Friday, May 16, 2008

[MT 7] Stochastic Assembly Processes

In our biophysical model of spontaneous bead diffusion, the persistent motion of the bead is eventually traced back to selfassembly processes in the stress fibers. It is therefore important to consider the statistics of such processes in a simple model.

Let's assume that the goal of a biochemical factory is to produce Functional Units (FU) of some kind. The fabrication of each new FU requires a fixed number of m Assembly Steps (AS) to be performed (successively). The ASs are accomplished at an average rate R, with exponentially distributed inter-event-times (Poisson process). We are interested in the probability P(n | m, k_av = R dt) that in a given time intervall dt, n complete FUs are fabricated.

Note that the case m=1 is described by the Poisson distribution. The cases m>1 can be viewed as a generalization of the Poisson distribution. The intervalls between successive FUs are completed are described by the Gamma distribution.

It is straight forward to compute P(n | m, k_av = R dt) in a Monte-Carlo simulation, where exponentially distributed random durations tau_i are added, until sum(i=1..n) tau_i >= dt. Always when this happens, the resulting random integer n is recorded in a histogram and the counting process restarts. It is also possible to treat n as a continuous random variable by adding to the integer n the fractional part of the next FU that is already fabricated at the end of intervall dt.

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