Mathematical models have contributed quantitative predictions of chemical reactions with much success, most notably through ordinary differential equation (ODE) models based on the law of mass action. At the cellular level, however, random fluctuations in the low copy numbers of reactant molecules contribute to the function of the system and must be included through stochastic modeling techniques. The discrete nature of kinetic processes is best described in terms of a stochastic Markov jump process and the chemical master equation (CME). The corresponding ODE model is the infinite volume limit of such a formulation.

This thesis compares deterministic and stochastic models of biochemical reaction systems and highlights stochastic dynamic behaviors that occur on the intracellular (mesoscopic) scale. We consider three well-known examples of homogenous chemical reactions, each representing a qualitatively different situation. First, an autocatalytic reaction introduced by Joel Keizer is used to investigate one dimensional logistic dynamics. Next, we study multistability and bifurcation in one dimension through Schlögl's model, a classical example of bistability. Finally, we examine limit cycle oscillations through a two dimensional example known as Schnakenberg's model.

Through these examples, we show how stable fixed points in an ODE model become quasistationary attractors under small volume conditions. Noise will push the system away from an attractor and, in the case of multistable systems, over the unstable state which separates basins of attraction. The amount of time spent near each attractor increases exponentially with the system volume and thus switching is unobservable in ODE models. In the CME, the quasistationary behavior leads to multiple time scales, which we are able to quantify through spectral analysis.

Oscillations become difficult to define in the stochastic regime, where trajectories no longer obey the restrictions of a typical autonomous ODE system. In two dimensions, the stochastic system is free to cross over itself in the phase plane and rotate against the direction of the vector field. We introduce a novel concept called the Poincaré-Hill Cycle Map to discern limit cycle oscillations under noisy conditions and find that stochastic oscillation can occur even when there is no deterministic rotation.