arXiv subjects

David F. Anderson

Publications and source records attributed to David F. Anderson.

At least 19 recordsLinked to original sources

arXiv2026

Instantaneous arithmetic computation via ratio-encoding in chemical reaction networks

We develop a general framework for instantaneous arithmetic computation using chemical reaction networks. Numerical values are encoded by ratios of species concentrations: an extended nonnegative value $a\in[0,+\infty]$ is represented by a computational pair $(A_0,A_1)$ through the ratio $a=a_1/a_0$. For this ratio encoding, our main theoretical result is a feedforward compositionality theorem: if a collection of modules satisfies two easily checked structural conditions, then any admissible finite feedforward composition computes the corresponding arithmetic expression instantaneously, meaning that every intermediate and output ratio is correct at every positive time. We make this framework concrete by constructing reaction network modules for four elementary operations---identification, inversion, multiplication, and addition---and verifying that each satisfies these structural conditions. As applications, we show that truncated power series and matrix products can be computed instantaneously using reaction networks whose size reflects the number of arithmetic operations in the underlying computation. We also extend the nonnegative ratio encoding to signed quantities by representing a real number as the difference of two nonnegative ratio-encoded values; this allows addition and multiplication over real-valued inputs. We discuss the practical implications of instantaneous computation, including its relationship to other resource constraints such as the number of chemical species required and the cost of reading out the final answer, which requires division and, in the signed case, rectified subtraction.

math.DS↗

arXiv2026

A general-purpose sensitivity method for multiple simultaneous parameter perturbations in stochastic reaction networks

Stochastic reaction networks are continuous-time Markov chain models for interacting populations, with applications in biochemistry, epidemiology, ecology, and related areas. We study finite-difference sensitivity estimation when a single estimator requires several nearby parameterized paths. Existing variance-reducing couplings are typically pairwise, so that repeated use is either inefficient or requires application-specific choices in multi-path settings. We introduce the multi-path stacked coupling (MSC), a space-time Poisson construction that jointly generates any finite collection of parameterized paths. Each pairwise marginal of MSC has the same law as the corresponding split coupling pair, allowing existing variance bounds to transfer directly; in finite-state settings, we also obtain first-order expansions for the mean and second moment of finite-difference numerators. We apply MSC in three settings of practical importance: estimating many first derivatives simultaneously, estimating a single first derivative using a wider finite-difference stencil, and estimating higher-order derivatives. Numerical experiments on a processive phosphorylation network demonstrate strong performance in each of the three application areas considered, consistent with the theoretical advantages of MSC: across all three applications, MSC achieves the smallest root mean square error (RMSE) among the methods considered over the tested computational budgets.

stat.ME↗

arXiv2026

Fluctuation impossibility results for stochastic burst networks

Stochastic reaction networks often involve components at low copy number, where individual production and degradation events generate substantial fluctuations. Yan et al.\ conjectured in 2019 that for networks with linear degradation and arbitrary cross-regulatory production rates, feedback cannot suppress the stationary fluctuations for each component below the fluctuations of its constant-rate counterpart. Their formulation allows random burst sizes $K_i$: the unit-birth case has $K_i\equiv1$, the biologically important burst model takes $K_i$ to be geometrically distributed, but more general positive integer-valued burst laws are also permitted. The conjecture was recently proved for unit births, $K_i\equiv1$. We show here that the conjecture is \textit{false in general} by constructing a two-component network with bounded production rates and burst sizes in $\{1,2\}$ for which both stationary Fano factors lie below their common constant-rate baseline. We then prove the conjecture for positive geometric bursts, the canonical burst model in stochastic gene expression. For arbitrary regulatory architecture and nonlinear cross-regulatory production rates, we prove an exact weighted tradeoff that rules out simultaneous suppression of every component below its geometric-burst baseline. We also prove a complementary structural impossibility result for arbitrary positive integer-valued burst laws with finite second moments: if the activating and inhibiting interactions have a globally consistent sign structure, in the sense that every cycle of the regulatory interaction graph contains an even number of negative interactions, then every component individually satisfies $F_{X_i}\ge B_i$, where $F_{X_i}$ is its stationary Fano factor and $B_i$ is its constant-rate burst baseline.

math.PR↗

arXiv2026

Noise Tradeoffs, Stationary Information Flow, and Structural Balance in Unit-Birth Networks

In 2019, Paulsson and collaborators conjectured that stochastic biochemical control networks have fundamental limits on how much intrinsic noise can be simultaneously suppressed across multiple components. Ripsman, Kell, and Hilfinger recently proposed a formal proof strategy for unit-birth models based on a stationary information-theoretic decomposition. Here, we provide a rigorous mathematical justification for this argument. We consider continuous-time Markov chains on $\Z^N_{\ge 0}$ in which each component is degraded linearly and produced in unit births at a state-dependent rate depending on the other components but not on itself. Noise in component $i$ is measured by the Fano factor $F_{X_i}$, the ratio of stationary variance to mean, with Poisson value $1$ as baseline. Our first contribution is to isolate explicit hypotheses on moments, mean birth rates, and total-rate growth under which the formal information-flow identities can be rigorously justified. Following the proof outline of Ripsman, Kell, and Hilfinger, we then prove the conjecture. Our second contribution is to make these hypotheses checkable: a uniform positive lower bound on the birth rates and at-most-linear total growth dominated by the weakest degradation rate suffices, via Foster--Lyapunov methods. Our third contribution is a structural strengthening. Under a signed monotonicity condition on the rate functions, satisfied by structurally balanced signed interaction networks, we prove that the stationary distribution is associated with respect to the corresponding signed partial order. This upgrades the global tradeoff to the termwise bound $F_{X_i}\ge 1$ for every $i$. Hence, within the signed-monotone subclass, sub-Poissonian noise requires a frustrated interaction topology.

math.PR↗

arXiv2026

Computing with reaction networks at input-independent speed: exponential and logarithmic functions

The concept of \textit{input-independent computational speed} for chemistry-based analog computers was introduced in Anderson-Joshi (2025), where it was shown that arithmetic operations can be computed with a convergence rate bounded below independently of the input values. Here, by inputs we mean the numerical values encoded by the initial concentrations of designated input species, with the underlying reaction network and rate constants held fixed. Combining these operations via power series to approximate transcendental functions is possible in principle, but the number of chemical species required grows with the number of terms retained, and achieving sufficient accuracy may demand many terms---a burden that is especially severe for slowly converging series such as the power series for the logarithm. In this paper, we begin the program of directly computing transcendental functions by chemical reaction networks by focusing on the exponential and logarithmic functions, two widely used transcendental functions, and constructing reaction network modules that compute them without relying on truncated power series. We show that the resulting modules are mass-action systems, and prove that they achieve arbitrary accuracy given sufficient time while operating at input-independent speed. Moreover, these modules can be combined with each other and with the arithmetic modules of Anderson-Joshi (2025): any such composite computation also runs at input-independent speed, with a convergence rate that does not depend on the length of the feedforward chain. Logarithm and exponential functions serve as foundational cases, and the constructions developed here are intended to serve as templates for the direct computation of more general transcendental functions by chemical reaction networks.

math.DS↗

arXiv2025

Stochastic Reaction Networks Within Interacting Compartments with Content-Dependent Fragmentation

Stochastic reaction networks with mass-action kinetics provide a useful framework for understanding processes -- biochemical and otherwise -- in homogeneous environments. However, cellular reactions are often compartmentalized, either at the cell level or within cells, and hence non-homogeneous. We investigate a model of compartmentalization in which the rate of fragmentation of a compartment depends on the abundance of some designated species inside that compartment. The particular model of study is part of a general framework for compartmentalized chemistry with dynamic compartments that was proposed in (Duso and Zechner, PNAS, 2020). This paper builds on (Anderson and Howells, Bull. Math. Biol., 2023) where the special case where the compartment dynamics do not depend on their contents was studied mathematically. In particular, we demonstrate that the explosivity characterization from (Anderson and Howells, Bull. Math. Biol., 2023) fails in this setting and provide new sufficient conditions for non-explosivity and positive recurrence, under the assumption that the underlying CRN admits a linear Lyapunov function. These results extend the theoretical foundation for modeling content-mediated compartment dynamics, with implications for systems such as cell division and intracellular transport.

math.PR↗

arXiv2025

Mathematical Analysis for a Class of Stochastic Copolymerization Processes

We study a stochastic model of a copolymerization process that has been extensively investigated in the physics literature. The main questions of interest include: (i) what are the criteria for transience, null recurrence, and positive recurrence in terms of the system parameters; (ii) in the transient regime, what are the limiting fractions of the different monomer types; and (iii) in the transient regime, what is the speed of growth of the polymer? Previous studies in the physics literature have addressed these questions using heuristic methods. Here, we utilize rigorous mathematical arguments to derive the results from the physics literature. Moreover, the techniques developed allow us to generalize to the copolymerization process with finitely many monomer types. We expect that the mathematical methods used and developed in this work will also enable the study of even more complex models in the future.

math.PR↗

arXiv2024

Parametric Sensitivity Analysis for Models of Reaction Networks within Interacting Compartments

Models of reaction networks within interacting compartments (RNIC) are a generalization of stochastic reaction networks. It is most natural to think of the interacting compartments as "cells" that can appear, degrade, split, and even merge, with each cell containing an evolving copy of the underlying stochastic reaction network. Such models have a number of parameters, including those associated with the internal chemical model and those associated with the compartment interactions, and it is natural to want efficient computational methods for the numerical estimation of sensitivities of model statistics with respect to these parameters. Motivated by the extensive work on computational methods for parametric sensitivity analysis in the context of stochastic reaction networks over the past few decades, we provide a number of methods in the basic RNIC setting. Provided methods include the (unbiased) Girsanov transformation method (also called the Likelihood Ratio method) and a number of coupling methods for the implementation of finite differences, each motivated by methods from previous work related to stochastic reaction networks. We provide several numerical examples comparing the various methods in the new setting. We find that the relative performance of each method is in line with its analog in the "standard" stochastic reaction network setting. We have made all of the Matlab code used to implement the various methods freely available for download.

q-bio.MN↗

arXiv2024

Chemical mass-action systems as analog computers: implementing arithmetic computations at specified speed

Recent technological advances allow us to view chemical mass-action systems as analog computers. In this context, the inputs to a computation are encoded as initial values of certain chemical species while the outputs are the limiting values of other chemical species. In this paper, we design chemical systems that carry out the elementary arithmetic computations of: identification, inversion, $m$th roots (for $m \ge 2$), addition, multiplication, absolute difference, rectified subtraction over non-negative real numbers, and partial real inversion over real numbers. We prove that these ``elementary modules'' have a speed of computation that is independent of the inputs to the computation. Moreover, we prove that finite sequences of such elementary modules, running in parallel, can carry out composite arithmetic over real numbers, also at a rate that is independent of inputs. Furthermore, we show that the speed of a composite computation is precisely the speed of the slowest elementary step. Specifically, the scale of the composite computation, i.e. the number of elementary steps involved in the composite, does not affect the overall asymptotic speed -- a feature of the parallel computing nature of our algorithm. Our proofs require the careful mathematical analysis of certain non-autonomous systems, and we believe this analysis will be useful in different areas of applied mathematics, dynamical systems, and the theory of computation. We close with a discussion on future research directions, including numerous important open theoretical questions pertaining to the field of computation with reaction networks.

math.DS↗

arXiv2024

Square-difference factor absorbing ideals of a commutative ring

Let $R$ be a commutative ring with $1 \neq 0$. A proper ideal $I$ of $R$ is a {\it square-difference factor absorbing ideal} (sdf-absorbing ideal) of $R$ if whenever $a^2 - b^2 \in I$ for $0 \neq a, b \in R$, then $a + b \in I$ or $a - b \in I$. In this paper, we introduce and investigate sdf-absorbing ideals.

math.AC↗

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

Stochastic reaction networks within interacting compartments

Stochastic reaction networks, which are usually modeled as continuous-time Markov chains on $\mathbb Z^d_{\ge 0}$, and simulated via a version of the "Gillespie algorithm," have proven to be a useful tool for the understanding of processes, chemical and otherwise, in homogeneous environments. There are multiple avenues for generalizing away from the assumption that the environment is homogeneous, with the proper modeling choice dependent upon the context of the problem being considered. One such generalization was recently introduced in (Duso and Zechner, PNAS, 2020), where the proposed model includes a varying number of interacting compartments, or cells, each of which contains an evolving copy of the stochastic reaction system. The novelty of the model is that these compartments also interact via the merging of two compartments (including their contents), the splitting of one compartment into two, and the appearance and destruction of compartments. In this paper we begin a systematic exploration of the mathematical properties of this model. We (i) obtain basic/foundational results pertaining to explosivity, transience, recurrence, and positive recurrence of the model, (ii) explore a number of examples demonstrating some possible non-intuitive behaviors of the model, and (iii) identify the limiting distribution of the model in a special case that generalizes three formulas from an example in (Duso and Zechner, PNAS, 2020).

math.PR↗

arXiv2022

Mixing times for two classes of stochastically modeled reaction networks

The past few decades have seen robust research on questions regarding the existence, form, and properties of stationary distributions of stochastically modeled reaction networks. When a stochastic model admits a stationary distribution an important practical question is: what is the rate of convergence of the distribution of the process to the stationary distribution? With the exception of \cite{XuHansenWiuf2022} pertaining to models whose state space is restricted to the non-negative integers, there has been a notable lack of results related to this rate of convergence in the reaction network literature. This paper begins the process of filling that hole in our understanding. In this paper, we characterize this rate of convergence, via the mixing times of the processes, for two classes of stochastically modeled reaction networks. Specifically, by applying a Foster-Lyapunov criteria we establish exponential ergodicity for two classes of reaction networks introduced in \cite{anderson2018some}. Moreover, we show that for one of the classes the convergence is uniform over the initial state.

math.PR↗

arXiv2022

Conditional Monte Carlo for Reaction Networks

Reaction networks are often used to model interacting species in fields such as biochemistry and ecology. When the counts of the species are sufficiently large, the dynamics of their concentrations are typically modeled via a system of differential equations. However, when the counts of some species are small, the dynamics of the counts are typically modeled stochastically via a discrete state, continuous time Markov chain. A key quantity of interest for such models is the probability mass function of the process at some fixed time. Since paths of such models are relatively straightforward to simulate, we can estimate the probabilities by constructing an empirical distribution. However, the support of the distribution is often diffuse across a high-dimensional state space, where the dimension is equal to the number of species. Therefore generating an accurate empirical distribution can come with a large computational cost. We present a new Monte Carlo estimator that fundamentally improves on the "classical" Monte Carlo estimator described above. It also preserves much of classical Monte Carlo's simplicity. The idea is basically one of conditional Monte Carlo. Our conditional Monte Carlo estimator has two parameters, and their choice critically affects the performance of the algorithm. Hence, a key contribution of the present work is that we demonstrate how to approximate optimal values for these parameters in an efficient manner. Moreover, we provide a central limit theorem for our estimator, which leads to approximate confidence intervals for its error.

math.NA↗

arXiv2021

Prevalence of deficiency zero reaction networks in an Erdos-Renyi framework

Reaction networks are commonly used within the mathematical biology and mathematical chemistry communities to model the dynamics of interacting species. These models differ from the typical graphs found in random graph theory since their vertices are constructed from elementary building blocks, i.e., the species. In this paper, we consider these networks in an Erd\H os-Rényi framework and, under suitable assumptions, derive a threshold function for the network to have a deficiency of zero, which is a property of great interest in the reaction network community. Specifically, if the number of species is denoted by $n$ and if the edge probability is denote by $p_n$, then we prove that the probability of a random binary network being deficiency zero converges to 1 if $\frac{p_n}{r(n)}\to 0$, as $n \to \infty$, and converges to 0 if $\frac{p_n}{r(n)}\to \infty$, as $n \to \infty$, where $r(n)=\frac{1}{n^3}$.

math.PR↗

arXiv2021

On reaction network implementations of neural networks

This paper is concerned with the utilization of deterministically modeled chemical reaction networks for the implementation of (feed-forward) neural networks. We develop a general mathematical framework and prove that the ordinary differential equations (ODEs) associated with certain reaction network implementations of neural networks have desirable properties including (i) existence of unique positive fixed points that are smooth in the parameters of the model (necessary for gradient descent), and (ii) fast convergence to the fixed point regardless of initial condition (necessary for efficient implementation). We do so by first making a connection between neural networks and fixed points for systems of ODEs, and then by constructing reaction networks with the correct associated set of ODEs. We demonstrate the theory by constructing a reaction network that implements a neural network with a smoothed ReLU activation function, though we also demonstrate how to generalize the construction to allow for other activation functions (each with the desirable properties listed previously). As there are multiple types of "networks" utilized in this paper, we also give a careful introduction to both reaction networks and neural networks, in order to disambiguate the overlapping vocabulary in the two settings and to clearly highlight the role of each network's properties.

cs.NE↗

arXiv2021

Deficiency zero for random reaction networks under a stochastic block model framework

Deficiency zero is an important network structure and has been the focus of many celebrated results within reaction network theory. In our previous paper \textit{Prevalence of deficiency zero reaction networks in an Erd\H os-Rényi framework}, we provided a framework to quantify the prevalence of deficiency zero among randomly generated reaction networks. Specifically, given a randomly generated binary reaction network with $n$ species, with an edge between two arbitrary vertices occurring independently with probability $p_n$, we established the threshold function $r(n)=\frac{1}{n^3}$ such that the probability of the random network being deficiency zero converges to 1 if $\frac{p_n}{r(n)}\to 0$ and converges to 0 if $\frac{p_n}{r(n)}\to\infty$, as $n \to \infty$. With the base Erd\H os-Rényi framework as a starting point, the current paper provides a significantly more flexible framework by weighting the edge probabilities via control parameters $α_{i,j}$, with $i,j\in \{0,1,2\}$ enumerating the types of possible vertices (zeroth, first, or second order). The control parameters can be chosen to generate random reaction networks with a specific underlying structure, such as "closed" networks with very few inflow and outflow reactions, or "open" networks with abundant inflow and outflow. Under this new framework, for each choice of control parameters $\{α_{i,j}\}$, we establish a threshold function $r(n,\{α_{i,j}\})$ such that the probability of the random network being deficiency zero converges to 1 if $\frac{p_n}{r(n,\{α_{i,j}\})}\to 0$ and converges to 0 if $\frac{p_n}{r(n,\{α_{i,j}\})}\to \infty$.

math.PR↗

arXiv2020

Bounded and finite factorization domains

An integral domain is atomic if every nonzero nonunit factors into irreducibles. Let $R$ be an integral domain. We say that $R$ is a bounded factorization domain if it is atomic and for every nonzero nonunit $x \in R$, there is a positive integer $N$ such that for any factorization $x = a_1 \cdots a_n$ of $x$ into irreducibles $a_1, \dots, a_n$ in $R$, the inequality $n \le N$ holds. In addition, we say that $R$ is a finite factorization domain if it is atomic and every nonzero nonunit in $R$ factors into irreducibles in only finitely many ways (up to order and associates). The notions of bounded and finite factorization domains were introduced by D. D. Anderson, D. F. Anderson, and M. Zafrullah in their systematic study of factorization in atomic integral domains. Here we provide a survey of some of the most relevant results on bounded and finite factorization domains.

math.AC↗