A reliable prediction of 3D protein structures from sequence data remains a big challenge due to both theoretical and computational difficulties. We have previously shown that our kinetostatic compliance method (KCM) implemented into the Protofold package can overcome some of the key difficulties faced by other de novo structure prediction methods, such as the very small time steps required by the molecular dynamics (MD) approaches or the very large number of samples needed by the Monte Carlo (MC) sampling techniques. In this article, we improve the free energy formulation used in Protofold by including the typically underrated entropic effects, imparted due to differences in hydrophobicity of the chemical groups, which dominate the folding of most water-soluble proteins. In addition to the model enhancement, we revisit the numerical implementation by redesigning the algorithms and introducing efficient data structures that reduce the expected complexity from quadratic to linear. Moreover, we develop and optimize parallel implementations of the algorithms on both central and graphics processing units (CPU GPU) achieving speed-ups up to two orders of magnitude on the GPU. Our simulations are consistent with the general behavior observed in the folding process in aqueous solvent, confirming the effectiveness of model improvements. We report on the folding process at multiple levels; namely, the formation of secondary structural elements and tertiary interactions between secondary elements or across larger domains. We also observe significant enhancements in running times that make the folding simulation tractable for large molecules.
 Molecular dynamics (MD) is an invaluable tool with which to study protein folding in silico. Although just a few years ago the dynamic behavior of a protein molecule could be simulated only in the neighborhood of the experimental conformation (or protein unfolding could be simulated at high temperature), the advent of distributed computing, new techniques such as replica-exchange MD, new approaches (based on, e.g., the stochastic difference equation), and physics-based reduced models of proteins now make it possible to study proteinfolding pathways from completely unfolded structures. In this review, we present algorithms for MD and their extensions and applications to protein-folding studies, using all-atom models with explicit and implicit solvent as well as reduced models of polypeptide chains. Abstract The rate of protein-protein association limits the response time due to protein-protein interactions. The bimolecular association rate may be diffusion-controlled or influenced, and in such cases, Brownian dynamics simulations of protein-protein diffusional association may be used to compute association rates. Here, we report Brownian dynamics simulations of the diffusional association of five different protein-protein pairs: barnase and barstar, acetylcholinesterase and fasciculin-2, cytochrome c peroxidase and cytochrome c, the HyHEL-5 antibody and hen egg lysozyme (HEL), and the HyHEL-10 antibody and HEL. The same protocol was used to compute the diffusional association rates for all the protein pairs in order to assess, by comparison to experimentally measured rates, whether the association of these proteins can be explained solely on the basis of diffusional encounter. The simulation protocol is similar to those previously derived for simulation of the association of barnase and barstar, and of acetylcholinesterase and fasciculin-2; these produced results in excellent agreement with experimental data for these protein pairs, with changes in association rate due to mutations reproduced within the limits of expected computational and modeling errors. Here, we find that for all protein pairs, the effects of mutations can be well reproduced by the simulations, even though the degree of the electrostatic translational and orientational steering varies widely between the cases. However, the absolute values of association rates for the acetylcholinesterase: fasciculin-2 and HyHEL-10 antibody: HEL pairs are overestimated. Comparison of bound and unbound protein structures shows that this may be due to gating resulting from protein flexibility in some of the proteins. This may lower the association rates compared to their bimolecular diffusional encounter rates. Computer experiments using particle models A one-dimensional plasma model The simulation program Time integration schemes The particle-mesh force calculation The solution of field equations Collisionless particle models Particle-particle particle-mesh algorithms Plasma simulation Semiconductor device simulation Astrophysics Solids, liquids and phase changes Fourier transforms Fourier series and finite Fourier transforms Bibliography Index Protein folding occurs on a time scale ranging from milliseconds to minutes for a majority of proteins. Computer simulation of protein folding, from a random configuration to the native structure, is nontrivial owing to the large disparity between the simulation and folding time scales. As an effort to overcome this limitation, simple models with idealized protein subdomains, e.g., the diffusion–collision model of Karplus and Weaver, have gained some popularity. We present here new results for the folding of a four-helix bundle within the framework of the diffusion–collision model. Even with such simplifying assumptions, a direct application of standard Brownian dynamics methods would consume 10,000 processor-years on current supercomputers. We circumvent this difficulty by invoking a special Brownian dynamics simulation. The method features the calculation of the mean passage time of an event from the flux overpopulation method and the sampling of events that lead to productive collisions even if their probability is extremely small (because of large free-energy barriers that separate them from the higher probability events). Using these developments, we demonstrate that a coarse-grained model of the four-helix bundle can be simulated in several days on current supercomputers. Furthermore, such simulations yield folding times that are in the range of time scales observed in experiments. We derive a class of efficient and stable algorithms of Brownian dynamics using a formula, derived by Suzuki, to express time-ordered operators. These algorithms are simpler than those derived by Helfand from Runge-Kutta algorithms and, like Helfand algorithms, can be combined with SHAKE to describe the Brownian dynamics of constrained systems. We applied an atomistic Brownian dynamics (BD) simulation with multiple time step method for the folding simulation of a 13-mer α-helical peptide and a 12-mer β-hairpin peptide, giving successful folding simulations. In this model, the driving energy contribution towards folding came from both electrostatic and van der Waals interactions for the α-helical peptide and from van der Waals interactions for the β-hairpin peptide. Although, many non-native structures having the same or lower energy than that of native structure were observed, the folded states formed the most populated cluster when the structures obtained by the BD simulations were subjected to the cluster analysis based on distance-based root mean square deviation of side-chains between different structures. This result indicates that we can predict the native structures from conformations sampled by BD simulation. Successful modeling of the processes of protein folding and aggregation may ultimately require accurate descriptions of proteins’ diffusive characteristics, which are expected to be influenced by hydrodynamic effects; a comprehensive study of the diffusion and folding of 11 model proteins with an established simulation model extended to include hydrodynamic interactions between residues has therefore been carried out. Molecular simulations that neglect hydrodynamic interactions are incapable of simultaneously reproducing the expected experimental translational and rotational diffusion coefficients of folded proteins, drastically underestimating both when reasonable hydrodynamic radii are employed. In contrast, simulations that include hydrodynamic interactions produce diffusion coefficients that match very well with the expected experimental values for translation and rotation and also correctly capture the significant decrease in translational diffusion coefficient that accompanies protein unfolding. The... We present a molecular dynamics computer simulation method for calculating equilibrium constants for the formation of physical clusters of molecules. The method is based on Hill’s formal theory of physical clusters. In the method, a molecular dynamics calculation is used to calculate the average potential energy of a cluster of molecules as a function of temperature, and the equilibrium constants are calculated from the integral of the energy with respect to reciprocal temperature. The method is illustrated by calculations of the equilibrium constants for the formation of clusters of two to five water molecules that interact with each other by an intermolecular potential devised by Watts. The method is compared with other procedures for calculating the thermodynamic properties of clusters. We discuss the general philosophy underlying the statistical behaviour, the dynamical evolution and the integration of the equations of motion for systems subject to constraints. We also show how a... Abstract A third-order algorithm for stochastic dynamics (SD) simulations is proposed, identical to the powerful molecular dynamics leap-frog algorithm in the limit of infinitely small friction coefficient γ. It belongs to the class of SD algorithms, in which the integration time step Δt is not limited by the condition Δt ≤ γ−1, but only by the properties of the systematic force. It is shown how constraints, such as bond length or bond angle constraints, can be incorporated in the computational scheme. It is argued that the third-order Verlet-type SD algorithm proposed earlier may be simplified without loosing its third-order accuracy. The leap-frog SD algorithm is proven to be equivalent to the verlet-type SD algorithm. Both these SD algorithms are slightly more economical on computer storage than the Beeman-type SD algorithm. Abstract Some multistep methods are presented that are particularly well suited for integrating the equations of motion describing the molecular dynamics of liquids and dense gases. Some tests and comparisons with other methods are described which demonstrate that a combination of a third-order explicit method for the calculation of positions and an implicit method for the calculation of velocities is particularly stable when used in computations of this type. Although Monte Carlo and molecular dynamics are the primary methods used for free energy simulations of molecular systems, their application to molecules that have multiple conformations separated by energy barriers of ≥ 3 kcal mol is problematic because of slow rates of convergence. In this article we introduce a hybrid simulation method termed MC-SD which mixes Monte Carlo (MC) and stochastic dynamics (SD). This new method generates a canonical ensemble via alternating MC and SD steps and combines the local exploration strengths of dynamics with the barrier-crossing ability of large-step Monte Carlo. Using calculations on double-well potentials and long simulations (108 steps of MC and 1 μs of SD) of the simple, conformationally flexible molecule n-pentane, we find that MC-SD simulations converage faster than either MC or SD alone and generate ensembles which are equivalent to those created by classical MC or SD. Using pure SD at 300 K, the conformational populations of n-pentane are shown to be poorly converged even after a full microsecond of simulation. © 1994 by John Wiley & Sons, Inc.