A central issue in the analysis of multi-stable systems is that of controlling the relative size of the basins of attraction of alternative states through suitable choices of system parameters. We are interested here mainly in the stochastic version of this problem, that of shaping the stationary probability distribution of a Markov chain so that various alternative modes become more likely than others. Although many of our results are more general, we were motivated by an important biological question, that of cell differentiation. In the mathematical modeling of cell differentiation, it is common to think of internal states of cells (quanfitied by activation levels of certain genes) as determining the different cell types. Specifically, we study here the "PU.l/GATA-l circuit" which is involved in the control of the development of mature blood cells from hematopoietic stem cells (HSCs). All mature, specialized blood cells have been shown to be derived from multipotent HSCs. Our first contribution is to introduce a rigorous chemical reaction network model of the PU.l/GATA-l circuit, which incorporates current biological knowledge. We then find that the resulting ODE model of these biomolecular reactions is incapable of exhibiting multistability, contradicting the fact that differentiation networks have, by definition, alternative stable steady states. When considering instead the stochastic version of this chemical network, we analytically construct the stationary distribution, and are able to show that this distribution is indeed capable of admitting a multiplicity of modes. Finally, we study how a judicious choice of system parameters serves to bias the probabilities towards different stationary states. We remark that certain changes in system parameters can be physically implemented by a biological feedback mechanism; tuning this feedback gives extra degrees of freedom that allow one to assign higher likelihood to some cell types over others.
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