Validity of delayed differential equations in biochemical reactions systems

It is well known that the time evolution of spatially homogeneous mixture composition consisting of molecules from N different species that can react through M chemical channels can be deterministically described by some set of ordinary differential equations. The method of generating stochastic simulations of such systems was developed by D.T. Gillespie. There is a high correspondence between quantitative results obtained by these two methods. Recently, to incorporate the fact that some reactions take some time or to reduce complexity of systems there were introduced delays in reactions. Therefore, ordinary differential equations have been reformulated as delay differential equations. To incorporate delay in the stochastic algorithm, some modifications of the Gillespie algorithm were introduced. We investigate validity of deterministic descriptions of two delayed reactions presented in some papers with the usage of stochastic algorithm. Comparison of deterministic and stochastic result revealed that in both delayed reactions the natural formulation of DDEs for those reactions brings assumptions which are not consistent with reactionsdescriptions. Therefore, we propose also another deterministic descriptions for those reactions from which the general idea of formulating deterministic descriptions for delayed reactions can be seen.