Exact and WKB-approximate distributions in a gene expression model with feedback in burst frequency, burst size, and protein stability

Pavol Bokes

doi:10.3934/dcdsb.2021126

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2022, Volume 27, Issue 4: 2129-2145. Doi: 10.3934/dcdsb.2021126

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Exact and WKB-approximate distributions in a gene expression model with feedback in burst frequency, burst size, and protein stability

Pavol Bokes

Department of Applied Mathematics and Statistics, Comenius University, Bratislava 842 48, Slovakia

Received: October 31, 2020

Revised: February 28, 2021

Early access: April 2021

Published: April 2022

The author is supported by the Slovak Research and Development Agency under the contract No. APVV-18-0308 and by the VEGA grant 1/0347/18

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Abstract

The expression of individual genes into functional protein molecules is a noisy dynamical process. Here we model the protein concentration as a jump-drift process which combines discrete stochastic production bursts (jumps) with continuous deterministic decay (drift). We allow the drift rate, the jump rate, and the jump size to depend on the protein level to implement feedback in protein stability, burst frequency, and burst size. We specifically focus on positive feedback in burst size, while allowing for arbitrary autoregulation in burst frequency and protein stability. Two versions of feedback in burst size are thereby considered: in the first, newly produced molecules instantly participate in feedback, even within the same burst; in the second, within-burst regulation does not occur due to the so-called infinitesimal delay. Without infinitesimal delay, the model is explicitly solvable; with its inclusion, an exact distribution to the model is unavailable, but we are able to construct a WKB approximation that applies in the asymptotic regime of small but frequent bursts. Comparing the asymptotic behaviour of the two model versions, we report that they yield the same WKB quasi-potential but a different exponential prefactor. We illustrate the difference on the case of a bimodal protein distribution sustained by a sigmoid feedback in burst size: we show that the omission of the infinitesimal delay overestimates the weight of the upper mode of the protein distribution. The analytic results are supported by kinetic Monte-Carlo simulations.

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Mathematics Subject Classification: Primary: 92C40; Secondary: 60J76, 45D05, 41A60.

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Figure 1. Upper Left: The model includes bursty protein production and continuous protein decay, and allows for feedback in burst frequency, burst size, and protein stability, as quantified by functions $ \alpha(x) $, $ \beta(x) $, and $ \gamma(x) $, respectively. Bottom Left: Bursts lead to an instantaneous increases in protein concentration; between bursts protein concentration decays continuously. Bottom Right: In the deterministic limit $ \varepsilon\to0 $, the concentration changes per unit time by the difference of the production rate $ \alpha(x)\beta(x) $ (solid line) and the decay rate $ \gamma(x) $ (dotted line). The intersections of the two are the fixed points (FPTs) of the deterministic model. Upper Right: The distribution potential (16) is a Lyapunov function of the deterministic model: it possesses minima/maxima where the deterministic model exhibits stable/unstable points. The depicted example pertains to feedback in burst size, with $ \gamma(x) = x $, $ \alpha(x)\equiv 1 $, $ \beta(x) = 0.4 + 1.6x^4/(1 + x^4) $

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Figure 2. Left: For small values of the noise parameter $ \varepsilon $, the sample paths of the stochastic model (coloured curves) are close to the solutions of the deterministic model (black curves). The stable/unstable fixed points of the deterministic model (14) are shown as dashed/dotted horizontal lines. Right: Transitions between the basins of attractions of the stable steady states occur on an extremely slow timescale. Parameter values: Feedback is in burst size, with $ \gamma(x) = x $, $ \alpha(x)\equiv 1 $, $ \beta(x) = 0.4 + 1.6x^4/(1 + x^4) $. The noise parameter is varied in the left panel and fixed to $ \varepsilon = 0.05 $ in the right panel

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Figure 3. Upper Panels: Simulation-based time-dependent distributions (coloured curves) approach, as simulation time increases, the WKB stationary distribution (dashed black curve). This is markedly different from the exact stationary distribution of the model without infinitesimal delay (dotted black curve). The initial condition is $ x_0 = 0 $ (left panel) and $ x_0 = 3 $ (right panel). Bottom Panels: Large-time simulation-based distributions (solid curve) are compared to the WKB approximation (dashed curve). Parameter values: Feedback is in burst size, with $ \gamma(x) = x $, $ \alpha(x)\equiv 1 $, $ \beta(x) = 0.4 + 1.6x^4/(1 + x^4) $, except the bottom right panel, where $ \beta(x) = 0.4 + 2.6x^4/(1 + x^4) $. The noise parameter is fixed to $ \varepsilon = 0.05 $ in the upper panels; in the bottom panels, it assumes values that are specified in the inset

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