Biochemical systems including gene regulatory networks are often too large and complex to be analyzed by standard mathematical methods. Therefore, these systems are usually reduced to a sufficiently small size to enable the application of mathematical tools such as the stability and bifurcation analysis. The standard way for reducing the mass action systems is base on the quasi-steady state assumptions. This approach often enables to reduce the system considerably while preserving the qualitative properties of the original system. We enhance this technique by delays which leads to a reduced system whose solution agrees with the original solution even quantitatively. In general, the delays are state dependent, but we provide an explicit formulas for them. This enables to determine what the delays depend on and to quantify their magnitude. Moreover, there is a possibility to reverse the derivation of the delays and derive a mass action system from a given delay system.

The resulting delayed system can be also interpreted stochastically and simulated by the Gillespie algorithm with delays. We will present examples such as the model of circadian rhythms and a model of transcription to show the possible degree of reduction and the accuracy in both the deterministic and stochastic cases.

model reduction; mass action; stochastic simulations; state dependent delays