Polly Yu


Office:   518 Van Vleck Hall

Department of Mathematics
University of Wisconsin-Madison
480 Lincoln Drive
Madison, WI, 53706

pollyyu [at] math [·] wisc [·] edu

--- research ---

I work on dynamical systems arising from network interactions, typically deterministic models for biochemical kinetics. A realistic model of the chemistry within cells is complex: with many chemical species, uncertain kinetic rate functions and unknown parameters. Understandably, analytical solutions of the dynamics is difficult to obtain. Instead of purely relying on numerical methods, I am interested in inferring dynamical properties from the network structure.

My work can roughly be seen in the light of two (nonorthogonal) directions: (1) applying classical results for mass-action systems in novel ways, and (2) studying how the network structure relates to possible dynamics under different kinetics assumptions.

  1. There are some very strong results for mass action systems from classical chemical reaction network theory. (Overview of mass action systems and reaction network.) I am looking into how these results can serve as tools in other contexts, as well as continuing to work with mass action systems.

  2. Motivated by the success of Chemical Reaction Network Theory, one may naturally try to understand what role the structure of the underlying network play in the dynamics. I am interested in answering the following question without costly computations: under other kinetic assumptions (i.e., not mass action), what dynamics are permitted or prohibited by the network structure.

--- project descriptions ---

Chemical reaction network theory

An aim of chemical reaction network theory (CRNT) is to relate network structure to dynamical properties. I am interested in open questions in CRNT, and extending some of the results to non-mass-action kinetics.

  • Weakly reversible system with infinitely many steady states. In preparation. (with Balázs Boros and Gheorghe Craciun)

Dynamical equivalence and flux systems

By balancing fluxes on a reaction network, we can study when different networks give rise to the same dynamical system, and use this flexibility in network realization to conclude dynamical properties that may not be at first obvious.

In theory, dynamically equivalent mass-action systems can have additional source complexes in the reaction network. We showed that a reversible, weakly reversible, detailed-balanced, or complex-balanced realization can be achieved with no additional complexes than those appearing in the monomials of the differential equations. Moreover, this is true of any kinetics, not only mass-action type.

We also investigated the parameter space for dynamical equivalence to complex-balanced. We give conditions for networks with positive deficiencies to admit a set of rate constants (with positive measure), where the mass-action system is dynamically equivalent to complex-balanced. In particular, the system with any of those rate constants inherit all the dynamical properties of complex-balancing, e.g., a globally defined Lyapunov function, uniqueness and stability of steady states.

Other related projects:

  • Uniqueness of weakly reversible deficiency zero realizations. In preparation. (with Gheorghe Craciun and Jiaxin Jin)

Injectivity and delay mass-action systems

We give an algebraic condition, and also conditions on the species-reaction graph of a reaction network that guarantees delay stability under mass-action kinetics for any choice of rate constants and delay parameters.

  • Absolute stability of delay mass-action systems. In preparation. (with Gheorghe Craciun, Maya Mincheva and Casian Pantea)

Generalized mass-action systems

We give a sign condition on vector spaces that guarantees the existence and uniqueness of a vertex-balanced steady state of a generalized mass-action system. This vertex-balanced steady state is a generalization of a Birch point - the intersection of a variety with an affine space.

-- publications ---
  1. Gheorghe Craciun, Jiaxin Jin, Polly Y. Yu. Uniqueness of weakly reversible deficiency zero realizations. In preparation.

  2. Balázs Boros, Gheorghe Craciun, Polly Y. Yu. Weakly reversible system with infinitely many steady states. In preparation.

  3. Gheorghe Craciun, Maya Mincheva, Casian Pantea, Polly Y. Yu. Absolute stability of delay mass-action systems. In preparation.

  4. Gheorghe Craciun, Polly Y. Yu. Robust global stability of complex-balanced systems. In preparation.

  5. Gheorghe Craciun, Matthew D. Johnston, Gábor Szederkényi, Elisa Tonello, János Tóth, Polly Y. Yu. Realizations of kinetic differential equations. Preprint, arXiv:1907.07266, 2019. Submitted.
    [ arXiv | pdf ]

  6. Gheorghe Craciun, Jiaxin Jin, Polly Y. Yu. An efficient characterization of complex-balanced, detailed-balanced, and weakly reversible systems. Preprint, arXiv:1812.06214, 2018. Submitted.
    [ arXiv | pdf ]

  7. 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. Preprint, arXiv:1802.06919, 2018. Accepted.
    [ arXiv | pdf ]

  8. Polly Y. Yu, Gheorghe Craciun. Mathematical analysis of chemical reaction systems. Israel Journal of Chemistry, 58, 2018.
    [ arXiv | url | pdf ]

-- presentations ---