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Transient Dynamics of Genetic Regulatory Networks

Matthew R. Bennett ^* , Dmitri Volfson ^* , Lev Tsimring ^* and Jeff Hasty ^* 

^* Institute for Nonlinear Science and Department of Bioengineering University of California at San Diego, La Jolla, California

Correspondence: Address reprint requests to J. Hasty, E-mail: hasty{at}bioeng.ucsd.edu.

We present an approximation scheme for deriving reaction rate equations of genetic regulatory networks. This scheme predicts the timescales of transient dynamics of such networks more accurately than does standard quasi-steady state analysis by introducing prefactors to the ODEs that govern the dynamics of the protein concentrations. These prefactors render the ODE systems slower than their quasi-steady state approximation counterparts. We introduce the method by examining a positive feedback gene regulatory network, and show how the transient dynamics of this network are more accurately modeled when the prefactor is included. Next, we examine the repressilator, a genetic oscillator, and show that the period, amplitude, and bifurcation diagram defining the onset of the oscillations are better estimated by the prefactor method. Finally, we examine the consequences of the method to the dynamics of reduced models of the phage lambda switch, and show that the switching times between the two states is slowed by the presence of the prefactor that arises from protein multimerization and DNA binding.