Gene regulation is a complex process involving many biochemical reactions with proteins being final products. It is usually assumed that reactions are instantaneous. However, they are rather slow. In addition, biochemical processes usually involve several sequential steps. It is a common procedure to replace these steps by time delays.

We will discuss simple stochastic models of gene expression and regulation with time-delayed reactions. A new methodology to deal with time delays in biological systems will be presented. We will use it to show that delayed degradation does not produce oscillations in cells.

We will also discuss a stochastic model of a self-repressed gene. A self-consistent mean field approximation can be used to treat it analytically. We will discuss an effect of delayed protein production in such a model. In particular, an analytical expressions for the variance of the number of proteins in the stationary state can be derived in the limit of small time delays.

Bibliography

J. Miękisz, J. Poleszczuk, M. Bodnar, and U. Foryś, Stochastic model of gene expression with delayed degradation, Bull. Math. Biol. 73: 2231-2247 (2011).

J. Miękisz, Stochasticity and time delays in gene expression and evolutionary game theory, Probabilistic Engineering Mechanics 26: 33–38 (2011).

J. Miękisz and P. Szymańska, Gene expression in self-repressing system with multiple gene copies, Bull. Math. Biol. 75: 317–330 (2013).