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Daniele Cappelletti

Publications and source records attributed to Daniele Cappelletti.

At least 19 recordsLinked to original sources
arXiv2026

Stochastic ordering tools for continuous-time Markov chains and applications to reaction network models

Stochastic reaction networks are mathematical models with a wide range of applications in biochemistry, ecology, and epidemiology, and are often complex to analyze. Except for some special cases, it is generally difficult to predict how the abundances of all considered species evolve over time. A possible approach to address this issue is to develop tools to compare the model under study with a similar one whose behavior is better understood. The main contribution of our work is to provide direct and computable conditions that can be used to ensure the existence of an ordered coupling between two stochastic reaction networks and to identify which parameter changes in a given model lead to an increase or decrease in the count of certain species. We also make available an algorithm that implements our theory, and we illustrate it with several applications.

math.PR↗
arXiv2025

Stability of randomly switching stochastic reaction networks with asymptotically linear transition rates

Stochastic reaction networks are mathematical models frequently used in, but not limited to, biochemistry. These models are continuous-time Markov chains whose transition rates depend on certain parameters called rate constants, which despite the name may not be constant in real-world applications. In this paper we study how random switching between different stochastic reaction networks with asymptotically linear rate functions affects the stability of the process. We give matrix conditions for both positive recurrence (indeed, exponentially ergodicity) and transience (indeed, evanescence) in both the regime with high switching rates and the regime with low switching rates. We then make use of these conditions to provide examples of processes whose stability behavior changes as the switching rate varies. We also explore what happens in the middle regime where the switching rates are neither high nor low and our theorems do not apply. Specifically, we show by examples that there can be arbitrarily many phase transitions between exponentially ergodicity and evanescence as the switching rate increases.

math.PR↗
arXiv2023

A new path method for exponential ergodicity of Markov processes on $\mathbb Z^d$, with applications to stochastic reaction networks

This paper provides a new path method that can be used to determine when an ergodic continuous-time Markov chain on $\mathbb Z^d$ converges exponentially fast to its stationary distribution in $L^2$. Specifically, we provide general conditions that guarantee the positivity of the spectral gap. Importantly, our results do not require the assumption of time-reversibility of the Markov model. We then apply our new method to the well-studied class of stochastically modeled reaction networks. Notably, we show that each complex-balanced model that is also ``open'' has a positive spectral gap, and is therefore exponentially ergodic. We further illustrate how our results can be applied for models that are not necessarily complex-balanced. Moreover, we provide an example of a detailed-balanced (in the sense of reaction network theory), and hence complex-balanced, stochastic reaction network that is not exponentially ergodic. We believe this to be the first such example in the literature.

math.PR↗
arXiv2023

Individual molecules dynamics in reaction network models

In a stochastic reaction network setting we consider the problem of tracking the fate of individual molecules. We show that using the classical large volume limit results, we may approximate the dynamics of a single tracked molecule in a simple and computationally efficient way. We give examples on how this approach may be used to obtain various characteristics of single-molecule dynamics (for instance, the distribution of the number of infections in a single individual in the course of an epidemic or the activity time of a single enzyme molecule). Moreover, we show how to approximate the overall dynamics of species of interest in the full system with a collection of independent single-molecule trajectories, and give explicit bounds for the approximation error in terms of the reaction rates. This approximation, which is well defined for all times, leads to an efficient and fully parallelizable simulation technique for which we provide some numerical examples.

math.PR↗
arXiv2022

Transition graph decomposition for complex balanced reaction networks with non-mass-action kinetics

Reaction networks are widely used models to describe biochemical processes. Stochastic fluctuations in the counts of biological macromolecules have amplified consequences due to their small population sizes. This makes it necessary to favor stochastic, discrete population, continuous time models. The stationary distributions provide snapshots of the model behavior at the stationary regime, and as such finding their expression in terms of the model parameters is of great interest. The aim of the present paper is to describe when the stationary distributions of the original model, whose state space is potentially infinite, coincide exactly with the stationary distributions of the process truncated to finite subsets of states, up to a normalizing constant. The finite subsets of states we identify are called copies and are inspired by the modular topology of reaction network models. With such a choice we prove a novel graphical characterization of the concept of complex balancing for stochastic models of reaction networks. The results of the paper hold for the commonly used mass-action kinetics but are not restricted to it, and are in fact stated for more general setting.

math.PR↗
arXiv2021

Private Attacks in Longest Chain Proof-of-stake Protocols with Single Secret Leader Elections

Single Secret Leader Elections have recently been proposed as an improved leader election mechanism for proof-of-stake (PoS) blockchains. However, the security gain they provide has not been quantified. In this work, we present a comparison of PoS longest-chain protocols that are based on Single Secret Leader Elections (SSLE) - that elect exactly one leader per round - versus those based on Probabilistic Leader Elections (PLE) - where one leader is elected on expectation. Our analysis shows that when considering the private attack - the worst attack on longest-chain protocols - the security gained from using SSLE is substantial: the settlement time is decreased by roughly 25% for a 33% or 25% adversary. Furthermore, when considering grinding attacks, we find that the security threshold is increased by 10% (from 0.26 in the PLE case to 0.36 inthe SSLE case) and the settlement time is decreased by roughly 70% for a 20% adversary in the SSLE case.

cs.CR↗
arXiv2020

Stochastically modeled weakly reversible reaction networks with a single linkage class

It has been known for nearly a decade that deterministically modeled reaction networks that are weakly reversible and consist of a single linkage class have trajectories that are bounded from both above and below by positive constants (so long as the initial condition has strictly positive components). It is conjectured that the stochastically modeled analogs of these systems are positive recurrent. We prove this conjecture in the affirmative under the following additional assumptions: (i) the system is binary and (ii) for each species, there is a complex (vertex in the associated reaction diagram) that is a multiple of that species. To show this result, a new proof technique is developed in which we study the recurrence properties of the $n-$step embedded discrete time Markov chain.

math.PR↗
arXiv2019

Addition of flow reactions preserving multistationarity and bistability

We consider the question whether a chemical reaction network preserves the number and stability of its positive steady states upon inclusion of inflow and outflow reactions. Often a model of a reaction network is presented without inflows and outflows, while in fact some of the species might be degraded or leaked to the environment, or be synthesized or transported into the system. We provide a sufficient and easy-to-check criterion based on the stoichiometry of the reaction network alone and discuss examples from systems biology.

q-bio.MN↗
arXiv2019

Long-time asymptotics of stochastic reaction systems

We study the stochastic dynamics of a system of interacting species in a stochastic environment by means of a continuous-time Markov chain with transition rates depending on the state of the environment. Models of gene regulation in systems biology take this form. We characterise the finite-time distribution of the Markov chain, provide conditions for ergodicity, and characterise the stationary distribution (when it exists) as a mixture of Poisson distributions. The mixture measure is uniquely identified as the law of a fixed point of a stochastic recurrence equation. This recursion is crucial for statistical computation of moments and other distributional features.

math.DS↗
arXiv2019

A hidden integral structure endows Absolute Concentration Robust systems with resilience to dynamical concentration disturbances

Biochemical systems that express certain chemical species of interest at the same level at any positive equilibrium are called "absolute concentration robust" (ACR). These species behave in a stable, predictable way, in the sense that their expression is robust with respect to sudden changes in the species concentration, regardless the new positive equilibrium reached by the system. Such a property has been proven to be fundamentally important in certain gene regulatory networks and signaling systems. In the present paper, we mathematically prove that a well-known class of ACR systems studied by Shinar and Feinberg in 2010 hides an internal integral structure. This structure confers these systems with a higher degree of robustness that what was previously unknown. In particular, disturbances much more general than sudden changes in the species concentrations can be rejected, and robust perfect adaptation is achieved. Significantly, we show that these properties are maintained when the system is interconnected with other chemical reaction networks. This key feature enables design of insulator devices that are able to buffer the loading effect from downstream systems - a crucial requirement for modular circuit design in synthetic biology.

q-bio.SC↗
arXiv2019

Tier structure of strongly endotactic reaction networks

Reaction networks are mainly used to model the time-evolution of molecules of interacting chemical species. Stochastic models are typically used when the counts of the molecules are low, whereas deterministic models are used when the counts are in high abundance. In 2011, the notion of `tiers' was introduced to study the long time behavior of deterministically modeled reaction networks that are weakly reversible and have a single linkage class. This `tier' based argument was analytical in nature. Later, in 2014, the notion of a strongly endotactic network was introduced in order to generalize the previous results from weakly reversible networks with a single linkage class to this wider family of networks. The point of view of this later work was more geometric and algebraic in nature. The notion of strongly endotactic networks was later used in 2018 to prove a large deviation principle for a class of stochastically modeled reaction networks. We provide an analytical characterization of strongly endotactic networks in terms of tier structures. By doing so, we shed light on the connection between the two points of view, and also make available a new proof technique for the study of strongly endotactic networks. We show the power of this new technique in two distinct ways. First, we demonstrate how the main previous results related to strongly endotactic networks, both for the deterministic and stochastic modeling choices, can be quickly obtained from our characterization. Second, we demonstrate how new results can be obtained by proving that a sub-class of strongly endotactic networks, when modeled stochastically, is positive recurrent. Finally, and similarly to recent independent work by Agazzi and Mattingly, we provide an example which closes a conjecture in the negative by showing that stochastically modeled strongly endotactic networks can be transient (and even explosive).

math.PR↗
arXiv2018

Stochastic Chemical Reaction Networks for Robustly Approximating Arbitrary Probability Distributions

We show that discrete distributions on the $d$-dimensional non-negative integer lattice can be approximated arbitrarily well via the marginals of stationary distributions for various classes of stochastic chemical reaction networks. We begin by providing a class of detailed balanced networks and prove that they can approximate any discrete distribution to any desired accuracy. However, these detailed balanced constructions rely on the ability to initialize a system precisely, and are therefore susceptible to perturbations in the initial conditions. We therefore provide another construction based on the ability to approximate point mass distributions and prove that this construction is capable of approximating arbitrary discrete distributions for any choice of initial condition. In particular, the developed models are ergodic, so their limit distributions are robust to a finite number of perturbations over time in the counts of molecules.

cs.CC↗
arXiv2018

Discrepancies between extinction events and boundary equilibria in reaction networks

Reaction networks are mathematical models of interacting chemical species that are primarily used in biochemistry. There are two modeling regimes that are typically used, one of which is deterministic and one that is stochastic. In particular, the deterministic model consists of an autonomous system of differential equations, whereas the stochastic system is a continuous time Markov chain. Connections between the two modeling regimes have been studied since the seminal paper by Kurtz (1972), where the deterministic model is shown to be a limit of a properly rescaled stochastic model over compact time intervals. Further, more recent studies have connected the long-term behaviors of the two models when the reaction network satisfies certain graphical properties, such as weak reversibility and a deficiency of zero. These connections have led some to conjecture a link between the long-term behavior of the two models exists, in some sense. In particular, one is tempted to believe that positive recurrence of all states for the stochastic model implies the existence of positive equilibria in the deterministic setting, and that boundary equilibria of the deterministic model imply the occurrence of an extinction event in the stochastic setting. We prove in this paper that these implications do not hold in general, even if restricting the analysis to networks that are bimolecular and that conserve the total mass. In particular, we disprove the implications in the special case of models that have absolute concentration robustness, thus answering in the negative a conjecture stated in the literature in 2014.

q-bio.MN↗
arXiv2018

Graphically balanced equilibria and stationary measures of reaction networks

The graph-related symmetries of a reaction network give rise to certain special equilibria (such as complex balanced equilibria) in deterministic models of dynamics of the reaction network. Correspondingly, in the stochastic setting, when modeled as a continuous-time Markov chain, these symmetries give rise to certain special stationary measures. Previous work by Anderson, Craciun and Kurtz identified stationary distributions of a complex balanced network; later Cappelletti and Wiuf developed the notion of complex balancing for stochastic systems. We define and establish the relations between reaction balanced measure, complex balanced measure, reaction vector balanced measure, and cycle balanced measure and prove that with mild additional hypotheses, the former two are stationary distributions. Furthermore, in spirit of earlier work by Joshi, we give sufficient conditions under which detailed balance of the stationary distribution of Markov chain models implies the existence of positive detailed balance equilibria for the related deterministic reaction network model. Finally, we provide a complete map of the implications between balancing properties of deterministic and corresponding stochastic reaction systems, such as complex balance, reaction balance, reaction vector balance and cycle balance.

math.PR↗
arXiv2017

Node Balanced Steady States: Unifying and Generalizing Complex and Detailed Balanced Steady States

We introduce a unifying and generalizing framework for complex and detailed balanced steady states in chemical reaction network theory. To this end, we generalize the graph commonly used to represent a reaction network. Specifically, we introduce a graph, called a reaction graph, that has one edge for each reaction but potentially multiple nodes for each complex. A special class of steady states, called node balanced steady states, is naturally associated with such a reaction graph. We show that complex and detailed balanced steady states are special cases of node balanced steady states by choosing appropriate reaction graphs. Further, we show that node balanced steady states have properties analogous to complex balanced steady states, such as uniqueness and asymptotical stability in each stoichiometric compatibility class. Moreover, we associate an integer, called the deficiency, to a reaction graph that gives the number of independent relations in the reaction rate constants that need to be satisfied for a positive node balanced steady state to exist. The set of reaction graphs (modulo isomorphism) is equipped with a partial order that has the complex balanced reaction graph as minimal element. We relate this order to the deficiency and to the set of reaction rate constants for which a positive node balanced steady state exists.

math.DS↗
arXiv2017

Uniform Approximation of Solutions by Elimination of Intermediate Species in Deterministic Reaction Networks

Chemical reactions often proceed through the formation and the consumption of intermediate species. An example is the creation and subsequent degradation of the substrate-enzyme complexes in an enzymatic reaction. In this paper we provide a setting, based on ordinary differential equations, in which the presence of intermediate species has little effect on the overall dynamics of a biological system. The result provides a method to perform model reduction by elimination of intermediate species. We study the problem in a multiscale setting, where the species abundances as well a the reaction rates scale to different orders of magnitudes. The different time and concentration scales are parameterised by a single parameter $N$. We show that a solution to the original reaction system is uniformly approximated on compact time intervals to a solution of a reduced reaction system without intermediates and to a solution of a certain limiting reaction systems, which does not depend on $N$. Known approximation techniques such as the theorems by Tikhonov and Fenichel cannot readily be used in this framework.

math.DS↗
arXiv2016

Finite time distributions of stochastically modeled chemical systems with absolute concentration robustness

Recent research in both the experimental and mathematical communities has focused on biochemical interaction systems that satisfy an "absolute concentration robustness" (ACR) property. The ACR property was first discovered experimentally when, in a number of different systems, the concentrations of key system components at equilibrium were observed to be robust to the total concentration levels of the system. Followup mathematical work focused on deterministic models of biochemical systems and demonstrated how chemical reaction network theory can be utilized to explain this robustness. Later mathematical work focused on the behavior of this same class of reaction networks, though under the assumption that the dynamics were stochastic. Under the stochastic assumption, it was proven that the system will undergo an extinction event with a probability of one so long as the system is conservative, showing starkly different long-time behavior than in the deterministic setting. Here we consider a general class of stochastic models that intersects with the class of ACR systems studied previously. We consider a specific system scaling over compact time intervals and prove that in a limit of this scaling the distribution of the abundances of the ACR species converges to a certain product-form Poisson distribution whose mean is the ACR value of the deterministic model. This result is in agreement with recent conjectures pertaining to the behavior of ACR networks endowed with stochastic kinetics, and helps to resolve the conflicting theoretical results pertaining to deterministic and stochastic models in this setting.

math.PR↗