Laplacian Dynamics and Kron Reduction in Species-Reaction Graphs of Chemical Reaction Networks.

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 reate to sets of disjoint cycles in this graph and conditions for qualitative dynamic behaviors can be inferred from its cycle structure. The Jacobian of chemical reaction systems decomposes into the product of two matrices, which allows more fine-grained analyses by studying a corresponding bipartite species-reaction graph. Several different bipartite graphs have been proposed and results on injectivity, multistationarity, and bifurcations have been derived. Here, we present a new definition of the species-reaction graph that directly connects the cycle structure with determinant expansion terms, principal minors, and the coefficients of the characteristic polynomial. It encompasses previous graph constructions as special cases. This graph has a direct relation to the interaction graph, and properties of cycles and sub-graphs can be translated in both directions. A simple equivalence relation enables simplified decomposition of determinant expansions and allows simpler and more direct proofs of previous results.

Networks of organic chemical reactions are important in life and probably played a central part in its origin. Network dynamics regulate cell division, circadian rhythms, nerve impulses and chemotaxis, and guide the development of organisms. Although out-of-equilibrium networks of chemical reactions have the potential to display emergent network dynamics such as spontaneous pattern formation, bistability and periodic oscillations, the principles that enable networks of organic reactions to develop complex behaviours are incompletely understood. Here we describe a network of biologically relevant organic reactions (amide formation, thiolate-thioester exchange, thiolate-disulfide interchange and conjugate addition) that displays bistability and oscillations in the concentrations of organic thiols and amides. Oscillations arise from the interaction between three subcomponents of the network: an autocatalytic cycle that generates thiols and amides from thioesters and dialkyl disulfides; a trigger that controls autocatalytic growth; and inhibitory processes that remove activating thiol species that are produced during the autocatalytic cycle. In contrast to previous studies that have demonstrated oscillations and bistability using highly evolved biomolecules (enzymes and DNA) or inorganic molecules of questionable biochemical relevance (for example, those used in Belousov-Zhabotinskii-type reactions), the organic molecules we use are relevant to metabolism and similar to those that might have existed on the early Earth. By using small organic molecules to build a network of organic reactions with autocatalytic, bistable and oscillatory behaviour, we identify principles that explain the ways in which dynamic networks relevant to life could have developed. Modifications of this network will clarify the influence of molecular structure on the dynamics of reaction networks, and may enable the design of biomimetic networks and of synthetic self-regulating and evolving chemical systems.

We discuss a method to describe the qualitative dynamics of chemical reaction networks in terms of symbolic dynamics. The method, that can be applied to mass-action reaction networks with separated timescales, uses solutions of the partial tropical equilibration problem as proxies for symbolic states. The partial tropical equilibration solutions are found algorithmically. These solutions also provide the scaling needed for slow-fast decomposition and model reduction. Any trace of the model can thus be represented as a sequence of local approximations of the full model. We illustrate the method using as case study a biochemical model of the cell cycle.

In this paper we derive a compact mathematical formulation describing the dynamics of chemical reaction networks that are complex-balanced and are governed by mass action kinetics. The formulation is based on the graph of (substrate and product) complexes and the stoichiometric information of these complexes, and crucially uses a balanced weighted Laplacian matrix. It is shown that this formulation leads to elegant methods for characterizing the space of all equilibria for complex-balanced networks and for deriving stability properties of such networks. We propose a method for model reduction of complex-balanced networks, which is similar to the Kron reduction method for electrical networks and involves the computation of Schur complements of the balanced weighted Laplacian matrix.

Understanding the emergent behavior of chemical reaction networks (CRNs) is a fundamental aspect of biology and its origin from inanimate matter. A closed CRN monotonically tends to thermal equilibrium, but when it is opened to external reservoirs, a range of behaviors is possible, including transition to a new equilibrium state, a nonequilibrium state, or indefinite growth. This study shows that slowly driven CRNs are governed by the conserved quantities of the closed system, which are generally far fewer in number than the species. Considering both deterministic and stochastic dynamics, a universal slow-dynamics equationis derived with singular perturbation methods and is shown to be thermodynamically consistent. The slow dynamics is highly robust against microscopic details of the network, which may be unknown in practical situations. In particular, nonequilibrium states of realistic large CRNs can be sought without knowledge of bulk reaction rates. The framework is successfully tested against a suite of networks of increasing complexity and argued to be relevant in the treatment of open CRNs as chemical machines.

Modeling dynamics of chemical reaction networks using electrical analogs: Application to autocatalytic reactions

The theory of chemical kinetics forms the basis to describe the dynamics of chemical reaction networks. Owing to physical and thermodynamic constraints, the networks possess various structures, which can be utilized to characterize important properties of the networks. In this work, we reveal the Hessian geometry which underlies chemical reaction networks and demonstrate how it originates from the interplay of stoichiometric and thermodynamic constraints. Our derivation is based on kinetics, we assume the law of mass action and characterize the equilibrium states by the detailed balance condition. The obtained geometric structure is then related to thermodynamics via the Hessian geometry appearing in a pure thermodynamic derivation. We demonstrate, based on the fact that both equilibrium and complex balanced states form toric varieties, how the Hessian geometric framework can be extended to nonequilibrium complex balanced steady states. We conclude that Hessian geometry provides a natural framework to capture the thermodynamic aspects of chemical reaction networks.

For control in biomolecular systems, the most basic objective of maintaining a small error in a target variable, say the expression level of some protein, is often difficult due to the presence of both large uncertainty of every type and intrinsic limitations on the controller's implementation. This paper explores the limits of biochemically plausible controller design for the problem of robust perfect adaptation (RPA), biologists' term for robust steady state tracking. It is well-known that for a large class of nonlinear systems, a system has RPA iff it has integral feedback control (IFC), which has been used extensively in real control systems to achieve RPA. However, we show that due to intrinsic physical limitations on the dynamics of chemical reaction networks (CRNs), cells cannot implement IFC directly in the concentration of a chemical species. This contrasts with electronic implementations, particularly digital, where it is trivial to implement IFC directly in a single state. Therefore, biomolecular systems have to achieve RPA by encoding the integral control variable into the network architecture of a CRN. We describe a general framework to implement RPA in CRNs and show that well-known network motifs that achieve RPA, such as (negative) integral feedback (IFB) and incoherent feedforward (IFF), are examples of such implementations. We also develop methods to designing integral feedback variables for unknown plants. This standard control notion is surprisingly nontrivial and relatively unstudied in biomolecular control. The methods developed here connect different existing fields and approaches on the problem of biomolecular control, and hold promise for systematic chemical reaction controller synthesis as well as analysis.

We consider the dynamics of chemical reaction networks under the assumption of mass-action kinetics. We show that there exist reaction networks \({\mathcal{R}}\) for which the reaction rate constants are not uniquely identifiable, even if we are given complete information on the dynamics of concentrations for all chemical species of \({\mathcal{R}}\) . Also, we show that there exist reaction networks \({\mathcal{R}}_1 \neq {\mathcal{R}}_2\) such that their dynamics are identical under appropriate choices of reaction rate constants, and present theorems that characterize the properties of \({\mathcal{R}}\) , \({\mathcal{R}}_1\) , \({\mathcal{R}}_2\) that make this possible. We use these facts to show how we can determine dynamical properties of some chemical networks by analyzing other chemical networks.

Concordant chemical reaction networks and the Species-Reaction Graph

An ecological framework for the analysis of prebiotic chemical reaction networks

This paper discusses the geometric formulation of the dynamics of chemical reaction networks within the port-Hamiltonian formalism [10, 9, 6]. The basic idea dates back to the innovative work of Oster, Perselson and Katchalsky [8, 7]. The main contribution concerns the formulation of a Dirac structure based on the stoichiometric matrix, which is underlying the port-Hamiltonian formulation. Interaction with the environment is modelled through the boundary metabolites and their boundary fluxes and affinities. This allows a compositional view on chemical reaction network dynamics.KeywordsReaction NetworkResistive RelationDirac StructureBond GraphStoichiometric MatrixThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We prove a sample path Large Deviation Principle (LDP) for a class of jump processes whose rates are not uniformly Lipschitz continuous in phase space. Building on it, we further establish the corresponding Wentzell–Freidlin (W-F) (infinite time horizon) asymptotic theory. These results apply to jump Markov processes that model the dynamics of chemical reaction networks under mass action kinetics, on a microscopic scale. We provide natural sufficient topological conditions for the applicability of our LDP and W-F results. This then justifies the computation of nonequilibrium potential and exponential transition time estimates between different attractors in the large volume limit, for systems that are beyond the reach of standard chemical reaction network theory.

The Species-Reaction Graph is a diagrammatic representation of a reaction network that closely resembles those commonly used to depict biochemical pathways. We shall see that a network’s Species-Reaction Graph often carries an extraordinary amount of far-from-obvious information about how the network might behave. In fact, the theorems in this chapter will tell us a great deal about behavior across the entire reaction network landscape, in particular about why dull, stable behavior is more prevalent than one might expect within a mathematical macrocosm so rife with nonlinearity.

The effect of conservation on the dynamics of chemical reaction networks