arXiv subjects

Christian Mazza

Publications and source records attributed to Christian Mazza.

At least 19 recordsLinked to original sources

arXiv2026

Stability and feasibility of Microbial Consumer-Resource Model

Microbial communities are ubiquitous in nature but how they grow on available resources is still poorly understood. Communities are complex systems harboring thousands of microbial species that interact through resource competition. The classical MacArthur consumer-resource model has been shown to underestimate formed community biomass. A recent new microbial consumer-resource model (MiCRM) considers the inclusion of inter-specific interactions mediated by metabolite exchange (cross-feeding) where the various bacterial growth byproducts can be reused by other species for their own growth. We study persistence, feasibility and stability for MiCRM under some simplifying assumptions using slow-fast approximation. We show e.g. the non-persistence of the microbial community when the number of resource species M is smaller than the number of consumer species S. We then study the stability of the slow steady state when the number or survivors S is smaller than M, and show that such equilibria are generically stable. We finally propose a stochastic slow-fast version of the model having fast Poisson steady state and study related extinction events.

q-bio.PE↗

arXiv2025

On the abelian structure of noncompetitive chemical reaction networks

Chemical reaction networks (CRNs) are foundational models for describing complex biochemical processes. We study noncompetitive CRNs, a class of networks whose static states, where the CRN is inactive, are rate independent, and that can implement ReLU neural networks. CRNs of interest in biochemistry and systems biology are embedded in complex networks so that CRNs have to respond to internal and environmental cues. We describe the network's response to such perturbations using a new Markov chain that we call CRN sandpile Markov chain, whose state space is the set of static states. The transition mechanism of the CRN sandpile Markov chain is defined by adding a molecule of a randomly chosen species to a static state, and then letting the CRN state evolve toward a new static state. A central contribution of the present work is the observation that one can associate a natural Abelian Network (AN) to each noncompetitive CRN, and use AN theory to get new mathematical results on noncompetitive CRNs. For noncompetitive CRNs on a finite state space, we use AN theory to get that only a fraction of the static states are recurrent for the CRN sandpile Markov chain. We obtain furthermore that the set of recurrent states is in one to one correspondence with the critical group of the AN, which plays a major role in AN theory. Overall, this work establishes a unified algebraic and probabilistic framework for analyzing the long-term behavior of noncompetitive CRNs. We focus on a special class of noncompetitive CRNs called generalized toppling networks, and obtain new mathematical results both for the CRN and AN settings.

q-bio.MN↗

arXiv2023

Complete mathematical characterization of two simple toggle-switch biological systems

Prokaryotic gene expression is dynamic and noisy, and can lead to phenotypic variations. Previous work has shown that the selection of such variation can be modeled through basic reaction networks described by O.D.E. displaying a bistable behavior. While previous mathematical studies have shown that mono- or bistable behavior depends on the rate of the reactions in the system, no analytical solution of the curves delineating the actual parameter conditions that result in mono or bistability has so far been provided. In this work we provide the first explicit analytical solution for the boundary curve that separates the parameter space defining domains where double positive and double negative feedback loops become bistable.

q-bio.MN↗

arXiv2020

Geometrical and statistical properties of M-estimates of scatter on Grassmann manifolds

We consider data from the Grassmann manifold $G(m,r)$ of all vector subspaces of dimension $r$ of $\mathbb{R}^m$, and focus on the Grassmannian statistical model which is of common use in signal processing and statistics. Canonical Grassmannian distributions $\mathbb{G}_Σ$ on $G(m,r)$ are indexed by parameters $Σ$ from the manifold $\mathcal{M}= Pos_{sym}^{1}(m)$ of positive definite symmetric matrices of determinant $1$. Robust M-estimates of scatter (GE) for general probability measures $\mathcal{P}$ on $G(m,r)$ are studied. Such estimators are defined to be the maximizers of the Grassmannian log-likelihood $-\ell_{\mathcal{P}}(Σ)$ as function of $Σ$. One of the novel features of this work is a strong use of the fact that $\mathcal{M}$ is a CAT(0) space with known visual boundary at infinity $\partial \mathcal{M}$. We also recall that the sample space $G(m,r)$ is a part of $\partial \mathcal{M}$, show the distributions $\mathbb{G}_Σ$ are $SL(m,\mathbb{R})$--quasi-invariant, and that $\ell_{\mathcal{P}}(Σ)$ is a weighted Busemann function. Let $\mathcal{P}_n =(δ_{U_1}+\cdots+δ_{U_n})/n$ be the empirical probability measure for $n$-samples of random i.i.d. subspaces $U_i\in G(m,r)$ of common distribution $\mathcal{P}$, whose support spans $\mathbb{R}^m$. For $Σ_n$ and $Σ_{\mathcal{P}}$ the GEs of $\mathcal{P}_n$ and $\mathcal{P}$, we show the almost sure convergence of $Σ_n$ towards $Σ$ as $n\to\infty$ using methods from geometry, and provide a central limit theorem for the rescaled process $C_n = \frac{m}{tr(Σ_{\mathcal{P}}^{-1} Σ_n)}g^{-1} Σ_n g^{-1}$, where $Σ=gg$ with $g\in SL(m,\mathbb{R})$ the unique symmetric positive-definite square root of $Σ$.

math.ST↗

arXiv2019

Stationary distributions and condensation in autocatalytic CRN

We investigate a broad family of non weakly reversible stochastically modeled reaction networks (CRN), by looking at their steady-state distributions. Most known results on stationary distributions assume weak reversibility and zero deficiency. We first give explicitly product-form steady-state distributions for a class of non weakly reversible autocatalytic CRN of arbitrary deficiency. Examples of interest in statistical mechanics (inclusion process), life sciences and robotics (collective decision making in ant and robot swarms) are provided. The product-form nature of the steady-state then enables the study of condensation in particle systems that are generalizations of the inclusion process.

math.PR↗

arXiv2018

Mean field repulsive Kuramoto models: Phase locking and spatial signs

The phenomenon of self-synchronization in populations of oscillatory units appears naturally in neurosciences. However, in some situations, the formation of a coherent state is damaging. In this article we study a repulsive mean-field Kuramoto model that describes the time evolution of n points on the unit circle, which are transformed into incoherent phase-locked states. It has been recently shown that such systems can be reduced to a three-dimensional system of ordinary differential equations, whose mathematical structure is strongly related to hyperbolic geometry. The orbits of the Kuramoto dynamical system are then described by a ow of Möbius transformations. We show this underlying dynamic performs statistical inference by computing dynamically M-estimates of scatter matrices. We also describe the limiting phase-locked states for random initial conditions using Tyler's transformation matrix. Moreover, we show the repulsive Kuramoto model performs dynamically not only robust covariance matrix estimation, but also data processing: the initial configuration of the n points is transformed by the dynamic into a limiting phase-locked state that surprisingly equals the spatial signs from nonparametric statistics. That makes the sign empirical covariance matrix to equal 1 2 id2, the variance-covariance matrix of a random vector that is uniformly distributed on the unit circle.

nlin.AO↗

arXiv2017

The phenomenon of growing surface interference explains the rosette pattern of jaguar

One possible mechanism to explain how animals got their coat patterns was proposed by Alan Turing. He assumed that two kinds of morphogens diffuse on a surface and interact with each other, generating a reaction-diffusion mechanism. We developed a new framework for pattern generation incorporating a non-diffusing transcription factor in the system. The diffusion factors (one inhibitor and one activator) acting on cell surface receptors modulate the activity of a transcription factor. The difference in the local concentration of diffusion factors is translated into the degree of activation of transcription factors. The speed of this process determines then pattern formation velocity, i.e. the elapsed time from an initial noisy situation to a final developed pattern. If the pattern formation velocity slows down compared to the growth of the surface, the phenomenon of "growing surface interference" occurs. We find that this phenomenon might explain the rosette pattern observed on different types of felids and the pale stripes found between the regular black stripes of zebras. We also investigate the dynamics between pattern formation velocity and growth and to what extent a pattern may freeze on growing domains.

q-bio.TO↗

arXiv2017

Extending the mathematical palette for developmental pattern formation: Piebaldism

Piebaldism usually manifests as white areas of fur, hair or skin due to the absence of pigment-producing cells in those regions. The distribution of the white and colored zones does not follow the classical Turing patterns. Here we present a modeling framework for pattern formation that enables to easily modify the relationship between three factors with different feedback mechanisms. These factors consist of two diffusing factors and a cell-autonomous immobile transcription factor. Globally the model allowed to distinguishing four different situations. Two situations result in the production of classical Turing patterns; regularly spaced spots and labyrinth patterns. Moreover, an initial slope in the activation of the transcription factor produces straight lines. The third situation does not lead to patterns, but results in different homogeneous color tones. Finally, the fourth one sheds new light on the possible mechanisms leading to the formation of piebald patterns exemplified by the random patterns on the fur of some cow strains and Dalmatian dogs. We demonstrate that these piebald patterns are of transient nature, develop from random initial conditions and rely on a system's bi-stability. The main novelty lies in our finding that the presence of a cell-autonomous factor not only expands the range of reaction diffusion parameters in which a pattern may arise, but also extends the pattern-forming abilities of the reaction-diffusion equations.

q-bio.TO↗

arXiv2016

The feasibility of equilibria in large ecosystems: a primary but neglected concept in the complexity-stability debate

The consensus that complexity begets stability in ecosystems was challenged in the seventies, a result recently extended to ecologically-inspired networks. The approaches assume the existence of a feasible equilibrium, i.e. with positive abundances. However, this key assumption has not been tested. We provide analytical results complemented by simulations which show that equilibrium feasibility vanishes in species rich systems. This result leaves us in the uncomfortable situation in which the existence of a feasible equilibrium assumed in local stability criteria is far from granted. We extend our analyses by changing interaction structure and intensity, and find that feasibility and stability is warranted irrespective of species richness with weak interactions. Interestingly, we find that the dynamical behaviour of ecologically inspired architectures is very different and richer than that of unstructured systems. Our results suggest that a general understanding of ecosystem dynamics requires focusing on the interplay between interaction strength and network architecture.

q-bio.PE↗

arXiv2016

Ultrasensitivity and sharp threshold theorems for multisite systems

We study the ultrasensitivity of multisite binding processes where ligand molecules can bind to several binding sites, considering more particularly recent models involving complex chemical reactions in phosphorylation systems such as allosteric phosphorylation processes, or substrate-catalyst chain reactions and nucleosome mediated cooperativity. New statistics based formulas for the Hill coefficient and the effective Hill coefficient are provided and necessary conditions for a system to be ultrasensitive are exhibited. We then assume that the binding process is described by a density dependent birth and death process. We provide precise large deviation results for the steady state distribution of the process, and show that switch-like ultrasensitive responses are strongly related to the multi-stability of the associated dynamical system. Ultrasensitivity occurs if and only if the entropy of the dynamical system has more than one global minimum for some critical ligand concentration. In this case, the Hill coefficient is proportional to the number of binding sites, and the systems is highly ultrasensitive. We also discuss the interpretation of an extension $I_q$ of the effective Hill coefficient $I_{0.9}$ for which we recommend the computation of a broad range of values of $q$ instead of just the standard one corresponding to the 10% to 90% variation in the dose-response. It is shown that this single choice can sometimes mislead the conclusion by not detecting ultrasensitivity. This new approach allows a better understanding of multisite ultrasensitive systems and provides new tools for the design of such systems.

q-bio.SC↗

arXiv2013

Matching-centrality decomposition and the forecasting of new links in networks

Networks play a prominent role in the study of complex systems of interacting entities in biology, sociology, and economics. Despite this diversity, we demonstrate here that a statistical model decomposing networks into matching and centrality components provides a comprehensive and unifying quantification of their architecture. First we show, for a diverse set of networks, that this decomposition provides an extremely tight fit to observed networks. Consequently, the model allows very accurate prediction of missing links in partially known networks. Second, when node characteristics are known, we show how the matching-centrality decomposition can be related to this external information. Consequently, it offers a simple and versatile tool to explore how node characteristics explain network architecture. Finally, we demonstrate the efficiency and flexibility of the model to forecast the links that a novel node would create if it were to join an existing network.

physics.soc-ph↗

arXiv2011

Pattern formation in auxin flux

The plant hormone auxin is fundamental for plant growth, and its spatial distribution in plant tissues is critical for plant morphogenesis. We consider a leading model of the polar auxin flux, and study in full detail the stability of the possible equilibrium configurations. We show that the critical states of the auxin transport process are composed of basic building blocks, which are isolated in a background of auxin depleted cells, and are not geometrically regular in general. The same model was considered recently through a continuous limit and a coupling to the von Karman equations, to model the interplay of biochemistry and mechanics during plant growth. Our conclusions might be of interest in this setting, since, for example, we establish the existence of Lyapunov functions for the auxin flux, proving in this way the convergence of pure transport processes toward the set of critical configurations.

q-bio.TO↗

arXiv2009

Phenotypic diversity and population growth in fluctuating environment: a MBPRE approach

Organisms adapt to fluctuating environments by regulating their dynamics, and by adjusting their phenotypes to environmental changes. We model population growth using multitype branching processes in random environments, where the offspring distribution of some organism having trait $t\in\cT$ in environment $e\in\cE$ is given by some (fixed) distribution $Υ_{t,e}$ on $\bbN$. Then, the phenotypes are attributed using a distribution (strategy) $π_{t,e}$ on the trait space $\cT$. We look for the optimal strategy $π_{t,e}$, $t\in\cT$, $e\in\cE$ maximizing the net growth rate or Lyapounov exponent, and characterize the set of optimal strategies. This is considered for various models of interest in biology: hereditary versus non-hereditary strategies and strategies involving or not involving a sensing mechanism. Our main results are obtained in the setting of non-hereditary strategies: thanks to a reduction to simple branching processes in random environment, we derive an exact expression for the net growth rate and a characterisation of optimal strategies. We also focus on typical genealogies, that is, we consider the problem of finding the typical lineage of a randomly chosen organism.

math.PR↗

arXiv2008

Stochastic models and numerical algorithms for a class of regulatory gene networks

Regulatory gene networks contain generic modules like those involving feedback loops, which are essential for the regulation of many biological functions. We consider a class of self-regulated genes which are the building blocks of many regulatory gene networks, and study the steady state distributions of the associated Gillespie algorithm by providing efficient numerical algorithms. We also study a regulatory gene network of interest in synthetic biology and in gene therapy, using mean-field models with time delays. Convergence of the related time-nonhomogeneous Markov chain is established for a class of linear catalytic networks with feedback loops

q-bio.SC↗

arXiv2008

Grassmannian Estimation

This paper discusses the family of distributions on the Grassmannian of the linear span of r central gaussian vectors parametrized by the covariance matrix. Our main result is an existence and uniqueness criterion for the maximum likelihood estimate of a sample.

math.ST↗

arXiv2008

Some remarks on Betti numbers of random polygon spaces

Polygon spaces like $M_\ell=\{(u_1,...,u_n)\in S^1\times... S^1 ;\ \sum_{i=1}^n l_iu_i=0\}/SO(2)$ or they three dimensional analogues $N_\ell$ play an important rôle in geometry and topology, and are also of interest in robotics where the $l_i$ model the lengths of robot arms. When $n$ is large, one can assume that each $l_i$ is a positive real valued random variable, leading to a random manifold. The complexity of such manifolds can be approached by computing Betti numbers, the Euler characteristics, or the related Poincaré polynomial. We study the average values of Betti numbers of dimension $p_n$ when $p_n\to\infty$ as $n\to\infty$. We also focus on the limiting mean Poincaré polynomial, in two and three dimensions. We show that in two dimensions, the mean total Betti number behaves as the total Betti number associated with the equilateral manifold where $l_i\equiv \bar l$. In three dimensions, these two quantities are not any more asymptotically equivalent. We also provide asymptotics for the Poincaré polynomials

math.PR↗

arXiv2006

Limiting dynamics for spherical models of spin glasses at high temperature

We analyze the coupled non-linear integro-differential equations whose solutions is the thermodynamical limit of the empirical correlation and response functions in the Langevin dynamics for spherical p-spin disordered mean-field models. We provide a mathematically rigorous derivation of their FDT solution (for the high temperature regime) and of certain key properties of this solution, which are in agreement with earlier derivations based on physical grounds.

math.PR↗

arXiv2005

Stochastic gene expression in switching environments

We study a stochastic model proposed recently in the genetic literature to explain the heterogeneity of cell populations or of gene products. Cells are located in two colonies, whose sizes fluctuate as birth and migration processes in switching environments. We prove that there is a range of parameters where heterogeneity induces a larger mean fitness

q-bio.PE↗