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

Rejection-free cluster Wang-Landau algorithm for hard-core lattice gases

We introduce a rejection-free, flat histogram, cluster algorithm to determine the density of states of hard-core lattice gases. We show that the algorithm is able to efficiently sample low entropy states that are usually difficult to access, even when the excluded volume per particle is large. The algorithm is based on simultaneously evaporating all the particles in a strip and reoccupying these sites with a new appropriately chosen configuration. We implement the algorithm for the particular case of the hard-core lattice gas in which the first k next-nearest neighbors of a particle are excluded from being occupied. It is shown that the algorithm is able to reproduce the known results for k=1,2,3 both on the square and cubic lattices. We also show that, in comparison, the corresponding flat histogram algorithms with either local moves or unbiased cluster moves are less accurate and do not converge as the system size increases.

Multidimensional Free Energy and Accelerated Quantum Library Methods Provide a Gateway to Glycoenzyme Conformational, Electronic, and Reaction Mechanisms

Enzyme reactions are complex to simulate accurately, and none more so than glycoenzymes (glycosyltransferase and glycosidases). A rigorous sampling of the protein frame and the conformationally plural carbohydrate substrate coupled with an unbiased treatment of the electron dynamics is needed to discover the true reaction landscapes. Here, we demonstrate the effectiveness of two computational methods ported in libraries that we have developed. The first is a flat histogram free energy method called FEARCF capable of multidimensional sampling and rapidly converging to a complete coverage of phase space. The second, the Quantum Supercharger Library (QSL), is a method that accelerates the computation of the ab initio electronic wave function as well as the integral derivatives on graphical processing units (GPUs). These QSL accelerated computations form the core components needed for direct quantum dynamics and QM/MM dynamics when coupled with legacy codes such as GAMESS and NWCHEM, making state of the art hyper-parallel electronic computations in chemistry and chemical biology possible. The combination of QSL (acceleration of ab initio QM computation) and FEARCF (multidimensional hyper-parallel reaction dynamics) makes the simulation of ab initio QM/MM reaction dynamics of enzyme catalysis feasible. Enzymes that process carbohydrates pose an added challenge as their pyranose ring substrates span multidimensional conformational space whose sampling is an intimate function of the catalytic mechanism. Here, we use the pairing of FEARCF and QSL to simulate the catalytic effect of the glycoenzyme β-N-acetylglucosamine transferase (OGT). The reaction mechanism is discovered from a variable three bond reaction surface using SCCDFTB. The role of the OGT in distorting the pyranose ring of β-N-acetylglucosamine (GlcNAc) away from the equilibrium 4C1 chair conformation toward the E3 envelope needed for the transition state is discovered from its pucker free energy hypersurfaces (or free energy volume, FEV). A complete GlcNAc ring pucker HF 6-31g FEV is constructed from ab initio QM dynamics in vacuum and ab initio QM/MM dynamics in the OGT catalytic domain. The OGT is shown to clearly lower the pathway toward the transition state E3 ring conformer as well as stabilize it by 1.63 kcal/mol. Illustrated here is the use of QSL accelerated ab initio QM/MM dynamics that thoroughly explores carbohydrate catalyzed reactions through a FEARCF multidimensional sampling of the interdependence between reaction and conformational space. This demonstrates how experimentally inaccessible molecular and electronic mechanisms that underpin enzyme catalysis can be discovered by directly modeling the dynamics of these complex reactions.

Partition-function-zero analysis of polymer adsorption for a continuum chain model

Polymer chains undergoing adsorption are expected to show universal critical behavior which may be investigated using partition function zeros. The focus of this work is the adsorption transition for a continuum chain, allowing for investigation of a continuous range of the attractive interaction and comparison with recent high-precision lattice model studies. The partition function (Fisher) zeros for a tangent-hard-sphere N-mer chain (monomer diameter σ) tethered to a flat wall with an attractive square-well potential (range λσ, depth ε) have been computed for chains up to N=1280 with 0.01≤λ≤2.0. In the complex-Boltzmann-factor plane these zeros are concentrated in an annular region, centered on the origin and open about the real axis. With increasing N, the leading zeros, w_{1}(N), approach the positive real axis as described by the asymptotic scaling law w_{1}(N)-y_{c}∼N^{-ϕ}, where y_{c}=e^{ε/k_{B}T_{c}} is the critical point and T_{c} is the critical temperature. In this work, we study the polymer adsorption transition by analyzing the trajectory of the leading zeros as they approach y_{c} in the complex plane. We use finite-size scaling (including corrections to scaling) to determine the critical point and the scaling exponent ϕ as well as the approach angle θ_{c}, between the real axis and the leading-zero trajectory. Variation of the interaction range λ moves the critical point, such that T_{c} decreases with λ, while the results for ϕ and θ_{c} are approximately independent of λ. Our values of ϕ=0.479(9) and θ_{c}=56.8(1.4)^{∘} are in agreement with the best lattice model results for polymer adsorption, further demonstrating the universality of these constants across both lattice and continuum models.

Reducing autocorrelation time in determinant quantum Monte Carlo using the Wang-Landau algorithm: Application to the Holstein model

When performing a Monte Carlo calculation, the running time should, in principle, be much longer than the autocorrelation time in order to get reliable results. Among different lattice fermion models, the Holstein model is notorious for its particularly long autocorrelation time. In this paper, we employ the Wang-Landau algorithm in the determinant quantum Monte Carlo to achieve the flat-histogram sampling in the "configuration weight space," which can greatly reduce the autocorrelation time by sacrificing some sampling efficiency. The proposal is checked in the Holstein model on both square and honeycomb lattices. Based on such a Wang-Landau assisted determinant quantum Monte Carlo method, some models with long autocorrelation times can now be simulated possibly.

Thermodynamics of hydrophobic-polar model proteins on the face-centered cubic lattice

The HP model, a coarse-grained protein representation with only hydrophobic (H) and polar (P) amino acids, has already been extensively studied on the simple cubic (SC) lattice. However, this geometry severely restricts possible bond angles, and a simple improvement is to instead use the face-centered cubic (fcc) lattice. In this paper, the density of states and ground state energies are calculated for several benchmark HP sequences on the fcc lattice using the replica-exchange Wang-Landau algorithm and a powerful set of Monte Carlo trial moves. Results from the fcc lattice proteins are directly compared with those obtained from a previous lattice protein folding study with a similar methodology on the SC lattice. A thermodynamic analysis shows comparable folding behavior between the two lattice geometries, but with a greater rate of hydrophobic-core formation persisting into lower temperatures on the fcc lattice.

The CH(X2Π) + H2O reaction: two transition state kinetics

The reaction of ground state methylidyne (CH) with water vapor (H2O) is theoretically re-investigated using high-level coupled cluster computations in combination with semi-classical transition state theory (SCTST) and two-dimensional master equation simulations. Insertion of CH into a H-O bond of H2O over a submerged barrier via a well-skipping mechanism yielding solely H and CH2O is characterized. The reaction kinetics is effectively determined by the formation of a pre-reaction van der Waals complex (PRC, HC-OH2) and its subsequent isomerization to activated CH2OH in competition with PRC re-dissociation. The tunneling effects are found to be minor, while variational effects in the PRC → CH2OH step are negligible. The calculated rate coefficient k(T) is nearly pressure-independent, but strongly depends on temperature with pronounced down-up behavior: a high value of 2 × 10-10 cm3 s-1 at 50 K, followed by a fairly steep decrease down to 8 × 10-12 cm3 s-1 at 900 K, but increasing again to 5 × 10-11 cm3 s-1 at 3500 K. Over the T-range of this work, k(T) can be expressed as: k(T, P = 0) = 2.31 × 10-11 (T/300 K)-1.615 exp(-38.45/T) cm3 s-1 for T = 50-400 K k(T, P = 0) = 1.15 × 10-12 (T/300 K)0.8637 exp(892.6/T) cm3 s-1 for T = 400-1000 K k(T, P = 0) = 4.57 × 10-15 (T/300 K)3.375 exp(3477.4/T) cm3 s-1 for T = 1000-3500 K.

Coupled Monte Carlo and Molecular Dynamics Simulations on Interfacial Properties of Antifouling Polymer Membranes

We use molecular simulation to study the wetting behavior of antifouling polymer-tethered membranes. We obtain the interfacial properties (e.g., contact angle) of water at various temperatures for five polymer membranes, including a base polysulfone (PSF) membrane and four other PSF membranes grafted with antifouling polymers (two poly(ethylene glycol) (PEG) tethers and two zwitterionic tethers). We implement a coupled Monte Carlo (MC)/molecular dynamics (MD) approach to determine the interface potentials of water on the membrane surfaces in an efficient manner. Within this method, short MC and MD simulations are performed in cycles to collect the surface excess free energy of a thin water film on polymer membrane surfaces. Simulation results show that the grafting of zwitterionic tethers provides a more significant enhancement in the hydrophilicity of the PSF membrane than that of the PEG tethers. Water completely wets the surface of zwitterionic polymer membranes.

Interfacial segregation of interacting vacancies and their role on the wetting critical properties of the Blume-Emery-Griffiths model

We study the wetting critical behavior of the three-state (s=±1,0) Blume-Emery-Griffiths model using numerical simulations. This model provides a suitable scenario for the study of the role of vacancies on the wetting behavior of a thin magnetic film. To this aim we study a system confined between parallel walls with competitive short-range surface magnetic fields (h_{L}=-|h_{1}|). We locate relevant critical curves for different values of the biquadratic interaction and use a thermodynamic integration method to calculate the surface tension as well as the interfacial excess energy and determine the wetting transition. Furthermore, we also calculate the local position of the interface along the film and its fluctuations (capillary waves), which are a measure of the interface width. To characterize the role played by vacancies on the interfacial behavior we evaluate the excess density of vacancies, i.e., the density difference between a system with and without interface. We also show that the temperature dependence of both the local position of the interface and its width can be rationalized in term of a finite-size scaling description, and we propose and successfully test the same scaling behavior for the average position of the center of mass of the vacancies and its fluctuations. This shows that the excess of vacancies can be associated to the presence of the interface that causes the observed segregation. This segregation phenomena is also evidenced by explicitly evaluating the interfacial free energy.

Phase diagram of the Potts model with the number of spin states q = 4 on a kagome lattice

The magnetic structures of the ground state, phase transitions, and the thermodynamic properties of a two-dimensional ferromagnetic Potts model with the number of spin states q = 4 on a kagome lattice are studied using the Wang-Landau algorithm of the Monte Carlo method, taking into account the interactions of the nearest and the next-nearest neighbors. The studies were carried out for the value of the interaction of the next-nearest neighbors in the range 0 ≤ r ≤ 1.0. It is shown that taking into account the antiferromagnetic interactions of the next-nearest neighbor leads to a violation of the magnetic ordering. A phase diagram of the dependence of the critical temperature on the value of the interaction of the next-nearest neighbor is constructed. The analysis of the character of phase transitions is carried out. It was found that in the ranges 0 ≤ r ≤ 0.5 and 0.5 ≤ r ≤ 1.0, a first-order phase transition is observed, and for r = 0.5, frustrations are observed in the system.

A Deep Graph Network-Enhanced Sampling Approach to Efficiently Explore the Space of Reduced Representations of Proteins

The limits of molecular dynamics (MD) simulations of macromolecules are steadily pushed forward by the relentless development of computer architectures and algorithms. The consequent explosion in the number and extent of MD trajectories induces the need for automated methods to rationalize the raw data and make quantitative sense of them. Recently, an algorithmic approach was introduced by some of us to identify the subset of a protein's atoms, or mapping, that enables the most informative description of the system. This method relies on the computation, for a given reduced representation, of the associated mapping entropy, that is, a measure of the information loss due to such simplification; albeit relatively straightforward, this calculation can be time-consuming. Here, we describe the implementation of a deep learning approach aimed at accelerating the calculation of the mapping entropy. We rely on Deep Graph Networks, which provide extreme flexibility in handling structured input data and whose predictions prove to be accurate and-remarkably efficient. The trained network produces a speedup factor as large as 105 with respect to the algorithmic computation of the mapping entropy, enabling the reconstruction of its landscape by means of the Wang-Landau sampling scheme. Applications of this method reach much further than this, as the proposed pipeline is easily transferable to the computation of arbitrary properties of a molecular structure.

General relations to obtain the time-dependent heat capacity from isothermal simulations

It is well-known that time-dependent correlation functions related to temperature and energy can crucially depend on the thermostatting mechanism used in computer simulations of molecular systems. We argue, however, that linear response functions must be considered as universal properties of physical systems. This implies that the classical fluctuation equation for the transient heat capacity, c_v(t), is not applicable to the thermostatted molecular dynamics (apart from long enough times). To improve on this point, we derive a number of exact general expressions for the frequency-dependent heat capacity in terms of energy correlation functions, valid for the Nosé-Hoover and some other thermostats. We also establish a general relation between auto- and cross correlation functions of energy and temperature. Recommendations on how to use these relations to maximize the numerical precision are provided. It is demonstrated that our approach allows us to obtain c_v(t) for a supercooled liquid system with high precision and over many decades in time reflecting all pertinent relaxation processes.

A thermodynamic scaling law for electrically perturbed lipid membranes: Validation with steepest entropy ascent framework

Experimental evidence has demonstrated the ability of transient pulses of electric fields to alter mammalian cell behavior. Strategies with these pulsed electric fields (PEFs) have been developed for clinical applications in cancer therapeutics, in-vivo decellularization, and tissue regeneration. Successful implementation of these strategies involve understanding how PEFs impact the cellular structures and, hence, cell behavior. The caveat, however, is that the PEF parameter space (i.e., comprising different pulse widths, amplitudes, number of pulses) is large, and design of experiments to explore all possible combinations of pulse parameters is prohibitive from a cost and time standpoint. In this study, a scaling law based on the Ising model is introduced to understand the impact of PEFs on the outer cell lipid membrane so that an understanding developed in one PEF pulse regime may be extended to another. Combining non-Markovian Monte Carlo techniques to determine density-of-states with a novel non-equilibrium thermodynamic framework based on the principle of steepest entropy ascent, the applicability of this scaling model to predict the behavior of both thermally quenched and electrically perturbed lipid membranes is demonstrated. A comparison of the predictions made by the steepest-entropy-ascent quantum thermodynamic (SEAQT) framework to experimental data is performed to validate the robustness of this computational methodology and the resulting scaling law.

Anharmonic Infrared Spectra of Thermally Excited Pyrene (C16H10): A Combined View of DFT-Based GVPT2 with AnharmonicCaOs, and Approximate DFT Molecular dynamics with DemonNano

The study of the Aromatic Infrared Bands (AIBs) in astronomical environments has opened interesting spectroscopic questions on the effect of anharmonicity on the infrared (IR) spectrum of hot polycyclic aromatic hydrocarbons (PAHs) and related species in isolated conditions. The forthcoming James Webb Space Telescope will unveil unprecedented spatial and spectral details in the AIB spectrum; significant advancement is thus necessary now to model the infrared emission of PAHs, their presumed carriers, with enough detail to exploit the information content of the AIBs. This requires including anharmonicity in such models, and to do so systematically for all species included, requiring a difficult compromise between accuracy and efficiency. We performed a benchmark study to compare the performances of two methods in calculating anharmonic spectra, comparing them to available experimental data. One is a full quantum method, AnharmoniCaOs, relying on an ab initio potential, and the other relies on Molecular Dynamics simulations using a Density Functional based Tight Binding potential. The first one is found to be very accurate and detailed, but it becomes computationally very expensive for increasing temperature; the second is faster and can be used for arbitrarily high temperatures, but is less accurate. Still, its results can be used to model the evolution with temperature of isolated bands. We propose a new recipe to model anharmonic AIB emission using minimal assumptions on the general behaviour of band positions and widths with temperature, which can be defined by a small number of empirical parameters. Modelling accuracy will depend critically on these empirical parameters, allowing for an incremental improvement in model results, as better estimates become gradually available.

Modeling Gas-Phase Unimolecular Dissociation for Bond Dissociation Energies: Comparison of Statistical Rate Models within RRKM Theory

The Rice-Ramsperger-Kassel-Marcus (RRKM) theory provides a simple yet powerful rate theory for calculating microcanonical rate constants. In particular, it has found widespread use in combination with gas-phase kinetic experiments of unimolecular dissociations to extract experimental bond dissociation energies (BDEs). We have previously found several discrepancies between the computed BDE values and the respective experimental ones, obtained with our empirical rate model, named L-CID. To investigate the reliability of our rate model, we conducted a theoretical analysis and comparison of the performance of conventional rate models and L-CID within the RRKM framework. Using the previously published microcanonical rate data as well as reaction cross-section data, we show that the BDE values obtained with the L-CID model agree with the ones from the other rate models within the expected uncertainty bounds. Based on this agreement, we discuss the possible rationalization of the good performance of the L-CID model.

Effects of lattice constraints in coarse-grained protein models

We compare and contrast folding behavior in several coarse-grained protein models, both on- and off-lattice, in an attempt to uncover the effect of lattice constraints in these kinds of models. Using modern, extended ensemble Monte Carlo methods-Wang-Landau sampling, multicanonical sampling, replica-exchange Wang-Landau sampling, and replica-exchange multicanonical sampling, we investigate the thermodynamic and structural behavior of the protein Crambin within the context of the hydrophobic-polar, hydrophobic-"neutral"-polar (H0P), and semi-flexible H0P model frameworks. We uncover the folding process in all cases; all models undergo, at least, the two major structural transitions observed in nature-the coil-globule collapse and the folding transition. As the complexity of the model increases, these two major transitions begin to split into multi-step processes, wherein the lattice coarse-graining has a significant impact on the details of these processes. The results show that the level of structural coarse-graining is coupled to the level of interaction coarse-graining.

Calculation of the residual entropy of Ice Ih by Monte Carlo simulation with the combination of the replica-exchange Wang-Landau algorithm and multicanonical replica-exchange method

We estimated the residual entropy of Ice Ih by the recently developed simulation protocol, namely, the combination of the replica-exchange Wang-Landau algorithm and multicanonical replica-exchange method. We employed a model with the nearest neighbor interactions on the three-dimensional hexagonal lattice, which satisfied the ice rules in the ground state. The results showed that our estimate of the residual entropy is in accordance with various previous results. In this article, we not only give our latest estimate of the residual entropy of Ice Ih but also discuss the importance of the uniformity of a random number generator in Monte Carlo simulations.

A journey through mapping space: characterising the statistical and metric properties of reduced representations of macromolecules

ABSTRACT

A mapping of a macromolecule is a prescription to construct a simplified representation of the system in which only a subset of its constituent atoms is retained. As the specific choice of the mapping affects the analysis of all-atom simulations as well as the construction of coarse-grained models, the characterisation of the mapping space has recently attracted increasing attention. We here introduce a notion of scalar product and distance between reduced representations, which allows the study of the metric and topological properties of their space in a quantitative manner. Making use of a Wang-Landau enhanced sampling algorithm, we exhaustively explore such space, and examine the qualitative features of mappings in terms of their squared norm. A one-to-one correspondence with an interacting lattice gas on a finite volume leads to the emergence of discontinuous phase transitions in mapping space, which mark the boundaries between qualitatively different reduced representations of the same molecule.

Entropy in Molecular Fluids: Interplay between Interaction Complexity and Criticality

Using flat-histogram simulations, we calculate the entropy of molecular fluids along the vapor-liquid phase boundary. Our simulation approach is based on the evaluation of the canonical and grand-canonical partition functions, which, in turn, provide access to entropy through the statistical mechanics formalism. The results allow us to determine the critical entropy of molecular fluids and to uncover that the transition occurs symmetrically from an entropic standpoint. This can best be seen through the patterns exhibited by the thermodynamic variables temperature and pressure when plotted against the entropy of the coexisting phases. This behavior is found to hold for apolar, quadrupolar, and dipolar fluids. Finally, we identify functional forms that characterize the relation between thermodynamic variables and entropy along the coexistence curve up to the critical point.

First-principles calculation of the configurational energy density of states for a solid-state ion conductor with a variant of the Wang and Landau algorithm

In this work, a variant of the Wang and Landau algorithm for calculation of the configurational energy density of states is proposed. The algorithm was developed for the purpose of using first-principles simulations, such as density functional theory, to calculate the partition function of disordered sublattices in crystal materials. The expensive calculations of first-principles methods make a parallel algorithm necessary for a practical computation of the configurational energy density of states within a supercell approximation of a solid-state material. The algorithm developed in this work is tested with the two-dimensional (2d) Ising model to bench mark the algorithm and to help provide insight for implementation to a materials science application. Tests with the 2d Ising model revealed that the algorithm has good performance compared to the original Wang and Landau algorithm and the 1/t algorithm, in particular the short iteration performance. A proof of convergence is presented within an adiabatic assumption, and the analysis is able to correctly predict the time dependence of the modification factor to the density of states. The algorithm was then applied to the lithium and lanthanum sublattice of the solid-state lithium ion conductor Li_{0.5}La_{0.5}TiO_{3}. This was done to help understand the disordered nature of the lithium and lanthanum. The results find, overall, that the algorithm performs very well for the 2d Ising model and that the results for Li_{0.5}La_{0.5}TiO_{3} are consistent with experiment while providing additional insight into the lithium and lanthanum ordering in the material. The primary result is that the lithium and lanthanum become more mixed between layers along the c axis for increasing temperature. In part, the simulation of the disordered Li_{0.5}La_{0.5}TiO_{3} system serves as a benchmark for what size systems are currently and in the near future practical to calculate with density functional theory methods.

First-Principles Atomistic Thermodynamics and Configurational Entropy

In most applications, functional materials operate at finite temperatures and are in contact with a reservoir of atoms or molecules (gas, liquid, or solid). In order to understand the properties of materials at realistic conditions, statistical effects associated with configurational sampling and particle exchange at finite temperatures must consequently be taken into account. In this contribution, we discuss the main concepts behind equilibrium statistical mechanics. We demonstrate how these concepts can be used to predict the behavior of materials at realistic temperatures and pressures within the framework of atomistic thermodynamics. We also introduce and discuss methods for calculating phase diagrams of bulk materials and surfaces as well as point defect concentrations. In particular, we describe approaches for calculating the configurational density of states, which requires the evaluation of the energies of a large number of configurations. The cluster expansion method is therefore also discussed as a numerically efficient approach for evaluating these energies.

Ising universality in the two-dimensional Blume-Capel model with quenched random crystal field

Using high-precision Monte Carlo simulations based on a parallel version of the Wang-Landau algorithm and finite-size scaling techniques, we study the effect of quenched disorder in the crystal-field coupling of the Blume-Capel model on a square lattice. We mainly focus on the part of the phase diagram where the pure model undergoes a continuous transition, known to fall into the universality class of a pure Ising ferromagnet. A dedicated scaling analysis reveals concrete evidence in favor of the strong universality hypothesis with the presence of additional logarithmic corrections in the scaling of the specific heat. Our results are in agreement with an early real-space renormalization-group study of the model as well as a very recent numerical work where quenched randomness was introduced in the energy exchange coupling. Finally, by properly fine tuning the control parameters of the randomness distribution we also qualitatively investigate the part of the phase diagram where the pure model undergoes a first-order phase transition. For this region, preliminary evidence indicate a smoothing of the transition to second-order with the presence of strong scaling corrections.

Effects of macromolecular crowding on the folding of a polymer chain: A Wang-Landau simulation study

A flexible polymer chain in the presence of inert macromolecular crowders will experience a loss of configurational entropy due to the crowder excluded volume. This entropy reduction will be most pronounced in good solvent conditions where the chain assumes an expanded coil conformation. For polymers that undergo a folding transition from a coil to a compact ordered state, as is the case for many globular proteins, macromolecular crowding is expected to stabilize the folded state and thereby shift the transition location. Here, we study such entropic stabilization effects for a tangent square-well sphere chain (monomer diameter σ) in the presence of hard-sphere (HS) crowders (diameter D ≥ σ). We use the Wang-Landau simulation algorithm to construct the density of states for this chain in a crowded environment and are thus able to directly compute the reduction in configurational entropy due to crowding. We study both a chain that undergoes all-or-none folding directly from the coil state and a chain that folds via a collapsed-globule intermediate state. In each case, we find an increase in entropic stabilization for the compact states with an increase in crowder density and, for fixed crowder density, with a decrease in crowder size (concentrated, small crowders have the largest effect). The crowder significantly reduces the average size for the unfolded states while having a minimal effect on the size of the folded states. In the athermal limit, our results directly provide the confinement free energy due to crowding for a HS chain in a HS solvent.

Large deviations of connected components in the stochastic block model

We study the stochastic block model, which is often used to model community structures and study community-detection algorithms. We consider the case of two blocks in regard to its largest connected component and largest biconnected component, respectively. We are especially interested in the distributions of their sizes including the tails down to probabilities smaller than 10^{-800}. For this purpose we use sophisticated Markov chain Monte Carlo simulations to sample graphs from the stochastic block model ensemble. We use these data to study the large-deviation rate function and conjecture that the large-deviation principle holds. Further we compare the distribution to the well-known Erdős-Rényi ensemble, where we notice subtle differences at and above the percolation threshold.

Statistical Physics Approach to Liquid Crystals: Dynamics of Mobile Potts Model Leading to Smectic Phase, Phase Transition by Wang-Landau Method

We study the nature of the smectic-isotropic phase transition using a mobile 6-state Potts model. Each Potts state represents a molecular orientation. We show that with the choice of an appropriate microscopic Hamiltonian describing the interaction between individual molecules modeled by a mobile 6-state Potts spins, we observe the smectic phase dynamically formed when we cool the molecules from the isotropic phase to low temperatures (T). In order to elucidate the order of the transition and the low-T properties, we use the high-performance Wang-Landau flat energy-histogram technique. We show that the smectic phase goes to the liquid (isotropic) phase by melting/evaporating layer by layer starting from the film surface with increasing T. At a higher T, the whole remaining layers become orientationally disordered. The melting of each layer is characterized by a peak of the specific heat. Such a succession of partial transitions cannot be seen by the Metropolis algorithm. The successive layer meltings/evaporations at low T are found to have a first-order character by examining the energy histogram. These results are in agreement with experiments performed on some smectic liquid crystals.

Predicting miRNA-disease associations using a hybrid feature representation in the heterogeneous network

BACKGROUND

Studies have found that miRNAs play an important role in many biological activities involved in human diseases. Revealing the associations between miRNA and disease by biological experiments is time-consuming and expensive. The computational approaches provide a new alternative. However, because of the limited knowledge of the associations between miRNAs and diseases, it is difficult to support the prediction model effectively.

METHODS

In this work, we propose a model to predict miRNA-disease associations, MDAPCOM, in which protein information associated with miRNAs and diseases is introduced to build a global miRNA-protein-disease network. Subsequently, diffusion features and HeteSim features, extracted from the global network, are combined to train the prediction model by eXtreme Gradient Boosting (XGBoost).

RESULTS

The MDAPCOM model achieves AUC of 0.991 based on 10-fold cross-validation, which is significantly better than that of other two state-of-the-art methods RWRMDA and PRINCE. Furthermore, the model performs well on three unbalanced data sets.

CONCLUSIONS

The results suggest that the information behind proteins associated with miRNAs and diseases is crucial to the prediction of the associations between miRNAs and diseases, and the hybrid feature representation in the heterogeneous network is very effective for improving predictive performance.

Topological transition in a two-dimensional three-vector model: A non-Boltzmann Monte Carlo study

Two-dimensional three-vector (d=2,n=3) lattice model of a liquid crystal (LC) system with order parameter space (R) described by the fundamental group Π_{1}(R)=Z_{2} was recently investigated based on non-Boltzmann Monte Carlo simulations. Its results indicated that the system did not undergo a topological transition condensing to a low temperature critical state as was reported earlier. Instead, a crossover to a nematic phase was observed, induced by the onset of a competing relevant length scale. This mechanism is further probed here by assigning a more restrictive R symmetry with Π_{1}(R)=Q (the discrete and non-Abelian group of quaternions), thus engaging the three spin degrees in the formation of point topological defects (disclinations). The results reported here indicate that such a choice of symmetry of the Hamiltonian with suitable model parameters leads to a defect-mediated transition to a low-temperature phase with topological order. It is characterized by a line of critical points with quasi-long-range order of its three spin degrees. The associated temperature-dependent power-law exponent decreases progressively and vanishes linearly as temperature tends to zero. The high-temperature disordered phase shows exponential spin correlations and their temperature-dependent lengths exhibit an essential singular divergence as the system approaches the topological transition point. Biaxial LC models have the required R symmetry owing to their tensor orientational orders and are suggested to serve as prototype examples to exhibit topological transition in (d=2,n=3) lattice models.

Flat-histogram method comparison on the two-dimensional Ising model

We compare the convergence of several flat-histogram methods applied to the two-dimensional Ising model, including the recently introduced stochastic approximation with a dynamic update factor (SAD) method. We compare this method to the Wang-Landau (WL) method, the 1/t variant of the WL method, and standard stochastic approximation Monte Carlo (SAMC). In addition, we consider a procedure WL followed by a "production run" with fixed weights that refines the estimation of the entropy. We find that WL followed by a production run does converge to the true density of states, in contrast to pure WL. Three of the methods converge robustly: SAD, 1/t-WL, and WL followed by a production run. Of these, SAD does not require a priori knowledge of the energy range. This work also shows that WL followed by a production run performs superior to other forms of WL while ensuring both ergodicity and detailed balance.

Lattice Glass Model in Three Spatial Dimensions

The understanding of thermodynamic glass transition has been hindered by the lack of proper models beyond mean-field theories. Here, we propose a three-dimensional lattice glass model on a simple cubic lattice that exhibits the typical dynamics observed in fragile supercooled liquids such as two-step relaxation, super-Arrhenius growth in the relaxation time, and dynamical heterogeneity. Using advanced Monte Carlo methods, we compute the thermodynamic properties deep inside the glassy temperature regime, well below the onset temperature of the slow dynamics. The specific heat has a finite jump towards the thermodynamic limit with critical exponents close to those expected from the hyperscaling and the random first-order transition theory for the glass transition. We also study an effective free energy of glasses, the Franz-Parisi potential, as a function of the overlap between equilibrium and quenched configurations. The effective free energy indicates the existence of a first-order phase transition, consistent with the random first-order transition theory. These findings strongly suggest that the glassy dynamics of the model has its origin in thermodynamics.

Emergence of cooperative bistability and robustness of gene regulatory networks

Gene regulatory networks (GRNs) are complex systems in which many genes regulate mutually to adapt the cell state to environmental conditions. In addition to function, the GRNs possess several kinds of robustness. This robustness means that systems do not lose their functionality when exposed to disturbances such as mutations or noise, and is widely observed at many levels in living systems. Both function and robustness have been acquired through evolution. In this respect, GRNs utilized in living systems are rare among all possible GRNs. In this study, we explored the fitness landscape of GRNs and investigated how robustness emerged in highly-fit GRNs. We considered a toy model of GRNs with one input gene and one output gene. The difference in the expression level of the output gene between two input states, "on" and "off", was considered as fitness. Thus, the determination of the fitness of a GRN was based on how sensitively it responded to the input. We employed the multicanonical Monte Carlo method, which can sample GRNs randomly in a wide range of fitness levels, and classified the GRNs according to their fitness. As a result, the following properties were found: (1) Highly-fit GRNs exhibited bistability for intermediate input between "on" and "off". This means that such GRNs responded to two input states by using different fixed points of dynamics. This bistability emerges necessarily as fitness increases. (2) These highly-fit GRNs were robust against noise because of their bistability. In other words, noise robustness is a byproduct of high fitness. (3) GRNs that were robust against mutations were not extremely rare among the highly-fit GRNs. This implies that mutational robustness is readily acquired through the evolutionary process. These properties are universal irrespective of the evolutionary pathway, because the results do not rely on evolutionary simulation.

Low-dimensional projection approach for efficient sampling of molecular recognition and polymer aggregation

The one-dimensional projection (ODP) approach is extended to two-dimensional umbrella sampling (TDUS) and is applied to three different complex systems in combination with a reactive force field (ReaxFF). TDUS is capable of showing detailed features of the free-energy surface (FES) of the double-proton transfer of the acetic acid dimer. It also revealed the direct relationship between the types of hydrogen bonding and binding strengths in the case of adrenaline molecular recognition by SIVSF (Serine, Isoleucine, Valine, Cysteine, and Phenylalanine). The study of polymer aggregation using TDUS shows that aggregation is preferred with a less-polar solvent, which is also consistent with the experimental observation of a tape-casting process. Therefore, TDUS can be generally useful in FES explorations from simple chemical reactions to complex processes of molecular recognition and polymer aggregation.

All-or-none folding of a flexible polymer chain in cylindrical nanoconfinement

Geometric confinement of a polymer chain results in a loss of conformational entropy. For a chain that can fold into a compact native state via a first-order-like transition, as is the case for many small proteins, confinement typically provides an entropic stabilization of the folded state, thereby shifting the location of the transition. This allows for the possibility of confinement (entropy) driven folding. Here, we investigate such confinement effects for a flexible square-well-sphere N-mer chain (monomer diameter σ) confined within a long cylindrical pore (diameter D) or a closed cylindrical box (height H = D). We carry out Wang-Landau simulations to construct the density of states, which provides access to the complete thermodynamics of the system. For a wide pore, an entropic stabilization of the folded state is observed. However, as the pore diameter approaches the size of the folded chain (D ∼ N^1/3σ), we find a destabilization effect. For pore diameters smaller than the native ground-state, the chain folds into a different, higher energy, ground state ensemble and the T vs D phase diagram displays non-monotonic behavior as the system is forced into different ground states for different ranges of D. In this regime, isothermal reduction of the confinement dimension can induce folding, unfolding, or crystallite restructuring. For the cylindrical box, we find a monotonic stabilization effect with decreasing D. Scaling laws for the confinement free energy in the athermal limit are also investigated.

Wang-Landau algorithm as stochastic optimization and its acceleration

We show that the Wang-Landau algorithm can be formulated as a stochastic gradient descent algorithm minimizing a smooth and convex objective function, of which the gradient is estimated using Markov chain Monte Carlo iterations. The optimization formulation provides us another way to establish the convergence rate of the Wang-Landau algorithm, by exploiting the fact that almost surely the density estimates (on the logarithmic scale) remain in a compact set, upon which the objective function is strongly convex. The optimization viewpoint motivates us to improve the efficiency of the Wang-Landau algorithm using popular tools including the momentum method and the adaptive learning rate method. We demonstrate the accelerated Wang-Landau algorithm on a two-dimensional Ising model and a two-dimensional ten-state Potts model.

Logarithmic finite-size scaling correction to the leading Fisher zeros in the p-state clock model: A higher-order tensor renormalization group study

We investigate the finite-size-scaling (FSS) behavior of the leading Fisher zero of the partition function in the complex temperature plane in the p-state clock models of p=5 and 6. We derive the logarithmic finite-size corrections to the scaling of the leading zeros which we numerically verify by performing the higher-order tensor renormalization group (HOTRG) calculations in the square lattices of a size up to 128×128 sites. The necessity of the deterministic HOTRG method in the clock models is noted by the extreme vulnerability of the numerical leading zero identification against stochastic noises that are hard to be avoided in the Monte Carlo approaches. We characterize the system-size dependence of the numerical vulnerability of the zero identification by the type of phase transition, suggesting that the two transitions in the clock models are not of an ordinary first- or second-order type. In the direct FSS analysis of the leading zeros in the clock models, we find that their FSS behaviors show excellent agreement with our predictions of the logarithmic corrections to the Berezinskii-Kosterlitz-Thouless ansatz at both of the high- and low-temperature transitions.

Adsorption of a Helical Filament Subject to Thermal Fluctuations

We consider semiflexible chains governed by preferred curvature and twist and their flexural and twist moduli. These filaments possess a helical rather than straight three-dimensional (3D) ground state and we call them helical filaments (H-filament). Depending on the moduli, the helical shape may be smeared by thermal fluctuations. Secondary superhelical structures are expected to form on top of the specific local structure of biofilaments, as is documented for vimentin. We study confinement and adsorption of helical filaments utilizing both a combination of numerical simulations and analytical theory. We investigate overall chain shapes, transverse chain fluctuations, loop and tail distributions, and energy distributions along the chain together with the mean square average height of the monomers 〈 z 2 〉 . The number fraction of adsorbed monomers serves as an order parameter for adsorption. Signatures of adsorbed helical polymers are the occurrence of 3D helical loops/tails and spiral or wavy quasi-flat shapes. None of these arise for the Worm-Like-Chain, whose straight ground state can be embedded in a plane.

Stochastic approximation Monte Carlo with a dynamic update factor

We present a Monte Carlo algorithm based on the stochastic approximation Monte Carlo (SAMC) algorithm for directly calculating the density of states. The proposed method is stochastic approximation with a dynamic update factor (SAD), which dynamically adjusts the update factor γ_{t} during the course of the simulation. We test this method on a square-well fluid and a 31-atom Lennard-Jones cluster and compare the convergence behavior of several related Monte Carlo methods. We find that both the SAD and 1/t-Wang-Landau (1/t-WL) methods rapidly converge to the correct density of states without the need for the user to specify an arbitrary tunable parameter t_{0} as in the case of SAMC. SAD requires as input the temperature range of interest, in contrast with 1/t-WL, which requires that the user identify the interesting range of energies. The convergence of the 1/t-WL method is very sensitive to the energy range chosen for the low-temperature heat capacity of the Lennard-Jones cluster. Thus, SAD is more powerful in the common case in which the range of energies is not known in advance.

Protein structure predictions by enhanced conformational sampling methods

In this Special Festschrift Issue for the celebration of Professor Nobuhiro Gō's 80th birthday, we review enhanced conformational sampling methods for protein structure predictions. We present several generalized-ensemble algorithms such as multicanonical algorithm, replica-exchange method, etc. and parallel Monte Carlo or molecular dynamics method with genetic crossover. Examples of the results of these methods applied to the predictions of protein tertiary structures are also presented.

Efficient simulation protocol for determining the density of states: Combination of replica-exchange Wang-Landau method and multicanonical replica-exchange method

By combining two generalized-ensemble algorithms, the replica-exchange Wang-Landau (REWL) method and the multicanonical replica-exchange method (MUCAREM), we propose an effective simulation protocol to determine the density of states with high accuracy. The new protocol is referred to as REWL-MUCAREM, and REWL is first performed and then MUCAREM is performed. In order to verify the effectiveness of our protocol, we performed simulations of a square-lattice Ising model using the three methods, namely REWL, MUCAREM, and REWL-MUCAREM. The results showed that the density of states obtained by REWL-MUCAREM is more accurate than that is estimated by the two methods separately.

Multidimensional virtual-system coupled canonical molecular dynamics to compute free-energy landscapes of peptide multimer assembly

An enhanced-sampling method termed multidimensional virtual-system coupled canonical molecular dynamics (mD-VcMD) method is developed. In many cases, generalized-ensemble methods realizing enhanced sampling, for example, adaptive umbrella sampling, apply an effective potential, which is derived from temporarily assumed canonical distribution as a function of one or more arbitrarily defined reaction coordinates. However, it is not straightforward to estimate the appropriate canonical distribution, especially for cases applying multiple reaction coordinates. The current method, mD-VcMD, does not rely on the form of the canonical distribution. Therefore, it is practically useful to explore a high-dimensional reaction-coordinate space. In this article, formulation of mD-VcMD and its evaluation with the simple molecular models consisting of three or four alanine peptides are presented. We confirmed that mD-VcMD efficiently searched 2D and 3D reaction-coordinate spaces defined as interpeptide distances. Direct comparisons with results of long-term canonical MD simulations revealed that mD-VcMD produces correct canonical ensembles. © 2019 Wiley Periodicals, Inc.

Simulation of fluid/gel phase equilibrium in lipid vesicles

Simulation of single component dipalmitoylphosphatidylcholine (DPPC) coarse-grained DRY-MARTINI lipid vesicles of diameter 10 nm (1350 lipids), 20 nm (5100 lipids) and 40 nm (17 600 lipids) is performed using statistical temperature molecular dynamics (STMD), to study finite size effects upon the order-disorder gel/fluid transition. STMD obtains enhanced sampling using a generalized ensemble, obtaining a flat energy distribution between upper and lower cutoffs, with little computational cost over canonical molecular dynamics. A single STMD trajectory of moderate length is sufficient to sample 20+ transition events, without trapping in the gel phase, and obtain well averaged properties. Phase transitions are analyzed via the energy-dependence of the statistical temperature, T_S(U). The transition temperature decreases with decreasing diameter, in agreement with experiment, and the transition changes from first order to borderline first-second order. The size- and layer-dependence of the structure of both stable phases, and of the pathway of the phase transition, are determined. It is argued that the finite size effects are primarily caused by the disruption of the gel packing by curvature. Inhomogeneous states with faceted gel patches connected by unusual fluid seams are observed at high curvature, with visually different structure in the inner and outer layers due to the different curvatures. Thus a simple physical picture describes phase transitions in nanoscale finite systems far from the thermodynamic limit.

Order-disorder transition in a two-dimensional associating lattice gas

We study an associating lattice gas (ALG) using Monte Carlo simulation on the triangular lattice and semianalytical solutions on Husimi lattices. In this model, the molecules have an orientational degree of freedom and the interactions depend on the relative orientations of nearest-neighbor molecules, mimicking the formation of hydrogen bonds. We focus on the transition between the high-density liquid (HDL) phase and the isotropic phase in the limit of full occupancy, corresponding to chemical potential μ→∞, which has not yet been studied systematically. Simulations yield a continuous phase transition at τ_{c}=k_{B}T_{c}/γ=0.4763(1) (where -γ is the bond energy) between the low-temperature HDL phase, with a nonvanishing mean orientation of the molecules, and the high-temperature isotropic phase. Results for critical exponents and the Binder cumulant indicate that the transition belongs to the three-state Potts model universality class, even though the ALG Hamiltonian does not have the full permutation symmetry of the Potts model. In contrast with simulation, the Husimi lattice analyses furnish a discontinuous phase transition, characterized by a discontinuity of the nematic order parameter. The transition temperatures (τ_{c}=0.51403 and 0.51207 for trees built with triangles and hexagons, respectively) are slightly higher than that found via simulation. Since the Husimi lattice studies show that the ALG phase diagram features a discontinuous isotropic-HDL line for finite μ, three possible scenarios arise for the triangular lattice. The first is that in the limit μ→∞ the first-order line ends in a critical point; the second is a change in the nature of the transition at some finite chemical potential; the third is that the entire line is one of continuous phase transitions. Results from other ALG models and the fact that mean-field approximations show a discontinuous phase transition for the three-state Potts model (known to possess a continuous transition) lends some weight to the third alternative.

Three-dimensional Baxter-Wu model

A classic three-dimensional spin model, based upon the Baxter-Wu scheme, is presented. It is found, by entropic sampling simulations, that the behavior of the energy and magnetization fourth-order cumulants points to a first-order phase transition. A finite-size procedure was performed, confirming that the system scales with the dimensionality d=3, yielding a high-resolution estimate of the transition temperature as T_{c}=11.377485(29).

Configurational contribution to the Soret effect of a protein ligand system : An investigation with density-of-states simulations

Many of the biological functions of proteins are closely associated with their ability to bind ligands and change conformations in response to changing conditions. Since binding state and conformation of a protein affect its response to a temperature gradient, they may be probed with thermophoresis. In recent years, thermophoretic techniques to investigate biomolecular interactions, quantify ligand binding, and probe conformational changes have become established. To develop a better understanding of the mechanisms underlying the thermophoretic behavior of proteins and ligands, we employ a simple, off-lattice model for a protein and ligand in explicit solvent. To investigate the partitioning of the particles in a temperature gradient, we perform Wang-Landau-type simulations in a divided simulation box and construct the density of states over a two-dimensional state space. This method gives us access to the entropy and energy of the divided system and allows us to estimate the configurational contribution to the Soret coefficient. In this work, we focus on dilute solutions of hydrophobic proteins and investigate the effect of ligand binding on their thermophoretic behavior. We find that our simple model captures important aspects of protein-ligand interactions and allows us to relate the binding energy to the change in Soret coefficient upon ligand binding.

Molecular simulations by generalized-ensemble algorithms in isothermal-isobaric ensemble

Generalized-ensemble algorithms are powerful techniques for investigating biomolecules such as protein, DNA, lipid membrane, and glycan. The generalized-ensemble algorithms were originally developed in the canonical ensemble. On the other hand, not only temperature but also pressure is controlled in experiments. Additionally, pressure is used as perturbation to study relationship between function and structure of biomolecules. For this reason, it is important to perform efficient conformation sampling based on the isothermal-isobaric ensemble. In this article, we review a series of the generalized-ensemble algorithms in the isothermal-isobaric ensemble: multibaric-multithermal, pressure- and temperature-simulated tempering, replica-exchange, and replica-permutation methods. These methods achieve more efficient simulation than the conventional isothermal-isobaric simulation. Furthermore, the isothermal-isobaric generalized-ensemble simulation samples conformations of biomolecules from wider range of temperature and pressure. Thus, we can estimate physical quantities more accurately at any temperature and pressure values. The applications to the biomolecular system are also presented.

Diagrams of States of Single Flexible-Semiflexible Multi-Block Copolymer Chains: A Flat-Histogram Monte Carlo Study

The combination of flexibility and semiflexibility in a single molecule is a powerful design principle both in nature and in materials science. We present results on the conformational behavior of a single multiblock-copolymer chain, consisting of equal amounts of Flexible (F) and Semiflexible (S) blocks with different affinity to an implicit solvent. We consider a manifold of macrostates defined by two terms in the total energy: intermonomer interaction energy and stiffness energy. To obtain diagrams of states (pseudo-phase diagrams), we performed flat-histogram Monte Carlo simulations using the Stochastic Approximation Monte Carlo algorithm (SAMC). We have accumulated two-Dimensional Density of States (2D DoS) functions (defined on the 2D manifold of macrostates) for a SF-multiblock-copolymer chain of length N=64 with block lengths b = 4, 8, 16, and 32 in two different selective solvents. In an analysis of the canonical ensemble, we calculated the heat capacity and determined its maxima and the most probable morphologies in different regions of the state diagrams. These are rich in various, non-trivial morphologies, which are formed without any specific interactions, and depend on the block length and the type of solvent selectivity (preferring S or F blocks, respectively). We compared the diagrams with those for the non-selective solvent and reveal essential changes in some cases. Additionally, we implemented microcanonical analysis in the “conformational” microcanonical (NVU, where U is the potential energy) and the true microcanonical (NVE, where E is the total energy) ensembles with the aim to reveal and classify pseudo-phase transitions, occurring under the change of temperature.

Parallel Implementation of Semiclassical Transition State Theory

This paper presents the parsctst code, an efficient parallel implementation of the semiclassical transition state theory (SCTST) for reaction rate constant calculations. Parsctst is developed starting from a previously presented approach for the computation of the vibrational density of states of fully coupled anharmonic molecules ( Nguyen et al. Chem. Phys. Lett. 2010 , 499 , 915 ). The parallel implementation makes it practical to tackle reactions involving more than 100 fully coupled anharmonic vibrational degrees of freedom and also includes multidimensional tunneling effects. After describing the pseudocode and demonstrating its computational efficiency, we apply the new code for estimating the rate constant of the proton transfer isomerization reaction of the 2,4,6-tri- tert-butylphenyl to 3,5-di- tert-butylneophyl. Comparison with both theoretical and experimental results is presented. Parsctst code is user-friendly and provides a significant computational time saving compared to serial calculations. We believe that parsctst can boost the application of SCTST as an alternative to the basic transition state theory for accurate kinetics modeling not only in combustion or atmospheric chemistry, but also in organic synthesis, where bigger reactive systems are encountered.