# American Institute of Mathematical Sciences

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

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

Received  November 2020 Revised  March 2021 Published  April 2021

Fund Project: 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

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.

Citation: Pavol Bokes. Exact and WKB-approximate distributions in a gene expression model with feedback in burst frequency, burst size, and protein stability. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021126
##### References:

show all references

##### References:
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 $\gamma(x) = x$, $\alpha(x)\equiv 1$, $\beta(x) = 0.4 + 1.6x^4/(1 + x^4)$
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
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
 [1] Daoyi Xu, Yumei Huang, Zhiguo Yang. Existence theorems for periodic Markov process and stochastic functional differential equations. Discrete & Continuous Dynamical Systems, 2009, 24 (3) : 1005-1023. doi: 10.3934/dcds.2009.24.1005 [2] Pavol Bokes. Maintaining gene expression levels by positive feedback in burst size in the presence of infinitesimal delay. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5539-5552. doi: 10.3934/dcdsb.2019070 [3] Ábel Garab. Unique periodic orbits of a delay differential equation with piecewise linear feedback function. Discrete & Continuous Dynamical Systems, 2013, 33 (6) : 2369-2387. doi: 10.3934/dcds.2013.33.2369 [4] Zhenyu Lu, Junhao Hu, Xuerong Mao. Stabilisation by delay feedback control for highly nonlinear hybrid stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4099-4116. doi: 10.3934/dcdsb.2019052 [5] Qi Wang, Lifang Huang, Kunwen Wen, Jianshe Yu. The mean and noise of stochastic gene transcription with cell division. Mathematical Biosciences & Engineering, 2018, 15 (5) : 1255-1270. doi: 10.3934/mbe.2018058 [6] Benoît Perthame, P. E. Souganidis. Front propagation for a jump process model arising in spacial ecology. Discrete & Continuous Dynamical Systems, 2005, 13 (5) : 1235-1246. doi: 10.3934/dcds.2005.13.1235 [7] Vincent Renault, Michèle Thieullen, Emmanuel Trélat. Optimal control of infinite-dimensional piecewise deterministic Markov processes and application to the control of neuronal dynamics via Optogenetics. Networks & Heterogeneous Media, 2017, 12 (3) : 417-459. doi: 10.3934/nhm.2017019 [8] Feng Jiao, Qiwen Sun, Genghong Lin, Jianshe Yu. Distribution profiles in gene transcription activated by the cross-talking pathway. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2799-2810. doi: 10.3934/dcdsb.2018275 [9] Xuan Zhang, Huiqin Jin, Zhuoqin Yang, Jinzhi Lei. Effects of elongation delay in transcription dynamics. Mathematical Biosciences & Engineering, 2014, 11 (6) : 1431-1448. doi: 10.3934/mbe.2014.11.1431 [10] Ralf Banisch, Carsten Hartmann. A sparse Markov chain approximation of LQ-type stochastic control problems. Mathematical Control & Related Fields, 2016, 6 (3) : 363-389. doi: 10.3934/mcrf.2016007 [11] Salah-Eldin A. Mohammed, Tusheng Zhang. Large deviations for stochastic systems with memory. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 881-893. doi: 10.3934/dcdsb.2006.6.881 [12] Xian Chen, Zhi-Ming Ma. A transformation of Markov jump processes and applications in genetic study. Discrete & Continuous Dynamical Systems, 2014, 34 (12) : 5061-5084. doi: 10.3934/dcds.2014.34.5061 [13] Yan Wang, Lei Wang, Yanxiang Zhao, Aimin Song, Yanping Ma. A stochastic model for microbial fermentation process under Gaussian white noise environment. Numerical Algebra, Control & Optimization, 2015, 5 (4) : 381-392. doi: 10.3934/naco.2015.5.381 [14] Marek Bodnar. Distributed delays in Hes1 gene expression model. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2125-2147. doi: 10.3934/dcdsb.2019087 [15] Qing-Qing Yang, Wai-Ki Ching, Wanhua He, Tak-Kuen Siu. Pricing vulnerable options under a Markov-modulated jump-diffusion model with fire sales. Journal of Industrial & Management Optimization, 2019, 15 (1) : 293-318. doi: 10.3934/jimo.2018044 [16] Wei Wang, Linyi Qian, Xiaonan Su. Pricing and hedging catastrophe equity put options under a Markov-modulated jump diffusion model. Journal of Industrial & Management Optimization, 2015, 11 (2) : 493-514. doi: 10.3934/jimo.2015.11.493 [17] Tian Zhang, Huabin Chen, Chenggui Yuan, Tomás Caraballo. On the asymptotic behavior of highly nonlinear hybrid stochastic delay differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5355-5375. doi: 10.3934/dcdsb.2019062 [18] Wei Mao, Yanan Jiang, Liangjian Hu, Xuerong Mao. Stabilization by intermittent control for hybrid stochastic differential delay equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021055 [19] Guangjun Shen, Xueying Wu, Xiuwei Yin. Stabilization of stochastic differential equations driven by G-Lévy process with discrete-time feedback control. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 755-774. doi: 10.3934/dcdsb.2020133 [20] Massimiliano Tamborrino. Approximation of the first passage time density of a Wiener process to an exponentially decaying boundary by two-piecewise linear threshold. Application to neuronal spiking activity. Mathematical Biosciences & Engineering, 2016, 13 (3) : 613-629. doi: 10.3934/mbe.2016011

2019 Impact Factor: 1.27