Brownian dynamics method

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. 

In the Brownian dynamics method, one solves the Smoluchowski equation by an MC method ... 

The contents of the review are as follows. The dynamics of rodlike polymers are reviewed in Section 2 followed by a review of previous experimental results of the polymerization kinetics of rodlike molecules in Section 3. Theoretical analyses of the problem following Smoluchowski s approach are discussed next (Section 4), and this is followed by a review of computational studies based on multiparticle Brownian dynamics in Section 5. The pairwise Brownian dynamics method is discussed in some detail in Section 6, and the conclusions of the review are given in Section 7. 

The above results illustrate the utility of multiparticle Brownian dynamics for the analysis of diffusion controlled polymerizations. The results presented here are, however, qualitative because of the assumption of a two-dimensional system, neglect of polymer-polymer interactions and the infinitely fast kinetics in which every collision results in reaction. While the first two assumptions may be easily relaxed, incorporation of slower reaction kinetics by which only a small fraction of the collisions result in reaction may be computationally difficult. A more computationally efficient scheme may be to use Brownian dynamics to extract the rate constants as a function of polymer difflisivities, and to incorporate these in population balance models to predict the molecular weight distribution [48-50]. We discuss such a Brownian dynamics method in the next section. 

The pairwise Brownian dynamics method is a combination of Brownian dynamics and the Smoluchowski [9] approach, and the effective rate constant is obtained from the reaction probability of a single molecule undergoingdiffusive motion in the neighbourhood of a stationary test molecule, so that only a pair of molecules is considered at a time. The method was first proposed by Northrup et al. [58], and the basis of the method is to obtain the steady state reaction flux (y) as the product of the first visit flux (Jq) to a surface (spherical) which envelopes the reaction zone and the probability (/ ) that a molecule starting from the surface reacts rather than escaping to the far field, that is, j = The first visit flux (Jq) is obtained analytically whereas... 

Pairwise Brownian dynamics has been primarily used for the analysis of diffusion controlled reactions involving the reaction between isotropic molecules with complex reactive sites. Since its introduction by Northrup et al. [58], the pairwise Brownian dynamics method has been considerably refined and modified. Some of the developments include the use of variable time steps to reduce computational times [61], efficient calculation methods for charge effects [63], and incorporation of finite rates of reaction [58,61,62]. We review in the following sections, application of the method to two example problems involving isotropic translational diffusion reaction of isotropic molecules with a spherical reaction surface containing reactive patches and the reaction between rodlike molecules in dilute solution. 

The pairwise Brownian dynamics method has several advantages over numerical methods based on Smoluchowski s [9] approach (e.g., finite element method), and we discuss these here. The primary advantage of the method is the ease of mathematical formulation even for cases involving complex reaction site geometries, hydrodynamic interactions, charge effects, anisotropic diffusion and flow fields. Furthermore the method obviates the need to solve complex diffusion equations to obtain the concentration field from which the rate constant is calculated in the Smoluchowski method. In contrast, the rate constant is obtained directly in the pairwise Brownian dynamics method. The effective rate constants for different reaction conditions may be obtained from a single simulation this is not possible using the finite element method. 

S. H. Northrup, S. A. Allison, S. A. Curvin and J. A. McCammon, J. Chem. Phys., Optimization of Brownian dynamics methods for diffusion influenced rate constant calculations, 84 (1986) 2196-2203. 

For this reason, computer simulation methods (Monte Carlo and molecular and Brownian dynamic methods) have been developed not only for solving the problems of polymer statistics, but also for the investigation of the dynamic properties and the intramolecular mobility of polymers. 

According to the fluctuation-dissipation theorem [1], the electrical polarizability of polyelectrolytes is related to the fluctuations of the dipole moment generated in the counterion atmosphere around the polyions in the absence of an applied electric field [2-4], Here we calculate the fluctuations by computer simulation to determine anisotropy of the electrical polarizability Aa of model DNA fragments in salt-free aqueous solutions [5-7]. The Metropolis Monte Carlo (MC) Brownian dynamics method [8-12] is applied to calculate counterion distributions, electric potentials, and fluctuations of counterion polarization. 

The Brownian dynamics method described above can be used to generate diffusional trajectories of a substrate in the field of an enzyme target. 

Despite the many assumptions and approximations inherent in such simulations, the Brownian dynamics method proved useful in investigating the ability of the peptide loops to gate the active sites of this enzyme. Interestingly, they showed that the motion of the loops does not cause a reduction in the rate of the reaction, suggesting that the loops, which provide the appropriate environment for catalysis, have evolved to minimize any loss in kinetic efficiency that might arise as a result of gating. 

Optimization of Brownian Dynamics Methods for Diffusion-Influenced Rate Constant Calculations. 

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 two main types of computational procedure which have been used to simulate colloidal dispersions the Monte Carlo method and the Brownian dynamics method. A Monte Carlo simulation gives equilibrium behaviour only, whereas a Brownian dynamics simulation gives both equilibrium and time-dependent behaviour. 

In certain enzyme-substrate encounters the reactivity of the enzyme may vary with time. This may be due to a necessary conformational change or the movement of flexible loops acting as trapdoors at the entrance of the active site. Simulation of these gated reactions can be studied implicitly and explicitly using Brownian dynamics methods. Implicitly, the dynamics of the gate can be described by the rate constants... 

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