arXiv subjects

Nicola Vassena

Publications and source records attributed to Nicola Vassena.

At least 19 recordsLinked to original sources

arXiv2026

Minimality in Reflexive and Stoichiometric Autocatalysis

Autocatalysis, the ability of a chemical subsystem to sustain its own constituents when supplied with sufficient food molecules, has been closely related to the origin of life on Earth. Emerging from Wilhelm Ostwald's considerations about an explicit autocatalytic reaction, different notions of autocatalysis have been developed over the years. The two most prominent are reflexively autocatalytic F-generated sets (RAFs) and stoichiometric autocatalysis. After having shown that each RAFs is, under reasonable conditions, in general stoichiometrically autocatalytic, we examine here the relationship between the two notions of minimality: irreducible RAFs and autocatalytic cores. To this end, we overcome the obstacle that RAFs and stoichiometric autocatalysis have been formalized in distinct systems of chemical reactions, i.e., catalytic reaction systems (CRS) and chemical reaction networks (CRNs), respectively. We show that reactions in a CRS constitute equivalence classes of reactions of the corresponding CRNs w.r.t. their specific catalyzations. Using the fact that CRN and CRS can be canonically identified whenever each CRS reaction is associated with a single catalyzation, we demonstrate that the Kőnig graph of a monocatalyzed, irreducible RAF is composed of strong blocks devoid of food and waste species that are separated by reaction vertices, each of which contains an autocatalytic core. In fact, a single irreducible RAF can, in general, contain exponentially many autocatalytic cores.

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arXiv2026

Sequential and distributive dual futile cycle: Hopf bifurcation can occur under parameter-rich kinetics but cannot occur under mass action kinetics

This paper establishes that the system of ordinary differential equations arising from the sequential and distributive dual futile cycle has the structural capacity for Hopf bifurcations, whenever it is endowed with general parameter-rich kinetics, but it loses such capacity if it is endowed with mass action kinetics. The proof of the latter fact relies on a Routh-Hurwitz approach, improved by few preliminary structural considerations, but a decisive contribution came from ChatGPT Sol 5.6, which provided a nontrivial positivity certificate. A second purpose of the paper is indeed to document such AI-contribution, exploiting this well-studied toy-model, and to offer an example of the utilizability of such tools in the context of computer algebra and positivity certificates for large polynomials.

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arXiv2026

Layered mixed matrices and reaction networks

The purpose of this work is twofold. In the first part, we consider layered mixed matrices introduced by Murota, relate them to existing notions in combinatorial commutative algebra, and investigate the irreducibility of their determinants. Furthermore, for a layered mixed matrix in combinatorial canonical form, we determine the sparsity structure of its inverse. That is, we characterize which entries of the inverse are nonzero. In the second part, we establish for the first time a formal connection between these algebraic results and the theory of buffering structures for reaction networks developed by Mochizuki and Okada. We identify the lattice of buffering structures with the lattice of order ideals of the block poset of the combinatorial canonical form of the associated layered mixed matrix. This allows us to characterize the reducibility of the symbolic Jacobian determinant as a polynomial in the reaction-rate derivatives, as well as the nonzero sensitivity responses of species concentrations to reaction-rate perturbations.

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arXiv2026

Bridging two theoretical frameworks of autocatalysis: RAF sets and stoichiometric autocatalysis

Autocatalysis lies at the heart of many (bio)chemical processes and is key to processes leading up to the origin of life. Two seemingly very different formalisms have emerged that define autocatalysis. Kauffman introduced collective autocatalysis to describe systems of molecules that mutually catalyze each other's formation, emphasizing the self-sustaining character of autocatalytic systems. This view is mathematically formalized in the theory of Reflexively Autocatalytic and Food-generated sets (RAF). In parallel, stoichiometric autocatalysis emerged from the theory of Chemical Reaction Networks (CRN), focusing on the net-productive, self-amplifying character of autocatalytic subnetworks. These two frameworks have coexisted independently in the literature, since RAF theory considers each reaction as explicitly catalyzed, while the CRN approach often excludes explicitly catalyzed reactions altogether. Nevertheless, both frameworks describe reaction networks and thus admit a common mathematical representation in terms of stoichiometric matrices. We highlight this connection and show that the two formalisms are less disparate than they might appear. To illustrate this point we prove that, under mild and general conditions, any RAF is stoichiometrically autocatalytic.

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arXiv2026

Global Continuation of Stable Periodic Orbits in Systems of Competing Predators

We develop a continuation technique to obtain global families of stable periodic orbits, delimited by transcritical bifurcations at both ends. To this end, we formulate a zero-finding problem whose zeros correspond to families of periodic orbits. We then define a Newton-like fixed-point operator and establish its contraction near a numerically computed approximation of the family. To verify the contraction, we derive sufficient conditions expressed as inequalities on the norms of the fixed-point operator, and involving the numerical approximation. These inequalities are then rigorously checked by the computer via interval arithmetic. To show the efficacy of our approach, we prove the existence of global families in an ecosystem with Holling's type II functional response, and thereby solve a stable connection problem proposed by Butler and Waltman in 1981. Our method does not rely on restricting the choice of parameters and is applicable to many other systems that numerically exhibit global families.

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arXiv2026

Enumeration of Autocatalytic Subsystems in Large Chemical Reaction Networks

Autocatalysis is an important feature of metabolic networks, contributing crucially to the self-maintenance of organisms. Autocatalytic subsystems of chemical reaction networks (CRNs) are characterized in terms of algebraic conditions on submatrices of the stoichiometric matrix. Here, we derive sufficient conditions for subgraphs supporting irreducible autocatalytic systems in the bipartite Kőnig representation of the CRN. On this basis, we develop an efficient algorithm to enumerate autocatalytic subnetworks and, as a special case, autocatalytic cores, i.e., minimal autocatalytic subnetworks, in full-size metabolic networks. The same algorithmic approach can also be used to determine autocatalytic cores only. As a showcase application, we provide a complete analysis of autocatalysis in the core metabolism of E. coli and enumerate irreducible autocatalytic subsystems of limited size in full-fledged metabolic networks of E. coli, human erythrocytes, and Methanosarcina barkeri (Archea). The mathematical and algorithmic results are accompanied by software enabling the routine analysis of autocatalysis in large CRNs.

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arXiv2026

Mathematical modeling and analysis of the Notch-Delta pathway

In this paper mathematical models for the evolutionary conserved Notch-Delta pathway are developed and analyzed in order to better understand how two neighboring biological cells can become different. We pursue a structure-based stoichiometric type of approach, such that no specific reaction kinetics have to be defined. Only their dependencies on the relevant species participating in the model network are taken into account. Reaction networks and their related systems of ODEs are presented and analyzed with respect to their capacity for symmetry-induced bifurcations. The possibility to obtain a singular Jacobian is analyzed symbolically. This approach is valid for parameter-rich kinetics, where the parametrization of the steady-state fluxes and of the first derivatives of the reaction rates evaluated at the steady state are independent. In this context, also with the help of abstract minimal models, we could mathematically identify some of the Notch pathway's features being more relevant than others.

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arXiv2026

Autocatalytic Cores in Reaction Networks with Explicit Catalysis

Autocatalytic cores are minimal units in reaction networks (RNs) responsible for the emergence of autocatalysis. In the absence of explicit catalysis, i.e., when an entity appears both as reactant and product in the same reaction, they are known to be encoded by square submatrices of the stoichiometric matrix whose columns can be reordered as an irreducible child-selection (CS) matrix with negative diagonal and nonnegative off-diagonal (Metzler matrix). In the bipartite Koenig graph representing the RN, these CS matrices can be identified by fluffles, i.e., strong blocks with an identical number of entity and reaction vertices that have out- and in-degree 1, respectively. Here, we adapt the concepts derived for autocatalytic cores to RNs with explicitly catalyzed reactions, which emerge as digons, i.e., elementary circuits in the Koenig graph of length 2. In this setting, we confirm that an inspection of the stoichiometric matrix alone is inconclusive concerning the presence and number of autocatalytic cores, requiring a more delicate algebraic analysis. Nevertheless, this generalization preserves both the graph and the matrix representation as fluffles and irreducible Metzler CS matrices, respectively, although the diagonal is no longer necessarily strictly negative. We introduce the notion of hard autocatalytic cores, i.e. those that do not yield other autocatalytic cores upon inclusion of all reverse reactions. Finally, we consider the case of unit stoichiometries and show that each autocatalytic core can be constructed as the superposition of at most 2 elementary circuits. In particular, autocatalytic cores involving explicitly catalyzed reactions always contain a spanning subgraph consisting of a single elementary circuit together with a simple entity-to-reaction chord. Moreover, we identify the essentially unique example for which at least two circuits are required.

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arXiv2025

BiRNe: Symbolic bifurcation analysis of reaction networks with Python

Computer algebra methods for analyzing reaction networks often rely on the assumption of mass-action kinetics, which transform the governing ODEs into polynomial systems amenable to techniques such as Gröbner basis computation and related algebraic tools. However, these methods face significant computational complexity, limiting their applicability to relatively small networks involving only a handful of species. In contrast, building on recent theoretical advances, we introduce here \textsc{BiRNe} (BIfurcations in Reaction NEtworks) Python module, which relies on a symbolic approach designed to detect bifurcations in larger reaction networks (up to 10-20 species, depending on the network's connectivity) equipped with parameter-rich kinetics. This class includes enzymatic kinetics such as Michaelis--Menten, ligand-binding kinetics like Hill functions, and generalized mass-action kinetics. For a given network, the current algorithm identifies all minimal autocatalytic subnetworks and fully characterizes the presence of bifurcations associated with zero eigenvalues, thus determining whether the network admits multistationarity. It also detects oscillatory bifurcations arising from positive-feedback structures, capturing a significant class of possible oscillations.

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arXiv2025

Stoichiometric recipes for periodic oscillations in reaction networks

Oscillatory chemical reactions are functional components in a variety of biological contexts. In chemistry, the construction and identification of even rudimentary oscillators remain elusive and lack a general framework. Using parameter-rich kinetics - a methodology enabling the disentanglement of parametric dependencies from structural analysis - we investigate the stoichiometry of chemical oscillators. We introduce the concept of oscillatory cores: minimal subnetworks that guarantee the potential for oscillations in any reaction network containing them. These cores fall into two classes, depending on whether they involve positive or negative feedback. In particular, the latter class unveils a family of oscillators - yet to be synthesized - that require a minimum number of reaction steps to exhibit oscillations, a phenomenon we refer to as the principle of length. We identify several mechanisms through which catalysis promotes oscillations: (I) furnishing instability (e.g. autocatalysis), (II) lifting dependencies, (III) lowering length thresholds. Notwithstanding this mechanistic ubiquity, we show that oscillators can also be realized without employing any catalysis. Our results highlight branches of chemistry where oscillators are likely to arise by chance, suggest new strategies for their design, and point to novel classes of oscillators yet to be realized experimentally.

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arXiv2025

Mass action systems: two criteria for Hopf bifurcation without Hurwitz

We state two sufficient criteria for periodic oscillations in mass action systems. Neither criterion requires a computation of the Hurwitz determinants. Instead, both criteria exploit the linear algebra concepts of $D$-stability and $P$-matrices. The criteria are complementary: the first is based on a stable matrix that is not a $P^-$ matrix, while the second is based on a $P^-$ matrix that is not stable. In analogy, a qualitatively different interpretation follows: the first criterion relates to positive feedback in the network, while the second concerns negative feedback. We present examples that showcase the applicability of both criteria. As a final independent remark, we prove that for the special case of fully-open networks, the capacity for Hopf bifurcation is just equivalent to the capacity for a steady-state with a complex pair of eigenvalues with positive-real part.

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arXiv2024

Hybrid Bifurcations: Periodicity from Eliminating a Line of Equilibria

We describe a new mechanism that triggers periodic orbits in smooth dynamical systems. To this end, we introduce the concept of hybrid bifurcations: Such bifurcations occur when a line of equilibria with an exchange point of normal stability vanishes. Our main result is the existence and stability criteria of periodic orbits that bifurcate from breaking a line of equilibria. As an application, we obtain stable periodic coexistent solutions in an ecosystem for two competing predators with Holling's type II functional response.

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arXiv2024

Finding bifurcations in mathematical epidemiology via reaction network methods

Mathematical Epidemiology (ME) shares with Chemical Reaction Network Theory (CRNT) the basic mathematical structure of its dynamical systems. Despite this central similarity, methods from CRNT have been seldom applied to solving problems in ME. We explore here the applicability of CRNT methods to find bifurcations at endemic equilibria of ME models. We adapt three CRNT methods to the features of ME. First, we prove that essentially all ME models admit Hopf bifurcations for certain monotone choices of the interaction functions. Second, we offer a parametrization of equilibria Jacobians of ME systems where few interactions are not in mass action form. Third, for a quite general class of models, we show that periodic oscillations in closed systems imply periodic oscillations when demography is added. Finally, we apply such results to two families of networks: a general SIR model with a nonlinear force of infection and treatment rate and a recent SIRnS model with a gradual increase in infectiousness. We give both necessary conditions and sufficient conditions for the occurrence of bifurcations at endemic equilibria of both families.

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arXiv2024

Unstable Cores are the source of instability in chemical reaction networks

In biochemical networks, complex dynamical features such as superlinear growth and oscillations are classically considered a consequence of autocatalysis. For the large class of parameter-rich kinetic models, which includes Generalized Mass Action kinetics and Michaelis-Menten kinetics, we show that certain submatrices of the stoichiometric matrix, so-called unstable cores, are sufficient for a reaction network to admit instability and potentially give rise to such complex dynamical behavior. The determinant of the submatrix distinguishes unstable-positive feedbacks, with a single real-positive eigenvalue, and unstable-negative feedbacks without real-positive eigenvalues. Autocatalytic cores turn out to be exactly the unstable-positive feedbacks that are Metzler matrices. Thus there are sources of dynamical instability in chemical networks that are unrelated to autocatalysis. We use such intuition to design non-autocatalytic biochemical networks with superlinear growth and oscillations.

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arXiv2023

Symbolic hunt of instabilities and bifurcations in reaction networks

The localization of bifurcations in large parametric systems is still a challenge where the combination of rigorous criteria and informal intuition is often needed. With this motivation, we address symbolically the Jacobian matrix of reaction networks with general kinetics. More specifically, we consider any nonzero partial derivative of a reaction rate as a free positive symbol. The main tool are the Child-Selections: injective maps that associate to a species $m$ a reaction $j$ where $m$ participates as reactant. Firstly, we employ a Cauchy-Binet analysis and we structurally express any coefficient of the characteristic polynomial of the Jacobian in terms of Child-Selections. In particular, we fully characterize sign-changes of any of the coefficients. Secondly, we prove that the (in)stability of the Jacobian is inherited from the (in)stability of simpler submatrices identified by the Child-Selections. Thirdly, we provide sufficient conditions for purely imaginary eigenvalues of the Jacobian, hinting at Hopf bifurcation and oscillatory behavior. All conditions are in terms of signs of integer stoichiometric submatrices identified by the Child-Selections {and do not require any Hurwitz-type computaton}. Finally, we focus on systems endowed with Michaelis-Menten kinetics and we show that any symbolic realization of the Jacobian matrix can be achieved at a fixed equilibrium by a proper choice of the kinetic constants.

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arXiv2023

Structural conditions for saddle-node bifurcations in chemical reaction networks

Motivated by investigating multistationarity in biochemical systems, we address saddle-node bifurcations for chemical reaction networks endowed with general kinetics. At positive equilibria, we identify structural network conditions that guarantee the bifurcation behavior, and we develop a method to identify the proper bifurcation parameters. As a relevant example, we explicitly provide such bifurcation parameters for Michaelis-Menten and Hill kinetics. Examples of applications include reversible feedback cycles, the central carbon metabolism of Escherichia coli, and autocatalytic networks.

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arXiv2022

Structural obstruction to the simplicity of the eigenvalue zero in chemical reaction networks

Multistationarity is the property of a system to exhibit two distinct equilibria (steady-states) under otherwise identical conditions, and it is a phenomenon of recognized importance for biochemical systems. Multistationarity may appear in the parameter space as a consequence of saddle-node bifurcations, which necessarily require a simple eigenvalue zero of the Jacobian, at the bifurcating equilibrium. Matrices with a simple eigenvalue zero are generic in the set of singular matrices: any system whose Jacobian has an algebraically multiple eigenvalue zero can be perturbed to a system whose Jacobian has a simple eigenvalue zero. Thus, one would expect that in applications singular Jacobians are always with a simple eigenvalue zero. However, chemical reaction networks typically consider a fixed network structure, while the freedom rests with the various and different choices of kinetics. Here we present an example of a chemical reaction network, whose Jacobian is either nonsingular or has an algebraically multiple eigenvalue zero. The structural obstruction to the simplicity of the eigenvalue zero is based on the network alone, and it is independent of the value of concentrations and the choice of kinetics. This in particular constitutes an obstruction to standard saddle-node bifurcations.

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arXiv2021

Sign-sensitivity of metabolic networks: which structures determine the sign of the responses

Perturbations are ubiquitous in metabolism. A central tool to understand and control their influence on metabolic networks is sensitivity analysis, which investigates how the network responds to external perturbations. We follow here a structural approach: the analysis is based on the network stoichiometry only and it does not require any quantitative knowledge of the reaction rates. We consider perturbations of reaction rates and metabolite concentrations, at equilibrium, and we investigate the responses in the network. For general metabolic systems, this paper focuses on the sign of the responses, i.e. whether a response is positive, negative or whether its sign depends on the parameters of the system. In particular, we identify and describe the subnetworks that are the main players in the sign description. These subnetworks are associated to certain kernel vectors of the stoichiometric matrix and are thus independent from the chosen kinetics.

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