Kinetics Brownian dynamics

Northrup S H and Erickson H P 1992 Kinetics of protein-protein association explained by Brownian dynamics computer simulation Proc. Natl Acad. Sci. USA 89 3338-42... 

Chirico, G. and Langowski, J. (1994) Kinetics of DNA supercoiling studied by Brownian dynamics simulation. Biopolymers 34, 415-433. 

A reaction looked at earlier simulates borate inhibition of serine proteinases.33 Resorufin acetate (234) is proposed as an attractive substrate to use with chymotrypsin since the absorbance of the product is several times more intense than that formed when the more usual p-nitrophcnyl acetate is used as a substrate. The steady-state values are the same for the two substrates, which is expected if the slow deacylation step involves a common intermediate. Experiments show that the acetate can bind to chymotrypsin other than at the active site.210 Brownian dynamics simulations of the encounter kinetics between the active site of an acetylcholinesterase and a charged substrate together with ah initio quantum chemical calculations using the 3-21G set to probe the transformation of the Michaelis complex into a covalently bound tetrahedral intermediate have been carried out.211 The Glu 199 residue located near the enzyme active triad boosts acetylcholinesterase activity by increasing the encounter rate due to the favourable modification of the electric field inside the enzyme and by stabilization of the TS for the first chemical step of catalysis.211... 

In order to determine a system thermodynamically, one has to specify some independent parameters (e.g. N, T, P or V) besides the composition of the system. The most common choice in MC simulation is to specify N, V and T resulting in the canonical ensemble, where the Helmholtz free energy A is the natural thermodynamical potential. However, MC calculations can be performed in any ensemble, where the suitable choice depends on the application. It is straightforward to apply the Metropolis MC algorithm to a simple electric double layer in the iVFT ensemble. It is however, not so efficient for polymers composed of more than a few tens of monomers. For long polymers other algorithms should be considered and the Pivot algorithm [21] offers an efficient alternative. MC simulations provide thermodynamic and structural information, but time-dependent properties are not accessible. If kinetic or time-dependent properties are of interest one has to use molecular dynamic or brownian dynamic simulations. 

An interesting application of the molecular dynamics technique on single chains is found in the work of Mattice et al. One paper by these authors is cited here because it is relevant to both RIS and DRIS studies and deals with the isomerization kinetics of alkane chains. The authors have computed the trajectories for linear polyethylene chains of sizes C,o to Cioo- The simulation was fully atomistic, with bond lengths, bond angles, and rotational states all being variable. Analysis of the results shows that for very short times, correlations between rotational isomeric transitions at bonds i and i 2 exist, which is something a Brownian dynamics simulation had shown earlier. 

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. 

Triose Phosphate Isomerase Diffusional Encounters with D-Glyceraldehyde-3-Phosphate In this section we use a real system, triose phosphate isomerase (TIM) and its substrate D-glyceraldehyde—3-phosphate (GAP) to demonstrate the capabilities of Brownian dynamics simulations with electrostatics. TIM is a glycolytic enzyme that catalyzes the interconversion of GAP and dihydroxy-acetone phosphate (DHAP). It has been described as an almost perfea catalyst because of its remarkable efficiency. Structurally, TIM is a dimeric enzyme consisting of two identical polypeptide chains of 247 amino acid residues. Each subunit consists of eight loop-p/loop-a units and contains one aaive site. Located near each aaive site is a peptide loop, which is mobile in the native enzyme and folds down to cover the active site when the substrate is bound. Kinetically, the reaction appears to be diffusion controlled and proceeds with a measured rate constant of 4.8 x 10 M s L TIM has consequently been the focus of many kinetic and struaural studies. ... 

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. 

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. 

The various procedures discussed above address the thermodynamic aspects of protein-protein interactions but say nothing about the dynamical aspects, such as the association kinetics and the mechanism. To gain insights into these latter aspects from computational methods, protein-protein association must actually be simulated at the molecular level. The most successful approach to such simulations has been Brownian dynamics (BD) (for recent reviews, see Gabdoulline and Wade, 1998, and Elcock et clL, 2001). 

Zhou, H.-X. Brownian dynamics study of the influences of electrostatic interaction and diffusion on protein-protein association kinetics. Biophys. J. 1993, 64,1711-26. 

Zhou, H.-X. Kinetics of diffusion-influenced reactions studied by Brownian dynamics. J. Phys. Chem. 1990, 94, 8794-800. 

S. H. Northrup and H. P. Erickson, Proc. Natl. Acad. Sci. USA, 89,3338 (1992). Kinetics of Protein-Protein Association Explained by Brownian Dynamics Computer Simulation. 

Figure 25.9 Kinetics of particle aggregation via Brownian dynamics simulations (simulation conditions diluted polymer dispersion in water with solids content of 0.65%,...

Rzepiela, A.A., van Opheusden, and van Vliet, T. (2001) Brownian dynamics simulation of aggregation kinetics of hard spheres with flexible bonds./. CdUoid Interface Sci., 244, 43. 

Hernandez, H.F. and Tauer, K. (2010) Radical desorption kinetics in emulsion polymerization 2. Brownian dynamics simulation of radical desorption in non-homogeneous polymer particles. Macromol. Theory Simul., 19, 249. 

This is known as Chandrasekhar s equation. For those of you familiar with the kinetic theory of gases, note the identity of the LHS with the streaming part of the Liouville equation for the reduced single particle distribution function, whereas the RHS can be viewed as the Brownian dynamics analogue of the collision terms in the Boltzmann equation. 

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