Gillespie algorithm

We adapt our model for numerical simulation with the help of the Gillespie algorithm [10], which enables the system to jump to the next event via the calculation of the waiting time before any event will occur. Following the approach suggested by us [50], we stochastically model the system where several events can happen with different probabilities. Suppose that in some moment of time we have a set of N probable events with rates Ri, where the i — th event has the rate Ri and i = 1... N. Then by generating two uniformly distributed in (0, 1) random numbers RN and RN2, we estimate the time T after which the next event would occur as ... 

Rao, C. V. Arkin, A. P. Stochastic chemical kinetics and the quasi-steady-slate assumption application to the Gillespie algorithm. / Chern Phys 2003,118 4999-5010. 

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 ... 

Figure 11.8 An example of the stochastic trajectory from Monte Carlo simulation according to the Gillespie algorithm for reaction system given in Equation (11.19) and corresponding master equation graph given in Figure 11.4. Here we set Ns = 100 and Nes = 0 at time zero and total enzyme number Ne = 10. (A) The fluctuating numbers of S and ES molecules as functions of time. (B) The stochastic trajectory in the phase space of (m, n).

After completing a step (a jump in state) the corresponding outward rate constants for the new state are now all different. To continue the simulation, we draw another two random numbers, make another move, and so on. A stochastic trajectory is thus obtained. One notes that the trajectory has randomly variable time steps, a feature indicative of the Gillespie algorithm. 

Given the simplicity of the current system, it was possible to analytically solve the resulting chemical master equation. However, this is not always the case and one is limited to simulating individual realizations of the stochastic process in order to reconstruct the probability distributions out from several simulations. Below, we introduce the celebrated Gillespie algorithm (Gillespie 1977) to simulate the stochastic evolution of continuous-time discrete-state stochastic processes, like the one analyzed in the present chapter. 

Fig. 4.3 Simulations of the production-decay process using the Gillespie algorithm and different values of the molecule synthesis and degradation rates...

The General Form of the Chemical Master Equation and Gillespie Algorithm... 

In this and the last chapter we have derived the chemical Master equation, as well as Gillespie algorithm, for a few particular examples. Along the book we are going to employ both of them even more. Therefore, rather than deriving the particular forms each time, it is convenient to have a general derivation. 

The above derived general forms of the chemical master equation and of Gillespie algorithm will be invoked in the following chapters to study some specific examples. 

The main goal of the present chapter was to study the synthesis and degradation of molecules by means of the formerly reviewed approaches. To do so it was necessary to introduce and analyze (via the chemical master equation, as well as Gillespie algorithm) three novel stochastic processes the Poisson process, the exponential... 

Carletti, T, Filisetti, A. The stochastic evolution of a protocell the Gillespie algorithm in a dynamically varying volume. Comput. Math. Methods Med. 2012, 12 (2012)... 

To describe a stochastic chemical system, the most often used approach is to employ Gillespie algorithm [21] this is an algorithm derived from collision theory that operates by selecting, by means of two suitably generated random numbers, which reaction r to execute at the current step, and the waiting time... 

There are several problems to address in order to have a detailed description of the PS in a Gillespies algorithm-based simulator. These problems arise from the not negligible differences between a common chemical system as hypothesized by Gillespie, and a cell-free system, which hosts several complex multi-step processes. For instance, two major problems are represented by the description of transcription and translation as they imply the presence of a nucleic acid (either DNA or RNA) with an extensive reaction machinery (either polymerases or ribosomes) bound to it. These complexes are spaced from one another by a certain minimum distance, and slide on the nucleic acid molecules moving one step only when the complex ahead has itself moved. 

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