Research

My research lies broadly in the areas of dynamical systems, stochastic processes, and mathematical biology. I am primarily interested in understanding the dynamical behavior of biochemical reaction systems such as enzymatic reactions, signal transduction cascades, and gene regulatory networks.

A particular emphasis of in my research in the relationship between the network of interactions and the admissible dynamical behaviors of the resulting dynamical system. In this formulation, networks are represented as directed graphs G(V,E) where the set of vertexes are the stoichiometrically distinct complexes (i.e. net inputs or net outputs of a reaction) and the set of edges are the reactions.

[Full research statement]

In preparation

  • On the connection between endotactic chemical reaction networks and canonical network forms (w/ G. Craciun and D. Anderson), in preparation

Publications

  1. A Computational Approach to Extinction Events in Chemical Reaction Networks with Discrete State Spaces, submitted, 2017. [ArXiv] [Python source code: DE_Modules.py, DE_Run.py; Example files: example_envz_ompr, example_envz_ompr.dat; Also requires Elisa Tonello’s crnpy package, available here]
  2. Conditions for Extinction Events in Chemical Reaction Networks with Discrete State Spaces (w/ D.F. Anderson, G. Craciun, and R. Brijder), submitted, 2017. [ArXiv]
  3. A Linear Programming Approach to Dynamical Equivalence, Linear Conjugacy, and the Deficiency One Theorem, J. Math. Chem., 54(8):1612-1631, 2016. [ArXiv]
  4. A computational approach to persistence, permanence, and endotacticity of biochemical reaction networks (w/ C. Pantea and P. Donnell), J. Math. Biol., 72(1):467-498, 2016. [ArXiv]
  5. A computational approach to steady state correspondence of regular and generalized mass action systems, Bull. Math. Biol., 77(6):1065-1100, 2015. [ArXiv]
  6. Stochastic analysis of chemical reaction networks with absolute concentration robustness (w/ D. Anderson and G. Enciso), J. R. Soc. Interface, 11(93), 20130943, 2014. [ArXiv]
  7. Translated chemical reaction networks, Bull. Math. Biol. 76(5), 1081-1116,2014. [ArXiv]
  8. Computing weakly reversible linearly conjugate chemical reaction networks with minimal deficiency (w/ D. Siegel and G. Szederkenyi), Math. Biosci., 241(1), 88-98, 2013. [ArXiv]
  9. Dynamical equivalence and linear conjugacy of chemical reaction networks: new results and methods (w/ D. Siegel and G. Szederkenyi), MATCH Commun. Math. Comput. Chem., 68(2), 443, 2012. [ArXiv]
  10. A linear programming approach to weak reversibility and linear conjugacy of chemical reaction networks (w/ D. Siegel and G. Szederkenyi), J. Math. Chem., 50(1), 274-288, 2012. [ArXiv]
  11. Weak dynamical non-emptiability and persistence of chemical kinetics systems (w/ D. Siegel), SIAM J. Appl. Math., 714), 1263-1279, 2011. [ArXiv]
  12. Linear conjugacy of chemical reaction networks (w/ D. Siegel), J. Math. Chem., 49(7), 1263-1282, 2011. [ArXiv]
  13. A stratum approach to global stability of complex balanced systems (w/ D. Siegel), Dyn. Syst., 26(2), 125-146, 2011. [ArXiv]
  14. Equilibria and periodic solutions of projected dynamical systems on sets with corners (w/ M.G. Cojocaru), Aust. J. Math. Anal. Appl., 5(2), 4-12, 2008.
  15. Dynamics of vaccination strategies via projected dynamical systems (w/ M.G. Cojocaru and C. Bauch), Bull. Math. Biol., 69(5), 1453-1476, 2007.

Theses

  • Topics in chemical reaction network theory, Ph.D. thesis, University of Waterloo, 2011. [download]
  • Case studies of existence of periodic solutions for projected dynamical systems on euclidean spaces, Master’s Thesis, University of Guelph, 2006.

Unpublished Notes

  • A note on “MAPK networks and their capacity for multistationarity due to toric steady states”, 2014. [Arxiv]
  • Linearization of Complex Balanced Reaction Systems, 2008. [download]
  • Quadratic Form Linearization of Nonlinear Differential Equations, 2008. [download]