Office: 518 Van Vleck Hall

Department of Mathematics

University of Wisconsin-Madison

480 Lincoln Drive

Madison, WI, 53706

pollyyu [at] math [·] wisc [·] edu

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.

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.

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.

## Injectivity and delay mass-action systems

We found 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.

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

## Dynamical equivalences 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.

We characterized when a mass-action system can be dynamically equivalent to a weakly reversible or complex-balanced system. The weakly reversible or complex-balanced realization can be achieved using only the source complexes of the original system. Indeed, this result holds for any kinetics assumptions.

An efficient characterization of complex-balanced, detailed-balanced, weakly reversible, and reversible systems. Preprint, arXiv:1812.06214, 2018. (with Gheorghe Craciun and Jiaxin Jin)

In a related work, we proved that a single-target reaction network under mass-action kinetics either has no positive steady state, or is dynamically equivalent to a detailed-balanced system for any choice of rate constants. The latter holds if and only if the target complex is in the interior of the convex hull of the source complexes.

Stability of single-target networks. In preparation. (with Gheorghe Craciun and Jiaxin Jin)

## Generalized mass-action systems

We found 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.

A generalization of Birch's theorem and vertex-balanced steady states for generalized mass-action systems. Preprint, arXiv:1802.06919, 2018. Submitted. (with Gheorghe Craciun, Stefan Müller, Casian Pantea)

Gheorghe Craciun, Jiaxin Jin, Polly Y. Yu.

*Stability of single-target networks.*In preparation.

Gheorghe Craciun, Maya Mincheva, Casian Pantea, Polly Y. Yu.

*Absolute stability of delay mass-action systems.*In preparation.

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

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. Submitted. [ arXiv | pdf ]

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