arXiv subjects

Gheorghe Craciun

Publications and source records attributed to Gheorghe Craciun.

At least 19 recordsLinked to original sources

arXiv2026

A flux-based approach for analyzing the disguised toric locus of reaction networks

Dynamical systems with polynomial right-hand sides are very important in various applications, e.g., in biochemistry and population dynamics. The mathematical study of these dynamical systems is challenging due to the possibility of multistability, oscillations, and chaotic dynamics. One important tool for this study is the concept of reaction systems, which are dynamical systems generated by reaction networks for some choices of parameter values. Among these, disguised toric systems are remarkably stable: they have a unique attracting fixed point, and cannot give rise to oscillations or chaotic dynamics. The computation of the set of parameter values for which a network gives rise to disguised toric systems (i.e., the disguised toric locus of the network) is an important but difficult task. We introduce new ideas based on network fluxes for studying the disguised toric locus. We prove, under mild assumptions, that the disguised toric locus of any network $G$ is a contractible manifold with boundary, and introduce an associated graph $G^{\max}$ that characterizes its interior. These theoretical tools allow us, for the first time, to compute the full disguised toric locus for many networks of interest.

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arXiv2026

Extinction in Reaction Network Models

In this paper, we study extinction in dynamical systems generated by reaction networks. We introduce two notions: weak extinction and strong extinction, and relate them to the structure of the underlying network through Lyapunov functions and LaSalle's invariance principle. In particular, for all deficiency-zero networks that are not weakly reversible, we provide a geometric construction of linear Lyapunov functions. Using these functions, we establish that if these networks have bounded invariant subspaces, then they must exhibit weak extinction within every such subspace. Also, for linear networks that are not weakly reversible, we show that every species outside a terminal strongly connected component undergoes strong extinction. Moreover, in order to further emphasize the difference between weak and strong extinction, we construct an example of a reaction system (based on the Ivanova network) that exhibits weak extinction for all the species, but does not exhibit strong extinction in any species.

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arXiv2025

Asymptotic Robustness in Biochemical Systems

Living systems maintain stable internal states despite environmental fluctuations. Absolute concentration robustness (ACR) is a striking homeostatic phenomenon in which the steady-state concentration of a species remains invariant despite changes in total supply. Although experimental studies have reported approximate-but not exact-robustness in steady-state concentrations, such behavior has often been attributed to exact ACR motifs perturbed by measurement noise or minor reactions, rather than recognized as a structural property of a network itself. Here, we introduce a previously underappreciated phenomenon, namely asymptotic ACR (aACR): approximate robustness can emerge solely from the network structure, without requiring exact ACR motifs or negligible parameters. We find that aACR is more pervasive than classical ACR, as demonstrated in systems such as the Escherichia coli EnvZ-OmpR osmoregulation system and a futile cycle. Furthermore, we prove that such ubiquity stems solely from network structure without fine-tuning of kinetic parameters. The notion of aACR provides a rigorous and practical tool to analyze robust responses in broad biochemical systems.

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arXiv2025

Planar chemical reaction systems with algebraic and non-algebraic limit cycles

The Hilbert number $H(n)$ is defined as the maximum number of limit cycles of a planar autonomous system of ordinary differential equations (ODEs) with right-hand sides containing polynomials of degree at most $n \in {\mathbb N}$. The dynamics of chemical reaction systems with two chemical species can be (under mass-action kinetics) described by such planar autonomous ODEs, where $n$ is equal to the maximum order of the chemical reactions in the system. Analogues of the Hilbert number $H(n)$ for three different classes of chemical reaction systems are investigated: (i) chemical systems with reactions up to the $n$-th order; (ii) systems with up to $n$-molecular chemical reactions; and (iii) weakly reversible chemical reaction networks. In each case (i), (ii) and (iii), the question on the number of limit cycles is considered. Lower bounds on the modified Hilbert numbers are provided for both algebraic and non-algebraic limit cycles. Furthermore, given a general algebraic curve $h(x,y)=0$ of degree $n_h \in {\mathbb N}$ and containing one or more ovals in the positive quadrant, a chemical system is constructed which has the oval(s) as its stable algebraic limit cycle(s). The ODEs describing the dynamics of the constructed chemical system contain polynomials of degree at most $n=2\,n_h+1.$ Considering $n_h \ge 4,$ the algebraic curve $h(x,y)=0$ can contain multiple closed components with the maximum number of ovals given by Harnack's curve theorem as $1+(n_h-1)(n_h-2)/2$, which is equal to 4 for $n_h=4.$ Algebraic curve $h(x,y)=0$ with $n_h=4$ and the maximum number of four ovals is used to construct a chemical system which has four stable algebraic limit cycles.

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arXiv2025

The Computation of the Disguised Toric Locus of Reaction Networks

Mathematical models of reaction networks can exhibit very complex dynamics, including multistability, oscillations, and chaotic dynamics. On the other hand, under some additional assumptions on the network or on parameter values, these models may actually be toric dynamical systems, which have remarkably stable dynamics. The concept of disguised toric dynamical system" was introduced in order to describe the phenomenon where a reaction network generates toric dynamics without actually being toric; such systems enjoy all the stability properties of toric dynamical systems but with much fewer restrictions on the networks and parameter values. The \emph{disguised toric locus} is the set of parameter values for which the corresponding dynamical system is a disguised toric system. Here we focus on providing a generic and efficient method for computing the dimension of the disguised toric locus of reaction networks. Additionally, we illustrate our approach by applying it to some specific models of biological interaction networks, including Brusselator-type networks, Thomas-type networks, and circadian clock networks.

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arXiv2025

Weakly reversible deficiency zero realizations of reaction networks

We prove that if a given reaction network $\mathcal{N}$ has a weakly reversible deficiency zero realization for all choice of rate constants, then there exists a $\textit{unique}$ weakly reversible deficiency zero network $\mathcal{N}'$ such that $\mathcal{N}$ is realizable by $\mathcal{N}'$. Additionally, we propose an algorithm to find this weakly reversible deficiency zero network $\mathcal{N}'$ when it exists.

q-bio.MN↗

arXiv2024

Generalized Lotka-Volterra Systems and Complex Balanced Polyexponential Systems

We study the global stability of generalized Lotka-Volterra systems with generalized polynomial right-hand side, without restrictions on the number of variables or the polynomial degree, including negative and non-integer degree. We introduce polyexponential dynamical systems, which are equivalent to the generalized Lotka-Volterra systems, and we use an analogy to the theory of mass-action kinetics to define and analyze complex balanced polyexponential systems, and implicitly analyze complex balanced generalized Lotka-Volterra systems. We prove that complex balanced generalized Lotka-Volterra systems have globally attracting states, up to standard conservation relations, which become linear for the associated polyexponential systems. In particular, complex balanced generalized Lotka-Volterra systems cannot give rise to periodic solutions, chaotic dynamics, or other complex dynamical behaviors. We describe a simple sufficient condition for complex balance in terms of an associated graph structure, and we use it to analyze specific examples.

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arXiv2024

The Dimension of the Disguised Toric Locus of a Reaction Network

Under mass-action kinetics, complex-balanced systems emerge from biochemical reaction networks and exhibit stable and predictable dynamics. For a reaction network $G$, the associated dynamical system is called $\textit{disguised toric}$ if it can yield a complex-balanced realization on a possibly different network $G_1$. This concept extends the robust properties of toric systems to those that are not inherently toric. In this work, we study the $\textit{disguised toric locus}$ of a reaction network - i.e., the set of positive rate constants that make the corresponding mass-action system disguised toric. Our primary focus is to compute the exact dimension of this locus. We subsequently apply our results to Thomas-type and circadian clock models.

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arXiv2023

The toric locus of a reaction network is a smooth manifold

We show that the toric locus of a reaction network is a smoothly embedded submanifold of the Euclidean space. More precisely, we prove that the toric locus of a reaction network is the image of an embedding and it is diffeomorphic to the product space between the affine invariant polyhedron of the network and its set of complex-balanced flux vectors. Moreover, we prove that within each affine invariant polyhedron, the complex-balanced equilibrium depends smoothly on the parameters (i.e., reaction rate constants). We also show that the complex-balanced equilibrium depends smoothly on the initial conditions.

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arXiv2023

Power-engine-load form for dynamic absolute concentration robustness

In a reaction network, the concentration of a species with the property of dynamic absolute concentration robustness (dynamic ACR) converges to the same value independent of the overall initial values. This property endows a biochemical network with output robustness and therefore is essential for its functioning in a highly variable environment. It is important to identify structure of the dynamical system as well as constraints required for dynamic ACR. We propose a power-engine-load form of dynamic ACR and obtain results regarding convergence to the ACR value based on this form.

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arXiv2023

On the Connectivity of the Disguised Toric Locus of a Reaction Network

Complex-balanced mass-action systems are some of the most important types of mathematical models of reaction networks, due to their widespread use in applications, as well as their remarkable stability properties. We study the set of positive parameter values (i.e., reaction rate constants) of a reaction network $G$ that, according to mass-action kinetics, generate dynamical systems that can be realized as complex-balanced systems, possibly by using a different graph $G'$. This set of parameter values is called the disguised toric locus of $G$. The $\mathbb{R}$-disguised toric locus of $G$ is defined analogously, except that the parameter values are allowed to take on any real values. We prove that the disguised toric locus of $G$ is path-connected, and the $\mathbb{R}$-disguised toric locus of $G$ is also path-connected. We also show that the closure of the disguised toric locus of a reaction network contains the union of the disguised toric loci of all its subnetworks.

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arXiv2023

A Lower Bound on the Dimension of the $\mathbb{R}$-Disguised Toric Locus of a Reaction Network

Polynomial dynamical systems (i.e. dynamical systems with polynomial right hand side) are ubiquitous in applications, especially as models of reaction networks and interaction networks. The properties of general polynomial dynamical systems can be very difficult to analyze, due to nonlinearity, bifurcations, and the possibility for chaotic dynamics. On the other hand, toric dynamical systems are polynomial dynamical systems that appear naturally as models of reaction networks, and have very robust and stable properties. A disguised toric dynamical system is a polynomial dynamical system generated by a reaction network $\mathcal N$ and some choice of positive parameters, such that (even though it may not be toric with respect to $\mathcal N$) it has a toric realization with respect to some network $\mathcal N'$. Disguised toric dynamical systems enjoy all the robust stability properties of toric dynamical systems. In this paper, we study a larger set of dynamical systems where the rate constants are allowed to take both positive and negative values. More precisely, we analyze the $\mathbb{R}$-disguised toric locus of a reaction network $\mathcal N$, i.e., the subset in the space rate constants (positive or negative) of $\mathcal N$ for which the corresponding polynomial dynamical system is disguised toric. We focus especially on finding a lower bound on the dimension of the $\mathbb{R}$-disguised toric locus.

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arXiv2023

The structure of the moduli space of toric dynamical systems of a reaction network

We consider toric dynamical systems, which are also called complex-balanced mass-action systems. These are remarkably stable polynomial dynamical systems that arise from the analysis of mathematical models of reaction networks when, under the assumption of mass-action kinetics, they can give rise to complex-balanced equilibria. Given a reaction network, we study the moduli space of toric dynamical systems generated by this network, also called the toric locus of the network. The toric locus is an algebraic variety, and we are especially interested in its topological properties. We show that complex-balanced equilibria depend continuously on the parameter values in the toric locus, and, using this result, we prove that the toric locus has a remarkable product structure: it is homeomorphic to the product of the set of complex-balanced flux vectors and the affine invariant polyhedron of the network. In particular, it follows that the toric locus is a contractible manifold. Finally, we show that the toric locus is invariant with respect to bijective affine transformations of the generating reaction network.

math.AG↗

arXiv2023

The structure of the moduli spaces of toric dynamical systems

We consider complex-balanced mass-action systems, or toric dynamical systems. They are remarkably stable polynomial dynamical systems arising from reaction networks seen as Euclidean embedded graphs. We study the moduli spaces of toric dynamical systems, called the toric locus: given a reaction network, we are interested in the topological structure of the set of parameters giving rise to toric dynamical systems. First we show that the complex-balanced equilibria depend continuously on the parameter values. Using this result, we prove that the toric locus of any toric dynamical system is connected. In particular, we emphasize its product structure: it is homeomorphic to the product of the set of complex-balanced flux vectors and the affine invariant polyhedron. Finally, we show that the toric locus is invariant with respect to bijective affine transformations of the generating reaction network.

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arXiv2023

Weakly reversible deficiency one realizations of polynomial dynamical systems: an algorithmic perspective

Given a dynamical system with polynomial right-hand side, can it be generated by a reaction network that possesses certain properties? This question is important because some network properties may guarantee specific dynamical properties, such as existence or uniqueness of equilibria, persistence, permanence, or global stability. Here we focus on this problem in the context of weakly reversible deficiency one networks. In particular, we describe an algorithm for deciding if a polynomial dynamical system admits a weakly reversible deficiency one realization, and identifying one if it does exist. In addition, we show that weakly reversible deficiency one realizations can be partitioned into mutually exclusive Type I and Type II realizations, where Type I realizations guarantee existence and uniqueness of positive steady states, while Type II realizations are related to stoichiometric generators, and therefore to multistability.

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arXiv2023

Weakly reversible single linkage class realizations of polynomial dynamical systems: an algorithmic perspective

Systems of differential equations with polynomial right-hand sides are very common in applications. In particular, when restricted to the positive orthant, they appear naturally (according to the law of mass-action kinetics) in ecology, population dynamics, as models of biochemical interaction networks, and models of the spread of infectious diseases. On the other hand, their mathematical analysis is very challenging in general; in particular, it is very difficult to answer questions about the long-term dynamics of the variables (species) in the model, such as questions about persistence and extinction. Even if we restrict our attention to mass-action systems, these questions still remain challenging. On the other hand, if a polynomial dynamical system has a weakly reversible single linkage class ($W\!R^1$) realization, then its long-term dynamics is known to be remarkably robust: all the variables are persistent (i.e., no species goes extinct), irrespective of the values of the parameters in the model. Here we describe an algorithm for finding $W\!R^1$ realizations of polynomial dynamical systems, whenever such realizations exist.

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arXiv2022

Foundations of Static and Dynamic Absolute Concentration Robustness

Absolute Concentration Robustness (ACR) was introduced by Shinar and Feinberg as robustness of equilibrium species concentration in a mass action dynamical system. Their aim was to devise a mathematical condition that will ensure robustness in the function of the biological system being modeled. The robustness of function rests on what we refer to as empirical robustness -- the concentration of a species remains unvarying, when measured in the long run, across arbitrary initial conditions. Even simple examples show that the ACR notion introduced in Shinar and Feinberg (here referred to as static ACR) is neither necessary nor sufficient for empirical robustness. To make a stronger connection with empirical robustness, we define dynamic ACR, a property related to long-term, global dynamics, rather than only to equilibrium behavior. We discuss general dynamical systems with dynamic ACR properties as well as parametrized families of dynamical systems related to reaction networks. We find necessary and sufficient conditions for dynamic ACR in complex balanced reaction networks, a class of networks that is central to the theory of reaction networks.

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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.

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