and some of their H/D isotopologues are studied by semiclassical transition state theory. Many high-level density functional, ab initio, and combinatory quantum chemical methods, including B3LYP, BB1K, MP2, MP4, CCSD(T), CBS-Q, and G4 methods, are employed to compute the energies and rovibrational properties of the stationary points for the title reactions. Xij vibrational anharmonicity coefficients, used in semiclassical transition state theory, are computed at the B3LYP, BB1K, and MP2 levels of theory. Thermal rate coefficients and kinetic isotope effects are computed over the temperature range from 200 to 2500 K and compared with available experimental data. The computed rate constants for the title reactions are represented as the equation k(T) = ATn exp[−E(T + T0)/(T2 + T02)].

Phase transitions in free water nanoparticles. Theoretical modeling of [H2O]48 and [H2O]118

Classical parallel-tempering Monte Carlo simulations of [H2O]48 and [H2O]118 have been performed in the isothermal-isobaric ensemble and a two-dimensional multiple-histogram method has been used to calculate the heat capacity of the two clusters. A semiempirical procedure is proposed for the inclusion of quantum effects and transformed heat capacity profiles are compared with state-of-the-art experimental data [C. Hock et al., Phys. Rev. Lett., 2009, 103, 073401]. A very good agreement is achieved. A detailed analysis of the simulation data is provided to gain an insight into the nature of the phase change which takes place in the two clusters at T ≈ 100 K.

Force-extension curves for broken-rod macromolecules: Dramatic effects of different probing methods for two and three rods

We study force-extension curves of a single semiflexible chain consisting of several rigid rods connected by flexible spacers. The atomic force microscopy and laser optical or magnetic tweezers apparatus stretching these rod-coil macromolecules are discussed. In addition, the stretching by external isotropic force is analyzed. The main attention is focused on computer simulation and analytical results. We demonstrate that the force-extension curves for rod-coil chains composed of two or three rods of equal length differ not only quantitatively but also qualitatively in different probe methods. These curves have an anomalous shape for a chain of two rods. End-to-end distributions of rod-coil chains are calculated by Monte Carlo method and compared with analytical equations. The influence of the spacer's length on the force-extension curves in different probe methods is analyzed. The results can be useful for interpreting experiments on the stretching of rod-coil block-copolymers.

An efficient algorithm to perform local concerted movements of a chain molecule

The devising of efficient concerted rotation moves that modify only selected local portions of chain molecules is a long studied problem. Possible applications range from speeding the uncorrelated sampling of polymeric dense systems to loop reconstruction and structure refinement in protein modeling. Here, we propose and validate, on a few pedagogical examples, a novel numerical strategy that generalizes the notion of concerted rotation. The usage of the Denavit-Hartenberg parameters for chain description allows all possible choices for the subset of degrees of freedom to be modified in the move. They can be arbitrarily distributed along the chain and can be distanced between consecutive monomers as well. The efficiency of the methodology capitalizes on the inherent geometrical structure of the manifold defined by all chain configurations compatible with the fixed degrees of freedom. The chain portion to be moved is first opened along a direction chosen in the tangent space to the manifold, and then closed in the orthogonal space. As a consequence, in Monte Carlo simulations detailed balance is easily enforced without the need of using Jacobian reweighting. Moreover, the relative fluctuations of the degrees of freedom involved in the move can be easily tuned. We show different applications: the manifold of possible configurations is explored in a very efficient way for a protein fragment and for a cyclic molecule; the "local backbone volume", related to the volume spanned by the manifold, reproduces the mobility profile of all-α helical proteins; the refinement of small protein fragments with different secondary structures is addressed. The presented results suggest our methodology as a valuable exploration and sampling tool in the context of bio-molecular simulations.

Error catastrophe and phase transition in the empirical fitness landscape of HIV

We have translated clinical sequence databases of the p6 HIV protein into an empirical fitness landscape quantifying viral replicative capacity as a function of the amino acid sequence. We show that the viral population resides close to a phase transition in sequence space corresponding to an "error catastrophe" beyond which there is lethal accumulation of mutations. Our model predicts that the phase transition may be induced by drug therapies that elevate the mutation rate, or by forcing mutations at particular amino acids. Applying immune pressure to any combination of killer T-cell targets cannot induce the transition, providing a rationale for why the viral protein can exist close to the error catastrophe without sustaining fatal fitness penalties due to adaptive immunity.

Parallel multicanonical study of the three-dimensional Blume-Capel model

We study the thermodynamic properties of the three-dimensional Blume-Capel model on the simple cubic lattice by means of computer simulations. In particular, we implement a parallelized variant of the multicanonical approach and perform simulations by keeping a constant temperature and crossing the phase boundary along the crystal-field axis. We obtain numerical data for several temperatures in both the first- and second-order regime of the model. Finite-size scaling analyses provide us with transition points and the dimensional scaling behavior in the numerically demanding first-order regime, as well as a clear verification of the expected Ising universality in the respective second-order regime. Finally, we discuss the scaling behavior in the vicinity of the tricritical point.

Extended canonical Monte Carlo methods: Improving accuracy of microcanonical calculations using a reweighting technique

Velazquez and Curilef [J. Stat. Mech. (2010); J. Stat. Mech. (2010)] have proposed a methodology to extend Monte Carlo algorithms that are based on canonical ensemble. According to our previous study, their proposal allows us to overcome slow sampling problems in systems that undergo any type of temperature-driven phase transition. After a comprehensive review about ideas and connections of this framework, we discuss the application of a reweighting technique to improve the accuracy of microcanonical calculations, specifically, the well-known multihistograms method of Ferrenberg and Swendsen [Phys. Rev. Lett. 63, 1195 (1989)]. As an example of application, we reconsider the study of the four-state Potts model on the square lattice L×L with periodic boundary conditions. This analysis allows us to detect the existence of a very small latent heat per site qL during the occurrence of temperature-driven phase transition of this model, whose size dependence seems to follow a power law qL(L)∝(1/L)z with exponent z≃0.26±0.02. Discussed is the compatibility of these results with the continuous character of temperature-driven phase transition when L→+∞.

Sequence determines degree of knottedness in a coarse-grained protein model

Knots are abundant in globular homopolymers but rare in globular proteins. To shed new light on this long-standing conundrum, we study the influence of sequence on the formation of knots in proteins under native conditions within the framework of the hydrophobic-polar lattice protein model. By employing large-scale Wang-Landau simulations combined with suitable Monte Carlo trial moves we show that even though knots are still abundant on average, sequence introduces large variability in the degree of self-entanglements. Moreover, we are able to design sequences which are either almost always or almost never knotted. Our findings serve as proof of concept that the introduction of just one additional degree of freedom per monomer (in our case sequence) facilitates evolution towards a protein universe in which knots are rare.

Important roles of hydrophobic interactions in folding and charge interactions in misfolding of α-helix bundle protein

An enhanced-sampling molecular dynamics simulation is presented to quantitatively demonstrate the important roles of hydrophobic and charge interactions in the folding and misfolding of α-helix bundle protein, respectively.

Basis function sampling: a new paradigm for material property computation

Wang-Landau sampling, and the associated class of flat histogram simulation methods have been remarkably helpful for calculations of the free energy in a wide variety of physical systems. Practically, convergence of these calculations to a target free energy surface is hampered by reliance on parameters which are unknown a priori. Here, we derive and implement a method built upon orthogonal functions which is fast, parameter-free, and (importantly) geometrically robust. The method is shown to be highly effective in achieving convergence. An important feature of this method is its ability to attain arbitrary levels of description for the free energy. It is thus ideally suited to in silico measurement of elastic moduli and other material properties related to free energy perturbations. We demonstrate the utility of such applications by applying our method to calculate the Frank elastic constants of the Lebwohl-Lasher model of liquid crystals.

Sampling the equilibrium kinetic network of Trp-cage in explicit solvent

We employed the single replica multiple state transition interface sampling (MSTIS) approach to sample the kinetic (un)folding network of Trp-cage mini-protein in explicit water. Cluster analysis yielded 14 important metastable states in the network. The MSTIS simulation thus resulted in a full 14 × 14 rate matrix. Analysis of the kinetic rate matrix indicates the presence of a near native intermediate state characterized by a fully formed alpha helix, a slightly disordered proline tail, a broken salt-bridge, and a rotated arginine residue. This intermediate was also found in recent IR experiments. Moreover, the predicted rate constants and timescales are in agreement with previous experiments and simulations.

The adaptive biasing force method: everything you always wanted to know but were afraid to ask

In the host of numerical schemes devised to calculate free energy differences by way of geometric transformations, the adaptive biasing force algorithm has emerged as a promising route to map complex free-energy landscapes. It relies upon the simple concept that as a simulation progresses, a continuously updated biasing force is added to the equations of motion, such that in the long-time limit it yields a Hamiltonian devoid of an average force acting along the transition coordinate of interest. This means that sampling proceeds uniformly on a flat free-energy surface, thus providing reliable free-energy estimates. Much of the appeal of the algorithm to the practitioner is in its physically intuitive underlying ideas and the absence of any requirements for prior knowledge about free-energy landscapes. Since its inception in 2001, the adaptive biasing force scheme has been the subject of considerable attention, from in-depth mathematical analysis of convergence properties to novel developments and extensions. The method has also been successfully applied to many challenging problems in chemistry and biology. In this contribution, the method is presented in a comprehensive, self-contained fashion, discussing with a critical eye its properties, applicability, and inherent limitations, as well as introducing novel extensions. Through free-energy calculations of prototypical molecular systems, many methodological aspects are examined, from stratification strategies to overcoming the so-called hidden barriers in orthogonal space, relevant not only to the adaptive biasing force algorithm but also to other importance-sampling schemes. On the basis of the discussions in this paper, a number of good practices for improving the efficiency and reliability of the computed free-energy differences are proposed.

Phase transition properties of the Bell-Lavis model

Using Monte Carlo calculations we analyze the order and the universality class of phase transitions into a low-density (honeycomb) phase of a triangular antiferromagnetic three-state Bell-Lavis model. The results are obtained in a whole interval of chemical potential μ corresponding to the honeycomb phase. Our results demonstrate that the phase transitions might be attributed to the three-state Potts universality class for all μ values except for the edges of the honeycomb phase existence. At the honeycomb phase and the low-density gas phase boundary the transitions become of the first order. At another, honeycomb-to-frustrated phase boundary, we observe the approach to the crossover from the three-state Potts to the Ising model universality class. We also obtain the Schottky anomaly in the specific heat close to this edge. We show that the intermediate planar phase, found in a very similar antiferromagnetic triangular Blume-Capel model, does not occur in the Bell-Lavis model.

Effect of single-site mutations on hydrophobic-polar lattice proteins

We developed a heuristic method for determining the ground-state degeneracy of hydrophobic-polar (HP) lattice proteins, based on Wang-Landau and multicanonical sampling. It is applied during comprehensive studies of single-site mutations in specific HP proteins with different sequences. The effects in which we are interested include structural changes in ground states, changes of ground-state energy, degeneracy, and thermodynamic properties of the system. With respect to mutations, both extremely sensitive and insensitive positions in the HP sequence have been found. That is, ground-state energies and degeneracies, as well as other thermodynamic and structural quantities, may be either largely unaffected or may change significantly due to mutation.

Mapping central α-helix linker mediated conformational transition pathway of calmodulin via simple computational approach

The effects of intrinsic structural flexibility of calmodulin protein on the mechanism of its allosteric conformational transition are investigated in this article. Using a novel in silico approach, the conformational transition pathways of intact calmodulin as well as the isolated N- and C- terminal domains are identified and energetically characterized. It is observed that the central α-helix linker amplifies the structural flexibility of intact Ca(2+)-free calmodulin, which might facilitate the transition of the two domains. As a result, the global conformational transition of Ca(2+)-free calmodulin is initiated by the barrierless transition of two domains and proceeds through the barrier associated unwinding and bending of the central α-helix linker. The binding of Ca(2+) cations to calmodulin further increases the structural flexibility of the C-terminal domain and results in a downhill transition pathway of which all regions transit in a concerted manner. On the other hand, the separation of the N- and C-terminal domains from calmodulin protein loses the mediating function of central α-helix linker, leading to more difficult conformational transitions of both domains. The present study provides novel insights into the correlation of the integrity of protein, the structural flexibility, and the mechanism of conformational transition of proteinlike calmodulin.

Scalable replica-exchange framework for Wang-Landau sampling

We investigate a generic, parallel replica-exchange framework for Monte Carlo simulations based on the Wang-Landau method. To demonstrate its advantages and general applicability for massively parallel simulations of complex systems, we apply it to lattice spin models, the self-assembly process in amphiphilic solutions, and the adsorption of molecules on surfaces. While of general current interest, the latter phenomena are challenging to study computationally because of multiple structural transitions occurring over a broad temperature range. We show how the parallel framework facilitates simulations of such processes and, without any loss of accuracy or precision, gives a significant speedup and allows for the study of much larger systems and much wider temperature ranges than possible with single-walker methods.

Identifying transitions in finite systems by means of partition function zeros and microcanonical inflection-point analysis: a comparison for elastic flexible polymers

For the estimation of transition points of finite elastic, flexible polymers with chain lengths from 13 to 309 monomers, we compare systematically transition temperatures obtained by the Fisher partition function zeros approach with recent results from microcanonical inflection-point analysis. These methods rely on accurate numerical estimates of the density of states, which have been obtained by advanced multicanonical Monte Carlo sampling techniques. Both the Fisher zeros method and microcanonical inflection-point analysis yield very similar results and enable the unique identification of transition points in finite systems, which is typically impossible in the conventional canonical analysis of thermodynamic quantities.

Multiple phase transitions in extended hard-core lattice gas models in two dimensions

We study the k-NN hard-core lattice gas model in which the first k next-nearest-neighbor sites of a particle are excluded from occupation by other particles on a two-dimensional square lattice. This model is the lattice version of the hard-disk system with increasing k corresponding to decreasing lattice spacing. While the hard-disk system is known to undergo a two-step freezing process with increasing density, the lattice model has been known to show only one transition. Here, based on Monte Carlo simulations and high-density expansions of the free energy and density, we argue that for k = 4,10,11,14,⋯, the lattice model undergoes multiple transitions with increasing density. Using Monte Carlo simulations, we confirm the same for k = 4,...,11. This, in turn, resolves an existing puzzle as to why the 4-NN model has a continuous transition against the expectation of a first-order transition.

Thermodynamic and conformational insights into the phase transition of a single flexible homopolymer chain using replica exchange Monte Carlo method

The phase transition of a single flexible homopolymer chain in the limit condition of dilute solution is systematically investigated using a coarse-grained model. Replica exchange Monte Carlo method is used to enhance the performance of the conformation space exploration, and thus detailed investigation of phase behavior of the system can be provided. With the designed potentials, the coil-globule transition and the liquid-solid-like transition are identified, and the transition temperatures are measured with the conformational and thermodynamic analyses. Additionally, by extrapolating the coil-globule transition temperature, T Θ , and the liquid-solid-like transition temperature T(L → S) to the thermodynamic limit, N → ∞, we found no "tri-critical" point in the current model.

Phase transition behaviours of a single dendritic polymer

Dendritic polymers with highly branching structures exhibit many unique properties. In this paper, a computational study using the Wang-Landau sampling technique is carried out to reveal the phase transition behaviours of dendritic homopolymers with various branching structures. Two types of dendritic homopolymers, dendrimers/dendrigrafts (D/D) and hyperbranched (HB) polymers are studied. It is found that with increasing degree of branching in the dendritic polymer, the liquid-solid (LS) transition temperature increases and the coil-globule (CG) transition becomes weak. Additionally, under similar degrees of branching and polymerization, D/D polymers have a higher LS transition temperature than HB polymers. The reason is that the D/D polymers have greater regularity in the radial distribution of the branching units, which facilitates monomer packing during the LS transition. The distinctive internal unit distribution at various temperatures is quantitatively analysed. Our results show the importance of dendritic polymer structure regularity in phase transition behaviours and are valuable in guiding the structural design of dendritic macromolecules for functionalization applications.

Detection of an intermediate biaxial phase in the phase diagram of biaxial liquid crystals: entropic sampling study

We investigate the phase sequence of biaxial liquid crystals, based on a general quadratic model Hamiltonian over the relevant parameter space, with a Monte Carlo simulation which constructs equilibrium ensembles of microstates, overcoming possible (free) energy barriers (combining entropic and frontier sampling techniques). The resulting phase diagram qualitatively differs from the universal phase diagram predicted earlier from mean-field theory (MFT), as well as the Monte Carlo simulations with the Metropolis algorithm. The direct isotropic-to-biaxial transition predicted by the MFT is replaced in certain regions of the space by the onset of an additional intermediate biaxial phase of very low order, leading to the sequence N(B)-N(B1)-I. This is due to inherent barriers to fluctuations of the components comprising the total energy, and may explain the difficulties in the experimental realization of these phases.

Molecular Dynamics in the Multicanonical Ensemble: Equivalence of Wang-Landau Sampling, Statistical Temperature Molecular Dynamics, and Metadynamics

We show a direct formal relationship between the Wang-Landau iteration [PRL 86, 2050 (2001)], metadynamics [PNAS 99, 12562 (2002)], and statistical temperature molecular dynamics (STMD) [PRL 97, 050601 (2006)] that are the major work-horses for sampling from generalized ensembles. We demonstrate that STMD, itself derived from the Wang-Landau method, can be made indistinguishable from metadynamics. We also show that Gaussian kernels significantly improve the performance of STMD, highlighting the practical benefits of this improved formal understanding.

Intramolecular Processes Revealed Using UV-Laser-Induced IR-Fluorescence: A New Perspective on the "Channel Three" of Benzene

Radiative relaxation in the infrared (IR) is common following nonradiative electronic relaxation processes, but it is rarely measured. We present ultraviolet laser-induced infrared fluorescence (UV-LIIRF) excitation spectroscopy and dispersed UV-LIIRF spectroscopy of gas phase benzene vibronically excited around the onset of channel 3, using a homemade spectrometer. We found that the vibrational IR fluorescence yield is clearly higher when benzene is excited above the onset than when it is excited below. Significant changes in dispersed IR emission profiles resulting from excitations below and above the onset of channel 3 were also observed. These results suggest that isomerization of benzene toward fulvene occurs efficiently below the opening of channel 3 and confirm that channel 3 involves a photophysical relaxation pathway that efficiently competes with isomerization.

Wang-Landau sampling: a criterion for halting the simulations

In this work we propose a criterion to finish the simulations of the Wang-Landau sampling. Instead of determining a final modification factor for all simulations and every sample size, we investigate the behavior of the temperature of the peak of the specific heat during the simulations and finish them when this value varies below a given limit. As a result, different runs stop at different final modification factors. We show that in place of the temperature of the peak of the specific heat one can adopt alternatively the integrated heat transfer as a reference quantity. We apply this technique to the two-dimensional Ising model and a homopolymer. We verify that for the Ising model the mean order of the final modification factors is roughly the same for all lattice sizes, but for the homopolymer the order of the final modification factors increases with increasing polymer sizes. The results show that the simulations can be halted much earlier than is conventional in Wang-Landau sampling, but manifold finite-size simulations are required in order to obtain accurate results. A brief application to the three-dimensional Ising model is also available.

Wang-Landau sampling with logarithmic windows for continuous models

We present a modified Wang-Landau sampling (MWLS) for continuous statistical models by partitioning the energy space into a set of windows with logarithmically shrinking width. To demonstrate its necessity and advantages, we apply this sampling to several continuous models, including the two-dimensional square XY spin model, triangular J1-J2 spin model, and Lennard-Jones cluster model. Given a finite number of bins for partitioning the energy space, the conventional Wang-Landau sampling may not generate sufficiently accurate density of states (DOS) around the energy boundaries. However, it is demonstrated that much more accurate DOS can be obtained by this MWLS, and thus a precise evaluation of the thermodynamic behaviors of the continuous models at extreme low temperature (kBT<0.1) becomes accessible. The present algorithm also allows efficient computation besides the highly reliable data sampling.