Mass Action Systems Research Articles

Understanding and Quantifying Network Robustness to Stochastic Inputs.

A variety of biomedical systems are modeled by networks of deterministic differential equations with stochastic inputs. In some cases, the network output is remarkably constant despite a randomly fluctuating input. In the context of biochemistry and cell biology, chemical reaction networks and multistage processes with this property are called robust. Similarly, the notion of a forgiving drug in pharmacology is a medication that maintains therapeutic effect despite lapses in patient adherence to the prescribed regimen. What makes a network robust to stochastic noise? This question is challenging due to the many network parameters (size, topology, rate constants) and many types of noisy inputs. In this paper, we propose a summary statistic to describe the robustness of a network of linear differential equations (i.e. a first-order mass-action system). This statistic is the variance of a certain random walk passage time on the network. This statistic can be quickly computed on a modern computer, even for complex networks with thousands of nodes. Furthermore, we use this statistic to prove theorems about how certain network motifs increase robustness. Importantly, our analysis provides intuition for why a network is or is not robust to noise. We illustrate our results on thousands of randomly generated networks with a variety of stochastic inputs.

Sufficient Conditions for Linear Stability of Complex-Balanced Equilibria in Generalized Mass-Action Systems

Sufficient Conditions for Linear Stability of Complex-Balanced Equilibria in Generalized Mass-Action Systems

A Decomposition-Based Approach for Deriving Positive Steady States of a Class of Chemical Reaction Networks with Non-Mass-Action Kinetics

Steady states are frequently used to investigate the long-term behaviors of (bio)chemical systems. Recently, there has been a growing interest in network-based approaches due to their efficiency in deriving parametrizations of positive steady states in systems with mass-action kinetics. In this study, we extend this approach to derive positive steady states in networks under non-mass-action kinetics, specifically mixed kinetics. In a system with mixed kinetics, some reactions {may follow} mass-action kinetics, while others in the same network follow different rate laws, such as quotient rate laws. An example of such complexity is evident in a mathematical model of the insulin signaling pathway in type 2 diabetes. To compute its positive {steady states}, we adapt our existing network decomposition approach, originally designed for mass-action kinetics, to handle networks with non-mass-action kinetics. This approach involves breaking down a given network into smaller, independent subnetworks to derive the positive steady states of each subnetwork separately. These individual steady states are then combined to obtain the positive steady states of the entire network. This strategy makes computations more manageable for complex and large networks. More importantly, this method could separate reactions with purely mass-action kinetics into certain subnetworks from those that follow different rate laws. We also present an illustrative example that provides insights into methods for transforming networks with mixed kinetics into their associated mass-action systems.

A Provably Convergent Control Closure Scheme for the Method of Moments of the Chemical Master Equation.

In this article, we introduce a novel moment closure scheme based on concepts from model predictive control (MPC) to accurately describe the time evolution of the statistical moments of the solution of the chemical master equation (CME). The method of moments, a set of ordinary differential equations frequently used to calculate the first nm moments, is generally not closed since lower-order moments depend on higher-order moments. To overcome this limitation, we interpret the moment equations as a nonlinear dynamical system, where the first nm moments serve as states, and the closing moments serve as the control input. We demonstrate the efficacy of our approach using three example systems and show that it outperforms existing closure schemes. For polynomial systems, which encompass all mass-action systems, we provide probability bounds for the error between true and estimated moment trajectories. We achieve this by combining the convergence properties of a priori moment estimates from stochastic simulations with guarantees for nonlinear reference tracking MPC. Our proposed method offers an effective solution to accurately predict the time evolution of moments of the CME, which has wide-ranging implications for many fields, including biology, chemistry, and engineering.

Oscillations in three-reaction quadratic mass-action systems.

It is known that rank-two bimolecular mass-action systems do not admit limit cycles. With a view to understanding which small mass-action systems admit oscillation, in this paper we study rank-two networks with bimolecular source complexes but allow target complexes with higher molecularities. As our goal is to find oscillatory networks of minimal size, we focus on networks with three reactions, the minimum number that is required for oscillation. However, some of our intermediate results are valid in greatergenerality. One key finding is that an isolated periodic orbit cannot occur in a three-reaction, trimolecular, mass-action system with bimolecular sources. In fact, we characterize all networks in this class that admit a periodic orbit; in every case, all nearby orbits are periodic too. Apart from the well-known Lotka and Ivanova reactions, we identify another network in this class that admits a center. This new network exhibits a vertical Andronov-Hopfbifurcation. Furthermore, we characterize all two-species, three-reaction, bimolecular-sourced networks that admit an Andronov-Hopf bifurcation with mass-action kinetics. These include two families of networks that admit a supercritical Andronov-Hopf bifurcation and hence a stable limit cycle. These networks necessarily have a target complex with a molecularity of at least four, and it turns out that there are exactly four such networks that aretetramolecular.

A New Decomposition of the Graph Laplacian and the Binomial Structure of Mass-Action Systems

We provide a new decomposition of the Laplacian matrix (for labeled directed graphs with strongly connected components), involving an invertible core matrix, the vector of tree constants, and the incidence matrix of an auxiliary graph, representing an order on the vertices. Depending on the particular order, the core matrix has additional properties. Our results are graph-theoretic/algebraic in nature. As a first application, we further clarify the binomial structure of (weakly reversible) mass-action systems, arising from chemical reaction networks. Second, we extend a classical result by Horn and Jackson on the asymptotic stability of special steady states (complex-balanced equilibria). Here, the new decomposition of the graph Laplacian allows us to consider regions in the positive orthant with given monomial evaluation orders (and corresponding polyhedral cones in logarithmic coordinates). As it turns out, all dynamical systems are asymptotically stable that can be embedded in certain binomial differential inclusions. In particular, this holds for complex-balanced mass-action systems, and hence, we also obtain a polyhedral-geometry proof of the classical result.

The smallest bimolecular mass-action system with a vertical Andronov–Hopf bifurcation

We present a three-dimensional differential equation, which robustly displays a degenerate Andronov–Hopf bifurcation of infinite codimension, leading to a center, i.e., an invariant two-dimensional surface that is filled with periodic orbits surrounding an equilibrium. The system arises from a three-species bimolecular chemical reaction network consisting of four reactions. In fact, it is, up to a natural equivalence, the only such mass-action system that admits a center via an Andronov–Hopf bifurcation.

Some minimal bimolecular mass-action systems with limit cycles

We discuss three examples of bimolecular mass-action systems with three species, due to Feinberg, Berner, Heinrich, and Wilhelm. Each system has a unique positive equilibrium which is unstable for certain rate constants and then exhibits stable limit cycles, but no chaotic behaviour. For some rate constants in the Feinberg–Berner system, a stable equilibrium, an unstabe limit cycle, and a stable limit cycle coexist. All three networks are minimal in some sense.By way of homogenising these three examples, we construct bimolecular mass-conserving mass-action systems with four species that admit a stable limit cycle. The homogenised Feinberg–Berner system and the homogenised Wilhelm–Heinrich system admit the coexistence of a stable equilibrium, an unstable limit cycle, and a stable limit cycle.

Positive equilibria of power law kinetics on networks with independent linkage classes

Studies about the set of positive equilibria (\(E_+\)) of kinetic systems have been focused on mass action, and not that much on power law kinetic (PLK) systems, even for PL-RDK systems (PLK systems where two reactions with identical reactant complexes have the same kinetic order vectors). For mass action, reactions with different reactants have different kinetic order rows. A PL-RDK system satisfying this property is called factor span surjective (PL-FSK). In this work, we show that a cycle terminal PL-FSK system with \(E_+\ne \varnothing \) and has independent linkage classes (ILC) is a poly-PLP system, i.e., \(E_+\) is the disjoint union of log-parametrized sets. The key insight for the extension is that factor span surjectivity induces an isomorphic digraph structure on the kinetic complexes. The result also completes, for ILC networks, the structural analysis of the original complex balanced generalized mass action systems (GMAS) by Müller and Regensburger. We also identify a large set of PL-RDK systems where non-emptiness of \(E_+\) is a necessary and sufficient condition for non-emptiness of each set of positive equilibria for each linkage class. These results extend those of Boros on mass action systems with ILC. We conclude this paper with two applications of our results. Firstly, we consider absolute complex balancing (ACB), i.e., the property that each positive equilibrium is complex balanced, in poly-PLP systems. Finally, we use the new results to study absolute concentration robustness (ACR) in these systems. In particular, we obtain a species hyperplane containment criterion to determine ACR in the system species.

Disguised toric dynamical systems

We study families of polynomial dynamical systems inspired by biochemical reaction networks. We focus on complex balanced mass-action systems, which have also been called toric. They are known or conjectured to enjoy very strong dynamical properties, such as existence and uniqueness of positive steady states, local and global stability, persistence, and permanence. We consider the class of disguised toric dynamical systems, which contains toric dynamical systems, and to which all dynamical properties mentioned above extend naturally. By means of (real) algebraic geometry we show that some reaction networks have an empty toric locus or a toric locus of Lebesgue measure zero in parameter space, while their disguised toric locus is of positive measure. We also propose some algorithms one can use to detect the disguised toric locus.

A Graph-Theoretic Condition for Delay Stability of Reaction Systems

Delay mass-action systems provide a model of chemical kinetics when past states influence the current dynamics. In this work, we provide a graph-theoretic condition for delay stability, i.e., linear stability independent of both rate constants and delay parameters. In particular, the result applies when the system has no delay, implying asymptotic stability for the ODE system. The graph-theoretic condition is about cycles in the directed species-reaction graph of the network, which encodes how different species in the system interact.

Weakly reversible CF-decompositions of chemical kinetic systems

This paper studies chemical kinetic systems which decompose into weakly reversible complex factorizable (CF) systems. Among power law kinetic systems, CF systems (denoted as PL-RDK systems) are those where branching reactions of a reactant complex have identical rows in the kinetic order matrix. Mass action and generalized mass action systems (GMAS) are well-known examples. Schmitz's global carbon cycle model is a previously studied non-complex factorizable (NF) power law system (denoted as PL-NDK). We derive novel conditions for the existence of weakly reversible CF-decompositions and present an algorithm for verifying these conditions. We discuss methods for identifying independent decompositions, i.e., those where the stoichiometric subspaces of the subnetworks form a direct sum, as such decompositions relate positive equilibria sets of the subnetworks to that of the whole network. We then use the results to determine the positive equilibria sets of PL-NDK systems which admit an independent weakly reversible decomposition into PL-RDK systems of PLP type, i.e., the positive equilibria are log-parametrized, which is a broad generalization of a Deficiency Zero Theorem of Fortun et al. (2019).

Limit cycles in mass-conserving deficiency-one mass-action systems

We present some simple mass-action systems with limit cycles that fall under the scope of the Deficiency-One Theorem. All the constructed examples are mass-conserving and their stoichiometric subspace is two-dimensional. Using the continuation software MATCONT, we depict the limit cycles in all stoichiometric classes at once. The networks are trimolecular and tetramolecular, and some exhibit two or even three limit cycles. Finally, we show that the associated mass-action system of a bimolecular reaction network with two-dimensional stoichiometric subspace does not admit a limit cycle.

Absolutely Complex Balanced Kinetic Systems

A complex balanced kinetic system is absolutely complex balanced (ACB) if every positive equilibrium is complex balanced. Two results on absolute complex balancing were foundational for modern chemical reaction network theory (CRNT): in 1972, M. Feinberg proved that any deficiency zero complex balanced system is absolutely complex balanced. In the same year, F. Horn and R. Jackson showed that the (full) converse of the result is not true: any complex balanced mass action system, regardless of its deficiency, is absolutely complex balanced. In this paper, we present initial results on the extension of the Horn and Jackson ACB Theorem. In particular, we focus on other kinetic systems with positive deficiency where complex balancing implies absolute complex balancing. While doing so, we found out that complex balanced power law reactant determined kinetic systems (PL-RDK) systems are not ACB. In our search for necessary and sufficient conditions for complex balanced systems to be absolutely complex balanced, we came across the so-called CLP systems (complex balanced systems with a desired "log parametrization" property). It is shown that complex balanced systems with bi-LP property are absolutely complex balanced. For non-CLP systems, we discuss novel methods for finding sufficient conditions for ACB in kinetic systems containing non-CLP systems: decompositions, the Positive Function Factor (PFF) and the Coset Intersection Count (CIC) and their application to poly-PL and Hill-type systems.

Single-target networks

<p style='text-indent:20px;'>Reaction networks can be regarded as finite oriented graphs embedded in Euclidean space. <i>Single-target networks</i> are reaction networks with an arbitrarily set of source vertices, but <i>only one</i> sink vertex. We completely characterize the dynamics of all mass-action systems generated by single-target networks, as follows: either <i>(i)</i> the system is globally stable for all choice of rate constants (in fact, is dynamically equivalent to a detailed-balanced system with a single linkage class), or <i>(ii)</i> the system has no positive steady states for any choice of rate constants and all trajectories must converge to the boundary of the positive orthant or to infinity. Moreover, we show that global stability occurs if and only if the target vertex of the network is in the relative interior of the convex hull of the source vertices.</p>

Analysis of mass-action systems by split network translation

We introduce the notion of corresponding a chemical reaction network to a split network translation, and use this novel process to extend the scope of existing network-based theory for characterizing the steady state set of mass-action systems. In the process of network splitting, the reactions of a network are divided into subnetworks, called slices, in such a way that, when summed across the slices, the stoichiometry of each reaction sums to that of the original network. This can produce a network with more desirable structural properties, such as weak reversibility and a lower deficiency, which can then be used to establish steady state properties of the original mass-action system such as multistationarity and absolute concentration robustness. We also present a computational implementation utilizing mixed-integer linear programming for determining whether a given chemical reaction network has a weakly reversible split network translation.

Uniqueness of weakly reversible and deficiency zero realizations of dynamical systems

A reaction network together with a choice of rate constants uniquely gives rise to a system of differential equations, according to the law of mass-action kinetics. On the other hand, different networks can generate the same dynamical system under mass-action kinetics. Therefore, the problem of identifying “the” underlying network of a dynamical system is not well-posed, in general. Here we show that the problem of identifying an underlying weakly reversible deficiency zero network is well-posed, in the sense that the solution is unique whenever it exists. This can be very useful in applications because from the perspective of both dynamics and network structure, a weakly reversible deficiency zero (WR0) realization is the simplest possible one. Moreover, while mass-action systems can exhibit practically any dynamical behavior, including multistability, oscillations, and chaos, WR0 systems are remarkably stable for any choice of rate constants: they have a unique positive steady state within each invariant polyhedron, and cannot give rise to oscillations or chaotic dynamics. We also prove that both of our hypotheses (i.e., weak reversibility and deficiency zero) are necessary for uniqueness.

Oscillations in Planar Deficiency-One Mass-Action Systems.

Whereas the positive equilibrium of a planar mass-action system with deficiency zero is always globally stable, for deficiency-one networks there are many different scenarios, mainly involving oscillatory behaviour. We present several examples, with centers or multiple limit cycles.

Uniformly accurate nonlinear transmission rate models arising from disease spread through pair contacts.

We derive and asymptotically analyze mass-action models for disease spread that include transient pair formation and dissociation. Populations of unpaired susceptible individuals and infected individuals are distinguished from the population of three types of pairs of individuals: both susceptible, one susceptible and one infected, and both infected. Disease transmission can occur only within a pair consisting of one susceptible individual and one infected individual. We use perturbation expansion to formally derive uniformly valid approximations for the dynamics of the total infected and susceptible populations under different conditions including combinations of fast association, fast transmission, and fast dissociation limits. The effective equations are derived from the fundamental mass-action system without implicitly imposing transmission mechanisms, such as those used in frequency-dependent models. Our results represent submodels that show how effective nonlinear transmission can arise from pairing dynamics and are juxtaposed with density-based mass-action and frequency-based models.

Positive equilibria of Hill-type kinetic systems

This work introduces a novel approach to study properties of positive equilibria of a chemical reaction network \({\mathscr {N}}\) endowed with Hill-type kinetics K, called a Hill-type kinetic (HTK) system \(\left( {\mathscr {N}},K\right)\), including their multiplicity and concentration robustness in a species. We associate a unique poly-PL kinetic (PYK) system \(\left( {{\mathscr {N}},{K_\text {PY}}} \right)\) to the given HTK system, where PYK is a positive linear combination of PL functions. The associated system has the key property that its equilibria sets coincide with those of the Hill-type system. This allows us to identify two novel subsets of the (HTKs), called PL-equilibrated and PL-complex balanced kinetics, to which recent results on absolute concentration robustness (ACR) of species and complex balancing at positive equilibria of PL kinetic systems can be applied. Our main results also include the Shinar–Feinberg ACR Theorem for PL-equilibrated HT-RDK systems (i.e., subset of complex factorizable HTK systems), which establishes a foundation for the analysis of ACR in HTK systems, and the extension of the results of Müller and Regensburger on generalized mass action systems to PL-complex balanced HT-RDK systems. In addition, we derive the theory of balanced concentration robustness in an analogous manner to ACR for PL-equilibrated systems. Finally, we provide further extensions of our results to a more general class of kinetics, which include quotients of poly-PL functions.