# American Institute of Mathematical Sciences

## Cell-type switches induced by stochastic histone modification inheritance

 Zhou Pei-Yuan Center for Applied Mathematics, MOE Key Laboratory of Bioinformatics, Tsinghua University, Beijing China, 100084

* Corresponding author: Jinzhi Lei

Received  September 2018 Revised  November 2018 Published  April 2019

Fund Project: This work was supported by the National Natural Science Foundation of China (NSFC 917030301)

Cell plasticity is important for tissue developments during which somatic cells may switch between distinct states. Genetic networks to yield multistability are usually required to yield multiple states, and either external stimuli or noise in gene expressions are trigger signals to induce cell-type switches. In many biological systems, cells show highly plasticity and can switch between different states spontaneously, but maintaining the dynamic equilibrium of the cell population. Here, we considered a mechanism of spontaneous cell-type switches through the combination between gene regulation network and stochastic epigenetic state transitions. We presented a mathematical model that consists of a standard positive feedback loop with changes of histone modifications during cell cycling. Based on the model, nucleosome state of an associated gene is a random process during cell cycling, and hence introduces an inherent noise to gene expression, which can automatically induce cell-type switches in cell cycling. Our model reveals a simple mechanism of spontaneous cell-type switches through a stochastic histone modification inheritance during cell cycle. This mechanism is inherent to the normal cell cycle process, and is independent to the external signals.

Citation: Rongsheng Huang, Jinzhi Lei. Cell-type switches induced by stochastic histone modification inheritance. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2019074
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Illustration of gene regulation combines a positive feedback with stochastic histone modification at the enhancer region. Histone modifications alter the 50% effective concentration (EC50) of the toggle switch to regulate gene expression (see Formulations for details)
The bifurcation diagrams with respect to $K$ and $u$. (A) The protein level at steady state as a function of $K$. Blue dashed lines show the saddle-node bifurcations ($K^*$ and $K^{**}$). (B) The heat map representation of joint density distribution of the protein level and $u$ for a typical trajectory under random inheritance of histone modifications and without the external noise. In the simulation, $\varphi(u_k) = a + b (u_k - a)$ and $a = 0.5, b = 0.8$ (to be detailed in Section 3.2)
Cell-type switches induced by external noise. (A) Sample dynamics of protein level in $10^5$ cycles. In the simulation, $K = 450$, $\sigma = 1.8$. (B) The heat map representation of the switching frequency with respect to $K$ and $\sigma$. (C) A sample dynamics of $\lambda_p$ with $\sigma = 1.5$. Inset is the probability distribution of $\ln(\lambda_p)$. (D) The probability distribution of the steady state lifetimes with respect to the trajectory in (A)
Dynamics of nucleosome states under random histone modification inheritance. (A) A sample dynamics of the nucleosome state along with cell cycles. (B) The distribution (blue) of the sequence $\{u_k\}$ with three sets of parameters of $a$ and $b$, and the Beta distribution density function (red) (16) with shape parameters $(\alpha, \beta)$ given by (17)
Frequencies of cell-type switches with histone modification to affect the transcription activities. (A) The heat map representation of the switching frequency with respect to $a$ and $b$ without the external noise. (B) The switching frequency as a function of $a$ with varying parameter $b$. (C) The heat map representation of the switching frequency with respect to $b$ and the noise strength $\sigma$ (here $a = 0.5$). (D) The heat map representation of the switching frequency with respect to $a$ and the noise strength $\sigma$ (here $b = 0.7$)
Dynamic equilibrium of cell regeneration. Figures show the probability distribution of the protein level at the end of cycle $0$, $5$, $10$, $15$, and $20$. Results for 4 sample simulations ((A)-(D)) are shown, each with different initial conditions. Parameters are $a = 0.5, b = 0.7$
Stationary distribution of proteins in a cell colony. (A)The probability distribution of the protein level at the end of $20$ cycles with fixed $b = 0.7$ and varied $a$: $a = 0.4$, $a = 0.5$, and $a = 0.6$. (B) The probability distribution of the protein level at the end of $20$ cycles with fixed $a = 0.5$ and varied $b$: $b = 0.6$, $b = 0.7$, and $b = 0.8$. We take $\sigma = 0.5$ in simulations
Potential landscapes. (A)The potential landscape (23) as a function of protein level with $K = 200$, $K = 450$, and $K = 750$. In the simulation, $\sigma = 0.1$. (B)The Waddington landscape of colony evolution from single cells. The potential was obtained from simulations results of 30 independent trajectories. Parameters are $a = 0.5, b = 0.7$
Dynamics of intermediate cell states. (A) A sample dynamics of transitions between three cell states based on the assumption (24). (B) The distribution of nucleosome state $u_k$. (C) The distribution of the protein levels along the trajectory. In the simulation, we took $m = 6$, $c = 0.25$, $d = 0.5$, $u_0 = 0.488$, $n = 2.5$, $\gamma_1 = 0.499$, $\gamma_2 = 6.471$, $\sigma = 0.5$, and the other parameters are the same as those in Table 1
Model parameters used in simulations
 Parameters Definitions Values Source$^*$ $\lambda_{m,1}$ basal transcriptional rate $2 h^{-1}$ [1] $\lambda_{m,2}$ maximum promotion in transcription $18 h^{-1}$ [1] due to the protein regulation $\lambda_p$ translational efficiency $2 h^{-1}$ [1] $d_m$ mRNA degradation rate $0.5 h^{-1}$ [1] $d_p$ protein degradation rate $0.08 h^{-1}$ [1] $N$ number of nucleosomes $60$ a.a. $n$ Hill coefficient $4$ a.a. $T$ duration of one cell cycle $20h$ a.a. $\gamma_1$ constant coefficient $6.780$ adjust $\gamma_2$ constant coefficient $9.433$ adjust * Here a.a. means arbitrary assigned, adjust means that we adjust the parameters to yield bistability.
 Parameters Definitions Values Source$^*$ $\lambda_{m,1}$ basal transcriptional rate $2 h^{-1}$ [1] $\lambda_{m,2}$ maximum promotion in transcription $18 h^{-1}$ [1] due to the protein regulation $\lambda_p$ translational efficiency $2 h^{-1}$ [1] $d_m$ mRNA degradation rate $0.5 h^{-1}$ [1] $d_p$ protein degradation rate $0.08 h^{-1}$ [1] $N$ number of nucleosomes $60$ a.a. $n$ Hill coefficient $4$ a.a. $T$ duration of one cell cycle $20h$ a.a. $\gamma_1$ constant coefficient $6.780$ adjust $\gamma_2$ constant coefficient $9.433$ adjust * Here a.a. means arbitrary assigned, adjust means that we adjust the parameters to yield bistability.
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