Brownian dynamic simulation

For calculating the time-dependent properties of biopolymers, the equations of motion of the molecule in a viscous medium (i.e., water) under the influence of thermal motion must be solved. This can be done numerically by the method of Brownian dynamics (BD) [83]. Allison and co-workers [61,62,84] and later others [85-88] have employed BD calculations to simulate the dynamics of linear and superhelical DNA BD models for the chromatin chain will be discussed below. 

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  

Pi = ijPj jkPiPjPk + ( 5y -PiPj)Fj t). We can introduce a vector qt that satisfies 

5 Flow-Induced Alignment in Short-Fiber Reinforced Polymers 

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 

Here we express the term by a white noise. For a general case where the interactions between fibers are allowed to be anisotropic (Phan-Thien et al. 2000), we have 

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Madura J D, Davis M E, Gilson, M K, Wade R C, Luty B A and McCammon J A 1994 Biological applications of electrostatic calculations and Brownian dynamics simulations Rev. Comput. Chem. 5 229-67... 

R. C. Wade, M. E. Davis, B. A. Luty, J. D. Madura, and J. A. McCammon. Gating of the active site of triose phosphate isomerase Brownian dynamics simulations of flexible peptide loops in the enzyme. Biophys. J., 64 9-15, 1993. 

Dissipative particle dynamics (DPD) is a technique for simulating the motion of mesoscale beads. The technique is superficially similar to a Brownian dynamics simulation in that it incorporates equations of motion, a dissipative (random) force, and a viscous drag between moving beads. However, the simulation uses a modified velocity Verlet algorithm to ensure that total momentum and force symmetries are conserved. This results in a simulation that obeys the Navier-Stokes equations and can thus predict flow. In order to set up these equations, there must be parameters to describe the interaction between beads, dissipative force, and drag. 

FIG. 1 Results of a Brownian dynamic simulation for a two-dimensional coulombic system with specific interactions [40]. 

Subsequent work by Johansson and Lofroth [183] compared this result with those obtained from Brownian dynamics simulation of hard-sphere diffusion in polymer networks of wormlike chains. They concluded that their theory gave excellent agreement for small particles. For larger particles, the theory predicted a faster diffusion than was observed. They have also compared the diffusion coefficients from Eq. (73) to the experimental values [182] for diffusion of poly(ethylene glycol) in k-carrageenan gels and solutions. It was found that their theory can successfully predict the diffusion of solutes in both flexible and stiff polymer systems. Equation (73) is an example of the so-called stretched exponential function discussed further later. 

Johansson, L Lofroth, J-E, Diffusion and Interaction in Gels and Solutions. 4 Hard Sphere Brownian Dynamics Simulations, Journal of Chemical Physics 98, 7471, 1993. 

Ediger, M. D. and Adolf, D, B. Brownian Dynamics Simulations of Local Polymer Dynamics. VoU 16, pp. 73-110. 

We have performed Brownian dynamics simulations (i.e., overdamped Langevin dynamics) on the potential surface V(x, A) with a time-dependent A. In the simulations, the position x was updated as... 

Huber, G.A. Kim, S., Weighted-ensemble Brownian dynamics simulations for protein association reactions, Biophys. 7. 1996, 70, 97-110... 

Jeffry D. Madura, Malcolm E. Davis, Michael K. Gilson, Rebecca C. Wade, Brock A. Luty, and J. Andrew McCammon, Biological Applications of Electrostatic Calculations and Brownian Dynamics Simulations. 

A more detailed view of the dynamies of a ehromatin chain was achieved in a recent Brownian dynamics simulation by Beard and Schlick [65]. Like in previous work, the DNA is treated as a segmented elastic chain however, the nueleosomes are modeled as flat cylinders with the DNA attached to the cylinder surface at the positions known from the crystallographic structure of the nucleosome. Moreover, the electrostatic interactions are treated in a very detailed manner the charge distribution on the nucleosome core particle is obtained from a solution to the non-linear Poisson-Boltzmann equation in the surrounding solvent, and the total electrostatic energy is computed through the Debye-Hiickel approximation over all charges on the nucleosome and the linker DNA. 

Merlitz, H., Rippe, K., Klenin, K.V., and Langowski, J. (1998) Looping dynamics of linear DNA molecules and the effect of DNA curvature a study by Brownian dynamics simulation. Biophys. J. 74, 773-779. 

Allison, S.A. and McCammon, J.A. (1984) Transport properties of rigid and fiexible macromolecules by Brownian dynamics simulation. Biopolymers 23, 167-187. 

Allison, S.A. (1986) Brownian dynamics simulation of wormlike chains. Fluorescence depolarization and depolarized light scattering. Macromolecules 19, 118-124. 

Allison, S.A., Sorlie, S.S., and Pecora, R. (1990) Brownian dynamics simulations of wormlike chains Dynamic light scattering from a 2311 base pair DNA fragment. Macromolecules 23, 1110-1118. 

Gross, E.L. and Pearson, D.C. Jr. Brownian dynamics simulations of the interaction of chlamydomonas cytochrome f with plastocyanin and... 

In addition to these experimental methods, there is also a role for computer simulation and theoretical modelling in providing understanding of structural and mechanical properties of mixed interfacial layers. The techniques of Brownian dynamics simulation and self-consistent-field calculations have, for example, been used to some advantage in this field (Wijmans and Dickinson, 1999 Pugnaloni et al., 2003a,b, 2004, 2005 Parkinson et al., 2005 Ettelaie et al., 2008). 

Pugnaloni, L.A., Ettelaie, R., Dickinson, E. (2003a). Growth and aggregation of surfactant islands during the displacement of an adsorbed protein monolayer a Brownian dynamics simulation study. Colloids and Surfaces B Biointerfaces, 31, 149-157. 

Wijmans, C.M., Dickinson, E. (1999). Brownian dynamics simulation of the displacement of a protein monolayer by competitive adsorption. Langmuir, 15, 8344-8348. 

Figure 8.2 Phase separation in binary mixtures of model spherical particles at a planar interface generated by Brownian dynamics simulation. The three 2-D images refer to systems in which (A) light particles form irreversible bonds, (B) light particles form reversible bonds, and (C) neither dark nor light particles form bonds, but they repel each other. Picture D shows a 3-D representation. Reproduced from Pugnaloni et al. (2003b) with permission.

Figure 8.9 Phase separation in a mixed layer of protein + surfactant from Brownian dynamic simulation. In the picture are cross-linked protein-like particles (black) and surfactant-like displacer particles (grey). Reproduced from Wijmans and Dickinson (1999b) with permission.

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