Polly Y. Yu

1. Polly Y. Yu. Global stability of perturbed complex-balanced systems. arXiv:2210.13633.
   [ arXiv | pdf ]
   Abstract
   A class of polynomial dynamical systems called complex-balanced are locally stable and conjectured to be globally stable. In general, complex-balancing is not a robust property, i.e., small changes in parameter values may result in the loss of the complex-balanced property. We show that robustly permanent complex-balanced systems are globally stable even after the rate constants have been perturbed.
2. Diego Rojas La Luz, Gheorghe Craciun, Polly Y. Yu. Generalized Lotka-Volterra Systems and Complex Balanced Polyexponential Systems. arXiv:2412.13367.
   [ arXiv | pdf ]
   Abstract
   We study the global stability of generalized Lotka-Volterra systems with generalized polynomial right-hand side, without restrictions on the number of variables or the polynomial degree, including negative and non-integer degree. We introduce polyexponential dynamical systems, which are equivalent to the generalized Lotka-Volterra systems, and we use an analogy to the theory of mass-action kinetics to define and analyze complex balanced polyexponential systems, and implicitly analyze complex balanced generalized Lotka-Volterra systems. We prove that complex balanced generalized Lotka-Volterra systems have globally attracting states, up to standard conservation relations, which become linear for the associated polyexponential systems. In particular, complex balanced generalized Lotka-Volterra systems cannot give rise to periodic solutions, chaotic dynamics, or other complex dynamical behaviors. We describe a simple sufficient condition for complex balance in terms of an associated graph structure, and we use it to analyze specific examples.
3. Oskar Henriksson, Carlos Améndola, Jose Israel Rodriguez, Polly Y. Yu. Maximum likelihood estimation of log-affine models using detailed-balanced reaction networks. arXiv:2411.07986.
   [ arXiv | pdf | Github ]
   Abstract
   A fundamental question in the field of molecular computation is what computational tasks a biochemical system can carry out. In this work, we focus on the problem of finding the maximum likelihood estimate (MLE) for log-affine models. We revisit a construction due to Gopalkrishnan of a mass-action system with the MLE as its unique positive steady state, which is based on choosing a basis for the kernel of the design matrix of the model. We extend this construction to allow for any finite spanning set of the kernel, and explore how the choice of spanning set influences the dynamics of the resulting network, including the existence of boundary steady states, the deficiency of the network, and the rate of convergence. In particular, we prove that using a Markov basis as the spanning set guarantees global stability of the MLE steady state.
4. Xingchi Yan, Polly Y. Yu, Arvind Srinivasan, Sohaib Abdul Rehman, Maxim B. Prigozhin. Identifying Intermolecular Interactions in Single-Molecule Localization Microscopy. bioRxiv doi: 10.1101/2024.05.10.593617.
   [ bioRxiv | pdf | Supplemental Info ]
   Abstract
   Intermolecular interactions underlie all cellular functions, yet visualizing these interactions at the single-molecule level remains challenging. Single-molecule localization microscopy (SMLM) offers a potential solution. Given a nanoscale map of two putative interaction partners, it should be possible to assign molecules either to the class of coupled pairs or to the class of non-coupled bystanders. Here, we developed a probabilistic algorithm that allows accurate determination of both the absolute number and the proportion of molecules that form coupled pairs. The algorithm calculates interaction probabilities for all possible pairs of localized molecules, selects the most likely interaction set, and corrects for any spurious colocalizations. Benchmarking this approach across a set of simulated molecular localization maps with varying densities (up to ∼50 molecules µm⁻²) and localization precisions (5 to 50 nm) showed typical errors in the identification of correct pairs of only a few percent. At molecular densities of ∼5-10 molecules µm⁻² and localization precisions of 20-30 nm, which are typical parameters for SMLM imaging, the recall was ∼90%. The algorithm was effective at differentiating between non-interacting and coupled molecules both in simulations and experiments. Finally, it correctly inferred the number of coupled pairs over time in a simulated reaction-diffusion system, enabling determination of the underlying rate constants. The proposed approach promises to enable direct visualization and quantification of intermolecular interactions using SMLM.
5. Polly Y. Yu, Eduardo D. Sontag. A necessary condition for non-monotonic dose response, with an application to a kinetic proofreading model. In 2024 63rd IEEE Conference on Decision and Control (CDC). Accepted 2024. Appendix in arXiv:2403.13862.
   [ arXiv | doi | pdf (with appendix) ]
   Abstract
   Steady state nonmonotonic ("biphasic") dose responses are often observed in experimental biology, which raises the control-theoretic question of identifying which possible mechanisms might underlie such behaviors. It is well known that the presence of an incoherent feedforward loop (IFFL) in a network may give rise to a nonmonotonic response. It has been conjectured that this condition is also necessary, i.e. that a nonmonotonic response implies the existence of an IFFL. In this paper, we show that this conjecture is false, and in the process prove a weaker version: that either an IFFL must exist or both a positive feedback loop and a negative feedback loop must exist. Towards this aim, we give necessary and sufficient conditions for when minors of a symbolic matrix have mixed signs. Finally, we study in full generality when a model of immune T-cell activation could exhibit a steady state nonmonotonic dose response.
6. Gheorghe Craciun, Jiaxin Jin, Polly Y. Yu. An algorithm for finding weakly reversible deficiency zero realizations of polynomial dynamical systems. SIAM Journal on Applied Mathematics, 83(4), pp. 1717-1737, 2023.
   [ arXiv | doi | pdf ]
   Abstract
   Systems of differential equations with polynomial right-hand sides are very common in applications. On the other hand, their mathematical analysis is very challenging in general, due to the possibility of complex dynamics: multiple basins of attraction, oscillations, and even chaotic dynamics. Even if we restrict our attention to mass-action systems, all of these complex dynamical behaviours are still possible. On the other hand, if a polynomial dynamical system has a weakly reversible deficiency zero (WR0) realization, then its dynamics is known to be remarkably simple: oscillations and chaotic dynamics are ruled out and, up to linear conservation laws, there exists a single positive steady state, which is asymptotically stable. Here we describe an algorithm for finding WR0 realizations of polynomial dynamical systems, whenever such realizations exist.
7. Sabina J. Haque, Matthew Satriano, Miruna-Ștefana Sorea, Polly Y. Yu. The disguised toric locus and affine equivalence of reaction networks. SIAM Journal on Applied Dynamical Systems, 22(2), pp. 1423-1444, 2023
   [ arXiv | doi | pdf ]
   Abstract
   Under the assumption of mass-action kinetics, a dynamical system may be induced by several different reaction networks and/or parameters. It is therefore possible for a mass-action system to exhibit complex-balancing dynamics without being weakly reversible or satisfying toric constraints on the rate constants; such systems are called disguised toric dynamical systems. We show that the parameters that give rise to such systems are preserved under invertible affine transformations of the network. We also consider the dynamics of arbitrary mass-action systems under affine transformations, and show that there is a bijection between their sets of positive steady states, although their qualitative dynamics can differ substantially.
8. Benjamin Nordick, Polly Y. Yu, Guangyuan Liao, Tian Hong. Nonmodular oscillator and switch based on RNA decay drive regeneration of multimodal gene expression. Nucleic Acid Research, 50(7), pp. 3693-3708, 2022.
   [ bioRxiv | doi | pdf | Supplemental Info ]
   Abstract
   Periodic gene expression dynamics are key to cell and organism physiology. Studies of oscillatory expression have focused on networks with intuitive regulatory negative feedback loops, leaving unknown whether other common biochemical reactions can produce oscillations. Oscillation and noise have been proposed to support mammalian progenitor cells’ capacity to restore heterogenous, multimodal expression from extreme subpopulations, but underlying networks and specific roles of noise remained elusive. We use mass-action-based models to show that regulated RNA degradation involving as few as two RNA species—applicable to nearly half of human protein-coding genes—can generate sustained oscillations without explicit feedback. Diverging oscillation periods synergize with noise to robustly restore cell populations’ bimodal expression on timescales of days. The global bifurcation organizing this divergence relies on an oscillator and bistable switch which cannot be decomposed into two structural modules. Our work reveals surprisingly rich dynamics of post-transcriptional reactions and a potentially widespread mechanism underlying development, tissue regeneration, and cancer cell heterogeneity.
9. Polly Y. Yu, Gheorghe Craciun, Maya Mincheva, Casian Pantea. A graph-theoretic condition for delay stability of reaction systems. SIAM Journal on Applied Dynamical Systems, 21(2), pp. 1092-1118, 2022.
   [ arXiv* | doi | pdf ]
   Abstract
   Delay mass-action systems provide a model of chemical kinetics when past states influence the current dynamics. In this work, we provide a graph-theoretic condition for delay stability, i.e., linear stability independent of both rate constants and delay parameters. In particular, the result applies when the system has no delay, implying asymptotic stability for the ODE system. The graph-theoretic condition is about cycles in the directed species-reaction graph of the network, which encodes how different species in the system interact.
10. Gheorghe Craciun, Abhishek Deshpande, Badal Joshi, Polly Y. Yu. Autocatalytic recombination systems: A reaction network perspective. Mathematical Biosciences, 345, 108784, 2022.
    [ arXiv* | doi | pdf ]
    Abstract
    Autocatalytic systems called hypercycles are very often incorporated in ‘‘origin of life" models. We investigate the dynamics of certain related models called bimolecular autocatalytic systems. In particular, we consider the dynamics corresponding to the relative populations in these networks, and show that it can be analyzed using well-chosen autonomous polynomial dynamical systems. Moreover, we use results from reaction network theory to prove persistence and permanence of several families of bimolecular autocatalytic systems called autocatalytic recombination systems.
11. Gheorghe Craciun, Jiaxin Jin, Polly Y. Yu. Single-target networks. Discrete & Continuous Dynamical Systems - B, 27(2), pp. 799-819, 2022.
    [ arXiv | doi | pdf ]
    Abstract
    We characterize the dynamics of all single-target networks under mass-action kinetics: either the system is (i) globally stable for all choice of rate constants (in fact, dynamically equivalent to a detailed-balanced system) or (ii) has no positive steady states for any choice of rate constants and all trajectories must converge to the boundary of the positive orthant or to infinity. Moreover, global stability occurs if and only if the target vertex of the network is in the relative interior of the convex hull of the source vertices.
12. Gheorghe Craciun, Jiaxin Jin, Polly Y. Yu. Uniqueness of weakly reversible and deficiency zero realizations of dynamical systems. Mathematical Biosciences, 342, 108720, 2021.
    [ arXiv | doi | pdf ]
    Abstract
    A reaction network together with a choice of rate constants uniquely gives rise to a system of differential equations, according to the law of mass-action kinetics. On the other hand, different networks can generate the same dynamical system under mass-action kinetics. Therefore, the problem of identifying "the" underlying network of a dynamical system is not well-posed, in general. Here we show that the problem of identifying an underlying weakly reversible deficiency zero network is well-posed, in the sense that the solution is unique whenever it exists. This can be very useful in applications because from the perspective of both dynamics and network structure, a weakly reversibly deficiency zero (WR0) realization is the simplest possible one. Moreover, while mass-action systems can exhibit practically any dynamical behavior, including multistability, oscillations, and chaos, WR0 systems are remarkably stable for any choice of rate constants: they have a unique positive steady state within each invariant polyhedron, and cannot give rise to oscillations or chaotic dynamics. We also prove that both of our hypotheses (i.e., weak reversibility and deficiency zero) are necessary for uniqueness.
13. Balázs Boros, Gheorghe Craciun, Polly Y. Yu. Weakly reversible mass-action systems with infinitely many positive steady states. SIAM Journal on Applied Mathematics, 80(4), pp. 1936-1946, 2020.
    [ arXiv | doi | pdf ]
    Abstract
    We show that weakly reversible mass-action systems can have a continuum of positive steady states, coming from the zeroes of a multivariate polynomial. Moreover, the same is true of systems whose underlying reaction network is reversible and has a single connected component. In our construction, we relate operations on the reaction network to the multivariate polynomial occurring as a common factor in the system of differential equations.
14. Gheorghe Craciun, Maya Mincheva, Casian Pantea, Polly Y. Yu. Delay stability of reaction systems. Mathematical Biosciences, 326, 108387, 2020.
    [ arXiv | doi | pdf ]
    Abstract
    Delay differential equations are used as a model when the effect of past states has to be taken into account. In this work we consider delay models of chemical reaction networks with mass action kinetics. We obtain a sufficient condition for absolute delay stability of equilibrium concentrations, i.e., local asymptotic stability independent of the delay parameters. Several interesting examples on sequestration networks with delays are presented.
15. Gheorghe Craciun, Jiaxin Jin, Polly Y. Yu. An efficient characterization of complex-balanced, detailed-balanced, and weakly reversible systems. SIAM Journal on Applied Mathematics, 80(1), pp. 183-205, 2020.
    [ arXiv | doi | pdf ]
    Abstract
    Very often, models in biology, chemistry, physics, and engineering are systems of polynomial or power-law ordinary differential equations, arising from a reaction network. Such dynamical systems can be generated by many different reaction networks. On the other hand, networks with special properties (such as reversibility or weak reversibility) are known or conjectured to give rise to dynamical systems that have special properties: existence of positive steady states, persistence, permanence, and (for well-chosen parameters) complex balancing or detailed balancing. These last two are related to thermodynamic equilibrium, and therefore the positive steady states are unique and stable. We describe a computationally efficient characterization of polynomial or power-law dynamical systems that can be obtained as complex-balanced, detailed-balanced, weakly reversible, and reversible mass-action systems.
16. Gheorghe Craciun, Matthew D. Johnston, Gábor Szederkényi, Elisa Tonello, János Tóth, Polly Y. Yu. Realizations of kinetic differential equations. Mathematical Biosciences and Engineering, 17(1), pp. 862-892, 2020.
    [ arXiv* | doi | pdf ]
    Abstract
    The induced kinetic differential equations of a reaction network endowed with mass action type kinetics is a system of polynomial differential equations. The problem studied here is: Given a system of polynomial differential equations, is it possible to find a network which induces these equations; in other words: is it possible to find a kinetic realization of this system of differential equations? If yes, can we find a network with some chemically relevant properties (implying also important dynamic consequences), such as reversibility, weak reversibility, zero deficiency, detailed balancing, complex balancing, mass conservation, etc.? The constructive answers presented to a series of questions of the above type are useful when fitting differential equations to datasets, or when trying to find out the dynamic behavior of the solutions of differential equations. It turns out that some of these results can be applied when trying to solve seemingly unrelated mathematical problems, like the existence of positive solutions to algebraic equations.
17. Gheorghe Craciun, Stefan Müller, Casian Pantea, Polly Y. Yu. A generalization of Birch's theorem and vertex-balanced steady states for generalized mass-action systems. Mathematical Biosciences and Engineering, 16(6), pp. 8243-8267, 2019.
    [ arXiv | doi | pdf ]
    Abstract
    Mass-action kinetics and its generalizations appear in mathematical models of (bio-)chemical reaction networks, population dynamics, and epidemiology. The dynamical systems arising from directed graphs are generally non-linear and difficult to analyze. One approach to studying them is to find conditions on the network which either imply or preclude certain dynamical properties. For example, a vertex-balanced steady state for a generalized mass-action system is a state where the net flux through every vertex of the graph is zero. In particular, such steady states admit a monomial parametrization. The problem of existence and uniqueness of vertex-balanced steady states can be reformulated in two different ways, one of which is related to Birch's theorem in statistics, and the other one to the bijectivity of generalized polynomial maps, similar to maps appearing in geometric modelling. We present a generalization of Birch's theorem, by providing a sufficient condition for the existence and uniqueness of vertex-balanced steady states.
18. Polly Y. Yu, Gheorghe Craciun. Mathematical analysis of chemical reaction systems. Israel Journal of Chemistry, 58(6-7), pp. 733-741, 2018.
    [ arXiv | doi | pdf ]
    Abstract
    The use of mathematical methods for the analysis of chemical reaction systems has a very long history, and involves many types of models: deterministic versus stochastic, continuous versus discrete, and homogeneous versus spatially distributed. Here we focus on mathematical models based on deterministic mass-action kinetics. These models are systems of coupled nonlinear differential equations on the positive orthant. We explain how mathematical properties of the solutions of mass-action systems are strongly related to key properties of the networks of chemical reactions that generate them, such as specific versions of reversibility and feedback interactions.

PhD thesis:   Algebra and Dynamics of Reaction Network Models