title: Equilibrium distributions of simple biochemical reaction systems for time-scale separation in stochastic reaction networks.
abstract: 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 (stationary) distribution on a time scale where the slowly varying molecular counts are unlikely to have changed. We derive analytic equilibrium distributions for various simple biochemical systems, such as enzymatic reactions and gene regulation models. These can be directly inserted into simulations of the slow time-scale dynamics. They also provide insight into the stimulus-response of these systems. An important model for which we derive the analytic equilibrium distribution is the binding of dimer transcription factors (TFs) that first have to form from monomers. This gene regulation mechanism is compared to the cases of the binding of simple monomer TFs to one gene or to multiple copies of a gene, and to the cases of the cooperative binding of two or multiple TFs to a gene. The results apply equally to ligands binding to enzyme molecules.
authors.affiliation: Department of Mathematical Stochastics, University of Freiburg, Eckerstraße 1, 79104 Freiburg, Germany Centre for Biological Systems Analysis (ZBSA), University of Freiburg, Habsburgerstraße 49, 79104 Freiburg, Germany melykuti@stochastik.uni-freiburg.de.
authors.forename: Bence
authors.initials: B
authors.lastname: Mélykúti
authors.affiliation: Electrical and Computer Engineering, University of California, Harold Frank Hall, Santa Barbara, CA 93106-9560, USA.
authors.forename: João P
authors.initials: JP
authors.lastname: Hespanha
authors.affiliation: Department of Biosystems Science and Engineering, ETH Zürich, Mattenstrasse 26, 4058 Basel, Switzerland.
authors.forename: Mustafa
authors.initials: M
authors.lastname: Khammash
mesh_terms: D000818:Animals; D003198:Computer Simulation; D004735:Energy Transfer / Q000235:genetics; D005786:Gene Expression Regulation / Q000235:genetics*; D006801:Humans; D008956:Models, Chemical; D008957:Models, Genetic; D015233:Models, Statistical; D015398:Signal Transduction / Q000235:genetics; D013269:Stochastic Processes; D014157:Transcription Factors / Q000737:chemistry* / Q000235:genetics*; D014158:Transcription, Genetic / Q000235:genetics*; D015533:Transcriptional Activation / Q000235:genetics*
chemical_list: D014157:Transcription Factors
keywords: analytic combinatorics; dimer transcription factor; gene regulation; quasi-steady-state assumption; stochastic reaction kinetics; time-scale separation