Stochastic boundary

A. Briinger, C. L. Brooks, III, and M. Karpins. Stochastic boundary conditions for molecular dynamics simulations of ST2 water. Chem. Phys. Lett., 105 495-500, 1982. 

Fig. 6.7 Division into reaction zone and reservoir regions in a simulation using stochastic boundary conditions.

Brunger A, C B Brooks and M Karplus 1984. Stochastic Boundary Conditions for Molecular Dynaniii Simulations of ST2 Water. Chemical Physics Letters 105 495-500. 

De, Loof H, S C Harvey, J P Segrest and R W Pastor 1991. Mean Field Stochastic Boundary Molecul Dynamics Simulation of a Phospholipid in a Membrane. Biochemistry 30 2099-2113. 

In finite boundary conditions the solute molecule is surrounded by a finite layer of explicit solvent. The missing bulk solvent is modeled by some form of boundary potential at the vacuum/solvent interface. A host of such potentials have been proposed, from the simple spherical half-harmonic potential, which models a hydrophobic container [22], to stochastic boundary conditions [23], which surround the finite system with shells of particles obeying simplified dynamics, and finally to the Beglov and Roux spherical solvent boundary potential [24], which approximates the exact potential of mean force due to the bulk solvent by a superposition of physically motivated tenns. 

The use of QM-MD as opposed to QM-MM minimization techniques is computationally intensive and thus precluded the use of an ab initio or density functional method for the quantum region. This study was performed with an AMi Hamiltonian, and the first step of the dephosphorylation reaction was studied (see Fig. 4). Because of the important role that phosphorus has in biological systems [62], phosphatase reactions have been studied extensively [63]. From experimental data it is believed that Cys-i2 and Asp-i29 residues are involved in the first step of the dephosphorylation reaction of BPTP [64,65]. Alaliambra et al. [30] included the side chains of the phosphorylated tyrosine, Cys-i2, and Asp-i 29 in the quantum region, with link atoms used at the quantum/classical boundaries. In this study the protein was not truncated and was surrounded with a 24 A radius sphere of water molecules. Stochastic boundary methods were applied [66]. 

Figure 7-2. Properties of CAII active site in the COHH state (zinc-bound hydroxide and protonated His 64). (a) Superposition of a few key residues from two stochastic boundary SCC-DFTB/MM simulations with the X-ray structure [87] (colored based on atom-types) the two sets of simulations did not have any cut-off for the electrostatic interactions between SCC-DFTB and MM atoms but used different treatments for the electrostatic interactions among MM atoms group-based extended electrostatics (in yellow) and atom-based force-shift cut-off (in green). Extended electrostatics simulations sampled configurations with the protonated His 64 too close to the zinc moiety while force-shift simulations consistently sampled the out configuration of His 64 in multiple trajectories, (b) Statistics for productive water-bridges (only from two and four shown here) between the zinc bound water and His 64 with different electrostatics protocols...

Variational electrostatic projection method. In some instances, the calculation of PMF profiles in multiple dimensions for complex chemical reactions might not be feasible using full periodic simulation with explicit waters and ions even with the linear-scaling QM/MM-Ewald method [67], To remedy this, we have developed a variational electrostatic projection (VEP) method [75] to use as a generalized solvent boundary potential in QM/MM simulations with stochastic boundaries. The method is similar in spirit to that of Roux and co-workers [76-78], which has been recently... 

We begin in Section II with a review of the fundamental concepts of hydrodynamics and boundary conditions. In Section III, we present some common descriptions of coupling, followed in Section IV by a discussion of viscoelastic adsorbate films and the so-called inner slip. In Section V, we consider with the concept of stochastic boundary conditions, which we believe will be an important topic in situations where random fluctuations are strong. Finally, in Section VI, we present our concluding ideas and discuss some areas for future study. 

In most problems involving boundary conditions, the boundary is assigned a specific empirical or deterministic behavior, such as the no-slip case or an empirically determined slip value. The condition is defined based on an averaged value that assumes a mean flow profile. This is convenient and simple for a macroscopic system, where random fluctuations in the interfacial properties are small enough so as to produce little noise in the system. However, random fluctuations in the interfacial conditions of microscopic systems may not be so simple to average out, due to the size of the fluctuations with respect to the size of the signal itself. To address this problem, we consider the use of stochastic boundary conditions that account for random fluctuations and focus on the statistical variability of the system. Also, this may allow for better predictions of interfacial properties and boundary conditions. 

Inner slip, between the solid wall and an adsorbed film, will also influence the surface-liquid boundary conditions and have important effects on stress propagation from the liquid to the solid substrate. Linked to this concept, especially on a biomolecular level, is the concept of stochastic coupling. At the molecular level, small fluctuations about the ensemble average could affect the interfacial dynamics and lead to large shifts in the detectable boundary condition. One of our main interests in this area is to study the relaxation time of interfacial bonds using slip models. Stochastic boundary conditions could also prove to be all but necessary in modeling the behavior and interactions of biomolecules at surfaces, especially with the proliferation of microfluidic chemical devices and the importance of studying small scales. 

In an investigation of the role of water in enzymic catalysis. Brooks and Karplus (1989) chose lysozyme for their study. Stochastic boundary molecular dynamics methodology was applied, with which it was possible to focus on a small part of the overall system (i.e., the active site, substrate, and surrounding solvent). It was shown that both structure and dynamics are affected by solvent. These effects are mediated through solvation of polar residues, as well as stabilization of like-charged ion pairs. Conversely, the effects of the protein on solvent dynamics and... 

For example, periodic boundary conditions can be applied [124] (although special consideration must be given to the interaction of the QM group with its images), or where only part of a protein can be included, the stochastic boundary method for dynamics can be applied [9,125]. 

The most common boundary representation is periodic boundary conditions which assumes that the system consists of a periodic array (or a crystal ) of identical systems [1], Another common method, developed for the simulation of biomacromolecules, is the stochastic boundary approach, in which the influences of the atoms outside the boundary are replaced by a simple boundary force [78, 79, 80], Warshel uses a Langevin dipoles model in which the solvent is explicitly replaced by a grid of polarizable dipoles. The energy is calculated in a similar way to the polarization energy in a molecular mechanics force field (see above) [15]. 

Brooks III, C.L. and Karplus, M. (1983) Deformable Stochastic Boundaries in Molecular Dynamics, J. Chem. Phys. 79, 6312-6325. 

Active site, stochastic boundary simulations (Brooks and Karplus 1983 Briinger et al. 1985 Brooks and Karplus 1989) of RNase complexed with a substrate, a cyclic phosphate intermediate and a product have been made to clarify the contribution of various amino acid residues to the enzymatic reaction (Haydock et al. 1990). The structure of the enzyme complexed with a true substrate, CpA, was built from the crystal results for the enzyme complexed with deoxy-CpA, the substrate analog, by introducing the 02 H hydroxyl group to include the deprotonated His 12, the protonated His potential was replaced by that for the neutral form His 119 was protonated in the simulations. 

Brooks, C. L. Ill Briinger, A. T. and Karplus, M. (1985) Active Site Dynamics in Protein Molecules A Stochastic Boundary Molecular-Dynamics Approach, Biopolymers 24, 843-865. 

Wang et al. (1994) analyzed by MD the roles of the "double catalytic triad" in papain catalysis, based on the structure of the enzyme, which is not completely known from crystallography (Kamphuis et al., 1984) due to the oxidation state of Cys-25 (present as cysteic acid in the crystal). Stochastic boundary MD (Brooks and Karplus, 1983) was carried out on the whole enzyme + 350 water molecules. Three "layers" were treated according to their distance from the sulfur atom of Cys-25 - atoms within 12A, atoms between 12-16A and the more distant atoms were kept fixed. CHARMM forcefield was employed. The active site geometry was examined as a function of pH, for various mutual states of S-/SH and Im/ImH+. In addition, the mutations of Asp-158 (Menard et al., 1991) were studied. 

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