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The cytoplasm of a cell is crowded with many different kinds of macromolecules. The macromolecular crowding affects the thermodynamics and kinetics of biological reactions in a living cell, such as protein folding, association, and diffusion. Theoretical and simulation studies using simplified models focus on the essential features of the crowding effects and provide a basis for analyzing experimental data. In most of the previous studies on the crowding effects, a uniform crowder size is assumed, which is in contrast to the inhomogeneous size distribution of macromolecules in a living cell. Here, we evaluate the free energy changes upon macromolecular association in a cell-like inhomogeneous crowding system via a theory of hard-sphere fluids and free energy calculations using Brownian dynamics trajectories. The inhomogeneous crowding model based on 41 different types of macromolecules represented by spheres with different radii mimics the physiological concentrations of macromolecules in the cytoplasm of Mycoplasma genitalium. The free energy changes of macromolecular association evaluated by the theory and simulations were in good agreement with each other. The crowder size distribution affects both specific and non-specific molecular associations, suggesting that not only the volume fraction but also the size distribution of macromolecules are important factors for evaluating in vivo crowding effects. This study relates in vitro experiments on macromolecular crowding to in vivo crowding effects by using the theory of hard-sphere fluids with crowder-size heterogeneity.

Hydrodynamic interactions (HI) give rise to collective motions between molecules, which are known to be important in the dynamics of random coil polymers and colloids. However, their role in the biological self-assembly of many molecule systems has not been investigated. Here, using Brownian dynamics simulations, we evaluate the importance of HI on the kinetics of self-assembly of lipid membranes. One-thousand coarse-grained lipid molecules in periodic simulation boxes were allowed to assemble into stable bilayers in the presence and absence of intermolecular HI. Hydrodynamic interactions reduce the monomer-monomer association rate by 50%. In contrast, the rate of association of lipid clusters is much faster in the presence of intermolecular HI. In fact, with intermolecular HI, the membrane self-assembly rate is 3–10 times faster than that without intermolecular HI. We introduce an analytical model to describe the size dependence of the diffusive encounter rate of particle clusters, which can qualitatively explain our simulation results for the early stage of the membrane self-assembly process. These results clearly suggest that HI greatly affects the kinetics of self-assembly and that simulations without HI will significantly underestimate the kinetic parameters of such processes.

To evaluate the enthalpic and entropic contributions of polar residues in alanine-based peptides to α-helical propensities of peptides, we applied a Brownian dynamics simulation, together with the umbrella sampling, to two alanine-based 21-residue peptides: one composed of alanine only (AAA), the other composed of 18 alanines and 3 arginines (ARA). Higher α-helical propensity of ARA than that of AAA was obtained. However, they showed similar conformational stability in enthalpy, considering the contribution of the solvation energy and the potential energy. As an evaluation of entropic effects, the fluctuation of a dihedral angle, ψ, was investigated. Arginine residue showed smaller fluctuation than alanine in elongated states. Higher α-helical propensity might originate from entropic effects. Furthermore, arginine seems to affect the α-helical propensity of alanines interacting with arginine.

To begin to elucidate the principles of intermolecular dynamics in the crowded environment of cells, employing Brownian dynamics (BD) simulations, we examined possible mechanism(s) responsible for the great reduction in diffusion constants of macromolecules in vivo from that at infinite dilution. In an Escherichia coli cytoplasm model comprised of 15 different macromolecule types at physiological concentrations, BD simulations of molecular-shaped and equivalent sphere representations were performed with a soft repulsive potential. At cellular concentrations, the calculated diffusion constant of GFP is much larger than experiment, with no significant shape dependence. Next, using the equivalent sphere system, hydrodynamic interactions (HI) were considered. Without adjustable parameters, the in vivo experimental GFP diffusion constant was reproduced. Finally, the effects of nonspecific attractive interactions were examined. The reduction in diffusivity is very sensitive to macromolecular radius with the motion of the largest macromolecules dramatically slowed down; this is not seen if HI dominate. In addition, long-lived clusters involving the largest macromolecules form if attractions dominate, whereas HI give rise to significant, size independent intermolecular dynamic correlations. These qualitative differences provide a testable means of differentiating the importance of HI vs. nonspecific attractive interactions on macromolecular motion in cells.

ブラウン動力学（Brownian Dynamics；BD）法は溶媒を明示的に扱わないため計算を高速化することができる。従来のBD法は水環境でのタンパク質シミュレーションを対象にしていたため、膜タンパク質には適用できなかった。本研究では膜環境をも非明示的に再現することによってBD法を膜タンパク質のシミュレーションに適用できるように拡張した。αヘリックス構造のポリアラニンペプチドやパピロマーウイルス由来のE5タンパク質、蜂毒のメリチンをモデルとしてシミュレーションした結果、疎水性であるポリアラニンとE5膜タンパク質は膜中で安定に存在した。また、両親媒性であるメリチンペプチドは膜表面に安定に結合していた。これらの結果から、本膜モデルを用いたBD法は膜タンパク質のシミュレーションに有効であると考えられる。

DNA binding proteins efficiently search for their cognitive sites on long genomic DNA by combining 3D diffusion and 1D diffusion (sliding) along the DNA. Recent experimental results and theoretical analyses revealed that the proteins show a rotation-coupled sliding along DNA helical pitch. Here, we performed Brownian dynamics simulations using newly developed coarse-grained protein and DNA models for evaluating how hydrodynamic interactions between the protein and DNA molecules, binding affinity of the protein to DNA, and DNA fluctuations affect the one dimensional diffusion of the protein on the DNA. Our results indicate that intermolecular hydrodynamic interactions reduce 1D diffusivity by 30%. On the other hand, structural fluctuations of DNA give rise to steric collisions between the CG-proteins and DNA, resulting in faster 1D sliding of the protein. Proteins with low binding affinities consistent with experimental estimates of non-specific DNA binding show hopping along the CG-DNA. This hopping significantly increases sliding speed. These simulation studies provide additional insights into the mechanism of how DNA binding proteins find their target sites on the genome., Author Summary DNA binding proteins efficiently search for their cognitive sites on long genomic DNA in cells to control biological activities. Recent experimental studies have revealed that the proteins use not only three-dimensional diffusion, but also one-dimensional diffusion (sliding) on DNA for this search process. For a better understanding of this biological process, we need to elucidate the mechanism of sliding. We report here molecular simulations using newly developed coarse-grained protein and DNA models for elucidating the nature of the sliding motions. Our simulation results show that: 1) hydrodynamic interactions between protein and DNA reduce sliding rate by 30%, 2) structural fluctuations of DNA give rise to steric collisions between proteins and DNA, which facilitate sliding motions, and 3) proteins with low binding affinities to DNA can hop along the DNA, resulting in a significant increase in sliding speed. These simulation studies provide additional insights into the mechanism of how DNA binding proteins find their target sites on the genome.

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.

A new simple implicit solvent model, effective charge (EC) model, was introduced into the Brownian dynamics algorithm based on AMBER united-atom force field. In the EC model, an atomic charge was decreased as a function of solvent-accessible surface area of the atom. We carried out the Brownian dynamics simulations of a 28-mer ββα fold peptide using four implicit solvent models: a generalized Born/solvent-accessible surface area (GB/SA) model, a solvent-accessible surface area (SA) based solvent model, a SA in combination with distance-dependent dielectric (DD/SA) and the EC combined with DD/SA (DD/SA/EC) model; and the calculated results on structure and dynamics of the peptide were compared with those of molecular dynamics simulation using explicit solvent model. Several artifacts were observed in the simulation using the GB/SA model. On the other hand, simulation using the DD/SA and DD/SA/EC implicit solvent models were free from such artifacts. Especially BD with the DD/SA/EC model gave the most stable trajectory as judged by root mean square deviations from the initial structure without large computational cost.

We report a multiple time step algorithm applied to an atomistic Brownian dynamics simulation for simulating the long time scale dynamics of biomolecules. The algorithm was based on the original multiple time step method; a short time step was used to keep faster motions in local equilibrium. When applied to a 28-mer # # ! folded peptide, the simulation gave stable trajectories and the computation time was reduced by a factor of 160 compared to a conventional molecular dynamics simulation using explicit water molecules. We applied it for the folding simulation of a 13-mer ! -helical peptide, giving a successful folding simulation. These results indicate that the Brownian dynamics with the multiple time step algorithm is useful for studies of biomolecular motions by long time simulation.

We report the implementation of an atomistic Brownian dynamics simulation of proteins. A protein was described by united-atom model with AMBER91 force field. The solvent was treated by distance-dependent dielectric/surface area model. The computation time of the Brownian dynamics and the calculated results on structure and dynamics of 28-mer ββα fold peptide with the implicit solvent model were compared with those of molecular dynamics simulation with explicit solvent. The Brownian dynamics simulation was 53 times faster than molecular dynamics simulation with explicit solvent. The simulation was stable and the artifacts often observed in simulations at vacuum condition were reduced. The results indicate that the Brownian dynamics with the implicit solvent model can be widely used in studies of protein dynamics by long time simulation in future.

The interiors of all living cells are highly crowded with macromolecules, which results in a considerable difference between the thermodynamics and kinetics of biological reactions in vivo from that in vitro. To begin to elucidate the principles of intermolecular dynamics in the crowded environment of cells, employing Brownian dynamics (BD) simulations, we examined possible mechanism(s) responsible for the great reduction in diffusion constants of macromolecules in vivo from that at infinite dilution. In an E. coli cytoplasm modelcomprised of 15 different macromolecule types at physiological concentrations, where macromolecules were represented by spheres with their Stokes radii, BD simulations were performed with and without hydrodynamic interactions (HI). Without HI, the calculated diffusion constant of green fluorescent protein (GFP) is much larger than experiment. On the other hand, when HI were considered, the in vivo experimental GFP diffusion constant is almost reproduced without adjustable parameters. In addition, HI give rise to significant, size independent intermolecular dynamic correlations. These results suggest that HI play an important role on macromolecular dynamics in vivo.

The interiors of all living cells are highly crowded with macromolecules, which differs considerably the thermodynamics and kinetics of biological reactions between in vivo and in vitro. For example, the diffusion of green fluorescent protein (GFP) in E. coli is ~10-fold slower than in dilute conditions. In this study, we performed Brownian dynamics (BD) simulations of rigid macromolecules in a crowded environment mimicking the cytosol of E. coli to study the motions of macromolecules. The simulation systems contained 35 70S ribosomes, 750 glycolytic enzymes, 75 GFPs, and 392 tRNAs in a 100 nm × 100 nm × 100 nm simulation box, where the macromolecules were represented by rigid-objects of one bead per amino acid or four beads per nucleotide models. Diffusion tensors of these molecules in dilute solutions were estimated by using a hydrodynamic theory to take into account the diffusion anisotropy of arbitrary shaped objects in the BD simulations. BD simulations of the system where each macromolecule is represented by its Stokes radius were also performed for comparison. Excluded volume effects greatly reduce the mobility of molecules in crowded environments for both molecular-shaped and equivalent sphere systems. Additionally, there were no significant differences in the reduction of diffusivity over the entire range of molecular size between two systems. However, the reduction in diffusion of GFP in these systems was still 4–5 times larger than for the in vivo experiment. We will discuss other plausible factors that might cause the large reduction in diffusion in vivo.

Hydrodynamic interactions play an important role in the dynamics of macromolecules. The most common way to take into account hydrodynamic effects in molecular simulations is in the context of a brownian dynamics simulation. However, the calculation of correlated brownian noise vectors in these simulations is computationally very demanding and alternative methods are desirable. This paper studies methods based on Krylov subspaces for computing brownian noise vectors. These methods are related to Chebyshev polynomial approximations, but do not require eigenvalue estimates. We show that only low accuracy is required in the brownian noise vectors to accurately compute values of dynamic and static properties of polymer and monodisperse suspension models. With this level of accuracy, the computational time of Krylov subspace methods scales very nearly as O(N(2)) for the number of particles N up to 10 000, which was the limit tested. The performance of the Krylov subspace methods, especially the "block" version, is slightly better than that of the Chebyshev method, even without taking into account the additional cost of eigenvalue estimates required by the latter. Furthermore, at N = 10,000, the Krylov subspace method is 13 times faster than the exact Cholesky method. Thus, Krylov subspace methods are recommended for performing large-scale brownian dynamics simulations with hydrodynamic interactions.

Self-organization is one of the most fundamental phenomena in biology. Elucidating the mechanisms of the self-assembly of biological molecules, such as protein folding/binding and formation of lipid bilayer membranes, have been a fascinating topic over decades in biological field. Hydrodynamic interactions (HIs) give rise to collective motions between molecules. Therefore, they may play an important role on dynamics of self-assembly. Although there are significant advances in theoretical and computational studies on thermodynamics and kinetics of self-assembling systems recent years, analysis on effects of HIs to the reactions are limited due to their high complexity. Here, by using a Brownian dynamics (BD) simulation technique, we evaluate the importance of HIs on kinetic of self-assembly in a set of examples, formation of lipid membrane and actin polymerization reaction. We first built coarse-grained models of lipid and actin monomer. A cluster of lipid molecules was represented by two particles, one is hydrophilic head group and the other is hydrophobic tail group, and attractive interaction between tail groups was applied. For actin polymerization system, each monomer was represented by two particles and an angle-dependent attractive interaction was applied to form long filaments. In BD simulations, the Rotne-Prager-Yamakawa tensor was used to account for HIs between particles. In the early stage of self-assembly, HIs between inter-monomer particles decelerated the association rate of monomers to form oligomers compared with that in the simulations without inter-monomer HIs in both systems. On the other hand, the HIs accelerate the reaction of oligomers to form a fully-assembled state or much longer polymers. This phenomenon could be explained by the Kirkwood-Riseman theory. Those results clearly suggest that HIs greatly affect kinetics of self-assembly reactions and considering the interactions is crucial for studying dynamics of biological systems.