Modeling stochastic noise in gene regulatory systems

doi:10.1007/s40484-014-0025-7

Quant. Biol.

2014, Vol. 2

Issue (1) : 1-29 https://doi.org/10.1007/s40484-014-0025-7

REVIEW

Modeling stochastic noise in gene regulatory systems

Arwen Meister,Chao Du,Ye Henry Li,Wing Hung Wong(

)

Computational Biology Lab, Bio-X Program, Stanford University, Stanford, CA 94305, USA

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Abstract

The Master equation is considered the gold standard for modeling the stochastic mechanisms of gene regulation in molecular detail, but it is too complex to solve exactly in most cases, so approximation and simulation methods are essential. However, there is still a lack of consensus about the best way to carry these out. To help clarify the situation, we review Master equation models of gene regulation, theoretical approximations based on an expansion method due to N.G. van Kampen and R. Kubo, and simulation algorithms due to D.T. Gillespie and P. Langevin. Expansion of the Master equation shows that for systems with a single stable steady-state, the stochastic model reduces to a deterministic model in a first-order approximation. Additional theory, also due to van Kampen, describes the asymptotic behavior of multistable systems. To support and illustrate the theory and provide further insight into the complex behavior of multistable systems, we perform a detailed simulation study comparing the various approximation and simulation methods applied to synthetic gene regulatory systems with various qualitative characteristics. The simulation studies show that for large stochastic systems with a single steady-state, deterministic models are quite accurate, since the probability distribution of the solution has a single peak tracking the deterministic trajectory whose variance is inversely proportional to the system size. In multistable stochastic systems, large fluctuations can cause individual trajectories to escape from the domain of attraction of one steady-state and be attracted to another, so the system eventually reaches a multimodal probability distribution in which all stable steady-states are represented proportional to their relative stability. However, since the escape time scales exponentially with system size, this process can take a very long time in large systems.

Keywords gene regulation stochastic modeling simulation Master equation Gillespie algorithm Langevin equation

Corresponding Author(s): Wing Hung Wong

Issue Date: 25 June 2014

Cite this article:

Arwen Meister,Chao Du,Ye Henry Li, et al. Modeling stochastic noise in gene regulatory systems[J]. Quant. Biol., 2014, 2(1): 1-29.

URL:

https://academic.hep.com.cn/qb/EN/10.1007/s40484-014-0025-7
https://academic.hep.com.cn/qb/EN/Y2014/V2/I1/1

Fig.1 Informal derivation of the Master equation for gene regulation. In a infinitesimal timestep, P(k; t), the probability of k RNA transcripts, increases by P(k-1, t) times the probability, F(k-1), of a transcription event (number of transcripts increases by one) plus P(k+1, t) times the probability, γ (k+1), of a degradation event (number of transcripts decreases by one). It decreases by P(k) times the probability of transcription plus P(k) time the probability of degradation.

Fig.2 Bistability in a stochastic system modeled by a Fokker-Planck equation of the form (28), corresponding to deterministic equation dxdt=dUdt. The deterministic function dUdt (left) has zeros at the three steady-states ?a≈1,?b≈4,?c≈8. The points ?a and ?c are stable, while ?b is unstable. The potential U(x) (center) has minima at ?a and ?c and a maximum at ?b, corresponding to low energy (favorable) at the two steady-states and high energy (unfavorable) at the unstable state. ?c is more stable than ?a since its potential well is deeper and wider. The stationary distribution (right), to which the stochastic system will eventually converge, is bimodal with peaks at ?a and ?c. The peak at ?c is higher since ?c is more stable than ?a.

Fig.3 Steady-state probability distributions of a one gene system with one steady-state (31) for increasing system sizes ? = 1,10,100. The distribution always peaks at the deterministic steady-state solution (y = 1), and the variance decreases as ? increases. For smaller values of ?, it is clear that the mean lies slightly above the deterministic solution, but as ? increases, the distribution becomes quite symmetric.

Fig.4 One gene system with one steady-state (31). Mean (left) and variance (right) trajectories via Master equation (black), van Kampen approximation (blue), and average of 100 trajectories of the Gillespie (red) and Langevin simulation (cyan) with ? = 1,10,100 (top to bottom, respectively). There is excellent agreement between simulations, van Kampen approximation, and exact Master equation for both mean and variance. Discrepancy between the stochastic mean and deterministic trajectory and magnitude of the variance are both O(?^-1).

Fig.5 Final probability distributions of the exact Master equation for a one gene system with one steady-state (31), with ? = 1,10,100. The probabilities converge to approximately Gaussian steady-state distributions peaked near the deterministic steady-state. For larger system sizes, the distribution is more Gaussian and the peak is sharper.

Fig.6 . Two gene system with one stable steady-state (34). Mean (left) and variance (right) trajectories of van Kampen approximation (blue) and average of 100 trajectories of Gillespie (red) and Langevin simulation (cyan) with ? = 1,10,100 (top to bottom, respectively). As with the one gene system, agreement between the simulations and the van Kampen approximation is excellent, and both the variance and the discrepancy between the mean and deterministic trajectory are O(?^-1). The only exception is for ? = 1, where slight inaccuracy of the Langevin simulation and van Kampen expansion arises from the non-Gaussianity of the probability distribution.

Fig.7 The deterministic function α₁(x) = f(x) -γx for the system (36), with ? = 1, has three zeros corresponding to the three deterministic steady-states, e₁, e₂, e₃. The derivative of the deterministic function is negative (dα₁ = dt<0) at the stable steady-states e₁, e₂, and positive at the unstable steady-state e₃. The stationary distribution (computed with Equation (12)) has a strong peak at e₁ and a weaker one at e₂. The system is much more likely to be in the domain of e₁ (x<e₃) than in the domain of e₂ (x>e₃): specifically, π₁ ≈ 0.75, and π₂ ≈ 0.25. The steady-state mean is given by π₁e₁ + π₂e₂ ≈ 2.78.

Fig.8 The deterministic function, effective potential, and (approximate) stationary distribution for system (36) with ? = 10, computed with the Fokker-Planck approximation and Equations (29,30. (The result is nearly identical to what we would have obtained with the explicit equation (12)). The deterministic function and stationary distribution have the same qualitative properties as they did with ? = 1, except that the e₁ peak in the stationary distribution is now even higher relative to e₂ (π₁ ≈ 97%; π₂ ≈ 3%), and the steady-state mean is shifted toward e₁: π₁e₁ + π₂e₂ ≈ 1.24. The effective potential has minima at the two stable steady-states e₁, e₂, and a maximum at e₃. The more stable steady-state, e₁, has lower “energy”.

Fig.9 One gene system with two stable deterministic steady-states (36), ? = 1. Mean (left) and variance (right) trajectories via the Master equation (black), the (improperly applied) van Kampen expansion (blue), and the average of 100 trajectories of the Gillespie (red) and Langevin simulation (cyan). Regardless of the starting point, the stochastic mean trajectory eventually converges to the weighted average of the two deterministic stable steady-states predicted by the analysis of Figure 7: π₁e_i ≈ + π₂e₂ ≈ 2.78. The (improperly applied) van Kampen expansion seriously underestimates the discrepancy between the mean and the deterministic trajectory since, as an expansion about e₁, it effectively ignores e₂, and vice versa; van Kampen's stability analysis is therefore the correct theoretical approach in this case.

Fig.10 Initial (left), intermediate (center), and final (right) probability distributions of the exact Master equation for the one gene system with two stable steady-states (36), starting from e₁ (top) or e₂ (bottom), with ? = 1. The probability distributions start out peaked at their respective initial conditions. Over time, some of the probability begins to flow from one deterministic steady-state to the other. Regardless of the initial condition, the system eventually reaches a single bimodal stochastic steady-state (the same distribution shown in Figure 7), with a stronger peak at e₁ (the more stable of the two points) and a weaker peak at e₂.

Fig.11 Escape time τ_2,1 versus system size ? for system (36) (left), computed as mean first-passage time via Equation (27). The plot of log(τ_2,1) vs. ? (right) is linear, confirming that the escape time grows exponentially with the system size.

Fig.12 One gene system with two stable steady-states (36), ? = 10. Just as in Figures 9 and 10, regardless of the initial condition, the probability converges to a bimodal distribution with a strong peak at e₁ (π₁ = 97%) and weaker peak at e₂ (π₂ = 3%), and the mean converges to the weighted average π₁e_i + π₂e₂ ≈ 1.24 predicted in our stability analysis for ? = 10.

Fig.13 One gene system with two stable steady-states (36), ? = 100. Mean (left) and variance (right) trajectories via (improperly applied) van Kampen approximation (blue) and average of 100 trajectories of the Gillespie (red) and Langevin simulation (cyan). The exact Master equation calculation suffered from instability (oscillations) so the trajectory is not shown here. Since escape time scales exponentially with the system size, the escape time for this system far exceeds the length of the simulation. Therefore the stochastic trajectories remain close to the deterministic steady-state where they originated for the duration of the simulation. Since the two deterministic steady-states operate mostly independently of each other in the simulation timeframe, the van Kampen approximation agrees quite well, unlike for smaller system sizes. The variance and difference between the mean and deterministic trajectory are both on the order of O(?^-1).

Fig.14 Two gene system with two stable steady-states (37), with ? = 10 (top) and ? = 1000 (bottom): mean and variance via van Kampen approximation (blue), and average over 100 simulations of the Gillespie (red) and Langevin simulation (cyan). For small systems (? = 10), the stochastic mean trajectory converges to a weighted average of e₁ and e₂ corresponding to a bimodal stochastic steady-state. Since escape time scales exponentially with system size, the escape time for the large system (? = 1000) is very long and the trajectories remain near their initial conditions for the duration of the simulation, hence the van Kampen approximation is quite accurate (though technically not applicable) and the variance and mean-deterministic discrepancy are both O(?^-1).

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