Class Of Reaction Networks Research Articles

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.

Nonideal Reaction-Diffusion Systems: Multiple Routes to Instability.

We develop a general classification of the nature of the instabilities yielding spatial organization in open nonideal reaction-diffusion systems, based on linear stability analysis. This encompasses dynamics where chemical species diffuse, interact with each other, and undergo chemical reactions driven out of equilibrium by external chemostats. We find analytically that these instabilities can be of two types: instabilities caused by intermolecular energetic interactions (E type), and instabilities caused by multimolecular out-of-equilibrium chemical reactions (R type). Furthermore, we identify a class of chemical reaction networks, containing unimolecular networks but also extending beyond them, that can only undergo E-type instabilities. We illustrate our analytical findings with numerical simulations on two reaction-diffusion models, each displaying one of the two types of instability and generating stable patterns.

Trade-offs between number fluctuations and response in nonequilibrium chemical reaction networks.

We study the response of chemical reaction networks driven far from equilibrium to logarithmic perturbations of reaction rates. The response of the mean number of a chemical species is observed to be quantitively limited by number fluctuations and the maximum thermodynamic driving force. We prove these trade-offs for linear chemical reaction networks and a class of nonlinear chemical reaction networks with a single chemical species. Numerical results for several model systems support the conclusion that these trade-offs continue to hold for a broad class of chemical reaction networks, though their precise form appears to sensitively depend on the deficiency of the network.

Programming Hydrogels with Complex Transient Behaviors via Autocatalytic Cascade Reactions.

It is challenging to design complex synthetic life-like systems that can show both autoevolution and fuel-driven transient behaviors. Here, we report a new class of chemical reaction networks (CRNs) to construct life-like polymer hydrogels. The CRNs are constituted of autocatalytic cascade reactions and fuel-driven reaction networks. The reactions start with only two compounds, that is, thiol of 4-arm-PEG-SH and thiuram disulfides, and undergo thiol oxidation (k1), disulfide metathesis (k2), and thionate hydrolysis-coupling reactions (k3) subsequently, leading to a four-state autonomous transition of sol(I) → soft gel → sol(II) → stiff gel. Moreover, thiuram disulfides can be applied as a fuel to drive the repeated occurrence of metathesis and hydrolysis-coupling reactions, generating dissipative stiff gel → sol(II) → stiff gel cycles. Systematic kinetics studies reveal that the event and lifetime of every transient state could be delicately tailored-up by varying the thiuram disulfide concentration, pH of the system, and thiuram structures. Since the consecutive transient behaviors are precisely predictable, we envision the strategy's potential in guiding the molecular designs of autonomous and adaptive materials for many fields.

On classes of reaction networks and their associated polynomial dynamical systems

In the study of reaction networks and the polynomial dynamical systems that they generate, special classes of networks with important properties have been identified. These include reversible, weakly reversible}, and, more recently, endotactic networks. While some inclusions between these network types are clear, such as the fact that all reversible networks are weakly reversible, other relationships are more complicated. Adding to this complexity is the possibility that inclusions be at the level of the dynamical systems generated by the networks rather than at the level of the networks themselves. We completely characterize the inclusions between reversible, weakly reversible, endotactic, and strongly endotactic network, as well as other less well studied network types. In particular, we show that every strongly endotactic network in two dimensions can be generated by an extremally weakly reversible network. We also introduce a new class of source-only networks, which is a computationally convenient property for networks to have, and show how this class relates to the above mentioned network types.

Lyapunov Function Partial Differential Equations for Stability Analysis of a Class of Chemical Reaction Networks

We investigate a broad family of chemical reaction networks (CRNs) assigned with mass action kinetics, called complex-balanced-produced-CRNs (CBP-CRNs), which are generated by any given complex balanced mass action system (MAS) and whose structures depend on the selection of producing matrices. Unluckily, the generally applied pseudo-Helmholtz free energy function may fail to act as a Lyapunov function for the CBP-CRNs. Inspired by the method of Lyapunov function partial differential equations (PDEs), we construct one solution of their corresponding Lyapunov function PDEs, termed as the generalized pseudo-Helmholtz free energy function, and we further show that solution can behave as a Lyapunov function to render the asymptotic stability for the CBP-CRNs. This work can be taken as an argument of the conjecture that Lyapunov function PDEs approach can serve for any MAS.

The Deficiency Zero Theorem and global asymptotic stability for a class of chemical reaction networks with arbitrary time delays

The Deficiency Zero Theorem (DZT) is extended to be applied to chemical reaction networks (CRNs) with time delays, the time evolution of concentrations of species in which networks is described by delay differential equations (DDEs). The extended DZT provides a sufficient condition for a positive equilibrium point of DDEs to exist and be locally asymptotically stable. Furthermore, it is proven that the absence of boundary equilibria is a sufficient condition for the positive equilibrium point to be globally asymptotically stable. These results are natural analogues of theorems proven for CRNs without time delays.

Distributed control of interconnected Chemical Reaction Networks with delay

This paper introduces a control approach for a class of Chemical Reaction Networks (CRNs) that are interconnected through a delayed convection network. First, a control-oriented model is proposed for interconnected CRNs. Second, based on this model, a distributed control method is introduced which assures that each CRN can be driven into a desired fixed point (setpoint) independently of the delay in the convection network. The proposed algorithm is also augmented with a disturbance attenuation term to compensate the effect of unknown input disturbances on setpoint tracking performance. The control design applies the theory of passive systems and methods developed for multi-agent systems. Simulation results are provided to show the applicability of the proposed control method.

Noise-induced mixing and multimodality in reaction networks

We analyse a class of chemical reaction networks under mass-action kinetics involving multiple time scales, whose deterministic and stochastic models display qualitative differences. The networks are inspired by gene-regulatory networks and consist of a slow subnetwork, describing conversions among the different gene states, and fast subnetworks, describing biochemical interactions involving the gene products. We show that the long-term dynamics of such networks can consist of a unique attractor at the deterministic level (unistability), while the long-term probability distribution at the stochastic level may display multiple maxima (multimodality). The dynamical differences stem from a phenomenon we call noise-induced mixing, whereby the probability distribution of the gene products is a linear combination of the probability distributions of the fast subnetworks which are ‘mixed’ by the slow subnetworks. The results are applied in the context of systems biology, where noise-induced mixing is shown to play a biochemically important role, producing phenomena such as stochastic multimodality and oscillations.

Finite Time Distributions of Stochastically Modeled Chemical Systems with Absolute Concentration Robustness

Recent research in both the experimental and mathematical communities has focused on biochemical interaction systems that satisfy an "absolute concentration robustness" (ACR) property. The ACR property was first discovered experimentally when, in a number of different systems, the concentrations of key system components at equilibrium were observed to be robust to the total concentration levels of the system. Followup mathematical work focused on deterministic models of biochemical systems and demonstrated how chemical reaction network theory can be utilized to explain this robustness. Later mathematical work focused on the behavior of this same class of reaction networks, though under the assumption that the dynamics were stochastic. Under the stochastic assumption, it was proven that the system will undergo an extinction event with a probability of one so long as the system is conservative, showing starkly different long-time behavior than in the deterministic setting. Here we consider a general class of stochastic models that intersects with the class of ACR systems studied previously. We consider a specific system scaling over compact time intervals and prove that in a limit of this scaling the distribution of the abundances of the ACR species converges to a certain product-form Poisson distribution whose mean is the ACR value of the deterministic model. This result is in agreement with recent conjectures pertaining to the behavior of ACR networks endowed with stochastic kinetics, and helps to resolve the conflicting theoretical results pertaining to deterministic and stochastic models in this setting.

New Approach to the Stability of Chemical Reaction Networks: Piecewise Linear in Rates Lyapunov Functions

Piecewise-Linear in Rates (PWLR) Lyapunov functions are introduced for a class of Chemical Reaction Networks (CRNs). In addition to their simple structure, these functions are robust with respect to arbitrary monotone reaction rates, of which mass-action is a special case. The existence of such functions ensures the convergence of trajectories towards equilibria, and guarantee their asymptotic stability with respect to the corresponding stoichiometric compatibility class. We give the definition of these Lyapunov functions, prove their basic properties, and provide algorithms for constructing them. Examples are provided, relationship with consensus dynamics are discussed, and future directions are elaborated.

Exact Moment Dynamics for Feedforward Nonlinear Chemical Reaction Networks

We introduce a class of chemical reaction networks for which all moments can be computed by finite-dimensional linear differential equations. This class allows second and higher order reactions, subject to certain assumptions on structure and/or conservation laws.

Violation of the mass-action law in dilute chemical systems

The mass-action law, which predicts the rates of chemical reactions, is widely used for modeling the kinetics of the chemical reactions and their stationary states, also for complex chemical reaction networks. However, violations of the mass-action equations have been reported in various cases: in confined systems with a small number of molecules, in non-ideally-stirred systems, when the reactions are limited by the diffusion, at high concentrations of reactants, or in chemical reaction networks with marginally stable mass-action equations. In this paper, I describe a new mechanism, leading to the violation of the mass-action equations, that takes place at a low concentration of at least one of the reactants; in this limit, the reaction rates can be easily inferred from the chemical reaction network. I propose that this mechanism underlies the replication stability of the hypercycles, a class of chemical reaction networks hypothetically connected with abiogenesis. I provide two simple examples of chemical reaction networks in which the mechanism leading to the violation of the mass-action law is present. I study the two chemical reaction networks by means of a simulation performed with a cellular automaton model. The results have a general validity and represent a limitation of the validity of the mass-action law, which has been overlooked up to now in the studies about the chemical reaction networks.

Local and Global Stability of Equilibria for a Class of Chemical Reaction Networks

A class of chemical reaction networks is described with the property that each positive equilibrium is locally asymptotically stable relative to its stoichiometry class, an invariant subspace on which it lies. The reaction systems treated are characterized primarily by the existence of a certain factorization of their stoichiometric matrix and strong connectedness of an associated graph. Only very mild assumptions are made about the rates of reactions, and, in particular, mass action kinetics are not assumed. In many cases, local asymptotic stability can be extended to global asymptotic stability of each positive equilibrium relative to its stoichiometry class. The results are proved via the construction of Lyapunov functions whose existence follows from the fact that the reaction networks define monotone dynamical systems with increasing integrals.

Concordant chemical reaction networks

We describe a large class of chemical reaction networks, those endowed with a subtle structural property called concordance. We show that the class of concordant networks coincides precisely with the class of networks which, when taken with any weakly monotonic kinetics, invariably give rise to kinetic systems that are injective – a quality that, among other things, precludes the possibility of switch-like transitions between distinct positive steady states. We also provide persistence characteristics of concordant networks, instability implications of discordance, and consequences of stronger variants of concordance. Some of our results are in the spirit of recent ones by Banaji and Craciun, but here we do not require that every species suffer a degradation reaction. This is especially important in studying biochemical networks, for which it is rare to have all species degrade.

Reachability, persistence, and constructive chemical reaction networks (part I): reachability approach to the persistence of chemical reaction networks

For a chemical reaction network, persistence is the property that no species tend to extinction if all species are initially present. We investigate the stronger property of vacuous persistence: the same asymptotic feature with a weaker requirement on initial states, namely that all species be implicitly present. By implicitly present, we mean for instance that if only water is present and the reaction network incorporates the information that water is made of hydrogen and oxygen, then hydrogen and oxygen are implicitly present. Persistence is inherently interesting and has implications for the global asymptotic stability of equilibrium states. Our main tools are the work of A. I. Vol’pert on the nullity and positivity of species concentrations, and the enabling notion of reachability. The main result states that a reaction network is vacuously persistent if and only if the set of all species is the only set of species that both is closed with respect to reachability and causes the implicit presence of all species. This paper is the first in a series of three articles. Two sequel papers introduce additional formalisms and use them to describe two large classes of reaction networks that are used as models in biochemistry and are vacuously persistent.

Graph-theoretic characterizations of monotonicity of chemical networks in reaction coordinates

This paper derives new results for certain classes of chemical reaction networks, linking structural to dynamical properties. In particular, it investigates their monotonicity and convergence under the assumption that the rates of the reactions are monotone functions of the concentrations of their reactants. This is satisfied for, yet not restricted to, the most common choices of the reaction kinetics such as mass action, Michaelis-Menten and Hill kinetics. The key idea is to find an alternative representation under which the resulting system is monotone. As a simple example, the paper shows that a phosphorylation/dephosphorylation process, which is involved in many signaling cascades, has a global stability property. We also provide a global stability result for a more complicated example that describes a regulatory pathway of a prevalent signal transduction module, the MAPK cascade.

Sensitivity and Robustness in Chemical Reaction Networks

For a wide class of chemical reaction networks, including all those governed by detailed balanced mass-action kinetics, we examine the robustness of equilibrium species concentrations against fluctuations in the overall reactant supply. In particular, we present lower bounds on the individual species-concentration sensitivities that derive from reaction network structure alone, independent of kinetic parameters or even of the particular equilibrium state at which sensitivities are calculated. These bounds suggest that, in the class of reaction networks considered here, very high robustness (i.e., very low sensitivities) should be expected only when the various molecules are constructed from a large number of distinct elemental building blocks that appear in high multiplicity or that combine gregariously. This situation is often encountered in biology.

The existence and uniqueness of steady states for a class of chemical reaction networks

The existence and uniqueness of steady states for a class of chemical reaction networks

On imperfect contacting in batch heterogeneous chemical reaction systems—II. The effective scalar diffusion coefficient for a class of multicomponent chemical networks

For a certain simple class of chemical reaction networks, viz. rank K = 1, the introduction of an effective scalar diffusion coefficient allows the vector integro-differential equation describing the performance of a multicomponent batch heterogeneous chemical reaction system to be reduced to the scalar integro-differential equation solved earlier in a study of isomerization networks. In particular if the reaction 1 ⇄ 2 takes place in an inert solvent then in the limit of infinite dilution the effective scalar diffusion coefficient reduces to 1/√ ¢D = X 1 * /√ D 22+ X 2 * /√ D 11. We translate into the framework of our model the experimental studies of slurry reaction systems thought to be chemically controlled, e.g., the studies of Satterfield and Kölbel and Maening. We estimate the degree of chemical disguise, finding it to be important in the intermediate to high temperature range of the latter.