Stochastic difference equation

K. Siva and R. Elber (2003) Ion permeation through the gramicidin chaimel Atomically detailed modeling by the stochastic difference equation. Proteins, Structure, Function and Genetics 50, pp. 63-80... 

A. Ghosh, R. Fiber, and H. Scheraga (2002) An atomically detailed study of the folding pathways of Protein A with the Stochastic Difference Equation. Proc. Natl. Acad. Sci. 99, pp. 10394-10398... 

A. Cardenas and R. Elber (2003) Atomically detailed simulations of helix formation with the stochastic difference equation. Biophysical Journal, 85, pp. 2919-2939... 

A. Stochastic Difference in Time Definition A Stochastic Model for a Trajectory Weights of Trajectories and Sampling Procedures Mean Field Approach, Fast Equilibration, and Molecular Labeling Stochastic Difference Equation in Length Fractal Refinement of Trajectories Parameterized by Length... 

Despite the similarity to the Gauss approach to classical mechanics, there is a key difference between the classical actions described above and the corresponding action of the stochastic difference equation. The classical actions are deterministic mechanical models the SDE is a nondeterministic approach that is based on stochastic modeling of the numerical errors introduced by the finite difference formula. 

If the errors are zero, we obtain the most probable trajectory within the framework of the stochastic difference equation. This trajectory is not exact and is within cr from the exact trajectory [Eq. (13)]. What are the approximations made In Section III.C we argue that the approximate trajectory is a solution of the slow modes in the system where the high-frequency modes are filtered out. 

The stochastic difference equation in length is conceptually similar to the stochastic difference in time. We therefore do not repeat all of the arguments and... 

We are using a similar approach to define the analogous action for the stochastic difference equation in length, Ssdel [Sr is defined in Eq. (10)] ... 

The present chapter is mostly methodological, presenting the conceptual framework behind the new technique of the stochastic difference equation. It is therefore appropriate to discuss numerical examples of small systems for which different aspects can be tested in greater details. On the other hand the numerical examples should be sufficiently complex so that nontrivial effects could be observed. So, despite the fact that the techniques were already applied to investigate much larger systems, we focus here on conformational transitions of smaller systems dipeptides. 

The AMBER/OPLS force field is implemented in MOIL [16] and is used throughout the calculations. No cutoffs were used for this small system, and the 1 4 scaling factor was 2 and 8 for electrostatic and van der Waals interactions. No constraints on fast vibration were used. However, the stochastic difference equation filters the bond vibrations anyway. In Fig. 7 we compare the energy content of the bond vibrations in Ssdel optimization with different step sizes. 

In this review a variant of the SDET algorithm is summarized. In this more recent formulation called SDEL (for stochastic difference equation in length) the trajectory is parameterized as a function of its arc length and a unique path is obtained connecting the two boundary conformations [45,54,55]. In the next section, we will describe the algorithm and details of its numerical implementation to obtain conformational changes of peptides and folding mechanisms of protein systems. 

Elber, R., Ghosh, A., Cardenas, A. Stochastic difference equation as a tool to compute long time dynamics, in Nielaba, R, Mareschal, M., Ciccotti, G., editors. Bridging the Time Scale Gap. Berlin Springer Verlag 2002. 

Arora, K., SchKck, T. Deoxyadenosine sugar puckering pathway simulated by the stochastic difference equation algorithm, Chem. Phys. Lett. 2003,378,1-8. 

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