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Stochastic modelling approaches
Two main approaches have been developed to modelling stochastic events in gene expression, stochastic differential equations and the stochastic simulation algorithm. Stochastic differential equations extend the standard differential equation description of the reaction dynamics to include a noise term
where 
(t) is an additive noise term. This equation, known as the Langevin equation, can be developed into an alternative formulation that describes the evolution of the probability density function. These equations are generally too complex to be solved using analytic or numerical techniques, therefore a Monte-Carlo approach is generally used.
A characteristic of stochastic differential equation approaches is that they treat molecular concentrations as continuous variables. As mentioned above, in many situations signal molecules may exist in very small numbers, therefore it may be more appropriate to model them as discrete entities. An alternative approach formulates an equation in terms of the probability that a molecule undergoes a transition in a particular small time slice. This approach, known as the master equation, produces equations that are mathematically simple, but for realistic systems, are too numerous and too large to be feasibly solved. Again, the approach typically taken is to simulate the system a number of times and estimate a probability density function. A number of approaches to the stochastic simulation of such equations have been developed [47,48,89].
A major problem with both of these approaches is efficiency. Running multiple simulations of systems involving large numbers of reactions is computationally expensive. An important area of further research is the development of multiscale approaches. These models would be able to use continuous representations where individual events are not important, but still allow for the possible occurrence of rare, but significant, events [25].
In addition, a number of other methods for modelling stochasticity have been developed including stochastic Petri Nets [51] and stochastic neural networks [129]. Some of the logical modelling approaches described in Section 4, such as the asynchronous Boolean model and the generalised logic formalism, also include an degree of noises arising from the non-deterministic timing of regulatory events.
Next: Further reading Up: Stochastic models Previous: Noise from within and Nic Geard 2004-05-06