Spectral solutions to stochastic models of gene expression with bursts and regulation

Andrew Mugler, Aleksandra M. Walczak, and Chris H. Wiggins

Phys. Rev. E 80, 041921 – Published 20 October 2009

Abstract

Signal-processing molecules inside cells are often present at low copy number, which necessitates probabilistic models to account for intrinsic noise. Probability distributions have traditionally been found using simulation-based approaches which then require estimating the distributions from many samples. Here we present in detail an alternative method for directly calculating a probability distribution by expanding in the natural eigenfunctions of the governing equation, which is linear. We apply the resulting spectral method to three general models of stochastic gene expression: a single gene with multiple expression states (often used as a model of bursting in the limit of two states), a gene regulatory cascade, and a combined model of bursting and regulation. In all cases we find either analytic results or numerical prescriptions that greatly outperform simulations in efficiency and accuracy. In the last case, we show that bimodal response in the limit of slow switching is not only possible but optimal in terms of information transmission.

Received 20 July 2009

DOI:https://doi.org/10.1103/PhysRevE.80.041921

Authors & Affiliations

Andrew Mugler^*

Department of Physics, Columbia University, New York, New York 10027, USA

Aleksandra M. Walczak^†

Princeton Center for Theoretical Science, Princeton University, Princeton, New Jersey 08544, USA

Chris H. Wiggins^‡

Department of Applied Physics and Applied Mathematics, Center for Computational Biology and Bioinformatics, Columbia University, New York, New York 10027, USA

^*ajm2121@columbia.edu
^†awalczak@princeton.edu
^‡chris.wiggins@columbia.edu

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Vol. 80, Iss. 4 — October 2009

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Images

Figure 1
An example of the spectral method for $Z = 3$ states. Dashed curves show the joint distribution $p_{n}^{z}$ for each of $z = 1$ , 2, and 3, and solid curves show the marginal distribution $p_{n}$ . The stochastic transition matrix is given in Eq. (45), and the setting of $ω$ in each panel is indicated in the upper-right corner. Production rates are $g_{1} = 0$ , $g_{2} = 3$ , and $g_{3} = 12$ for all panels. Distributions are calculated via the spectral decomposition in Eq. (43) with $\bar{g} = ⟨ g_{z} ⟩ = 5$ .Reuse & Permissions
Figure 2
Summary of the bases discussed in Sec. 2A (black) and their gauge freedoms (barred parameters in gray). In the top row, neither the $m$ nor the $n$ sector is expanded in an eigenbasis; in the middle row, one sector is expanded; and in the bottom row, both sectors are expanded. The $| j, k ⟩$ basis can be viewed as a special case of the $| j, k_{j} ⟩$ or $| j, k_{n} ⟩$ basis with ${\bar{q}}_{j} = \bar{q}$ or ${\bar{q}}_{n} = \bar{q}$ , respectively. The $| j, m ⟩$ basis is not discussed as it is not useful in simplifying the problem.Reuse & Permissions
Figure 3
Error vs runtime for the spectral method and stochastic simulation. Error is the Jensen-Shannon divergence [41] between $p_{n m}$ obtained using the $| n, m ⟩$ basis (via iterative solution of the original master equation) and that obtained using the $| j, k ⟩$ basis [circles; cf. Eq. (80)], the $| j, k_{j} ⟩$ basis [triangles; cf. Eq. (90)], the $| n, k_{n} ⟩$ basis [squares; cf. Eq. (100)], the $| j, k_{n} ⟩$ basis [diamonds; cf. Eq. (114)], or stochastic simulation [28] (dots). Runtimes are scaled by that of the iterative solution, 150 s (in MATLAB). Spectral basis data is obtained by varying $K$ , the cutoff in the eigenmode number $k$ of the second gene; simulation data is obtained by varying the integration time. The input distribution $p_{n} = π_{1} e^{- λ_{1}} λ_{1}^{n} / n! + (1 - π_{1}) e^{- λ_{2}} λ_{2}^{n} / n!$ [from which $g_{n}$ is calculated via Eq. (59)] is a mixture of two Poisson distributions with $λ_{1} = 0.5$ , $λ_{2} = 15$ , and $π_{1} = 0.5$ . The regulation function $q_{n} = q_{-} + (q_{+} - q_{-}) n^{ν} / (n^{ν} + n_{0}^{ν})$ is a Hill function with $q_{-} = 1$ , $q_{+} = 11$ , $n_{0} = 7$ , and $ν = 2$ . The gauge choices used (cf. Fig. 2) are $\bar{g} = \sum_{n} p_{n} g_{n}$ , $\bar{q} = \sum_{n} p_{n} q_{n}$ , ${\bar{q}}_{n} = q_{n}$ , and ${\bar{q}}_{j} = \sum_{n} ⟨ j | n ⟩ q_{n} ⟨ n | j ⟩$ . The cutoffs used are $J = 80$ for the eigenmode number $j$ of the first gene and $N = 50$ for the protein numbers $n$ and $m$ . Inset: the joint probability distribution $p_{n m}$ . The peak at low protein number extends to $p_{00} \approx 0.1$ .Reuse & Permissions
Figure 4
Protein distributions for a gene regulated by a threshold function [dots; calculated via Eq. (75)] and a gene with two stochastic states [circles; calculated via Eq. (43)]. The relationship between regulation parameters and state transition rates is given by Eq. (136). In all panels the input to the regulation is a Poisson distribution with mean $g = 7$ , and the regulation rates [cf. Eq. (116)] are $q_{-} = 2$ and $q_{+} = 15$ . In the first column the threshold is $n_{0} = 4$ making $π_{+} = 0.827 > π_{-} = 0.173$ ; in the second column $n_{0} = 8$ making $π_{+} = 0.271 < π_{-} = 0.729$ . In the first row the ratio of the second gene’s degradation rate to that of the first is $ρ = 0.1$ ; in the second row $ρ = 10$ .Reuse & Permissions
Figure 5
(A) Mutual information $I$ [cf. Eq. (158)] versus stiffness $σ$ [cf. Eq. (160)] for fixed gain [ $γ = 16$ , cf. Eq. (159)] obtained by optimizing Eq. (161) for $λ$ values between $10^{- 3}$ and $10^{1}$ . Squares denote solutions whose joint distribution $p_{n m}$ has one peak (cf. B, lower left inset), and dots denote solutions for which $p_{n m}$ has two peaks (cf. B, upper right inset). Solid lines show the convex hulls of the one- and two-peaked solutions. Dotted lines indicate the stiffness value at which the hulls intersect and the stiffness values of the hull points to the left and right of the intersection. (B) Phase diagram between one- and two-peaked optimal solutions in the gain-stiffness plane. Circles and left and right error bars at each gain are determined by the stiffness values at the intersection of the one- and two-peak convex hulls and at the hull points to the left and right of the intersection, respectively (see dotted lines for the example case in A). Solid line shows a line of best fit. Insets show examples of one-(lower left) and two-peaked (upper right) optimal distributions $p_{n m}$ .Reuse & Permissions

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