Multistationarity in Structured Reaction Networks

Multistationarity questions in reduced versus extended biochemical networks

We address several questions in reduced versus extended networks via the elimination or addition of intermediate complexes in the framework of chemical reaction networks with mass-action kinetics. We clarify and extend advances in the literature concerning multistationarity in this context, mainly from Feliu and Wiuf (J R Soc Interface 10:20130484, 2013), Sadeghimanesh and Feliu (Bull Math Biol 81:2428-2462, 2019), Pérez Millán and Dickenstein (SIAM J Appl Dyn Syst 17(2):1650-1682, 2018), Dickenstein et al. (Bull Math Biol 81:1527-1581, 2019). We establish general results about MESSI systems, which we use to compute the circuits of multistationarity for significant biochemical networks.

Counting Real Roots in Polynomial-Time via Diophantine Approximation

Suppose A = { a 1 , … , a n + 2 } ⊂ Z n has cardinality n + 2 , with all the coordinates of the a j having absolute value at most d, and the a j do not all lie in the same affine hyperplane. Suppose F = ( f 1 , … , f n ) is an n × n polynomial system with generic integer coefficients at most H in absolute value, and A the union of the sets of exponent vectors of the f i . We give the first algorithm that, for any fixed n, counts exactly the number of real roots of F in time polynomial in log ( d H ) . We also discuss a number-theoretic hypothesis that would imply a further speed-up to time polynomial in n as well.

Dynamics of ERK regulation in the processive limit

We consider a model of extracellular signal-regulated kinase regulation by dual-site phosphorylation and dephosphorylation, which exhibits bistability and oscillations, but loses these properties in the limit in which the mechanisms underlying phosphorylation and dephosphorylation become processive. Our results suggest that anywhere along the way to becoming processive, the model remains bistable and oscillatory. More precisely, in simplified versions of the model, precursors to bistability and oscillations (specifically, multistationarity and Hopf bifurcations, respectively) exist at all "processivity levels". Finally, we investigate whether bistability and oscillations can exist together.

Inferring reaction network structure from single-cell, multiplex data, using toric systems theory

The goal of many single-cell studies on eukaryotic cells is to gain insight into the biochemical reactions that control cell fate and state. In this paper we introduce the concept of Effective Stoichiometric Spaces (ESS) to guide the reconstruction of biochemical networks from multiplexed, fixed time-point, single-cell data. In contrast to methods based solely on statistical models of data, the ESS method leverages the power of the geometric theory of toric varieties to begin unraveling the structure of chemical reaction networks (CRN). This application of toric theory enables a data-driven mapping of covariance relationships in single-cell measurements into stoichiometric information, one in which each cell subpopulation has its associated ESS interpreted in terms of CRN theory. In the development of ESS we reframe certain aspects of the theory of CRN to better match data analysis. As an application of our approach we process cytomery- and image-based single-cell datasets and identify differences in cells treated with kinase inhibitors. Our approach is directly applicable to data acquired using readily accessible experimental methods such as Fluorescence Activated Cell Sorting (FACS) and multiplex immunofluorescence.

We introduce a new notion, which we call the effective stoichiometric space (ESS), that elucidates network structure from the covariances of single-cell multiplex data. The ESS approach differs from methods that are based on purely statistical models of data: it allows a completely new and data-driven translation of the theory of toric varieties in geometry and specifically their role in chemical reaction networks (CRN). In the process, we reframe certain aspects of the theory of CRN. As illustrations of our approach, we find stoichiometry in different single-cell datasets, and pinpoint dose-dependence of network perturbations in drug-treated cells.

Oscillations and bistability in a model of ERK regulation

This work concerns the question of how two important dynamical properties, oscillations and bistability, emerge in an important biological signaling network. Specifically, we consider a model for dual-site phosphorylation and dephosphorylation of extracellular signal-regulated kinase (ERK). We prove that oscillations persist even as the model is greatly simplified (reactions are made irreversible and intermediates are removed). Bistability, however, is much less robust-this property is lost when intermediates are removed or even when all reactions are made irreversible. Moreover, bistability is characterized by the presence of two reversible, catalytic reactions: as other reactions are made irreversible, bistability persists as long as one or both of the specified reactions is preserved. Finally, we investigate the maximum number of steady states, aided by a network's "mixed volume" (a concept from convex geometry). Taken together, our results shed light on the question of how oscillations and bistability emerge from a limiting network of the ERK network-namely, the fully processive dual-site network-which is known to be globally stable and therefore lack both oscillations and bistability. Our proofs are enabled by a Hopf bifurcation criterion due to Yang, analyses of Newton polytopes arising from Hurwitz determinants, and recent characterizations of multistationarity for networks having a steady-state parametrization.

Multistationarity in the Space of Total Concentrations for Systems that Admit a Monomial Parametrization

We apply tools from real algebraic geometry to the problem of multistationarity of chemical reaction networks. A particular focus is on the case of reaction networks whose steady states admit a monomial parametrization. For such systems, we show that in the space of total concentrations multistationarity is scale invariant: If there is multistationarity for some value of the total concentrations, then there is multistationarity on the entire ray containing this value (possibly for different rate constants)-and vice versa. Moreover, for these networks it is possible to decide about multistationarity independent of the rate constants by formulating semi-algebraic conditions that involve only concentration variables. These conditions can easily be extended to include total concentrations. Hence, quantifier elimination may give new insights into multistationarity regions in the space of total concentrations. To demonstrate this, we show that for the distributive phosphorylation of a protein at two binding sites multistationarity is only possible if the total concentration of the substrate is larger than either the total concentration of the kinase or the total concentration of the phosphatase. This result is enabled by the chamber decomposition of the space of total concentrations from polyhedral geometry. Together with the corresponding sufficiency result of Bihan et al., this yields a characterization of multistationarity up to lower-dimensional regions.