Bence Mélykúti

Equilibrium distributions of simple biochemical reaction
systems for time-scale separation in stochastic reaction
networks


I will study the equilibrium distribution of
continuous-time Markov processes on simple, discrete state
spaces. The motivation is that these Markov processes model
important biochemical motifs, such as enzymatic reactions
and ones that play roles in the control of gene expression.

Many biochemical reaction networks are inherently
multiscale in time and in the counts of participating
molecular species. A standard technique to treat different
time scales in the stochastic kinetics framework is
averaging or quasi-steady-state analysis: it is assumed
that the fast dynamics reaches its equilibrium distribution
on the time scale where the slowly varying molecular counts
are unlikely to have changed. I derive analytic equilibrium
distributions for various simple biochemical systems. These
can be directly inserted into simulations of the slow
time-scale dynamics. They also provide insight into the
stimulus-response of these systems analogously to Hill
functions. Among the models there is the cooperative
binding of two ligands to a macromolecule, or equivalently,
the cooperative binding of two transcription factors to a
gene. (The resulting formula is an easy consequence of a
little-known technical report.) This gene regulation
mechanism is compared to the cases of the binding of single
transcription factors to one gene (very easy) or to
multiple copies of a gene, and to the case of the binding
of single dimer transcription factors that first have to
form from monomers (difficult calculation).

The calculation for dimer transcription factors rests on
product-form stationary distributions and uses complex
analysis, the saddle-point method.

[1] Bence Mélykúti, João P. Hespanha and Mustafa Khammash.
Equilibrium distributions of simple biochemical reaction
systems for time-scale separation in stochastic reaction
networks. J. R. Soc. Interface, 2014, 11, 20140054.