arXiv subjects

Laurent Tournier

Publications and source records attributed to Laurent Tournier.

18 recordsLinked to original sources

arXiv2026

A Kuramoto phase model to explore the synchronisation of a network of circadian clocks

We propose a model of the circadian clock in a population of cells based on a network of oscillators, derived from the Kuramoto model. The coupling between oscillators is described by a global interaction term but we introduce a phase-dependent coupling mechanism, such that oscillators interact only within a specific interval of the cycle corresponding to a specific stage of the circadian cycle. We analytically demonstrate that this modified system achieves complete asymptotic phase synchronisation, provided specific conditions on the initial phase distribution and the coupling window length are met. To bridge this theoretical framework with experimental observations, we introduce a signal processing procedure based on wavelet decomposition to extract quantitative oscillatory features from Per2::luciferase reporter traces. We then calibrate the model against datasets from both wild type and Cry2 knockout hepatocyte spheroids using a two-step quasi-Monte Carlo filtering algorithm. The comparative analysis reveals significant phenotypic divergence, showing that the Cry2KO condition alters the identifiability landscape of the model's parameters and introduces new compensatory mechanisms that confound the initial population heterogeneity with long-term macroscopic signal decay.

math.DS↗

arXiv2021

Three-speed ballistic annihilation: phase transition and universality

We consider ballistic annihilation, a model for chemical reactions first introduced in the 1980's physics literature. In this particle system, initial locations are given by a renewal process on the line, motions are ballistic - i.e. each particle is assigned an i.i.d. constant velocity - and collisions between pairs of particles result in mutual annihilation. We focus on the case when the velocities are symmetrically distributed among three values, i.e. particles either remain static (with given probability~$p$) or move at constant velocity uniformly chosen among $\pm1$. We establish that this model goes through a phase transition at $p_c=1/4$ between a subcritical regime where every particle eventually annihilates, and a supercritical regime where a positive density of static particles is never hit, confirming 1990s predictions of Droz et al. for the particular case of a Poisson process. Our result encompasses cases where triple collisions can happen; these are resolved by annihilation of one static and one randomly chosen moving particle. Our arguments, of combinatorial nature, show that, although the model is not completely solvable, certain large scale features can be explicitly computed, and are universal, i.e. insensitive to the distribution of the initial point process. In particular, in the critical and subcritical regimes, the asymptotics of the time decay of the densities of each type of particle is universal (among exponentially integrable interdistance distributions) and, in the supercritical regime, the distribution of the ``skyline'' process, i.e. the process restricted to the last particles to ever visit a location, has a universal description. We also prove that an alternative model introduced by Burdinski, Gupta and Junge does not share the same universality as our model, and find numerical bounds on its critical probability.

math.PR↗

arXiv2020

Combinatorial universality in three-speed ballistic annihilation

We consider a one-dimensional system of particles, moving at constant velocities chosen independently according to a symmetric distribution on $\{-1,0,+1\}$, and annihilating upon collision -- with, in case of triple collision, a uniformly random choice of survivor among the two moving particles. When the system contains infinitely many particles, whose starting locations are given by a renewal process, a phase transition was proved to happen (see arXiv:1811.08709) as the density of static particles crosses the value $1/4$. Remarkably, this critical value, along with certain other statistics, was observed not to depend on the distribution of interdistances. In the present paper, we investigate further this universality by proving a stronger statement about a finite system of particles with fixed, but randomly shuffled, interdistances. We give two proofs, one by an induction allowing explicit computations, and one by a more direct comparison. This result entails a new nontrivial independence property that in particular gives access to the density of surviving static particles at a given time in the infinite model. Finally, in the asymmetric case, further similar independence properties are proved to keep holding, including a striking property of gamma distributed interdistances that contrasts with the general behavior.

math.PR↗

arXiv2019

Automated generation of bacterial resource allocation models

Resource Balance Analysis (RBA) is a computational method based on resource allocation, which performs accurate quantitative predictions of whole-cell states (i.e. growth rate, meta-bolic fluxes, abundances of molecular machines including enzymes) across growth conditions. We present an integrated workflow of RBA together with the Python package RBApy. RBApy builds bacterial RBA models from annotated genome-scale metabolic models by add-ing descriptions of cellular processes relevant for growth and maintenance. The package in-cludes functions for model simulation and calibration and for interfacing to Escher maps and Proteomaps for visualization. We demonstrate that RBApy faithfully reproduces results ob-tained by a hand-curated and experimentally validated RBA model for Bacillus subtilis. We also present a calibrated RBA model of Escherichia coli generated from scratch, which ob-tained excellent fits to measured flux values and enzyme abundances. RBApy makes whole-cell modeling accessible for a wide range of bacterial wild-type and engineered strains, as il-lustrated with a CO2-fixing Escherichia coli strain.

q-bio.QM↗

arXiv2019

A deterministic walk on the randomly oriented Manhattan lattice

Consider a randomly-oriented two dimensional Manhattan lattice where each horizontal line and each vertical line is assigned, once and for all, a random direction by flipping independent and identically distributed coins. A deterministic walk is then started at the origin and at each step moves diagonally to the nearest vertex in the direction of the horizontal and vertical lines of the present location. This definition can be generalized, in a natural way, to larger dimensions, but we mainly focus on the two dimensional case. In this context the process localizes on two vertices at all large times, almost surely. We also provide estimates for the tail of the length of paths, when the walk is defined on the two dimensional lattice. In particular, the probability of the path to be larger than $n$ decays sub-exponentially in $n$. It is easy to show that higher dimensional paths may not localize on two vertices but will still eventually become periodic, and are therefore bounded.

math.PR↗

arXiv2018

The ballistic annihilation threshold is 1/4

We consider a system of annihilating particles where particles start from the points of a Poisson process on the line, move at constant i.i.d. speeds symmetrically distributed in {-1,0,+1} and annihilate upon collision. We prove that particles with speed 0 vanish almost surely if and only if their initial density is smaller than or equal to 1/4, and give an explicit formula for the probability of survival of a stationary particle, which is in accordance with the predictions of [Droz et al. 1995]. The present proof relies essentially on an identity proved in a recent paper by J. Haslegrave.

math.PR↗

arXiv2018

Non-fixation for Biased Activated Random Walks

We prove that the model of Activated Random Walks on Z^d with biased jump distribution does not fixate for any positive density, if the sleep rate is small enough, as well as for any finite sleep rate, if the density is close enough to 1. The proof uses a new criterion for non-fixation. We provide a pathwise construction of the process, of independent interest, used in the proof of this non-fixation criterion.

math.PR↗

arXiv2018

Greedy Clearing of Persistent Poissonian Dust

Given a Poisson point process on R, assign either one or two marks to each point of this process, independently of the others. We study the motion of a particle that jumps deterministically from its current location to the nearest point of the Poisson point process which still contains at least one mark, and removes one mark per each visit. A point of the Poisson point process which is left with no marks is removed from the system. We prove that the presence of any positive density of double marks leads to the eventual removal of every Poissonian point.

math.PR↗

arXiv2017

Note on a one-dimensional system of annihilating particles

We consider a system of annihilating particles where particles start from the points of a Poisson process on either the full-line or positive half-line and move at constant i.i.d. speeds until collision. When two particles collide, they annihilate. We assume the law of speeds to be symmetric. We prove almost sure annihilation of positive-speed particles started from the positive half-line, and existence of a regime of survival of zero-speed particles on the full-line in the case when speeds can only take 3 values.

math.PR↗

arXiv2017

Uniqueness of the infinite noodle

Consider the graph obtained by superposition of an independent pair of uniform infinite non-crossing perfect matchings of the set of integers. We prove that this graph contains at most one infinite path. Several motivations are discussed.

math.PR↗

arXiv2016

Random walks in Dirichlet environment: an overview

Random Walks in Dirichlet Environment (RWDE) correspond to Random Walks in Random Environment (RWRE) on $\Bbb{Z}^d$ where the transition probabilities are i.i.d. at each site with a Dirichlet distribution. Hence, the model is parametrized by a family of positive weights $(α_i)_{i=1, \ldots, 2d}$, one for each direction of $\Bbb{Z}^d$. In this case, the annealed law is that of a reinforced random walk, with linear reinforcement on directed edges. RWDE have a remarkable property of statistical invariance by time reversal from which can be inferred several properties that are still inaccessible for general environments, such as the equivalence of static and dynamic points of view and a description of the directionally transient and ballistic regimes. In this paper we give a state of the art on this model and several sketches of proofs presenting the core of the arguments. We also present new computation of the large deviation rate function for one dimensional RWDE.

math.PR↗

arXiv2013

Quenched limits for the fluctuations of transient random walks in random environment on Z

We consider transient nearest-neighbor random walks in random environment on Z. For a set of environments whose probability is converging to 1 as time goes to infinity, we describe the fluctuations of the hitting time of a level n, around its mean, in terms of an explicit function of the environment. Moreover, their limiting law is described using a Poisson point process whose intensity is computed. This result can be considered as the quenched analog of the classical result of Kesten, Kozlov and Spitzer [Compositio Math. 30 (1975) 145-168].

math.PR↗

arXiv2013

Cell death and life in cancer: mathematical modeling of cell fate decisions

Tumor development is characterized by a compromised balance between cell life and death decision mechanisms, which are tighly regulated in normal cells. Understanding this process provides insights for developing new treatments for fighting with cancer. We present a study of a mathematical model describing cellular choice between survival and two alternative cell death modalities: apoptosis and necrosis. The model is implemented in discrete modeling formalism and allows to predict probabilities of having a particular cellular phenotype in response to engagement of cell death receptors. Using an original parameter sensitivity analysis developed for discrete dynamic systems, we determine the critical parameters affecting cellular fate decision variables that appear to be critical in the cellular fate decision and discuss how they are exploited by existing cancer therapies.

q-bio.MN↗

arXiv2012

Asymptotic direction of random walks in Dirichlet environment

In this paper we generalize the result of directional transience from [SabotTournier10]. This enables us, by means of [Simenhaus07], [ZernerMerkl01] and [Bouchet12] to conclude that, on Z^d (for any dimension d), random walks in i.i.d. Dirichlet environment, or equivalently oriented-edge reinforced random walks, have almost-surely an asymptotic direction equal to the direction of the initial drift, unless this drift is zero. In addition, we identify the exact value or distribution of certain probabilities, answering and generalizing a conjecture of [SaTo10].

math.PR↗

arXiv2010

Stable fluctuations for ballistic random walks in random environment on Z

We consider transient random walks in random environment on Z in the positive speed (ballistic) and critical zero speed regimes. A classical result of Kesten, Kozlov and Spitzer proves that the hitting time of level $n$, after proper centering and normalization, converges to a completely asymmetric stable distribution, but does not describe its scale parameter. Following a previous article by three of the authors, where the (non-critical) zero speed case was dealt with, we give a new proof of this result in the subdiffusive case that provides a complete description of the limit law. The case of Dirichlet environment turns out to be remarkably explicit.

math.PR↗

arXiv2009

A note on the recurrence of edge reinforced random walks

We give a short proof of Theorem 2.1 from [MR07], stating that the linearly edge reinforced random walk (ERRW) on a locally finite graph is recurrent if and only if it returns to its starting point almost surely. This result was proved in [MR07] by means of the much stronger property that the law of the ERRW is a mixture of Markov chains. Our proof only uses this latter property on finite graphs, in which case it is a consequence of De Finetti's theorem on exchangeability.

math.PR↗

arXiv2008

Integrability of exit times and ballisticity for random walks in Dirichlet environment

We consider random walks in Dirichlet environment, introduced by Enriquez and Sabot in 2006. As this distribution on environments is not uniformly elliptic, the annealed integrability of exit times out of a given finite subset is a non-trivial property. We provide here an explicit equivalent condition for this integrability to happen, on general directed graphs. Such integrability problems arise for instance from the definition of Kalikow auxiliary random walk. Using our condition, we prove a refined version of the ballisticity criterion given by Enriquez and Sabot.

math.PR↗

arXiv2005

Approximation of dynamical systems using S-systems theory : application to biological systems

In this paper we propose a new symbolic-numeric algorithm to find positive equilibria of a n-dimensional dynamical system. This algorithm implies a symbolic manipulation of ODE in order to give a local approximation of differential equations with power-law dynamics (S-systems). A numerical calculus is then needed to converge towards an equilibrium, giving at the same time a S-system approximating the initial system around this equilibrium. This algorithm is applied to a real biological example in 14 dimensions which is a subsystem of a metabolic pathway in Arabidopsis Thaliana.

cs.SC↗