arXiv subjects

Gheorghe Craciun

Publications and source records attributed to Gheorghe Craciun.

At least 37 records · Page 2Linked to original sources

arXiv2022

Reaction Network Motifs for Static and Dynamic Absolute Concentration Robustness

Networks with absolute concentration robustness (ACR) have the property that a translation of a coordinate hyperplane either contains all steady states (static ACR) or attracts all trajectories (dynamic ACR). The implication for the underlying biological system is robustness in the concentration of one of the species independent of the initial conditions as well as independent of the concentration of all other species. Identifying network conditions for dynamic ACR is a challenging problem. We lay the groundwork in this paper by studying small reaction networks, those with 2 reactions and 2 species. We give a complete classification by ACR properties of these minimal reaction networks. The dynamics is rich even within this simple setting. Insights obtained from this work will help illuminate the properties of more complex networks with dynamic ACR.

math.DS↗

arXiv2022

An algorithm for finding weakly reversible deficiency zero realizations of polynomial dynamical systems

Systems of differential equations with polynomial right-hand sides are very common in applications. On the other hand, their mathematical analysis is very challenging in general, due to the possibility of complex dynamics: multiple basins of attraction, oscillations, and even chaotic dynamics. Even if we restrict our attention to mass-action systems, all of these complex dynamical behaviours are still possible. On the other hand, if a polynomial dynamical system has a weakly reversible deficiency zero ($WR_0$) realization, then its dynamics is known to be remarkably simple: oscillations and chaotic dynamics are ruled out and, up to linear conservation laws, there exists a single positive steady state, which is asymptotically stable. Here we describe an algorithm for finding $WR_0$ realizations of polynomial dynamical systems, whenever such realizations exist.

math.DS↗

arXiv2022

Multistationarity in cyclic sequestration-transmutation networks

We consider a natural class of reaction networks which consist of reactions where either two species can inactivate each other (i.e., sequestration), or some species can be transformed into another (i.e., transmutation), in a way that gives rise to a feedback cycle. We completely characterize the capacity of multistationarity of these networks. This is especially interesting because such networks provide simple examples of "atoms of multistationarity", i.e., minimal networks that can give rise to multiple positive steady states

math.DS↗

arXiv2022

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.

math.AG↗

arXiv2021

Minimal invariant regions and minimal globally attracting regions for variable-k reaction systems

The structure of invariant regions and globally attracting regions is fundamental to understanding the dynamical properties of reaction network models. We describe an explicit construction of the minimal invariant regions and minimal globally attracting regions for dynamical systems consisting of two reversible reactions, where the rate constants are allowed to vary in time within a bounded interval.

math.DS↗

arXiv2021

A reaction network approach to the convergence to equilibrium of quantum Boltzmann equations for Bose gases

When the temperature of a trapped Bose gas is below the Bose-Einstein transition temperature and above absolute zero, the gas is composed of two distinct components: the Bose-Einstein condensate and the cloud of thermal excitations. The dynamics of the excitations can be described by quantum Boltzmann models. We establish a connection between quantum Boltzmann models and chemical reaction networks. We prove that the discrete differential equations for these quantum Boltzmann models converge to an equilibrium point. Moreover, this point is unique for all initial conditions that satisfy the same conservation laws. In the proof, we then employ a toric dynamical system approach, similar to the one used to prove the global attractor conjecture, to study the convergence to equilibrium of quantum kinetic equations, derived in [49,50].

math.AP↗

arXiv2021

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.

math.DS↗

arXiv2021

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 reversibly deficiency zero ($\textit{WR}_\textit{0}$) realization is the simplest possible one. Moreover, while mass-action systems can exhibit practically any dynamical behavior, including multistability, oscillations, and chaos, $WR_0$ 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.

math.DS↗

arXiv2021

Homeostasis and injectivity: a reaction network perspective

Homeostasis is a mechanism by which a feature can remain invariant with change in external parameters. We adopt the definition of homeostasis in the context of singularity theory. We make a connection between homeostasis and the theory of injective reaction networks. In particular, we show that a reaction network cannot exhibit homeostasis if a modified reaction network (which we call the homeostasis-associated reaction network) is injective. We provide examples of reaction networks which can or cannot exhibit homeostasis by analyzing the injectivity of the homeostasis-associated reaction network.

math.DS↗

arXiv2021

Autocatalytic Networks: An Intimate Relation between Network Topology and Dynamics

We study a family of networks of autocatalytic reactions, which we call hyperchains, that are a generalization of hypercycles. Hyperchains, and the associated dynamical system called replicator equations, are a possible mechanism for macromolecular evolution and proposed to play a role in abiogenesis, the origin of life from prebiotic chemistry. The same dynamical system also occurs in evolutionary game dynamics, genetic selection, and as Lotka-Volterra equations of ecology. An arrow in a hyperchain encapsulates the enzymatic influence of one species on the autocatalytic replication of another. We show that the network topology of a hyperchain, which captures all such enzymatic influences, is intimately related to the dynamical properties of the mass action system it generates. Dynamical properties such as existence, uniqueness and stability of a positive equilibrium as well as permanence, are determined by graph-theoretic properties such as existence of a spanning linear subgraph, being unrooted, being cyclic, and Hamiltonicity.

math.DS↗

arXiv2021

Single-Target Networks

We characterize the dynamics of all single-target networks under mass-action kinetics: either the system is (i) globally stable for all choice of rate constants (in fact, dynamically equivalent to a detailed-balanced system) or (ii) 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, 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.

math.DS↗

arXiv2020

Autocatalytic systems and recombination: a reaction network perspective

Autocatalytic systems are very often incorporated in the "origin of life" models, a connection that has been analyzed in the context of the classical hypercycles introduced by Manfred Eigen. We investigate the dynamics of certain networks called bimolecular autocatalytic systems. In particular, we consider the dynamics corresponding to the relative populations in these networks, and show that they can be analyzed by studying well-chosen autonomous polynomial dynamical systems. Moreover, we find that one can use results from reaction network theory to prove persistence and permanence of several types of bimolecular autocatalytic systems called autocatalytic recombination networks.

math.DS↗

arXiv2020

Weakly reversible mass-action systems with infinitely many positive steady states

We show that weakly reversible mass-action systems can have a continuum of positive steady states, coming from the zeroes of a multivariate polynomial. Moreover, the same is true of systems whose underlying reaction network is reversible and has a single connected component. In our construction, we relate operations on the reaction network to the multivariate polynomial occurring as a common factor in the system of differential equations.

q-bio.MN↗

arXiv2020

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.

math.DS↗

arXiv2020

Minimal invariant regions and minimal globally attracting regions for toric differential inclusions

Toric differential inclusions occur as key dynamical systems in the context of the Global Attractor Conjecture. We introduce the notions of minimal invariant regions and minimal globally attracting regions for toric differential inclusions. We describe a procedure for constructing explicitly the minimal invariant and minimal globally attracting regions for two-dimensional toric differential inclusions. In particular, we obtain invariant regions and globally attracting regions for two-dimensional weakly reversible or endotactic dynamical systems (even if they have time-dependent parameters).

math.DS↗

arXiv2020

Delay stability of reaction systems

Delay differential equations are used as a model when the effect of past states has to be taken into account. In this work we consider delay models of chemical reaction networks with mass action kinetics. We obtain a sufficient condition for absolute delay stability of equilibrium concentrations, i.e., local asymptotic stability independent of the delay parameters. Several interesting examples on sequestration networks with delays are presented.

math.DS↗

arXiv2020

A reaction network approach to the theory of acoustic wave turbulence

We propose a new approach to study the long time dynamics of the wave kinetic equation in the statistical description of acoustic turbulence. The approach is based on rewriting the discrete version of the wave kinetic equation in the form of a chemical reaction network, then employing techniques used to study the Global Attractor Conjecture to investigate the long time dynamics of the newly obtained chemical system. We show that the solution of the chemical system converges to an equilibrium exponentially in time. In addition, a resonance broadening modification of the acoustic wave kinetic equation is also studied with the same technique. For the near-resonance equation, if the resonance broadening frequency is larger than a threshold, the solution of the system goes to infinity as time evolves.

math-ph↗

arXiv2019

An efficient characterization of complex-balanced, detailed-balanced, and weakly reversible systems

Very often, models in biology, chemistry, physics, and engineering are systems of polynomial or power-law ordinary differential equations, arising from a reaction network. Such dynamical systems can be generated by many different reaction networks. On the other hand, networks with special properties (such as reversibility or weak reversibility) are known or conjectured to give rise to dynamical systems that have special properties: existence of positive steady states, persistence, permanence, and (for well-chosen parameters) complex balancing or detailed balancing. These last two are related to thermodynamic equilibrium, and therefore the positive steady states are unique and stable. We describe a computationally efficient characterization of polynomial or power-law dynamical systems that can be obtained as complex-balanced, detailed-balanced, weakly reversible, and reversible mass-action systems.

math.DS↗