Parameter regions that give rise to 2[n/2] +1 positive steady states in the n-site phosphorylation system.

The distributive sequential n-site phosphorylation/dephosphorylation system is an important building block in networks of chemical reactions arising in molecular biology, which has been intensively studied. In the nice paper of Wang and Sontag (2008) it is shown that for certain choices of the reaction rate constants and total conservation constants, the system can have 2[n/2] +1 positive steady states (that is, n+1 positive steady states for n even and n positive steady states for n odd). In this paper we give open parameter regions in the space of reaction rate constants and total conservation constants that ensure these number of positive steady states, while assuming in the modeling that roughly only 1/4 of the intermediates occur in the reaction mechanism.

Multistationarity in Structured Reaction Networks.

Many dynamical systems arising in biology and other areas exhibit multistationarity (two or more positive steady states with the same conserved quantities). Although deciding multistationarity for a polynomial dynamical system is an effective question in real algebraic geometry, it is in general difficult to determine whether a given network can give rise to a multistationary system, and if so, to identify witnesses to multistationarity, that is, specific parameter values for which the system exhibits multiple steady states. Here we investigate both problems.

How far is complex balancing from detailed balancing?

We clarify the relation between the algebraic conditions that must be satisfied by the reaction constants in general (mass-action) kinetics systems for the existence of detailed or complex balancing equilibria. These systems have a wide range of applications in chemistry and biology. Their main properties have been set by Horn, Jackson and Feinberg.