General Simulation Techniques

Although different simulation techniques are appropriate for different models and lines of inquiry, common concerns originate with respect to all of them. One concern of paramount importance is assessing the quality of the data that have been collected. For simulations to appropriately capture protein behavior at a particular temperature (or for a particular dielectric constant, solvent, or other parameter), they must sample available protein conformations adequately. To ensure that the conformations being sampled are representative of a particular parameter set, sampling must be performed over periods of time that allow the system to relax to thermodynamic equilibrium. Determining this equilibrium can be difficult, particularly if sampling is performed at or near a transition temperature. 

One way of determining thermodynamic equilibrium is through the calculation of protein relaxation times. By choosing an appropriate variable to monitor folding progress (examples of which may be Rg, the radius of gyration, or Q, the number of native contacts formed) and monitoring its 

If A is saved at equally spaced time steps throughout the simulation, then Caa( t) can be calculated at any one of these points, labeled x. As x increases, Caa( i) should decay to 0 at x i, because the value of A at long times should be uncorrelated with its initial value. To collect data for a system in thermodynamic equilibrium, the total simulation time must be 10 to 100 times longer than Xre. For a more complete discussion of this topic, as well as a good general discussion of block analysis, an alternative method for determining equilibrium sampling, we refer the reader to the classic book by Allen and Tildesley.  

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We review here results of computer simulations of monolayers, with an emphasis on those models that include significant molecular detail to the surfactant molecule. We start with a focus on hydrocarbon chains and simple head groups (typically a COOH group in either the neutral or the ionized state) and a historical focus. A less comprehensive review follows on simulations of surfactants of other types, either nonhydrocarbon chains or different head groups. More detailed descriptions of the general simulation techniques discussed here are available in a book dedicated to simulation techniques, for example, Allen and Tildesley [338] or Frenkel and Smit [339],... 

A number of textbooks and review articles are available which provide background and more-general simulation techniques for fluids, beyond the calculations of the present chapter. In particular, the book by Frenkel and Smit [1] has comprehensive coverage of molecular simulation methods for fluids, with some emphasis on algorithms for phase-equilibrium calculations. General review articles on simulation methods and their applications - e.g., [2-6] - are also available. Sections 10.2 and 10.3 of the present chapter were adapted from [6]. The present chapter also reviews examples of the recently developed flat-histogram approaches described in Chap. 3 when applied to phase equilibria. 

The Monte Carlo simulation method (MCS) is a general simulation technique, i.e. it is applicable to linear and non linear problems indifferently. Moreover, its efficiency is independent of the number of random variables involved in the problem under analysis. The basic idea behind MCS is to generate N samples of 0 which are distributed according to h 0). Then, the failure probability can be estimated as ... 

Expanding this to the case of a more general coordinate and to the case when this coordinate is unconstrained has been discussed by Darve and Pohorille. We point out that calculating the PMF is just one example of free energy calculation methods that have been developed and discussed extensively in the literature. The reader can consult the many excellent sources on general simulation techniques mentioned earlier in this chapter and others,as well as a chapter in this book series dedicated to this topic. " ... 

In this chapter we shall discuss some of the general principles involved in the two most common simulation techniques used in molecular modelling the molecular dynamics and the Monte Carlo methods. We shall also discuss several concepts that are common to both of these methods. A more detailed discussion of the two simulation methods can be found in Chapters 7 and 8. 

K. Binder. General aspects of computer simulation techniques and their applications in polymer physics. In K. Binder, ed. Monte Carlo and Molecular Dynamics Simulations in Polymer Science. New York Oxford University Press, 1995, pp. 3 1. 

Studying the dynamics of systems in the time domain involves direct solutions of differential equations. The computer simulation techniques of Part II are very general in the sense that they can give solutions to very complex nonlinear problems. However, they are also very specific in the sense that they provide a solution to only the particular numerical case fed into the computer. 

Lobster hemocyanin in glycine-sodium hydroxide buffer at pH 9.6 undergoes a reversible whole molecule-half molecule dissociation which is very sensitive to the level of free calcium ion present (12). This dissociation process is also very sensitive to pH (12). Cann and colleagues have developed, by simulation techniques (13,14), a general picture of the kinds of partial or complete boundary resolution that may be expected for such coupled systems in various types of transport experiments. Kinetic investigations of the lobster hemocyanin system under the conditions of the present study were developed by using stopped-flow... 

Simulations may be grouped according to the general uses for which they were developed. As such, the first to be developed, and the simplest simulation techniques, are those investigating the structure of aggregates. These models first appeared in 1963 [18], and after Mandelbrot s seminal work on fractal geometry [45], development of these types of simulations increased dramatically, spurred by the well-known Witten-Sander model [46]. 

Some simulation techniques make use of points along x, and indeed sometimes along f, that are spaced unevenly, either in some smooth transformational progression or more or less arbitrarily. A general treatment is given in this section, as well as a few particular algebraic solutions. 

There is an inherently stretched grid implementation in the simulation technique called orthogonal collocation, to be discussed in Chap. 9. It will be seen that this can be extremely efficient but it suffers, as all fixed stretched grids do, from inflexibility, as is noted in general in Sect. 7.3. 

In this section we discuss first in general the techniques used to simulate the chemical reactions with relatively high activation barriers. After that we provide a more detailed discussion of the constrained dynamics approach. Special emphasis will be put on MD along the intrinsic reaction coordinate (IRC), illustrated by a few simple examples.33... 

A recent QMCF MD and LAXS (large angle X-ray scattering) study [76] of the sulfate ion has once more demonstrated the reliability of this simulation technique for the description of composite solutes and an accuracy equivalent to best experimental methods. Precise predictions of vibrational spectra as well as the solvation energy of this anion [77] have clearly indicated the ability of the QMCF MD approach to investigate a variety of properties in a general and comprehensive way. 

Four novel approaches to contemporary studies of suspensions are briefly reviewed in this final section. Addressed first is Stokesian dynamics, a newly developed simulation technique. Surveyed next is a recent application of generalized Taylor dispersion theory (Brenner, 1980a, 1982) to the study of momentum transport in suspensions. Third, a synopsis is provided of recent studies in the general area of fractal suspensions. Finally, some novel properties (e.g., the existence of antisymmetric stresses) of dipolar suspensions are reviewed in relation to their applications to magnetic and electrorheolog-ical fluid properties. 

The liquid phase of molecular matter is usually isotropic at equilibrium but becomes birefringent in response to an externally applied torque. The computer can be used to simulate (1) the development of this birefringence —the rise transient (2) the properties of the liquid at equilibrium under the influence of an arbitrarily strong torque and (3) the return to equilibrium when the torques are removed instantaneously—the fall transient. Evans initially considered the general case of the asymmetric top (C2 symmetry) diffusing in three-dimensional space and made no assumptions about the nature of the rotational and translational motion other than those inherent in the simulation technique itself. A sample of 108 such molecules was taken, each molecule s orientation described by three unit vectors, e, Cg, and parallel to its principal moment-of-inertia axes. 

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