Stochastic Biology: Master Equation & Gillespie
Stochastic Biology: Master Equation & Gillespie
When molecule numbers are small — a gene may have just one or two copies — deterministic ODEs fail. Stochastic chemical kinetics treats each reaction as a probabilistic event with propensity $a_\mu(\mathbf{n})$ equal to the rate times the number of ways reactants can meet.
The chemical master equation (CME) governs $P(\mathbf{n},t)$. Because the CME is rarely analytically tractable, the Gillespie algorithm (1977) generates exact stochastic trajectories by sampling inter-reaction times from an exponential distribution.
Chemical Master Equation
$$\frac{\partial P(\mathbf{n},t)}{\partial t}=\sum_\mu\bigl[a_\mu(\mathbf{n}-\mathbf{v}_\mu)P(\mathbf{n}-\mathbf{v}_\mu,t)-a_\mu(\mathbf{n})P(\mathbf{n},t)\bigr].$$
Gillespie Direct Method
1. Compute $a_0=\sum_\mu a_\mu$. 2. Draw $\tau\sim\text{Exp}(a_0)$; advance $t\to t+\tau$. 3. Select reaction $\mu$ with probability $a_\mu/a_0$; update state. Repeat.
Example 1
Production rate $k_p=5\,\text{min}^{-1}$, degradation $\gamma=0.1\,\text{min}^{-1}$. Find mean and variance of protein number at steady state.
Solution: Poisson: $\langle n\rangle=k_p/\gamma=50$; $\text{Var}(n)=50$ (Fano = 1).
Example 2
Why does bursty gene expression increase noise above the Poisson level?
Solution: Proteins produced in correlated bursts inflate variance relative to the mean, giving Fano factor $>1$ (super-Poissonian noise).
Practice
- Write the propensity function for the bimolecular reaction $A+B\to C$.
- When does the CME reduce to a deterministic ODE (mean-field limit)?
- Describe the tau-leaping approximation and its regime of validity.
- How does the Fano factor quantify noise in gene expression?
Show Answer Key
1. For $A + B \to C$: $a(\mathbf{x}) = k \cdot x_A \cdot x_B$ (mass-action stochastic kinetics). In a volume $\Omega$: $a = (k/\Omega) \cdot n_A \cdot n_B$ where $n_A, n_B$ are molecule counts.
2. In the thermodynamic limit ($\Omega \to \infty$, concentrations fixed), fluctuations become negligible ($\sim 1/\sqrt{\Omega}$). The CME reduces to deterministic rate equations (ODEs) by the law of large numbers / Kurtz's theorem.
3. Tau-leaping groups multiple reaction events in a time step $\tau$, drawing Poisson random numbers for each reaction. Valid when propensities change negligibly during $\tau$ (leap condition: $|\Delta a_j| \ll a_j$ for all $j$). Bridges exact Gillespie (slow, exact) and ODE (fast, deterministic).
4. Fano factor $F = \text{Var}(X)/\text{E}(X)$. For a Poisson process, $F = 1$. In gene expression: $F > 1$ indicates super-Poissonian noise (bursty transcription), $F < 1$ indicates sub-Poissonian (negative feedback). The Fano factor separates intrinsic from extrinsic noise contributions.