Program
& Abstracts

We discuss Feller's renewal theorem and present an elementary proof.

Structural identifiability answers a theoretical question about which parameter combinations of a mathematical model are uniquely recovered from ideal, noise-free input–output data. Identifiability has been long studied on linear compartment models, and is closely related to the algebraic structure of the input-output equation obtained from the model. In this work, we investigate how structural identifiability is affected under singular limits in which a single parameter of a model is sent to infinity. Such limits induce a collapse of the underlying linear compartment graph that resembles an edge contraction and results in a reduced model. Using the transfer function and differential algebra approaches, we show that the transfer function of the original model converges to the transfer function of the collapsed model. From this result, we can identify which parameter combinations survive the singular limit and characterize the resulting loss of identifiability. We illustrate these results on several families of linear compartment models, including mammillary, catenary, and cyclic networks, and discuss connections to model reduction and the geometry of parameter space under asymptotic limits.

In this talk I will introduce our recent progress on the numerical estimation of the speed of convergence of a Poisson-driven chemical reaction system towards its steady state or its quasi-stationary distribution (QSD). Our approach is based on the coupling method, which is a well-established approach for stochastic differential equations. In order to couple chemical reaction systems on high dimensional lattices, we developed a novel algorithm based on the strong approximation between a Wiener process and a centralized Poisson process. The diffusion approximation of a chemical reaction system, driven by Wiener processes, can “guide” the Poisson-driven chemical reaction system to couple. Then the speed of convergence is implied by the coupling inequality.

We address the problem of estimating the state of a chemical reaction network based on observed time trajectories of some components of the state. In the deterministic setting, our goal is the design of an observer with a tunable parameter which converges exponentially fast to the system state trajectory. The exponential convergence is guaranteed for all initial conditions inside a ball whose diameter increases indefinitely as the tunable parameter is increased. We provide an observer design methodology that provably works for a certain class of networks which includes oscillatory and even chaotic networks. We illustrate our approach via several numerical examples. We also present numerical examples where observations are corrupted by noise. 

Chemical reaction networks admit descriptions at multiple levels — stoichiometric, thermodynamic, kinetic, mechanistic — yet no principled account explains why each is necessary. We construct a canonical bottom-up tower of categorical structures in which each level is the unique minimal extension whose automorphism group separates networks the previous level conflates, unifying existing results in CRNT as special cases of a single forced hierarchy.

Cells often make discrete decisions from complex chemical inputs, such as when lectin signaling pathways decode high-dimensional patterns of glycan-bound receptor activity. These decisions are implemented by biochemical reaction networks, yet the structural features that determine their computational power remain unclear. To address this, we present a mathematical framework for understanding the computations performed by steady states of chemical reaction networks. We show that computational expressivity is governed by nonlinear kinetic features, as well as cross-talk among conserved moiety groups which we quantify as the number of directed interaction paths in the reaction graph. Our results reveal a simple design principle: classification performance can be enhanced either by increasing the diversity of receptor inputs or by increasing cross-talk within the downstream network. We numerically illustrate this by optimizing the rates of signaling networks, demonstrating that highly cross-linked networks can accurately separate chemical classes even in low-dimensional activation spaces, and furthermore such networks can flexibly support multiple classification tasks without changing kinetic parameters. These findings identify chemical cross-talk as a key structural feature underlying computational flexibility in biochemical networks.

Multistationarity underlies biochemical switching and cellular decision-making. In this talk we study how multistationarity in the sequential n-site phosphorylation–dephosphorylation cycle is affected when only some species are open, meaning allowed to exchange with the environment (so-called semi-open networks). For n>1, opening any nonempty subset of substrate species preserves nondegenerate multistationarity, while opening the enzymes (kinase and phosphatase), even together with substrates, always destroys it. The latter follows from a reduction that combines detection of absolute concentration robustness (ACR) with projection onto a monostationary subsystem; biologically, enzyme exchange acts as a robust “off” switch while substrate exchange preserves switching and modular coupling. We further illustrate the general method on multi-layer cascade variants.

Chemical Reaction Networks provide a fundamental framework for modeling the stochastic dynamics of biochemical systems, where molecular species evolve through discrete and random noise reaction events. Parameter inference in Chemical Reaction Networks is a central problem in systems biology, but traditional methods such as maximum likelihood estimation are often intractable due to computational complexity and the lack of continuous-time data. In this study, we introduce a statistically grounded and computationally efficient estimator for reaction rate parameters using high-frequency discrete-time observations. Modeling the system as a Continuous-Time Markov Chain, our method handles general kinetics, including non-mass-action and higher-order reactions. Validation on synthetic and experimental datasets demonstrates its accuracy and robustness. This approach offers a simple and reliable framework for parameter inference in complex stochastic systems.

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 an algorithm available that implements our theory and we illustrate it with several applications.

Glycolytic oscillations are a well-studied example of biological rhythms, yet their underlying biochemical networks are sufficiently complex that analysis often relies on reduced ordinary differential equation (ODE) models. Most such reductions are derived from deterministic descriptions, even though the true dynamics arise from stochastic reaction events. In this talk, I present a continuous-time Markov chain formulation of a classical glycolytic pathway model and show how a rigorous law of large numbers limit yields a reduced ODE system through multiscale stochastic averaging. I will also discuss parameter inference when only slow variables are observable and outline results establishing statistical consistency of the resulting estimates. Together, these results provide a mathematically grounded link between stochastic reaction networks, reduced dynamical models, and data-driven inference.

This is joint work with Arnab Ganguly.

Toric dynamical systems show simple dynamics: their unique positive equilibrium attracts every positive solution. Motivated by the fact that a number of non-toric mass-action systems behave similarly neatly, disguised toric systems were introduced a couple of years ago. By definition, the mass-action differential equation of such a system is dynamically identical to that of a toric mass-action system. In many instances, this concept allows us to conclude the global stability of reaction networks for a set of parameter values significantly larger than their toric variety.

In this talk, we take another step forward. We say that a mass-action system is disguised toric in the general sense if there exists a coordinate change that makes it disguised toric. Even the simplest coordinate transformation, a linear diagonal change of variables, leads to a powerful method. We illustrate the strength of this new concept via several examples.

Joint work with Gheorghe Craciun, Oskar Henriksson, Jiaxin Jin, and Diego Rojas La Luz.

Recent work has revealed that biochemical networks with bifunctional enzymes can display remarkably rich dynamics, including ultrasensitivity, switch-like responses, concentration robustness, and even exhaustion of species. This talk presents a dynamical systems analysis of such networks which sheds light on the subtle architectural differences which produces these vast differences in functional behavior. We employ the next-generation matrix method---only recently adapted to biochemical reaction networks---to characterize previously incomputable thresholds for the stability of boundary steady states. These thresholds are critical for determining when a mechanism will proceed and when it will shut down. Using bifurcation analysis, we further establish conditions for multistationarity, showing how multiple positive steady states can arise within a single stoichiometric compatibility class. This properties is theorized to underlie toggle-switch behavior in genetic networks. We also consider the capacity of such networks for absolute concentration robustness--an essential feature of metabolic regulation in which a species maintains a fixed steady-state concentration despite fluctuations in other species---and explore the connection between robustness and boundary stability.

The steady state equations of chemical reaction networks are vertically parametrized systems under common kinetic assumptions, which, in many ways, make them extremely well-behaved from an algebraic point of view. In this talk, I'll follow up on my Sunday tutorial on algebraic reaction network theory by giving an introduction to the vertically parametrized perspective, and our latest results on what it mean for multistationarity, absolute concentration robustness and steady-state invariants. This is joint work with Elisenda Feliu and Beatriz Pascual Escudero, and is primarily based on the preprints 2412.17798 and 2411.15134.

Living systems maintain stable internal states despite environmental fluctuations. Absolute concentration robustness (ACR) is a striking homeostatic phenomenon in which the steady-state concentration of a species remains invariant despite changes in total supply. In this talk, we introduce a previously underappreciated phenomenon, namely asymptotic ACR (aACR): approximate robustness can emerge solely from the network structure, without requiring exact ACR motifs or negligible parameters. We find that aACR is more pervasive than classical ACR and prove that this ubiquity stems solely from network structure. This notion of aACR would provide a rigorous and practical tool to analyze robust responses in broad biochemical systems. This is joint work with Diego Rojas La Luz and Gheorge Craciun.

Stochastic chemical reaction networks satisfying topological conditions, such as weak reversibility and deficiency zero, and kinetic conditions like propensity factorizability, admit product-form stationary distributions. In addition, the network translation solves some violations of topological conditions whenever translated networks hold the kinetic condition.

In this talk, I will introduce the framework of the "dummy species extension" to solve some violations of kinetic conditions. By adding an additional species, originally non-propensity factorizable network systems that satisfy certain requirements are transformed into propensity factorizable network systems within this framework, enabling the analytic derivation of their stationary distributions. Furthermore, this study categorizes network systems and establishes the necessary conditions for a network system to be dummy representable.

I will present a rigorous mathematical framework for analyzing a class of stochastic copolymerization processes, where finitely many types of monomers attach and detach at the tip of a polymer chain. These dynamics are modeled as a continuous-time Markov chain on an infinite tree-like state space. Sharp criteria for transience, null recurrence, and positive recurrence in terms of the attachment and detachment rates are established. In the transient regime, explicit formulas for the almost sure asymptotic composition of the polymer and its growth velocity are provided. The framework also naturally extends to models allowing finite blocks of monomers to attach or detach and the block-memory case, and is expected to facilitate the analysis of more complex polymerization mechanisms in future work.

Many biological systems operate far from thermodynamic equilibrium by sustained energy consumption – here, we focus on a system of dynamic microtubules, energy-utilizing biopolymers made of the protein tubulin. At the molecular level, microtubule dynamics is governed by a stochastic chemical reaction network consisting of polymerization, depolymerization, and nucleotide hydrolysis reactions at filament ends. Irreversible GTP hydrolysis breaks detailed balance in the network, generating sustained probability currents that propagate to system-level observables and give rise to multiple distinct nonequilibrium steady states (NESS). To characterize these nonequilibrium steady states, we model the molecular-scale reaction network as a continuous-time Markov process and simulate its dynamics using exact stochastic simulation (Gillespie) methods. This framework provides access to both trajectory-level fluctuations and steady-state probability distributions under broken detailed balance. We then analyze the resulting steady states using stochastic thermodynamics, quantifying probability currents, entropy production, and system-level observables such as time to reach polymer-mass NESS (tNESS), concentration of polymerized tubulin at NESS ([polymerized tubulin]), emergent from the underlying reaction network. We find that tNESS exhibits systematic and nontrivial dependence on both free tubulin concentration and the rate of GTP hydrolysis, the latter controlling the strength of irreversibility in the molecular-scale reaction network. Altogether we show how local irreversibility at the molecular reaction-network level governs both – the transient state and the nonequilibrium steady states at the system level.

Chemical reaction networks (CRNs) provide a natural model for analog computation in which inputs and outputs are encoded by molecular abundances. Classical studies of CRN-based computation have mainly focused on deterministic mass-action systems. Yet in biomolecular implementations using DNA and protein-based circuits, low copy-number effects and intrinsic fluctuations can be non-negligible, so stochastic models are often more faithful than concentration-based deterministic models. In this research, we develop a framework for computation in stochastic CRNs at the level of the mean of stationary distributions. Specifically, we construct elementary arithmetic modules such as identification, addition, and multiplication, and analyze their ergodicity and mixing times. We then study how these modules can be interconnected to form composite circuits and examine how their computational behavior interacts. Our results provide a systematic framework for computation in stochastic chemical reaction networks.

Given a reaction network, how can we systematically use data science and machine learning to understand its dynamics? More fundamentally, how do we generate the data?

In this preliminary study, we propose a data-driven exploration framework that integrates Latin hypercube sampling of kinetic parameters, conservation-consistent initial conditions within stoichiometric compatibility classes, large-scale numerical simulation, and automated dynamical classification.

This work is an alternative step toward addressing the CRNT parameter-space localization problem: Where do complex dynamical properties arise in high-dimensional parameter space?

To improve discovery efficiency for rare behaviors, we formulate the detection as a sequential learning problem over parameter space and employ ridge-regularized, class-imbalance-aware logistic regression as an interpretable surrogate model. The surrogate is exploited in two complementary ways: (i) probability-guided sampling, which concentrates evaluations in regions with high predicted regime likelihood, and (ii) $\beta$-guided sampling, which uses learned coefficient directions to drive geometry-aware exploration while retaining a fixed fraction of global coverage.

We apply the method to two biochemical networks to illustrate its performance and practical value: (1) a two-protein gene transcription network and (2) a methylation-regulated circadian clock model. The framework estimates the prevalence of each dynamical regime and identifies the factors driving multistationarity and oscillations under a fixed simulation budget.

The proposed method complements structural CRNT results by identifying where behaviors occur in parameter space, revealing key parameter influences, and enabling efficient discovery of rare dynamical regimes in mass-action systems.

An important research problem for chemical reaction networks is determining the maximum number of steady states for a network. Many researchers used the mixed volume bound to give an upper bound for the maximum number of steady states, however, the Newton-Okounkov bound (introduced by Obatake and Walker) is known to be able to give sharper bounds in some cases. In this poster we investigate a class of networks for which the Newton-Okounkov bound will always be sharper than the mixed volume bound; we call such networks Full Volume. We show that checking whether a network is Full Volume can be done easily using the newton polytope, and introduce reduced networks, a class of networks that requires significantly fewer checks to verify Full Volume.

In chemical reaction network theory, a common area of study is the analysis of steady state solutions. Steady states can be described by a variety in high dimensional parameter space, which is partitioned into connected components that give different numbers of steady state solutions. Computing this discriminant can be computationally difficult as our chemical reaction networks increase in complexity, but numerical methods allows us to analyze the connected components of the complement of the discriminant without computing the defining equation.

In population dynamics, the Allee effect refers to the phenomenon where a population has a higher growth rate at higher densities. It has been thoroughly studied and until recently the number of steady states was not well understood. I will present a case study of how numerical methods can be used to describe the number of positive steady states of the Allee effect.

[1] Breiding, Paul, John Cobb, Aviva K. Englander, Nayda Farnsworth, Jonathan D. Hauenstein, Oskar Henriksson, David K. Johnson, Jordy Lopez Garcia, and Deepak Mundayur. "Elimination Without Eliminating: Computing Complements of Real Hypersurfaces Using Pseudo-Witness Sets." arXiv preprint arXiv:2601.04383 (2026).

[2] Englander, Aviva K. and Jose Israel Rodriguez. "Towards Learning the Positive Real Discriminant of the Wnt Signaling Pathway Shuttle Model." ACM Communications in Computer Algebra 58, no. 3 (2025): 85-88.

[3] Song, Kuo, and Xiaoxian Tang. "Steady State Classification of Allee Effect System." arXiv preprint arXiv:2501.19062 (2025).