Multistationarity in mass action networks with applications to ERK activation

Abstract

Ordinary Differential Equations (ODEs) are an important tool in many areas of Quantitative Biology. For many ODE systems multistationarity (i.e. the existence of at least two positive steady states) is a desired feature. In general establishing multistationarity is a difficult task as realistic biological models are large in terms of states and (unknown) parameters and in most cases poorly parameterized (because of noisy measurement data of few components, a very small number of data points and only a limited number of repetitions). For mass action networks establishing multistationarity hence is equivalent to establishing the existence of at least two positive solutions of a large polynomial system with unknown coefficients. For mass action networks with certain structural properties, expressed in terms of the stoichiometric matrix and the reaction rate-exponent matrix, we present necessary and sufficient conditions for multistationarity that take the form of linear inequality systems. Solutions of these inequality systems define pairs of steady states and parameter values. We also present a sufficient condition to identify networks where the aforementioned conditions hold. To show the applicability of our results we analyse an ODE system that is defined by the mass action network describing the extracellular signal-regulated kinase (ERK) cascade (i.e. ERK-activation).

Catalytic constants enable the emergence of bistability in dual phosphorylation

“…Multistationarity refers to the existence of at least two positive steady states within the same set v c [13] and bistability to the existence of two 'stable' steady states in v c . In the light of the previous discussion, the existence of multistationarity (and hence bistability) requires the existence of a positive vector c [ R…”

MAPK’s networks and their capacity for multistationarity due to toric steady states

“…However, there is no agreement about these values and differences of more than two orders of magnitude have appeared [28]. In this sense, in our case, any nontrivial solution of the linear inequality system defined by (2) gives two different steady sates and a set of rate constants for which the system has those steady states, and both the steady states and the constants are determined explicitly. Our results highlight that the robustness of the topology also tolerates changes in protein concentrations and rate constants, allowing a similar overall behavior of the network.…”

“…α ∈ Rowspan(A) and σ ∈ Columnspan(N) with sign(α) = sign(σ ), (2) then two steady states x 1 and x 2 and a reaction rate constant vector k that witness multistationarity arise in the following way:…”

“…In [2], the authors present necessary and sufficient conditions for multistationarity for mass-action networks with certain structural properties and they also apply their results on some simple ERK cascade networks. They translate the question of multistationarity into a system of linear inequalities although they do not consider the possibility of these networks having toric steady states, while we do take this characteristic into account thus simplifying the way to prove the existence of more than one steady state.…”

ATP production optimization in biochemical mass action models with protein turnover

“…Even with such strict constraints on reaction rates (and without any quasi steady-state assumption), mass action systems have been shown to be still capable of producing complex dynamical phenomena like multistability. Thus, several conditions and methods have been formulated corresponding to the analysis of this property [Conradi andFlockerzi, 2012, Craciun andFeinberg, 2006]. As emphasized in [Sorribas et al, 2010], mass action and generalized mass action models do not necessary use steady state assumptions, and so are able to describe both transient and steady-state responses of metabolites and fluxes to changes in the environment of the model, and provide a more accurate description of processes than techniques based merely on the stochiometric matrix, like Flux Balance Analysis [Orth et al, 2010].…”