One important question that interests those who work in chemical reaction network theory (CRNT) is this: Does the system obtained from a reaction network admit a positive equilibrium and if it does, can there be more than one within a stoichiometric class? The higher deficiency algorithm (HDA) of Ji and Feinberg provided a method of determining the multistationarity capacity of a CRN with mass action kinetics (MAK). An extension of this, called Multistationarity Algorithm (MSA), recently came into the scene tackling CRNs with power law kinetics (PLK), a kinetic system which is more general (having MAK systems as a special case). For this paper, we provide a computational approach to study the multistationarity feature of reaction networks endowed with kinetics which are non-negative linear combinations of power law functions called poly-PL kinetics (PYK). The idea is to use MSA and combine it with a transformation called STAR-MSC (i.e., $S$-invariant Termwise Addition of Reactions via Maximal Stoichiometric Coefficients) producing PLKs that are dynamically equivalent to PYKs. This leads us to being able to determinine the multistationarity capacity of a much larger class of kinetic systems. We show that if the transformed dynamically equivalent PLK system is multistationary for a stoichiometric class for a set of particular rate constants, then so is its original corresponding PYK system. Moreover, the monostationarity property of the transformed PLK system also implies the monostationarity property of the original PYK system.
This paper presents a computational solution to determine if a chemical reaction network endowed with power-law kinetics (PLK system) has the capacity for multistationarity, i.e., whether there exist positive rate constants such that the corresponding differential equations admit multiple positive steady states within a stoichiometric class. The approach, which is called the "Multistationarity Algorithm for PLK systems" (MSA), combines (i) the extension of the "higher deficiency algorithm" of Ji and Feinberg for mass action to PLK systems with reactant-determined interactions, and (ii) a metho…
The fundamental decomposition of a chemical reaction network (also called its "$\mathscr{F}$-decomposition") is the set of subnetworks generated by the partition of its set of reactions into the "fundamental classes" introduced by Ji and Feinberg in 2011 as the basis of their "higher deficiency algorithm" for mass action systems. The first part of this paper studies the properties of the $\mathscr{F}$-decomposition, in particular, its independence (i.e., the network's stoichiometric subspace is the direct sum of the subnetworks' stoichiometric subspaces) and its incidence-independence (i.e., t…
We present determinant criteria for the preclusion of non-degenerate multiple steady states in networks of interacting species. A network is modeled as a system of ordinary differential equations in which the form of the species formation rate function is restricted by the reactions of the network and how the species influence each reaction. We characterize families of so-called power-law kinetics for which the associated species formation rate function is injective within each stoichiometric class and thus the network cannot exhibit multistationarity. The criterion for power-law kinetics is d…
Multistationarity in molecular systems underlies switch-like responses in cellular decision making. Determining whether and when a system displays multistationarity is in general a difficult problem. In this work we completely determine the set of kinetic parameters that enable multistationarity in a ubiquitous motif involved in cell signaling, namely a dual phosphorylation cycle. In addition we show that the regions of multistationarity and monostationarity are both path connected. We model the dynamics of the concentrations of the proteins over time by means of a parametrized polynomial ordi…
We study how the properties of allowing multiple positive nondegenerate equilibria (MPNE) and multiple positive linearly stable equilibria (MPSE) are inherited in chemical reaction networks (CRNs). Specifically, when is it that we can deduce that a CRN admits MPNE or MPSE based on analysis of its subnetworks? Using basic techniques from analysis we are able to identify a number of situations where MPNE and MPSE are inherited as we build up a network. Some of these modifications are known while others are new, but all results are proved using the same basic framework, which we believe will yiel…
Multisite phosphorylation is a signaling mechanism well known to give rise to multiple steady states, a property termed multistationarity. When phosphorylation occurs in a sequential and distributive manner, we obtain a family of networks indexed by the number of phosphorylation sites $n$. This work addresses the problem of understanding the parameter region where this family of networks displays multistationarity, by focusing on the projection of this region onto the set of kinetic parameters. The problem is phrased in the context of real algebraic geometry and reduced to studying whether a p…
We provide a short supplement to the paper "MAPK networks and their capacity for multistationarity due to toric steady states" by Pérez Millán and Turjanski. We show that the capacity for toric steady states in the three networks analyzed in that paper can be derived using the process of network translation, which corresponds the original mass action system to a generalized mass action system with the same steady states. In all three cases, the translated chemical reaction network is proper, weakly reversible, and has both a structural and kinetic deficiency of zero. This is sufficient to guar…
The fundamental decomposition of a chemical reaction network (CRN) is induced by partitioning the reaction set into "fundamental classes". It was the basis of the Higher Deficiency Algorithm for mass action systems of Ji and Feinberg, and the Multistationarity Algorithm for power-law kinetic systems of Hernandez et al. In addition to our previous work, we provide important properties of the independence (i.e., the network's stoichiometric subspace is the direct sum of the subnetworks' stoichiometric subspaces) and the incidence-independence (i.e., the image of the network's incidence map is th…
Much attention has been focused in recent years on the following algebraic problem arising from applications: which chemical reaction networks, when taken with mass-action kinetics, admit multiple positive steady states? The interest behind this question is in steady states that are stable. As a step toward this difficult question, here we address the question of multiple nondegenerate positive steady states. Mathematically, this asks whether certain families of parametrized, real, sparse polynomial systems ever admit multiple positive real roots that are simple. Our main results settle this p…
Parametrized polynomial ordinary differential equation systems are broadly used for modeling, specially in the study of biochemical reaction networks under the assumption of mass-action kinetics. Understanding the qualitative behavior of the solutions with respect to the parameter values gives rise to complex problems within real algebraic geometry, concerning the study of the signs of multivariate polynomials over the positive orthant. In this work we provide further insight into the number of positive steady states of a benchmark model, namely the two-site phosphorylation cycle. In particula…
Given a real sparse polynomial system, we present a general framework to find explicit coefficients for which the system has more than one positive solution, based on the recent article by Bihan, Santos and Spaenlehauer. We apply this approach to find explicit reaction rate constants and total conservation constants in biochemical reaction networks for which the associated dynamical system is multistationary.
Approximating Parameter Regions for Multistationarity and Multistability in Mass-Action Systems
Institute of Mathematical Sciences, University of the Philippines Los Baños（菲律宾大学洛斯巴尼奥斯分校数学科学研究所）；Data Science Program, College of Science, University of the Philippines Diliman（菲律宾大学迪利曼分校理学院数据科学项目）；Institute of Mathematics, University of the Philippines Diliman（菲律宾大学迪利曼分校数学研究所）
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 multistati…
Bistability and multistationarity are properties of reaction networks linked to switch-like responses and connected to cell memory and cell decision making. Determining whether and when a network exhibits bistability is a hard and open mathematical problem. One successful strategy consists of analyzing small networks and deducing that some of the properties are preserved upon passage to the full network. Motivated by this we study chemical reaction networks with few chemical complexes. Under mass-action kinetics the steady states of these networks are described by fewnomial systems, that is po…
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. First, we build on work of Conradi…
Multistationarity is the property of a system to exhibit two distinct equilibria (steady-states) under otherwise identical conditions, and it is a phenomenon of recognized importance for biochemical systems. Multistationarity may appear in the parameter space as a consequence of saddle-node bifurcations, which necessarily require a simple eigenvalue zero of the Jacobian, at the bifurcating equilibrium. Matrices with a simple eigenvalue zero are generic in the set of singular matrices: any system whose Jacobian has an algebraically multiple eigenvalue zero can be perturbed to a system whose Jac…
Chemical reaction networks taken with mass-action kinetics are dynamical systems that arise in chemical engineering and systems biology. In general, determining whether a chemical reaction network admits multiple steady states is difficult, as this requires determining existence of multiple positive solutions to a large system of polynomials with unknown coefficients. However, in certain cases, various easy criteria can be applied. One such test is the Jacobian Criterion, due to Craciun and Feinberg, which gives sufficient conditions for ruling out the possibility of multiple steady states. A …
University of Copenhagen（哥本哈根大学）；Max Planck Institute of Molecular Cell Biology and Genetics（马克斯·普朗克分子细胞生物学与遗传学研究所）；Universidad Politécnica de Madrid（马德里理工大学）
Reaction networks taken with mass-action kinetics arise in many settings, from epidemiology to population biology to systems of chemical reactions. Bistable reaction networks are posited to underlie biochemical switches, which motivates the following question: which reaction networks have the capacity for multiple steady states? Mathematically, this asks: among certain parametrized families of polynomial systems, which admit multiple positive roots? No complete answer is known. This work analyzes the smallest networks, those with only a few chemical species or reactions. For these "smallest" n…
Some results are presented on how oscillation is inherited by chemical reaction networks (CRNs) when they are built in natural ways from smaller oscillatory networks. The main results describe four important ways in which a CRN can be enlarged while preserving its capacity for oscillation. The results are for general CRNs, not necessarily fully open, but lead to an important corollary for fully open networks: if a fully open CRN R with mass action kinetics admits a nondegenerate (resp., linearly stable) periodic orbit, then so do all such CRNs which include R as an induced subnetwork. This cla…
Mass-action chemical reaction systems are frequently used in Computational Biology. The corresponding polynomial dynamical systems are often large (consisting of tens or even hundreds of ordinary differential equations) and poorly parametrized (due to noisy measurement data and a small number of data points and repetitions). Therefore, it is often difficult to establish the existence of (positive) steady states or to determine whether more complicated phenomena such as multistationarity exist. If, however, the steady state ideal of the system is a binomial ideal, then we show that these questi…
We present conditions which guarantee a parametrization of the set of positive equilibria of a generalized mass-action system. Our main results state that (i) if the underlying generalized chemical reaction network has an effective deficiency of zero, then the set of positive equilibria coincides with the parametrized set of complex-balanced equilibria and (ii) if the network is weakly reversible and has a kinetic deficiency of zero, then the equilibrium set is nonempty and has a positive, typically rational, parametrization. Via the method of network translation, we apply our results to class…
Poly-PL kinetic systems are kinetic systems consisting of nonnegative linear combinations of power law functions. In this contribution, we analyze these kinetic systems using two main approaches: (1) we define a canonical power law representation of a poly-PL system, and (2) we transform a poly-PL system into a dynamically equivalent power law kinetic system that preserves the stoichiometric subspace of the system. These approaches led us to establish results that concern important dynamical properties of poly-PL systems such as existence and parametrization of complex balanced steady states, …
Poly-PL kinetic systems (PYK) are kinetic systems consisting of nonnegative linear combinations of power law functions. In this contribution, we analyze these kinetic systems using two main approaches: (1) we define a canonical power law representation of a poly-PL system, and (2) we transform a poly-PL system into a dynamically equivalent power law kinetic system that preserves the stoichiometric subspace of the system. These approaches led us to establish results that concern important dynamical properties of poly-PL systems that extend known results for generalized mass actions systems (GMA…
Which reaction networks, when taken with mass-action kinetics, have the capacity for multiple steady states? There is no complete answer to this question, but over the last 40 years various criteria have been developed that can answer this question in certain cases. This work surveys these developments, with an emphasis on recent results that connect the capacity for multistationarity of one network to that of another. In this latter setting, we consider a network $N$ that is embedded in a larger network $G$, which means that $N$ is obtained from $G$ by removing some subsets of chemical specie…
The Jacobian matrix of a dynamic system and its principal minors play a prominent role in the study of qualitative dynamics and bifurcation analysis. When interpreting the Jacobian as an adjacency matrix of an interaction graph, its principal minors correspond to sets of disjoint cycles in this graph and conditions for various dynamic behaviors can be inferred from its cycle structure. For chemical reaction systems, more fine-grained analyses are possible by studying a bipartite species-reaction graph. Several results on injectivity, multistationarity, and bifurcations of a chemical reaction s…
