Through both observation and theory, it is now generally accepted that the dynamics of biological systems, especially gene networks, are a noisy process. It is also generally agreed that a good description of the dynamics is either a jump Markov process with a corresponding Master equation or a continuous Markov process with a corresponding Langevin or Fokker-Planck equation.

But what sort of solutions do these Master, Fokker-Planck, or Langevin equations generate? How do parameters of the model affect the solution? How are solutions created or destroyed? Are the solutions stable? What does stability mean in the context of a stochastic system? What does the noise do to the stability of each solution? How may we quantify these effects?

The study of these questions is bifurcation analysis and its application to stochastic systems (called random dynamical systems in the mathematics literature) is a new field. We aim at a conceptual presentation of how bifurcation analysis of random dynamical systems differs from deterministic ones and why it is important to know about these differences. We present the key concepts of random dynamical systems without a rigorous treatment of the mathematics.

We use gene networks to demonstrate how the tools from this field may be applied to biological systems with great effect, especially in understanding how bistability, hysteresis, and oscillations arise in a random dynamical system. By drawing upon our recent research in computing the stable and unstable random attractors of arbitrary chemical Master equations, we can show how mesoscopic solutions to the Master equation emerge, grow in stability, lose stability, and collapse when parameter values are changed. We also compute the Lyapunov exponents of these random attractors using the Furstenberg-Khasminskii formulae.

In our example gene network, we consider how experimentally changable parameters, such as DNA-binding affinities or protein production rates, cause the solution to bifurcate from monostability to bistability to oscillations. We show how the unstable random attractor tells us how fast the bistable system will spontaneously escape from one stable state to another. The analysis helps us to understand the underlying dynamics of the system and what type of solutions we might expect from experimental observations.