Stochastic simulation Brownian dynamics

One aspect of MD simulations is that all molecules, including the solvent, are specified in full detail. As detailed above, much of the CPU time in such a simulation is used up by following all the solvent (water) molecules. An alternative to the MD simulations is Brownian dynamics (BD) simulation. In this method, the solvent molecules are removed from the simulations. The effects of the solvent molecules are then reintroduced into the problem in an approximate way. Firstly, of course, the interaction parameters are adjusted, because the interactions should now include the effect of the solvent molecules. Furthermore, it is necessary to include a fluctuating force acting on the beads (atoms). These fluctuations represent the stochastic forces that result from the collisions of solvent molecules with the atoms. We know of no results using this technique on lipid bilayers. 

Having specified the interactions (i.e., the model of the system), the actual simulation then constructs a sequence of states (or the system trajectory) in some statistical mechanical ensemble. Simulations can be stochastic (Monte Carlo (MC)) or deterministic (MD), or they can combine elements of both, such as force-biased MC, Brownian dynamics, or generalized Lan-gevin dynamics. It is usually assumed that the laws of classical mechanics (i.e., Newton s second law) may adequately describe the atoms and molecules in the physical system. 

The computationally intensity of the MD methods led to the development of Brownian Dynamics methods (BD), which use approximate expressions in place of the exact equations of motion. BD methods use randomness to simulate Brownian motion, and thus are stochastic in nature. Over sufficiently long time periods, Brownian motion appears random, and is amenable to this type of treatment. 

Brownian Dynamics (BD) methods treat the short-term behavior of particles influenced by Brownian motion stochastically. The requirement must be met that time scales in these simulations are sufficiently long so that the random walk approximation is valid. Simultaneously, time steps must be sufficiently small such that external force fields can be considered constant (e.g., hydrodynamic forces and interfacial forces). Due to the inclusion of random elements, BD methods are not reversible as are the MD methods (i.e., a reverse trajectory will not, in general, be the same as the forward using BD methods). BD methods typically proceed by discretization and integration of the equation for motion in the Langevin form... 

Now that we have settled on a model, one needs to choose the appropriate algorithm. Three methods have been used to study polymers in the continuum Monte Carlo, molecular dynamics, and Brownian dynamics. Because the distance between beads is not fixed in the bead-spring model, one can use a very simple set of moves in a Monte Carlo simulation, namely choose a monomer at random and attempt to displace it a random amount in a random direction. The move is then accepted or rejected based on a Boltzmann weight. Although this method works very well for static and dynamic properties in equilibrium, it is not appropriate for studying polymers in a shear flow. This is because the method is purely stochastic and the velocity of a mer is undefined. In a molecular dynamics simulation one can follow the dynamics of each mer since one simply solves Newton s equations of motion for mer i,... 

Molecular Dynamics simulation is one of many methods to study the macroscopic behavior of systems by following the evolution at the molecular scale. One way of categorizing these methods is by the degree of determinism used in generating molecular positions [134], On the scale from the completely stochastic method of Metropolis Monte Carlo to the pure deterministic method of Molecular Dynamics, we find a multitude and increasingly diverse number of methods to name just a few examples Force-Biased Monte Carlo, Brownian Dynamics, General Langevin Dynamics [135], Dissipative Particle Dynamics [136,137], Colli-sional Dynamics [138] and Reduced Variable Molecular Dynamics [139]. 

In the following, we briefly describe the techniques commonly employed in computer simulation studies of lipid assemblies (and of other biomole-cules " ), namely, Monte Carlo (MC) and dynamic simulations such as molecular dynamics (MD), Brownian dynamics and stochastic boundary mo-... 

Other dynamic simulations commonly used are Brownian dynamics- " and stochastic boundary molecular dynamics." These techniques are suitable when interest is limited to a small portion of a large system and the molecular details of the rest of the system are not of concern. Under such situations, MD will be an inefficient choice. For example, if the effect of a solvent on the dynamics of a solute molecule can be obtained by a suitable choice of parameters in the potential function, one can study the dynamics in more detail for longer times. In Brownian dynamics, the forces acting on a solute molecule have a component from intramolecular interactions in the solute and/or any external field, a component arising from the solvent friction, and a third random component to model the thermal fluctuations of the solvent molecules ... 

Simpler BGK kinetic theory models have, however, been applied to the study of isomerization dynamics. The solutions to the kinetic equation have been carried out either by expansions in eigenfunctions of the BGK collision operator (these are similar in spirit to the discussion in Section IX.B) or by stochastic simulation of the kinetic equation. The stochastic trajectory simulation of the BGK kinetic equation involves the calculation of the trajectories of an ensemble of particles as in the Brownian dynamics method described earlier. 

There are basically two ways of simulating a many-body system through a stochastic process, sueh as the Monte Carlo (MC) simulation, or through a deterministic process, such as a Molecular Dynamics (MD) simulation. Numerical simulations are also performed in a hybridized form, like the Langevin dynamics which is similar to MD except for the presence of a random dissipative force, or the Brownian dynamics, which is based on the condition that the acceleration is balanced out by drifting and random dissipative forces. 

Instead of solving the evolution equation in terms of the orientation tensor, one can simulate the stochastic equation such as Eq. 5.7 for the orientation vector p without the need of closure approximations, using the numerical technique for the simulation of stochastic processes (Ottinger 1996) known as the Brownian dynamics simulation. Once trajectories for aU fibers are obtained, the orientation tensor can be calculated in terms of the ensemble average of the discrete form ... 

Different from the molecular dynamics (MD) simulation method (Sect. 4.5), the Brownian dynamics approach does not directly simulate the inter-particle collision. Instead, in the Brownian dynamics, the pseudorandom motion characteristic of the effect of particle-particle interactions is mimicked by a stochastic force generated from random numbers. This makes the Brownian dynamics more efficient than the... 

Ottinger (1996) combined the Brownian dynamics simulation technique with finite elements to solve polymer flow problems. The approach has been introduced as the CONNFFESSIT (Calculation of Non-Newtonian Flow Finite Element and Stochastic Simulation Techniques) approach. Hulsen et al. (1997) extended the approach to the so-called Brownian configuration field (BCF) method, which treats the stochastic equation as the stochastic field equation, and hence avoids the difficulties associated with individual molecule tracking. If the BCF is applied to fiber suspension flows, the vectors p and q will be functions of space and time. The discrete equation for the time evolution is... 

Siettos, C., M.D. Graham, and LG. Kevrekidis. 2003. Coarse Brownian dynamics for nematic liquid crystals bifurcation, projective integration and control via stochastic simulation. Journal of Chemical Physics 118(22) 10149-10156. 

What is the relationship between the molecular dynamics simulations of a continuous model and an isothermal Monte Carlo trajectory of an otherwise similar discretized (or lattice) model When only local (and small distance) moves are applied in a properly controlled random (or rather pseudorandom) scheme, the discrete models mimic the coarse-grained Brownian dynamics of the chain. The Monte Carlo trajectory could be then interpreted as the numerical solution to a stochastic equation of motion. Of course, the short-time dynamics... 

We now consider probability theory, and its applications in stochastic simulation. First, we define some basic probabihstic concepts, and demonstrate how they may be used to model physical phenomena. Next, we derive some important probability distributions, in particular, the Gaussian (normal) and Poisson distributions. Following this is a treatment of stochastic calculus, with a particular focus upon Brownian dynamics. Monte Carlo methods are then presented, with apphcations in statistical physics, integration, and global minimization (simulated annealing). Finally, genetic optimization is discussed. This chapter serves as a prelude to the discussion of statistics and parameter estimation, in which the Monte Carlo method will prove highly usefiil in Bayesian analysis. 

Next follows a detailed discussion of probability theory, stochastic simulation, statistics, and parameter estimation. As engineering becomes more focused upon the molecular level, stochastic simulation techniques gain in importance. Particular attention is paid to Brownian dynamics, stochastic calculus, and Monte Carlo simulation. Statistics and parameter estimation are addressed from a Bayesian viewpoint, in which Monte Carlo simulation proves a powerful and general tool for making inferences and testing hypotheses from experimental data. 

Kramers idea was to give a more realistic description of the dynamics in the reaction coordinate by including dynamical effects of the solvent. Instead of giving a deterministic description, which is only possible in a large-scale molecular dynamics simulation, he proposed to give a stochastic description of the motion similar to that of the Brownian motion of a heavy particle in a solvent. From the normal coordinate analysis of the activated complex, a reduced mass pi has been associated with the motion in the reaction coordinate, so the proposal is to describe the motion in that coordinate as that of a Brownian particle of mass g in the solvent. 

The general principle of BD is based on Brownian motion, which is the random movement of solute molecules in dilute solution that result from repeated collisions of the solute with solvent molecules. In BD, solute molecules diffuse under the influence of systematic intermolecular and intramolecular forces, which are subject to frictional damping by the solvent, and the stochastic effects of the solvent, which is modeled as a continuum. The BD technique allows the generation of trajectories on much longer temporal and spatial scales than is feasible with molecular dynamics simulations, which are currently limited to a time of about 10 ns for medium-sized proteins. 

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