| Abstract | Many biochemical processes in living cells take place in small volumes and involve small number of molecules. Such systems are usually modeled by birth and death processes where products of various reactions appear or degrade immediately after corresponding reactions are triggered. However, many such reactions take a considerable amount of time. Therefore to describe them we have to introduce models with time delays. Reactions with delays are of two kinds: non-consuming and consuming. Reactants of unfinished consuming reactions cannot participate in new reactions, reactants of non-consuming reactions can participate in new reactions. We will discuss simple models of gene expression with time delays. We will analyze both kinetic rate equations and corresponding birth and death processes with both types of time delays. Many kinetic rate equations with non-consuming reactions undergo the Hopf bifurcation when the delay increases and crosses a critical value. For small time delays the system evolves into its stationary state with damped oscillations observed in transient states. We will show that such effects are not present in the case of consuming reactions where for all values of time delay the unique stationary state is asymptotically stable. In the stochastic models corresponding to deterministic rate equations, the variance of the number of protein molecules and autocorrelation functions will be calculated analytically. To deal with more complex models, we will develop a small delay approximation. We compare our results with those obtained earlier by Bratsun, Volfson, Tsimring, and Hasty, for models with delayed degradation and repression, PNAS 102: 14593-14598 (2005). |
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