Enzyme sequestration by the substrate: An analysis in the deterministic and stochastic domains

This paper is concerned with the potential multistability of protein concentrations in the cell. That is, situations where one, or a family of, proteins may sit at one of two or more different steady state concentrations in otherwise identical cells, and in spite of them being in the same environment. For models of multisite protein phosphorylation for example, in the presence of excess substrate, it has been shown that the achievable number of stable steady states can increase linearly with the number of phosphosites available. In this paper, we analyse the consequences of adding enzyme docking to these and similar models, with the resultant sequestration of phosphatase and kinase by the fully unphosphorylated and by the fully phosphorylated substrates respectively. In the large molecule numbers limit, where deterministic analysis is applicable, we prove that there are always values for these rates of sequestration which, when exceeded, limit the extent of multistability. For the models considered here, these numbers are much smaller than the affinity of the enzymes to the substrate when it is in a modifiable state. As substrate enzyme-sequestration is increased, we further prove that the number of steady states will inevitably be reduced to one. For smaller molecule numbers a stochastic analysis is more appropriate, where multistability in the large molecule numbers limit can manifest itself as multimodality of the probability distribution; the system spending periods of time in the vicinity of one mode before jumping to another. Here, we find that substrate enzyme sequestration can induce bimodality even in systems where only a single steady state can exist at large numbers. To facilitate this analysis, we develop a weakly chained diagonally dominant M-matrix formulation of the Chemical Master Equation, allowing greater insights in the way particular mechanisms, like enzyme sequestration, can shape probability distributions and therefore exhibit different behaviour across different regimes.Author summary: Models of multisite protein phosphorylation have been of great interest to the systems biology community, largely due to their ability to exhibit multistable behaviour. In the presence of excess substrate it has been shown that the number of stable steady states achieved can increase linearly with the number of phosphosites available. In this paper, we provide a quantitative mathematical analysis of the effect that enzyme docking, and the consequent phosphatase and kinase sequestration by the unphosphorylated and the fully phosphorylated substrates respectively, has on a multisite protein phosphorylation system. The analysis is done in both the deterministic and the stochastic domains, for large and small molecule numbers respectively. We prove, by finding sufficient conditions, that in the deterministic domain substrate enzyme-sequestration must inevitably limit the extent of multistability, ultimately to one steady state, even for systems with arbitrary processivity or sequentiality (i.e. where multiple phosphorylations or dephosphorylations can happen per reaction and in any order). In contrast, in the stochastic domain it can provide bimodality even in cases where bistability is not possible for large molecule numbers.