arXiv subjects

Carsten Wiuf

Publications and source records attributed to Carsten Wiuf.

At least 19 recordsLinked to original sources

arXiv2025

Inhomogeneous branching trees with symmetric and asymmetric offspring and their genealogies

We define symmetric and asymmetric branching trees, a class of processes particularly suited for modeling genealogies of inhomogeneous populations where individuals may reproduce throughout life. In this framework, a broad class of Crump-Mode-Jagers processes can be constructed as (a)symmetric Sevast'yanov processes, which count the branches of the tree. Analogous definitions yield reduced (a)symmetric Sevast'yanov processes, which restrict attention to branches that lead to extant progeny. We characterize their laws through generating functions. The genealogy obtained by pruning away branches without extant progeny at a fixed time is shown to satisfy a branching property, which provides distributional characterizations of the genealogy.

math.PR↗

arXiv2024

Asymptotic analysis for stationary distributions of scaled reaction networks

We study stationary distributions in the context of stochastic reaction networks. In particular, we are interested in complex balanced reaction networks and reduction of such networks by assuming a set of species (called non-interacting species) are degraded fast (and therefore essentially absent in the network), implying some reaction rates are large compared to others. Technically, we assume these reaction rates are scaled by a common parameter $N$ and let $N\to\infty$. The limiting stationary distribution as $N\to\infty$ is compared to the stationary distribution of the reduced reaction network obtained by algebraic elimination of the non-interacting species. In general, the limiting stationary distribution might differ from the stationary distribution of the reduced reaction network. We identify various sufficient conditions for when these two distributions are the same, including when the reaction network is detailed balanced and when the set of non-interacting species consists of intermediate species. In the latter case, the limiting stationary distribution essentially retains the form of the complex balanced distribution. This finding is particularly surprising given that the reduced reaction network might be non-weakly reversible and exhibit unconventional kinetics.

math.PR↗

arXiv2023

Stationary measures of continuous time Markov chains with applications to stochastic reaction networks

We study continuous-time Markov chains on the non-negative integers under mild regularity conditions (in particular, the set of jump vectors is finite and both forward and backward jumps are possible). Based on the so-called flux balance equation, we derive an iterative formula for calculating stationary measures. Specifically, a stationary measure $\pi(x)$ evaluated at $x\in\N_0$ is represented as a linear combination of a few generating terms, similarly to the characterization of a stationary measure of a birth-death process, where there is only one generating term, $\pi(0)$. The coefficients of the linear combination are recursively determined in terms of the transition rates of the Markov chain. For the class of Markov chains we consider, there is always at least one stationary measure (up to a scaling constant). We give various results pertaining to uniqueness and non-uniqueness of stationary measures, and show that the dimension of the linear space of signed invariant measures is at most the number of generating terms. A minimization problem is constructed in order to compute stationary measures numerically. Moreover, a heuristic linear approximation scheme is suggested for the same purpose by first approximating the generating terms. The correctness of the linear approximation scheme is justified in some special cases. Furthermore, a decomposition of the state space into different types of states (open and closed irreducible classes, and trapping, escaping and neutral states) is presented. Applications to stochastic reaction networks are well illustrated.

math.PR↗

arXiv2023

Any Stochastic Reaction Network has a Stationary Measure

In this note, we use a result by Harris (1957) to show that there always exists a stationary measure (not necessarily a distribution) on a closed irreducible component of a stochastic reaction network. This measure might not be unique. In particular, any weakly reversible stochastic reaction network has a stationary measure on all closed irreducibe components, irrespective whether it is compelx balanced or not.

q-bio.MN↗

arXiv2023

Evaluation of population structure inferred by principal component analysis or the admixture model

Principal component analysis (PCA) is commonly used in genetics to infer and visualize population structure and admixture between populations. PCA is often interpreted in a way similar to inferred admixture proportions, where it is assumed that individuals belong to one of several possible populations or are admixed between these populations. We propose a new method to assess the statistical fit of PCA (interpreted as a model spanned by the top principal components) and to show that violations of the PCA assumptions affect the fit. Our method uses the chosen top principal components to predict the genotypes. By assessing the covariance (and the correlation) of the residuals (the differences between observed and predicted genotypes), we are able to detect violation of the model assumptions. Based on simulations and genome wide human data we show that our assessment of fit can be used to guide the interpretation of the data and to pinpoint individuals that are not well represented by the chosen principal components. Our method works equally on other similar models, such as the admixture model, where the mean of the data is represented by linear matrix decomposition.

stat.ME↗

arXiv2023

Reaction cleaving and complex-balanced distributions for chemical reaction networks with general kinetics

Reaction networks have become a major modelling framework in the biological sciences from epidemiology and population biology to genetics and cellular biology. In recent years, much progress has been made on stochastic reaction networks (SRNs), modelled as continuous time Markov chains (CTMCs) and their stationary distributions. We are interested in complex-balanced stationary distributions, where the probability flow out of a complex equals the flow into the complex. We characterise the existence and the form of complex-balanced distributions of SRNs with arbitrary transition functions through conditions on the cycles of the reaction graph (a digraph). Furthermore, we give a sufficient condition for the existence of a complex-balanced distribution and give precise conditions for when it is also necessary. The sufficient condition is also necessary for mass-action kinetics (and certain generalisations of that) or if the connected components of the digraph are cycles. Moreover, we state a deficiency theorem, a generalisation of the deficiency theorem for stochastic mass-action kinetics to arbitrary stochastic kinetics. The theorem gives the co-dimension of the parameter space for which a complex-balanced distribution exists. To achieve this, we construct an iterative procedure to decompose a strongly connected reaction graph into disjoint cycles, such that the corresponding SRN has equivalent dynamics and preserves complex-balancedness, provided the original SRN had so. This decomposition might have independent interest and might be applicable to edge-labelled digraphs in general.

math.PR↗

arXiv2022

Estimation of the covariance structure from SNP allele frequencies

We propose two new statistics, V and S, to disentangle the population history of related populations from SNP frequency data. If the populations are related by a tree, we show by theoretical means as well as by simulation that the new statistics are able to identify the root of a tree correctly, in contrast to standard statistics, such as the observed matrix of F2-statistics (distances between pairs of populations). The statistic V is obtained by averaging over all SNPs (similar to standard statistics). Its expectation is the true covariance matrix of the observed population SNP frequencies, offset by a matrix with identical entries. In contrast, the statistic S is put in a Bayesian context and is obtained by averaging over pairs of SNPs, such that each SNP is only used once. It thus makes use of the joint distribution of pairs of SNPs. In addition, we provide a number of novel mathematical results about old and new statistics, and their mutual relationship.

stat.ME↗

arXiv2021

Fast reactions with non-interacting species in stochastic reaction networks

We consider stochastic reaction networks modeled by continuous-time Markov chains. Such reaction networks often contain many reactions, potentially occurring at different time scales, and have unknown parameters (kinetic rates, total amounts). This makes their analysis complex. We examine stochastic reaction networks with non-interacting species that often appear in examples of interest (e.g. in the two-substrate Michaelis Menten mechanism). Non-interacting species typically appear as intermediate (or transient) chemical complexes that are depleted at a fast rate. We embed the Markov process of the reaction network into a one-parameter family under a two time-scale approach, such that molecules of non-interacting species are degraded fast. We derive simplified reaction networks where the non-interacting species are eliminated and that approximate the scaled Markov process in the limit as the parameter becomes small. Then, we derive sufficient conditions for such reductions based on the reaction network structure for both homogeneous and time-varying stochastic settings, and study examples and properties of the reduction.

math.PR↗

arXiv2021

Critical parameters for singular perturbation reductions of chemical reaction networks

We are concerned with polynomial ordinary differential systems that arise from modelling chemical reaction networks. For such systems, which may be of high dimension and may depend on many parameters, it is frequently of interest to obtain a reduction of dimension in certain parameter ranges. Singular perturbation theory, as initiated by Tikhonov and Fenichel, provides a path toward such reductions. In the present paper we discuss parameter values that lead to singular perturbation reductions (so-called Tikhonov-Fenichel parameter values, or TFPVs). An algorithmic approach is known, but it is feasible for small dimensions only. Here we characterize conditions for classes of reaction networks for which TFPVs arise by turning off reactions (by setting rate parameters to zero), or by removing certain species (which relates to the classical quasi-steady state approach to model reduction). In particular, we obtain definitive results for the class of complex balanced reaction networks (of deficiency zero) and first order reaction networks.

math.DS↗

arXiv2021

Fiber decomposition of deterministic reaction networks with applications

Deterministic reaction networks (RNs) are tools to model diverse biological phenomena characterized by particle systems, when there are abundant number of particles. Examples include but are not limited to biochemistry, molecular biology, genetics, epidemiology, and social sciences. In this chapter we propose a new type of decomposition of RNs, called fiber decomposition. Using this decomposition, we establish lifting of mass-action RNs preserving stationary properties, including multistationarity and absolute concentration robustness. Such lifting scheme is simple and explicit which imposes little restriction on the reaction networks. We provide examples to illustrate how this lifting can be used to construct RNs preserving certain dynamical properties.

q-bio.MN↗

arXiv2021

On the sum of chemical reactions

It is standard in chemistry to represent a sequence of reactions by a single overall reaction, often called a complex reaction in contrast to an elementary reaction. Photosynthesis $6 \text{CO}_2+6 \text{H}_2\text{O} \to \ \text{C}_6\text{H}_{12}\text{O}_6$ $+\ 6 \text{O}_2$ is an example of such complex reaction. We introduce a mathematical operation that corresponds to summing two chemical reactions. Specifically, we define an associative and non-communicative operation on the product space $\mathbb{N}_0^n\times \mathbb{N}_0^n$ (representing the reactant and the product of a chemical reaction, respectively). The operation models the overall effect of two reactions happening in succession, one after the other. We study the algebraic properties of the operation and apply the results to stochastic reaction networks, in particular to reachability of states, and to reduction of reaction networks.

math.CO↗

arXiv2020

Classification and threshold dynamics of stochastic reaction networks

Stochastic reaction networks (SRNs) provide models of many real-world networks. Examples include networks in epidemiology, pharmacology, genetics, ecology, chemistry, and social sciences. Here, we model stochastic reaction networks by continuous time Markov chains (CTMCs) and pay special attention to one-dimensional mass-action SRNs (1-d stoichiometric subspace). We classify all states of the underlying CTMC of 1-d SRNs. In terms of (up to) four parameters, we provide sharp checkable criteria for various dynamical properties (including explosivity, recurrence, ergodicity, and the tail asymptotics of stationary or quasi-stationary distributions) of SRNs in the sense of their underlying CTMCs. As a result, we prove that all 1-d endotactic networks are non-explosive, and positive recurrent with an ergodic stationary distribution with Conley-Maxwell-Poisson (CMP)-like tail, provided the state space of the associated CTMCs consists of closed communicating classes. In particular, we prove the recently proposed positive recurrence conjecture in one dimension: Weakly reversible mass-action SRNs with 1-d stoichiometric subspaces are positive recurrent. The proofs of the main results rely on our recent work on CTMCs with polynomial transition rate functions.

math.PR↗

arXiv2020

The asymptotic tails of limit distributions of continuous time Markov chains

This paper investigates tail asymptotics of stationary distributions and quasi-stationary distributions (QSDs) of continuous-time Markov chains on subsets of the non-negative integers. Based on the so-called flux-balance equation, we establish identities for stationary measures and QSDs, which we use to derive tail asymptotics. In particular, continuous-time Markov chains with asymptotic power law transition rates, tail asymptotics for stationary distributions and QSDs are classified into three types using three easily computable parameters: (i) super-exponential distributions, (ii) exponential-tailed distributions, and (iii) sub-exponential distributions. Our approach to establish tail asymptotics of stationary distributions is different from the classical semimartingale approach, and we do not impose ergodicity nor moment bound conditions. In particular, the results also hold for explosive Markov chains, for which multiple stationary distributions may exist. Furthermore, our results on tail asymptotics of QSDs seem new. We apply our results to biochemical reaction networks, a general single-cell stochastic gene expression model, an extended class of branching processes, and stochastic population processes with bursty reproduction, none of which are birth-death processes. The approach together with the identities easily extends to discrete time Markov chains.

math.PR↗

arXiv2020

Dynamics of continuous time Markov chains with applications

This paper contributes an in-depth study of properties of continuous time Markov chains (CTMCs) on non-negative integer lattices $\N_0^d$, with particular interest in one-dimensional CTMCs with polynomial transitions rates. Such stochastic processes are abundant in applications, in particular in biology. We characterize the structure of the state space of general CTMCs on $\N_0^d$ in terms of the set of jump vectors and their corresponding transition rate functions. For CTMCs on $\N_0$ with polynomial transition rate functions, we provide threshold criteria in terms of easily computable parameters for various dynamical properties such as explosivity, recurrence, transience, certain absorption, positive/null recurrence, implosivity, and existence and non-existence of moments of hitting times. In particular, simple sufficient conditions for exponential ergodicity of stationary distributions and quasi-stationary distributions are obtained, and the few gap cases are well-illustrated by examples. Subtle differences in conditions for different dynamical properties are revealed in terms of examples. Finally, we apply our results to stochastic reaction networks, an extended class of branching processes, a general bursty single-cell stochastic gene expression model, and population processes which are not birth-death processes.

math.PR↗

arXiv2020

Full classification of dynamics for one-dimensional continuous time Markov chains with polynomial transition rates

This paper provides full classification of dynamics for continuous time Markov chains (CTMCs) on the non-negative integers with polynomial transition rate functions. Such stochastic processes are abundant in applications, in particular in biology. More precisely, for CTMCs of bounded jumps, we provide necessary and sufficient conditions in terms of calculable parameters for explosivity, recurrence vs transience, certain absorption, positive recurrence vs null recurrence, and implosivity. Simple sufficient conditions for exponential ergodicity of stationary distributions and quasi-stationary distributions as well as existence and non-existence of moments of hitting times are also obtained. Similar simple sufficient conditions for the aforementioned dynamics together with their opposite dynamics are established for CTMCs with unbounded jumps. The results generalize respective criteria for birth-death processes by Karlin and McGregor in the 1960s. Finally, we apply our results to stochastic reaction networks, an extended class of branching processes, a general bursty single-cell stochastic gene expression model, and population processes, none of which are birth-death processes. The approach is based on a mixture of Lyapunov-Foster type results, semimartingale approach, as well as estimates of stationary measures.

math.PR↗

arXiv2020

Structural classification of continuous time Markov chains with applications

This paper is motivated by examples from stochastic reaction network theory. The $Q$-matrix of a stochastic reaction network can be derived from the reaction graph, an edge-labelled directed graph encoding the jump vectors of an associated continuous time Markov chain on the invariant space $\mathbb{N}^d_0$. An open question is how to decompose the space $\mathbb{N}^d_0$ into neutral, trapping, and escaping states, and open and closed communicating classes, and whether this can be done from the reaction graph alone. Such general continuous time Markov chains can be understood as natural generalizations of birth-death processes, incorporating multiple different birth and death mechanisms. We characterize the structure of $\mathbb{N}^d_0$ imposed by a general $Q$-matrix generating continuous time Markov chains with values in $\mathbb{N}^d_0$, in terms of the set of jump vectors and their corresponding transition rate functions. Thus the setting is not limited to stochastic reaction networks. Furthermore, we define structural equivalence of two $Q$-matrices, and provide sufficient conditions for structural equivalence. Examples are abundant in applications. We apply the results to stochastic reaction networks, a Lotka-Volterra model in ecology, the EnvZ-OmpR system in systems biology, and a class of extended branching processes, none of which are birth-death processes.

math.PR↗

arXiv2020

Stationary distributions of systems with Discreteness Induced Transitions

We provide a theoretical analysis of some autocatalytic reaction networks exhibiting the phenomenon of discretely induced transitions (DITs). The family of networks that we address includes the celebrated Togashi and Kaneko model. We prove positive recurrence, finiteness of all moments, and geometric ergodicity of the models in the family. For some parameter values, we find the analytic expression for the stationary distribution, and discuss the effect of volume scaling on the stationary behavior of the chain. We find the exact critical value of the volume for which DITs disappear.

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↗