In probability theory, the Gillespie algorithm generates a statistically correct trajectory (possible solution) of a stochastic equation system for which the reaction rates are known. Mathematically, it is a variant of a dynamic Monte Carlo method and similar to the kinetic Monte Carlo methods.

• Gillespie Algorithm
  • 概览
  • 算法
    • 思想
    • 步骤
    • 相关方程

Gillespie Algorithm

概览

Time evolution of the model has been studied by computersimulation with the Gillespie algorithm (direct method), taking into account gene replication and cell division events. Kierzek A, Zhou L, Wanner B, Stochastic Kinetics Model of TCS

However, an equation-free analysis of the stochastic model, using the Gillespie algorithm with tau-leaping as a black-box time-stepper, in order to find stationary states for the mean of an ensemble of stochastic trajectories, reveals the long-term persistence of an approximate basal solution that combines with the graded response to produce the mixed mode. Hoyle RB, Avitabile D, Kierzek AM (2012) Equation-Free Analysis of Two-Component System Signalling Model Reveals the Emergence of Co-Existing Phenotypes in the Absence of Multistationarity.

对于生物体内的化学反应来说，反应物分子远小于普通化学反应，速度也很慢，所以发生反应的随机性就很明显。这些随机性是由两方面造成的，一方面是因为反应物碰撞后才可能发生反应，而当分子数很少时，碰撞概率也很小；另一方面是热力学涨落，即使反应物发生碰撞，也需要有足够大的活化能才能发生相应的反应，而活化能是受热涨落影响的，具有显著的随机性。

ref: 哪位大神能详细解释一下Gillespie algorithm 和chemical system？

算法

思想

若时刻$t$系统状态为$x$，下一次反应在$(t+\tau)$时刻发生，而且所发生的反应是第$\mu$个反应通道$R_{\mu}$，则系统状态在$(t, \tau)$时间区间内为$x$，而$(t+\tau)$时刻变为$x+v_{\mu}$，因此根据当前时刻状态$X(t)=x$计算出下一个反应发生的时间$(t+\tau)$和相应的反应通道$R_{\mu}$，就得到了系统状态随时间的变化，即一个样本轨道。

步骤

1. 初始化$X(0)=x_0$，并令初始时间$t_0=0$
2. 计算$a_v=a_v(x)(v=1, \cdots, M)$，并令$a_0=\sum_{v=1}^Ma_v$
3. 产生一组随机数$(\tau, \mu)$，使其分布满足概率密度函数$P(\tau, \mu;x)$: 给定$t$时刻系统状态$X(t)=x$，系统中下一个反应在无穷小时间区间$d\tau$内发生，且发生的是反应通道$R_{\mu}$的概率
4. 令$t=t+\tau$，并根据反应通道$R_{\mu}$更新分子数目$X_i\rightarrow X_i+v_{ui}$
5. 重复2~4步

ref: 12_stochastic_simulation

相关方程