Abstract
A deterministic modeling approach – as the one explained in Chap. 2 assumes to deal with systems isolated from any source of noise, or randomness, that could spoil model predictions. In classical physics this approximation is generally valid. Biological systems, however, are intrinsically noisy, which makes deterministic models inappropriate to simulate, at least, certain kinds of synthetic gene circuits. The stochastic simulation (or Gillespie) algorithm is a purely computational approach to calculate the temporal evolution of biological systems. In principle, it may give a more realistic description of gene circuit dynamics. However, it becomes computationally too demanding if applied to large synthetic networks. In this Chapter, we will explain the theoretical foundations, working, and recent improvements of this algorithm. Moreover, we will discuss under which conditions a deterministic model can provide faithful predictions, despite the complexity of the system, and when, in contrast, stochastic simulations are preferable.
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Notes
- 1.
In general, \(\langle s_T \rangle ^2 \neq \langle s_T^2 \rangle \).
- 2.
The standard deviation of x – over a set of N data – is calculated as: \(\sigma (x) = \sqrt {\frac {1}{N}\sum _{i=1}^N (x_i - \langle x \rangle )^2}\). It represents the average error associated with each measurement.
- 3.
If we call CF i and Y F i the cyan and yellow fluorescence measured from the ith cell over a population of N samples, the average intrinsic noise corresponds to
$$\displaystyle \begin{aligned} \eta_{int} =\frac{1}{N}\sum_{i=1}^N | CF_i - YF_i | \,\mbox{.} {} \end{aligned} $$ - 4.
The concentration of LacI in the Repressilator oscillates over time – see Chap. 6. The Repressilator was inserted into lacI − cells.
- 5.
In a deterministic framework, the initial conditions (x 0, t 0) determine unequivocally the ODE solutions. In a stochastic framework, (x 0, t 0) determine unequivocally the probability that the system lies in x at time t.
- 6.
In the phase space every point corresponds to a different system configuration (parameter values and species concentrations). A trajectory in the phase space gives the temporal evolution of a system from its initial state to the equilibrium.
- 7.
The Poisson distribution \(\mathcal {P}(n,\lambda )\) gives the probability that a discrete random variable K is equal to n, where n = 0, 1, 2, … It is defined as
$$\displaystyle \begin{aligned} Prob(K=n)=\mathcal{P}(n, \lambda)= \frac{\lambda^n e^{-\lambda}}{n!} \mbox{,} \end{aligned} $$(4.22)where λ is greater than zero and represents both the expected value (the mean) and the variance of the distribution. In other words, Prob(K = n) is maximal when n = λ. In the tau-leaping method, K = k j and λ = a j(x)τ. Each k j can be computed with the inversion method after drawing a random number r 2 from the uniform distribution between zero and one and setting r 2 = Prob(k j = n).
- 8.
In some variants of the tau-leaping algorithm τ is fixed.
- 9.
A Normal distribution with mean value μ and variance σ 2 is defined as \(\mathcal {N}(\mu ,\sigma ^2)=\frac {1}{\sqrt {2\pi }\sigma }e^{-\frac {(x-\mu )^2}{2\sigma ^2}}\), where x is a continuous (real) variable. Notice that \(\mathcal {N}(\mu ,\sigma ^2)=\mu + \sigma \mathcal {N}(0,1)\).
- 10.
τ is renamed as dt and the limit dt → 0 is then considered.
- 11.
It holds that \(\langle \xi _i(t) \xi _j(t^{'}) \rangle =\delta _{ij}\delta (t-t^{'})\).
References
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Marchisio, M.A. (2018). Stochastic Modeling. In: Introduction to Synthetic Biology. Learning Materials in Biosciences. Springer, Singapore. https://doi.org/10.1007/978-981-10-8752-3_4
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DOI: https://doi.org/10.1007/978-981-10-8752-3_4
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