Stochastic and deterministic modelling.

Stochastic and deterministic modelling.

1. Purposes of stochastic modelling

Deterministic modelling assumes the systems to be continuous and evolve deterministically. The behaviour of the system can be described using ODEs, which are then solved. However, such models ignore the phenomena that occur due to the fact that each system consists of a finite number of discrete particles, such as random fluctuations. For systems with very small particle numbers the deterministic models are not even appropriate because the concentrations are not continuous.

Stochastic modelling takes into account the fact that each system is composed of a finite and countable number of particles and considers the number of those particles similar to the way the deterministic system considers concentrations.

2. Drawbacks of stochastic modelling

2.1. Limits on particle numbers

Considering the fact that the number of particles in the system is very large, computational modelling of a stochastic method is very demanding and developing an algorithm is a complex task.

2.2. Lack of analysis methods

Stochastic modelling does not have such rigorously developed analysis methods as metabolic control analysis for deterministic modelling.

3. Drawbacks of deterministic modelling

3.1. Systems with small particle numbers

Stochastic methods consider random fluctuations which lead to significant change in system behaviour when the number of particles is small. Species are allowed to become extinct. In deterministic models the fluctuations are not accounted for and species concentrations never fall to zero. Therefore, in linear processes, the deterministic model behaviour will only be determined by difference in concentrations. The stochastic models can behave differently. This remains true even if stochastic systems have the same marginal distribution of system states.

3.2. Bi-Stable systems

Under deterministic simulation the system which is bi-stable will converge to the same stable steady state if the initial concentrations remain the same. Under stochastic simulation the system will converge to one of the two stable states, and it can not be predicted to which one. The probability of the system converging to each state, however, can be calculated.

4. Difference between the deterministic solution and the mean of stochastic solutions

It should be noted that if we repeat the stochastic simulation many times and calculate the mean, we will not end up with the same solution as the deterministic. This is only true for linear systems, but the solutions for nonlinear systems can be totally different.

5. Conclusion

Stochastic modelling should definitely be chosen when the particle numbers are in range where the concept of continuous concentration is no longer applicable or when the stochastic phenomena are themselves the object of research. The limit on the application of stochastic model is generally enforced at a certain particle number where computation becomes not feasible.

References

Pahle J, Biochemical simulations: stochastic, approximate stochastic and hybrid approaches, Briefings in Bioinformatics 2009, 10(1), pp 53-64

by Evgeny. Also posted on my website