N-site phosphorylation systems with 2N-1 steady states

Multisite protein phosphorylation plays a prominent role in intracellular processes like signal transduction, cell-cycle control and nuclear signal integration. Many proteins are phosphorylated in a sequential and distributive way at more than one phosphorylation site. Mathematical models of $n$-site sequential distributive phosphorylation are therefore studied frequently. In particular, in {\em Wang and Sontag, 2008,} it is shown that models of $n$-site sequential distributive phosphorylation admit at most $2n-1$ steady states. Wang and Sontag furthermore conjecture that for odd $n$, there are at most $n$ and that, for even $n$, there are at most $n+1$ steady states. This, however, is not true: building on earlier work in {\em Holstein et.al., 2013}, we present a scalar determining equation for multistationarity which will lead to parameter values where a $3$-site system has $5$ steady states and parameter values where a $4$-site system has $7$ steady states. Our results therefore are counterexamples to the conjecture of Wang and Sontag. We furthermore study the inherent geometric properties of multistationarity in $n$-site sequential distributive phosphorylation: the complete vector of steady state ratios is determined by the steady state ratios of free enzymes and unphosphorylated protein and there exists a linear relationship between steady state ratios of phosphorylated protein

Switching in mass action networks based on linear inequalities

Many biochemical processes can successfully be described by dynamical systems allowing some form of switching when, depending on their initial conditions, solutions of the dynamical system end up in different regions of state space (associated with different biochemical functions). Switching is often realized by a bistable system (i.e. a dynamical system allowing two stable steady state solutions) and, in the majority of cases, bistability is established numerically. In our point of view this approach is too restrictive, as, one the one hand, due to predominant parameter uncertainty numerical methods are generally difficult to apply to realistic models originating in Systems Biology. And on the other hand switching already arises with the occurrence of a saddle type steady state (characterized by a Jacobian where exactly one Eigenvalue is positive and the remaining eigenvalues have negative real part). Consequently we derive conditions based on linear inequalities that allow the analytic computation of states and parameters where the Jacobian derived from a mass action network has a defective zero eigenvalue so that -- under certain genericity conditions -- a saddle-node bifurcation occurs. Our conditions are applicable to general mass action networks involving at least one conservation relation, however, they are only sufficient (as infeasibility of linear inequalities does not exclude defective zero eigenvalues).Comment: in revision SIAM Journal on Applied Dynamical System

Multistationarity in sequential distributed multisite phosphorylation networks

Multisite phosphorylation networks are encountered in many intracellular processes like signal transduction, cell-cycle control or nuclear signal integration. In this contribution networks describing the phosphorylation and dephosphorylation of a protein at $n$ sites in a sequential distributive mechanism are considered. Multistationarity (i.e.\ the existence of at least two positive steady state solutions of the associated polynomial dynamical system) has been analyzed and established in several contributions. It is, for example, known that there exist values for he rate constants where multistationarity occurs. However, nothing else is known about these rate constants. Here we present a sign condition that is necessary and sufficient for multistationarity in $n$-site sequential, distributive phosphorylation. We express this sign condition in terms of linear systems and show that solutions of these systems define rate constants where multistationarity is possible. We then present, for $n\geq 2$, a collection of {\em feasible} linear systems and hence give a new and independent proof that multistationarity is possible for $n\geq 2$. Moreover, our results allow to explicitly obtain values for the rate constants where multistationarity is possible. Hence we believe that, for the first time, a systematic exploration of the region in parameter space where multistationarity occurs has become possible.One consequence of our work is that, for any pair of steady states, the ratio of the steady state concentrations of kinase-substrate complexes equals that of phosphatase-substrate complexes

The roles of the subunits in the function of the calcium channel

Dihydropyridine-sensitive voltage-dependent L-type calcium channels are critical to excitation-secretion and excitation-contraction coupling. The channel molecule is a complex of the main, pore-forming subunit alpha 1 and four additional subunits: alpha 2, delta, beta, and gamma (alpha 2 and delta are encoded by a single messenger RNA). The alpha 1 subunit messenger RNA alone directs expression of functional calcium channels in Xenopus oocytes, and coexpression of the alpha 2/delta and beta subunits enhances the amplitude of the current. The alpha 2, delta, and gamma subunits also have pronounced effects on its macroscopic characteristics, such as kinetics, voltage dependence of activation and inactivation, and enhancement by a dihydropyridine agonist. In some cases, specific modulatory functions can be assigned to individual subunits, whereas in other cases the different subunits appear to act in concert to modulate the properties of the channel