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Reaction Coordinates for the Flipping of Genetic Switches

Marco J. Morelli ^*, Sorin Tnase-Nicola ^*, Rosalind J. Allen  and Pieter Rein ten Wolde ^*

^* FOM Institute for Atomic and Molecular Physics, Kruislaan, Amsterdam, The Netherlands; and Scottish Universities Physics Alliance, School of Physics, University of Edinburgh, James Clerk Maxwell Building, The King's Buildings, Edinburgh, United Kingdom

Correspondence: Address reprint requests to Marco J. Morelli, E-mail: morelli{at}amolf.nl.

We present a detailed analysis, based on the forward flux sampling simulation method, of the switching dynamics and stability of two models of genetic toggle switches, consisting of two mutually repressing genes encoding transcription factors (TFs); in one model (the exclusive switch), the two transcription factors mutually exclude each other's binding, while in the other model (general switch), the two TFs can bind simultaneously to the shared operator region. We assess the role of two pairs of reactions that influence the stability of these switches: TF-TF homodimerization and TF-DNA association/dissociation. In both cases, the switch flipping rate increases with the rate of TF dimerization, while it decreases with the rate of TF-operator binding. We factorize the flipping rate k into the product of the probability (q*) of finding the system at the dividing surface (separatrix) between the two stable states, and a kinetic prefactor R. In the case of the exclusive switch, the rate of TF-operator binding affects both (q*) and R, while the rate of TF dimerization affects only R. The general switch displays a higher flipping rate than the exclusive switch, and both TF-operator binding and TF dimerization affect k, R, and (q*). To elucidate this, we analyze the transition state ensemble. For the exclusive switch, the transition state ensemble is strongly affected by the rate of TF-operator binding, but unaffected by varying the rate of TF-TF binding. Thus, varying the rate of TF-operator binding can drastically change the pathway of switching, while changing the rate of dimerization changes the switching rate without altering the mechanism. The switching pathways of the general switch are highly robust to changes in the rate constants of both TF-operator and TF-TF binding, even though these rate constants do affect the flipping rate; this feature is unique for nonequilibrium systems.