Figure : Network for the EnvZ-OmpR signaling system in E. coli.
If the counts of the constituent species in a system associated with a reaction network are high, then their concentrations are typically modeled deterministically by a system of ordinary differential equation. However, if the abundances are low, then the randomness inherent in the molecular interactions is important to the system dynamics, and the abundances are modeled stochastically as a discrete-space, continuous-time Markov chain.
One of the most challenging issues facing researchers who study biological systems is the often extraordinarily complicated structure of their interaction networks. hus, how to characterize network structures that induce emergent phenotypes (characteristic behaviors) of the system dynamics is one of the major open questions in systems biology.
This type of theoretical mathematics falls into the broad area known as chemical reaction network theory, which dates back at least to [6,7], where graphical characteristics of networks were shown to ensure stability of the steady states for deterministically modeled systems. Since that time, much of the research (such as that aimed at resolving the Global Attractor Conjecture) of chemical reaction network theory has focused on discovering the qualitative properties of deterministiccally modeled reaction networks. However, there is now a large literature demonstrating that the fluctuations arising from the effective randomness of individual interactions in cellular systems can have significant consequences on the behavior of the system. nce, analytic results related to stochastic models are essential if these systems are to be well understood.
In the deterministic modeling regime, a number of network conditions have been produced, each of which characterizes a different form of stability. Conversely, there are very few results relating the dynamics of stochastically modeled reaction networks with their associated network structure.
In my thesis work and followup papers [1,2,3], I discovered novel network conditions that guarantee the associated Markov models are positive recurrent, and therefore admit a stationary distribution. Moreover, I characterized their mixing times (rates of convergence to the stationary distribution). The main analytical tools I utilized to carry out this research included (i) the Foster-Lyapunov criteria of Meyn and Tweedie, and (ii) a generalization I made to the ``tier'' style of analysis first developed in  and  for deterministic models
to the stochastic setting. Importantly, the main results I have proven hold regardless of the choice of system parameters for the model
 David F. Anderson, Daniele Cappelletti, Wai-Tong (Louis) Fan, and Jinsu Kim. Mixing times for some classes of stochastically modeled reaction networks. In preparation.
 David F. Anderson, Daniele Cappelletti, and Jinsu Kim. Some network conditions for positive recurrence of stochastically modeled reaction networks II. In preparation.
 David F. Anderson. Boundedness of trajectories for weakly reversible, single linkage class reaction systems. Journal of Mathematical Chemistry, 49(10):2275-2290, 2011.
 David F. Anderson. A proof of the global attractor conjecture in the single linkage class case. SIAM J. Appl. Math, 71(4):1487-1508, 2011.
 Friedrich Josef Maria Horn. Necessary and sufficient conditions for complex balancing in chemical kinetics. Arch. Rat. Mech. Anal., 49(3):172-186, 1972.
 Friedrich Josef Maria Horn and Roy Jackson. General Mass Action Kinetics. Arch. Rat. Mech. Anal., 47:81- 116, 1972.