Stochastic boundary conditions

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. 

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. 

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. 

C. MOLECULAR DYNAMICS WITH STOCHASTIC BOUNDARY CONDITIONS... 

Safety deals with stochastic events, for example the moment of occurrence of an accident, and stochastic boundary conditions (e.g. the weather at that moment). These together with lacks of knowledge about some of the phenomena to be described and imperfections in models and input data lead to uncertainties, which are normally compensated by safety factors and often lead to procedures based on conventions. 

The methods of calculation employed are those which are used as well for deterministic analyses. The difference is that stochastic boundary conditions, which are closer to reality, are used for the calculations. For example, instead of a fixed leak size a whole spectrum of leak sizes is treated with pertinent expected frequencies of occurrence being assigned to the different leak sizes. Instead of calculating the dispersion of a toxic substance based on a specific weather situation, different possible weather situations with their corresponding probabilities of occurrence are accounted for. This is reasonable, since the instant in time of the accident and the weather condition, which then prevails, are not known beforehand. 

To overcome problems arising fi-om the finite system size used in MC or MD simulation, boundary conditions are imposed using periodic-stochastic approximations or continuum models. In particular, in stochastic boundary conditions the finite system is not duplicated but a boundary force is applied to interact with atoms of the system. This force is set as to reproduce the solvent regions that have been neglected. Anyway, in general any of the methods used to impose boundary conditions in MC or MD can be used in the QM/MM approach. 

M. Berkowitz and J. A. McCammon, Chem. Phys. Lett., 90, 215 (1982). Molecular Dynamics with Stochastic Boundary Conditions. 

An alternative approach is to employ stochastic boundary conditions where the finite molecular system employed in the simulation is not duplicated, but rather, a boundary force is applied to interact with atoms of the sys-tem. 2-67 Yhe boundary force is chosen to mimic the bulk solvent regions that have been neglected. However, a difficulty with the use of this method is due to the ambiguity associated with the definition and parameterization of the boundary forces. Thus, there is sometimes the impression that the method is not rigorous. Recently, Beglov and Roux provided a formal definition and suggested a useful implementation of the boundary forces. Their results are very promising. 

Any of the methods used in classical Monte Carlo and molecular dynamics simulations may be borrowed in the combined QM/MM approach. However, the use of a finite system in condensed phase simulations is always a severe approximation, even when appropriate periodic or stochastic boundary conditions are employed. A further complication is the use of potential function truncation schemes, particular in ionic aqueous solutions where the long-range Coulombic interactions are significant beyond the cutoff distance.Thus, it is alluring to embed a continuum reaction field model in the quantum mechanical calculations in addition to the explicit solute—solvent interaaions to include the dielectric effect beyond the cutoff distance. - uch an onion shell arrangement has been used in spherical systems, whereas Lee and Warshel introduced an innovative local reaction field method for evaluation of long-... 

MARESCHAL - In the Benard problem, the thermal boundaries are simulated along the ways developped in non-equilibrium molecular dynamics, using stochastic boundary conditions (see G. Ciccotti). The boundary layer does not extend over more than a mean free path in the system and can hardly be seen in our measurements. 

Kantorovich, L., 8cRompotis, N. (2008). Generalized Langevin equation for solids. II. Stochastic boundary conditions for nonequilibrium molecular dynamics simulations. Physical Review B, 78, 094305. 

Brunger A, Brooks CL, Karplus M (1984) Stochastic boundary-conditions for molecular-dynamics simulatirais of St2 water. Chem Phys Lett 105 495-500... 

Figure 10.4 Overview of the simulation system used to study proton transfer in ferre-doxin I (Fdl) (a). Solvent water is treated with stochastic boundary conditions. A magnified view of the relevant PT motif is shown in (c). Panel (c) reports the free energy profile to move the proton from the protein side chain Aspl5(-COOH) to the [3Fe-4S] cluster.

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