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This paper resurrects and archives an unpublished original Cornell Laboratory of Nuclear Studies report by Yuko Okamoto and Richard Talman, "Rational Approximation of the Complex Error Function and the Electric Field of a Two-Dimensional Gaussian Charge Distribution" CBN 80-13, dating from September 1980, during the start-up period of the CESR-CLEO $e^+e^-$ collider. This code has played a significant role in the calculation of the beam-beam interaction (in particular the beam-beam tune shift) for subsequent storage ring colliders. Electronic access to the (refactored) original codes is provided by active links., Comment: 43 pages, 2 figures

We propose a method to extend the fast on-the-fly weight determination scheme for simulated tempering to two-dimensional space including not only temperature but also pressure. During the simulated tempering simulation, weight parameters for temperature-update and pressure-update are self-updated independently according to the trapezoidal rule. In order to test the effectiveness of the algorithm, we applied our proposed method to a peptide, chignolin, in explicit water. After setting all weight parameters to zero, the weight parameters were quickly determined during the simulation. The simulation realised a uniform random walk in the entire temperature-pressure space., Comment: 11 pages, (Revtex4-1), 4 figures

We estimated the residual entropy of ice Ih by the recently developed simulation protocol, namely, the combination of 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 found to be within 0.038 % of series expansion estimate by Nagle and within 0.000077 % of PEPS algorithm by Vanderstraeten. 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 MC simulations., Comment: 15 pages, , (Revtex4-1), 12 figures

By combining two generalized-ensemble algorithms, Replica-Exchange Wang-Landau (REWL) method and 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 next. In order to verify the effectiveness of our protocol, we performed simulations of a square-lattice Ising model by the three methods, namely, REWL, MUCAREM, and REWL-MUCAREM. The results showed that the density of states obtained by the REWL-MUCAREM is more accurate than that is estimated by the two methods separately., Comment: 12 pages, , (Revtex4-1), 5 figures

We combined the genetic crossover, which is one of the operations of genetic algorithm, and replica-exchange method in parallel molecular dynamics simulations. The genetic crossover and replica-exchange method can search the global conformational space by exchanging the corresponding parts between a pair of conformations of a protein. In this study, we applied this method to an $\alpha$-helical protein, Trp-cage mini protein, which has 20 amino-acid residues. The conformations obtained from the simulations are in good agreement with the experimental results., Comment: 5 pages, 3 figures, Molecular Simulation 41 (2015), in press (the title is changed). arXiv admin note: substantial text overlap with arXiv:1505.05709

Many proteins carry out their biological functions by forming the characteristic tertiary structures. Therefore, the search of the stable states of proteins by molecular simulations is important to understand their functions and stabilities. However, getting the stable state by conformational search is difficult, because the energy landscape of the system is characterized by many local minima separated by high energy barriers. In order to overcome this difficulty, various sampling and optimization methods for conformations of proteins have been proposed. In this study, we propose a new conformational search method for proteins by using genetic crossover and Metropolis criterion. We applied this method to an $\alpha$-helical protein. The conformations obtained from the simulations are in good agreement with the experimental results., Comment: 4 pages, 3 figures

We propose a new implementation of the replica-exchange method (REM) in which replicas follow a pre-planned route in temperature space instead of a random walk. Our method satisfies the detailed balance condition in the proposed route. The method forces tunneling events between the highest and lowest temperatures to happen with an almost constant period. The number of tunneling counts is proportional to that of the random-walk REM multiplied by the square root of moving distance in temperature space. We applied this new implementation to two kinds of REM and compared the results with those of the conventional random-walk REM. The test system was a two-dimensional Ising model, and our new method reproduced the results of the conventional random-walk REM and improved the tunneling counts by three times or more than that of the random-walk REM., Comment: 6 pages, 3 figures, 1 table; methods are the same before, but results and some references are added, table is updated

We propose a replica-exchange method (REM) which does not use pseudo random numbers. For this purpose, we first give a conditional probability for Gibbs sampling replica-exchange method (GSREM) based on the heat bath method. In GSREM, replica exchange is performed by conditional probability based on the weight of states using pseudo random numbers. From the conditional probability, we propose a new method called deterministic replica-exchange method (DETREM) that produces thermal equilibrium distribution based on a differential equation instead of using pseudo random numbers. This method satisfies the detailed balance condition using a conditional probability of Gibbs heat bath method and thus results can reproduce the Boltzmann distribution within the condition of the probability. We confirmed that the equivalent results were obtained by REM and DETREM with two-dimensional Ising model. DETREM can avoid problems of choice of seeds in pseudo random numbers for REM using a differential equation., Comment: 12 pages, 1 table, 11 figures, auto-correlation function and time series of internal state are added

We propose an improved prediction method of the tertiary structures of $\alpha$-helical membrane proteins based on the replica-exchange method by taking into account helix deformations. Our method allows wide applications because transmembrane helices of native membrane proteins are often distorted. In order to test the effectiveness of the present method, we applied it to the structure predictions of glycophorin A and phospholamban. The results were in accord with experiments., Comment: 11 pages, (Revtex4-1), 10 figures

We propose a novel method for refining force-field parameters of protein systems. In this method, the agreement of the secondary-structure stability and instability between the protein conformations obtained by experiments and those obtained by molecular dynamics simulations is used as a criterion for the optimization of force-field parameters. As an example of the applications of the present method, we refined the force-field parameter set of the AMBER ff99SB force field by searching the torsion-energy parameter spaces of $\psi$ (N-C$^{\alpha}$-C-N) and $\zeta$ (C$^{\beta}$-C$^{\alpha}$-C-N) of the backbone dihedral angles. We then performed folding simulations of $\alpha$-helical and $\beta$-hairpin peptides, using the optimized force field. The results showed that the new force-field parameters gave structures more consistent with the experimental implications than the original AMBER ff99SB force field., Comment: 10 pages, (Revtex4.1), 6 figures. arXiv admin note: substantial text overlap with arXiv:1208.6150, arXiv:1206.3909

We applied the simulated tempering and magnetizing (STM) method to the two-dimensional three-state Potts model in an external magnetic field in order to perform further investigations of the STM's applicability. The temperature as well as the external field are treated as dynamical variables updated during the STM simulations. After we obtained adequate information for several lattice sizes $L$ (up to $160\times 160$), we also performed a number of conventional canonical simulations of large lattices, especially in order to illustrate the crossover behavior of the Potts model in external field with increasing $L$. The temperature and external field for larger lattice size simulations were chosen by extrapolation of the detail information obtained by STM. We carefully analyzed the crossover scaling at the phase transitions with respect to the lattice size as well as the temperature and external field. The crossover behavior is clearly observed in the simulations in agreement with theoretical predictions., Comment: 21 pages, 21 figures

In this Chapter we review our works on force fields for molecular simulations of protein systems. We first discuss the functional forms of the force fields and present some extensions of the conventional ones. We then present various methods for force-field parameter optimizations. Finally, some examples of our applications of these parameter optimization methods are given and they are compared with the results from the existing force-fields., Comment: 56 pages, (Springer Verlag global LaTeX2e support for multi authored books), 28 figures, 13 tables

Many commonly used force fields for protein systems such as AMBER, CHARMM, GROMACS, OPLS, and ECEPP have amino-acid-independent force-field parameters of main-chain torsion-energy terms. Here, we propose a new type of amino-acid-dependent torsion-energy terms in the force fields. As an example, we applied this approach to AMBER ff03 force field and determined new amino-acid-dependent parameters for $\psi$ and $\psi'$ angles for each amino acid by using our optimization method, which is one of the knowledge-based approach. In order to test the validity of the new force-field parameters, we then performed folding simulations of $\alpha$-helical and $\beta$-hairpin peptides, using the optimized force field. The results showed that the new force-field parameters gave structures more consistent with the experimental implications than the original AMBER ff03 force field., Comment: 10 pages, (Revtex4.1), 3 tables, 6 figures

A new simulated tempering method, which is referred to as simulated tempering umbrella sampling, for calculating the free energy of chemical reactions is proposed. First principles molecular dynamics simulations with this simulated tempering were performed in order to study the intramolecular proton transfer reaction of malonaldehyde in aqueous solution. Conformational sampling in reaction coordinate space can be easily enhanced with this method, and the free energy along a reaction coordinate can be calculated accurately. Moreover, the simulated tempering umbrella sampling provides trajectory data more efficiently than the conventional umbrella sampling method., Comment: 5 pages, (Revtex4-1), 3 figures

We performed two-dimensional simulated tempering (ST) simulations of the two-dimensional Ising model with different lattice sizes in order to investigate the two-dimensional ST's applicability to dealing with phase transitions and to study the crossover of critical scaling behavior. The external field, as well as the temperature, was treated as a dynamical variable updated during the simulations. Thus, this simulation can be referred to as "Simulated Tempering and Magnetizing (STM)." We also performed the "Simulated Magnetizing" (SM) simulations, in which the external field was considered as a dynamical variable and temperature was not. As has been discussed by previous studies, the ST method is not always compatible with first-order phase transitions. This is also true in the magnetizing process. Flipping of the entire magnetization did not occur in the SM simulations under $T_\mathrm{c}$ in large lattice-size simulations. However, the phase changed through the high temperature region in the STM simulations. Thus, the dimensional extension let us eliminate the difficulty of the first-order phase transitions and study wide area of the phase space. We then discuss how frequently parameter-updating attempts should be made for optimal convergence. The results favor frequent attempts. We finally study the crossover behavior of the phase transitions with respect to the temperature and external field. The crossover behavior was clearly observed in the simulations in agreement with the theoretical implications., Comment: 15 pages, (Revtex4-1), 23 figures, 1 video (link)

In biomolecular systems (especially all-atom models) with many degrees of freedom such as proteins and nucleic acids, there exist an astronomically large number of local-minimum-energy states. Conventional simulations in the canonical ensemble are of little use, because they tend to get trapped in states of these energy local minima. Enhanced conformational sampling techniques are thus in great demand. A simulation in generalized ensemble performs a random walk in potential energy space and can overcome this difficulty. From only one simulation run, one can obtain canonical-ensemble averages of physical quantities as functions of temperature by the single-histogram and/or multiple-histogram reweighting techniques. In this article we review uses of the generalized-ensemble algorithms in biomolecular systems. Three well-known methods, namely, multicanonical algorithm, simulated tempering, and replica-exchange method, are described first. Both Monte Carlo and molecular dynamics versions of the algorithms are given. We then present various extensions of these three generalized-ensemble algorithms. The effectiveness of the methods is tested with short peptide and protein systems., Comment: 42 pages, 7 figures, to be published in "Biomolecular simulations: methods and protocols" edited by Luca Monticelli and Emppu Salonen (Humana Press, 2011)

In the replica-exchange molecular dynamics method, where constant-temperature molecular dynamics simulations are performed in each replica, one usually rescales the momentum of each particle after replica exchange. This rescaling method had previously been worked out only for the Gaussian constraint method. In this letter, we present momentum rescaling formulae for four other commonly used constant-temperature algorithms, namely, Langevin dynamics, Andersen algorithm, Nos\'{e}-Hoover thermostat, and Nos\'{e}-Poincar\'{e} thermostat. The effectiveness of these rescaling methods is tested with a small biomolecular system, and it is shown that proper momentum rescaling is necessary to obtain correct results in the canonical ensemble., Comment: 6 pages, (Revtex4), 4 figures

We present generalized-ensemble algorithms for isobaric-isothermal molecular simulations. In addition to the multibaric-multithermal algorithm and replica-exchange method for the isobaric-isothermal ensemble, which have already been proposed, we propose a simulated tempering method for this ensemble. We performed molecular dynamics simulations with these algorithms for an alanine dipeptide system in explicit water molecules to test the effectiveness of the algorithms. We found that these generalized-ensemble algorithms are all useful for conformational sampling of biomolecular systems in the isobaric-isothermal ensemble., Comment: 6 pages, (Revtex4), 4 figures

We present a new type of the Hamiltonian replica-exchange method, in which not temperatures but the van der Waals radius parameter is exchanged. By decreasing the van der Waals radii that control spatial sizes of atoms, this Hamiltonian replica-exchange method overcomes the steric restrictions and energy barriers. Furthermore, the simulation based on this method escapes from the local-minimum free-energy states and realizes effective sampling in the conformational space. We applied this method to an alanine dipeptide in aqueous solution and showed the effectiveness of the method by comparing the results with those obtained from the conventional canonical method., Comment: 14 pages, (Revtex4), 11 figures

We discuss multi-dimensional generalizations of multicanonical algorithm, simulated tempering, and replica-exchange method. We generalize the original potential energy function $E_0$ by adding any physical quantity $V$ of interest as a new energy term with a coupling constant $\lambda$. We then perform a multi-dimensional multicanonical simulation where a random walk in $E_0$ and $V$ space is realized. We can alternately perform a multi-dimensional simulated tempering simulation where a random walk in temperature $T$ and parameter $\lambda$ is realized. The results of the multi-dimensional replica-exchange simulations can be used to determine the weight factors for these multi-dimensional multicanonical and simulated tempering simulations., Comment: 4 pages, (Revtex4), 2 figures

In a recent paper we introduced two Potts-like models in three dimensions, which share the following properties: (A) One of the ice rules is always fulfilled (in particular also at infinite temperature). (B) Both ice rules hold for groundstate configurations. This allowed for an efficient calculation of the residual entropy of ice I (ordinary ice) by means of multicanonical simulations. Here we present the thermodynamics of these models. Despite their similarities with Potts models, no sign of a disorder-order phase transition is found., Comment: 5 pages, 7 figures

Conventional simulations of complex systems in the canonical ensemble suffer from the quasi-ergodicity problem. A simulation in generalized ensemble overcomes this difficulty by performing a random walk in potential energy space and other parameter space. From only one simulation run, one can obtain canonical-ensemble averages of physical quantities as functions of temperature by the single-histogram and/or multiple-histogram reweighting techniques. In this article we review the generalized-ensemble algorithms. Three well-known methods, namely, multicanonical algorithm, simulated tempering, and replica-exchange method, are described first. Both Monte Carlo and molecular dynamics versions of the algorithms are given. We then present further extensions of the above three methods., Comment: 37 pages, (LaTeX2e), 5 figures

We propose explicit symplectic integrators of molecular dynamics (MD) algorithms for rigid-body molecules in the canonical and isothermal-isobaric ensembles. We also present a symplectic algorithm in the constant normal pressure and lateral surface area ensemble and that combined with the Parrinello-Rahman algorithm. Employing the symplectic integrators for MD algorithms, there is a conserved quantity which is close to Hamiltonian. Therefore, we can perform a MD simulation more stably than by conventional nonsymplectic algorithms. We applied this algorithm to a TIP3P pure water system at 300 K and compared the time evolution of the Hamiltonian with those by the nonsymplectic algorithms. We found that the Hamiltonian was conserved well by the symplectic algorithm even for a time step of 4 fs. This time step is longer than typical values of 0.5-2 fs which are used by the conventional nonsymplectic algorithms., Comment: 23 pages, (Revtex4), 4 figures

We propose a new generalized-ensemble algorithm, which we refer to as the multicanonical-multioverlap algorithm. By utilizing a non-Boltzmann weight factor, this method realizes a random walk in the multi-dimensional, energy-overlap space and explores widely in the configurational space including specific configurations, where the overlap of a configuration with respect to a reference state is a measure for structural similarity. We apply the multicanonical-multioverlap molecular dynamics method to a penta peptide, Met-enkephalin, in vacuum as a test system. We also apply the multicanonical and multioverlap molecular dynamics methods to this system for the purpose of comparisons. We see that the multicanonical-multioverlap molecular dynamics method realizes effective sampling in the configurational space including specific configurations more than the other two methods. From the results of the multicanonical-multioverlap molecular dynamics simulation, furthermore, we obtain a new local-minimum state of the Met-enkephalin system., Comment: 15 pages, (Revtex4), 9 figures

We introduce two simple models with nearest neighbor interactions on 3D hexagonal lattices. Each model allows one to calculate the residual entropy of ice I (ordinary ice) by means of multicanonical simulations. This gives the correction to the residual entropy derived by Linus Pauling in 1935. Our estimate is found to be within less than 0.1% of an analytical approximation by Nagle, which is an improvement of Pauling's result. We pose it as a challenge to experimentalists to improve on the accuracy of a 1936 measurement by Giauque and Stout by about one order of magnitude, which would allow one to identify corrections to Pauling's value unambiguously. It is straightforward to transfer our methods to other crystal systems., Comment: 3 pages, 2 figures. One reference added and changes after comments by John Nagle

We propose a new backbone-torsion-energy term in the force field for protein systems. This torsion-energy term is represented by a double Fourier series in two variables, the backbone dihedral angles phi and psi. It gives a natural representation of the torsion energy in the Ramachandran space in the sense that any two-dimensional energy surface periodic in both phi and psi can be expanded by the double Fourier series. We can then easily control secondary-structure-forming tendencies by modifying the torsion-energy surface. For instance, we can increase/decrease the alpha-helix-forming-tendencies by lowering/raising the torsion-energy surface in the alpha-helix region and likewise increase/decrease the beta-sheet-forming tendencies by lowering/raising the surface in the beta-sheet region in the Ramachandran space. We applied our approach to AMBER parm94 and AMBER parm96 force fields and demonstrated that our modifications of the torsion-energy terms resulted in the expected changes of secondary-structure-forming-tendencies by performing folding simulations of alpha-helical and beta-hairpin peptides., Comment: 13 pages, (Revtex4), 5 figures