In many cases of biological phenomena the resulting cell phenotype is a direct result of the randomness of collisions between sparse interacting or reacting biological species. Consequently, the classical, well known and well studied continuous-deterministic modeling approach can be distinctive false and incapable of capturing the importance of intrinsic noise fluctuations. Instead a discrete-stochastic theoretical approach is more appropriate. The disadvantage is that the latter is less studied and more computationally demanding. In the classical approach the reaction networks used to describe cell functions are modeled using mass action kinetics resulting in systems of ordinary differential equations (ODEs). On the other hand, in the stochastic regime chemical kinetics models are described as jump (discrete) Markov processes governed by the chemical Master equation (CME). The question is when these two descriptions are equivalent, in which case the deterministic approach is preferable. In the simplest of cases, linear kinetics, it has been established that the mean of the probability distribution corresponds to the solution of the corresponding ODEs [1]. The two approaches become equivalent at the thermodynamic limit where the variance of the probability distribution vanishes. Conversely, for non-linear systems the correspondence of the mean of the probability distribution to the solution of the ODEs is just an approximation under the assumption that higher probability distribution moments (variance, third, fourth and so on moments) become negligible as the system's size increases, i.e. approaches the thermodynamic limit. Therefore the question reduces to when higher moments indeed become negligible. The challenge is that there are no analytical expressions for computing higher moments, except for the simplest of cases where the CME can be analytically solved. Thus a numerical method is a necessity for approximating their values. Our approach is based on the definition of jump moments or derivatives moments as originally called by Moyal [2]. We use a series of numerical experiments to compute (estimate) the values of jump moments for different biochemical networks and convey on the validity of the classical approach. Additionally, we try to relate our findings to the number of reacting species and probabilistic reaction rates and define thresholds above which higher moments of the probability distribution can be ignored for all practical purposes. Ultimately, we would like to explore the general applicability of such conditions, where one can determine for arbitrary networks a priori which regime, stochastic or deterministic, is appropriate.

[1]. Van Kampen NG: Stochastic Processes in Physics and Chemistry. Revised and enlarged edn. Amsterdam: Elsevier; 2004.

[2]. Moyal JE: Stochastic Processes and Statistical Physics. Journal of the Royal Statistical Society Series B (Methodological) 1949, 11:150-210.