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Jan Poleszczuk, "Validity of delayed differential equations in biochemical reactions systems." Short abstract: 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. Unfortunately, with this method only evolution of average densities of molecules from each species can be obtained, so it does not include random fluctuations that occurs during experiments. In~\\cite{Gill} the method of generating stochastic simulations of such systems was developed. There is high correspondence between quantitative results obtained by this two methods.\\\\ Recently, to reduce complexity of some systems there was introduced a delay in some reactions. Therefore, ordinary differential equations has been reformulated as delay differential equations. To incorporate delay in the stochastic algorithm the modification of algorithm developed in~\\cite{Gill} was introduced in~\\cite{Cai}.\\\\ At first, we considered a simple reaction of delayed degradation. Suppose we have only one species $X$ migrating to the container. Molecule of that species can only degrade in non-delayed or delayed way. In~\\cite{Pnas} the following delayed differential equation to describe that process was proposed: \\begin{equation}\\label{r1} \\dot{X}(t)=A-BX(t)-CX(t-\\tau). \\end{equation} Eq.~(\\ref{r1}) is a natural modification of the ordinary differential equation for the same non-delayed process, that is Eq.~(\\ref{r1}) with $\\tau=0$. However, stochastic simulation gives the same quantitative and qualitative results only when we assume that the same molecule can degrade several times. If we change that assumption, we should rewrite the equation and we get diametrically opposite solutions. We investigated that in almost each case delay differential equations cannot be such natural modifications of ODEs as Eq.~(\\ref{r1}). We developed a method of writing such equations correctly. |