Team:Grenoble/Modeling/Amplification/Stochastic

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Goal


Statistic modeling is a technique of presenting data or predicting outcomes that takes into account a certain degree of randomness or unpredictability. The stochastic process is often used to represent the evolution of some random value, or system, over time.

It is the probabilistic counterpart to a deterministic process.

Why


Gene expression is a stochastic process due to the inherent unpredictability of molecular collisions resulting from Brownian motion : the binding or unbinding of RNA polymerase to a promotor is partially random. In biology systems, introducing stochastic noise has been found to help improve the signal strength of the internal feedback loops for balance and other vestibular communication.

How


Rather than using fixed variables such as in other mathematical modeling, a stochastic model incorporates random variations to predict future conditions and to see what they might be like.
To introduce that randomness we use a new function : propensities.


For example we take four possible reactions. Each reaction has a probability to happen in the next amount of time.


We randomly chose the next reaction regarding the propensities.
The possibility of one random variation implies that many could occur. For this reason, stochastic models are not run just once, but hundreds or even thousands of times.


Instead of describing a process which can only evolve in one way, in a stochastic or random process there is some indeterminacy : even if the initial condition is known, there are several directions in which the process may evolve.

To model that randomness we use a Gillepsie algorithm or Stoachastic Simulation Algorithm (SSA).

Gillespie _ Stochastic Simulation Algorithm


The Gillespie algorithm generates a statistically correct trajectory of a stochastic equation.

Traditional continuous and deterministic biochemical rate equations do not accurately predict cellular reactions since they rely on bulk reactions that require the interactions of millions of molecules. In contrast, the Gillespie algorithm allows a discrete and stochastic simulation of a system with few reactants because every direction is explicitly simulated. When simulated, a Gillespie realization represents a random walk that exactly represents the distribution of the master equation.

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