Conformational changes averages computation

The classical microscopic description of molecular processes leads to a mathematical model in terms of Hamiltonian differential equations. In principle, the discretization of such systems permits a simulation of the dynamics. However, as will be worked out below in Section 2, both forward and backward numerical analysis restrict such simulations to only short time spans and to comparatively small discretization steps. Fortunately, most questions of chemical relevance just require the computation of averages of physical observables, of stable conformations or of conformational changes. The computation of averages is usually performed on a statistical physics basis. In the subsequent Section 3 we advocate a new computational approach on the basis of the mathematical theory of dynamical systems we directly solve a... 

An improvement is to use a windowed RMSD function as a measure of the rate of conformation change. Specifically, for a given window length (e.g., 10 consecutive trajectory snapshots), the average of the all of the pairwise RMSDs (or alternatively, the average deviation from the average over that interval) is computed as a function of time. This yields a measure of conformational diversity over time, and can more readily reveal conformational transitions. 

The information about the existence of the multiple intermediate conformational states involving the enzymatic active complex formation and a detailed characterization of the energy landscape (Fig. 24.7) of the complex formation process cannot be obtained either by only an ensemble-averaged experiment, only a single-molecule experiment, or a solely computational approach. The combined approach demonstrated here is essential to achieve the potential of both single-molecule spectroscopy and MD simulations for studies of slow enzymatic reactions and protein conformational change dynamics. 

A very important theorem due to Widom [42] allows expression of the excess chemical potential of a component in a fluid (i.e., the chemical potential of the component minus the chemical potential it would have if it were an ideal gas under the temperature and molar density with which it is present in the fluid) as an ensemble average, computable through simulation. The Widom theorem considers the virtual addition of a test particle (test molecule) of the component of interest in the fluid at a random position, orientation, and conformation. If is the potential energy change that would result from this addition. 

One of the most important phenomena in the polymer solvation is the change in the overall size of the polymer chain upon solvation. In fact at equilibrium the average size of isolated polymer molecules in solution is a function of solvent quality and varies from expanded conformations in good solvents to random walk conformations in poor solvents. This is referred to as collapse transition and was first predicted by Stockmayer [82] more than 45 years ago. The phenomenon was observed by Nishio et al. [83] and Swislow et al. [84] more than 25 years ago and is still a subject of much experimental, computational, and theoretical research today. So far many investigators have tried to study the chain size with solvation using a variety of methods. 

When considering an isolated polymer molecule it is not possible to assign a unique value of r because the chain conformation (and hence r) is continuously changing due to rotation of backbone bonds. Since the single polymer chain can take any of an infinite number of conformations, an average magnitude of r over all possible conformations is computed from the mean of the squares of end-to-end distances and is called the root mean square (RMS) end-to-end distance, represented by where ()... 

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