A non-linear biochemical reaction system with concentration fluctuations

2 A non-linear biochemical reaction system with concentration fluctuations 

The previous example involved a two-dimensional system (involving two independent dynamic species). Thus the CME followed from the two-dimensional reaction diagram. For systems with more species, the dimension of the problem grows accordingly. For a system with three species, say A, B, and C, the CME tracks the three-dimensional probability of A molecules, m B molecules, and n C molecules present at time t. In general, the mathematical description of an A-dimensional system is the joint probability distribution 

As a third example, let us consider a simple non-linear chemical reaction system 

Stochastic biochemical systems and the chemical master equation 

It is usually not possible (and never easy ) to solve equations such as Equation (11.27) analytically. So computational simulation of the stochastic trajectories are necessary. The numerical method to obtain stochastic trajectories by Monte Carlo sampling, which we shall discuss in Section 11.4.4, is known as the Gillespie algorithm [68], However, it happens that the steady state of Equation (11.27) can be obtained in closed form. This is because in steady state, the probability of leaving state 0, v0po has to exactly balance the probability of entering state 0 from state 1, wopi. Similarly, since v0po = vjqp, we have v p = w p2, and so on  

---