Deterministic Molecular Models

There are three main steps in a typical deterministic MD limitation, which are discussed in the following sections. 

In this step, a set of molecules (N) are introduced in a two- or three-dimensional regular lattice, and each molecule is assigned a random velocity. To achieve faster equilibration, atoms can be assigned velocities with the equilibrium velocity, that is. Maxwell distribution at a specified temperature. 

For ease in computational effort, the L-J potential equation is also used in a modified form as 

r is the cutoff rstdius with typical values in the range 2.2a-2.5(j. In this equation, the first term (with c, ) represents a short-range repulsive force and the second term (rf,/ represents 

The dynamics of the wall boundary can be described using the same potential with different values of the energy, length, and cutoff radius. In flows with active boundaries, for example, microchannels with elastomeric walls, appropriate reflnements should be employed. 

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Molecular dynamics-deterministic molecular modeling tool that evaluates the forces on individnal atoms using an energy forcefield, then uses Newton s classical eqnation of motion to compute new atomic positions after a short time interval (on the order of a femtosecond) snccessive evaluation for a large number of time steps provides a time-dependent trajectory of all atomic motions. 

A major difficulty is that such hierarchies of molecular models are not exactly known. Recent work by Gillespie (2000, 2002) has established such a hierarchy for stochastic models of chemical reactions in a well-mixed batch reactor. This hierarchy is depicted in Fig. 3b. In particular, it was shown that the chemical master equation is deduced to a chemical Langevin equation when the population sizes are relatively large. Finally, the deterministic behavior can be... 

Following the construction of the model is the calculation of a sequence of states (or a trajectory of the system). This step is usually referred to as the actual simulation. Simulations can be stochastic (Monte Carlo) or deterministic (Molecular Dynamics) or they can combine elements of both, like force-biased Monte Carlo, Brownian dynamics or general Langevin dynamics (see Ref. 16 for a discussion). It is usually assumed that the physical system can be adequately described by the laws of classical mechanics. This assumption will alsq be made throughout the present work. 

Often, the strict definition of ergodicity must be relaxed to accommodate the realistic modelling setting for example only some portion of the eno gy surface may be accessible by trajectories in the time-interval available, due to the cost of computing steps with a numerical method. For a more detailed discussion of these and other ergodicity issues in the context of deterministic molecular dynamics, see the articles of Tupper [379, 381]. 

Molecular Dynamics and Monte Carlo Simulations. At the heart of the method of molecular dynamics is a simulation model consisting of potential energy functions, or force fields. Molecular dynamics calculations represent a deterministic method, ie, one based on the assumption that atoms move according to laws of Newtonian mechanics. Molecular dynamics simulations can be performed for short time-periods, eg, 50—100 picoseconds, to examine localized very high frequency motions, such as bond length distortions, or, over much longer periods of time, eg, 500—2000 ps, in order to derive equiUbrium properties. It is worthwhile to summarize what properties researchers can expect to evaluate by performing molecular simulations ... 

Only deterministic models for cellular rhythms have been discussed so far. Do such models remain valid when the numbers of molecules involved are small, as may occur in cellular conditions Barkai and Leibler [127] stressed that in the presence of small amounts of mRNA or protein molecules, the effect of molecular noise on circadian rhythms may become significant and may compromise the emergence of coherent periodic oscillations. The way to assess the influence of molecular noise on circadian rhythms is to resort to stochastic simulations [127-129]. Stochastic simulations of the models schematized in Fig. 3A,B show that the dynamic behavior predicted by the corresponding deterministic equations remains valid as long as the maximum numbers of mRNA and protein molecules involved in the circadian clock mechanism are of the order of a few tens and hundreds, respectively [128]. In the presence of molecular noise, the trajectory in the phase space transforms into a cloud of points surrounding the deterministic limit cycle. 

Biological media are inhomogeneous, and the simplest way to capture structural and functional heterogeneity is to operate at a molecular level. First, one has to model the time spent by each particle in the process and second, to statistically compile the molecular behaviors. As will be shown in Section 9.3.4, this compilation generates a process uncertainty that did not exist in the deterministic model, and this uncertainty is the expression of process heterogeneity. 

Scheme 1.1 The molecular information system modeling the chemical bond between two basis functions /=(o,b) and its entropy/information descriptors. In Panel b, the corresponding nonbonding (deterministic) channel due to the lone-pair hybrid 6° is shown. For the molecular input p = (P, Q), the orbital channel of Panel a gives the bond entropy-covalency represented by the binary entropy function H[P). For the promolecular input p° = (1/2,1/2), when both basis functions contribute a single electron each to form the chemical bond, one thus predicts H[p°] = 1 and the bond information ionicity / = 1 — H(P). Hence, these two bond components give rise to the conserved (P-independent) value of the single overall bond multiplicity N = I + S = 1.

Due to the changes in the dynamics, a general relationship for stochastic dynamics is not available like it is for deterministic dynamics. However, for mesoscopic systems, a mesoscopic FR is useful. Therefore, there has been much work on developing stochastic models with different conditions. Andrieux and Gaspard developed a stochastic fluctuation relation for nonequilibrium systems whose dynamics can be described by Schnakenberg s network theory (e.g. mesoscopic electron transport, biophysical models of ion transport and some chemical reactions). Due to early experimental work on protein unfolding and related molecular motors, and their ready treatment by stochastic dynamics, a number of papers have appeared that model these systems and test the or JE for these. FR... 

The rate constants used in the deterministic model, expressed in terms of molecular concentrations, are related to the constants for the stochastic model, expressed in terms of molecule numbers by taking into account the volume, V, of the system as in the earlier expressions for A and [23]. 

Besides the above simplified models, more interesting is the understanding of the anomalous diffusion in incompressible velocity fields or deterministic maps. In this direction, Avellaneda, Majda, and Vergassola [22, 23] obtained a very important and general result about the character of the asymptotic diffusion in an incompressible velocity field u(x). If the molecular diffusivity D is nonzero and the infrared contribution to the velocity field are weak enough, namely,... 

Hydrate crystal decomposition, like the hydrate growth, is a deterministic process. Hence, the process is amenable to study experimentally and modeling. Although some studies have been undertaken at the molecular level, most of them on the hydrate decomposition kinetics are based on a macroscopic approach. The hydrate decomposition is a heterogeneous process where liquid water and gas are released as the solid... 

In a series of impressive publications. Maxwell [65] [66] [67] [68] provided most of the fundamental concepts constituting the statistical theory recognizing that the molecular motion has a random character. When the molecular motion is random, the absolute molecular velocity cannot be described deterministically in accordance with a physical law so a probabilistic (stochastic) model is required. 

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