# American Institute of Mathematical Sciences

December  2017, 14(5&6): 1379-1397. doi: 10.3934/mbe.2017071

## A mathematical model of stem cell regeneration with epigenetic state transitions

 Zhou Pei-Yuan Center for Applied Mathematics, Tsinghua University, Beijing 100084, China

* Corresponding author: Jinzhi Lei

Received  May 18, 2016 Accepted  January 15, 2017 Published  May 2017

Fund Project: This work is supported by National Natural Science Foundation of China (91430101 and 11272169).

In this paper, we study a mathematical model of stem cell regeneration with epigenetic state transitions. In the model, the heterogeneity of stem cells is considered through the epigenetic state of each cell, and each epigenetic state defines a subpopulation of stem cells. The dynamics of the subpopulations are modeled by a set of ordinary differential equations in which epigenetic state transition in cell division is given by the transition probability. We present analysis for the existence and linear stability of the equilibrium state. As an example, we apply the model to study the dynamics of state transition in breast cancer stem cells.

Citation: Qiaojun Situ, Jinzhi Lei. A mathematical model of stem cell regeneration with epigenetic state transitions. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1379-1397. doi: 10.3934/mbe.2017071
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Model illustration. During stem cell regeneration, cells in the resting phase either enter the proliferating phase with a rate $\beta$, or be removed from the resting pool with a rate $\gamma$. The proliferating cells undergo apoptosis with a probability $\mu$. Each daughter cell generated from mitosis is either a differentiated cell (with a probability $\kappa$) or a stem cell (with a probability $(1-\kappa)$)
Transition dynamics. (A) The cell population dynamics. (B) The percentage of epigenetic-state cells at the equilibrium state. Here, results of $\nu = 0$ (green), $20$ (blue), and $200$ (red) are shown. In simulations, the initial cell population is taken as $N(0) = 300$, and $N(0, X_i)=1, (i=1, \cdots, 300)$
Simulation of cell-state dynamics. Dynamics of cell-state proportion with different initial states (left: $(S, B, L)=(99.9, 0.05, 0.05)$, middle: $(S, B, L)=(0.05, 0.05, 99.9)$, right: $(S, B, L)=(0.05, 99.9, 0.05)$). Markers are data taken from [13]. Parameters are listed in Table 1
Parameter values in the model of cancer cell state transition. Left: the probabilities $\gamma, \kappa, \mu$ for cells of these three states. Right: the transition matrix $p(X, Y), X, Y\in \Omega$
 Parameter S B L S B L $\gamma$ 0.95 0.7 0.65 S 0.58 0.04 0.01 $\kappa$ 0.02 0.03 0 B 0.07 0.47 0 $\mu$ 0.1 0.1 0.1 L 0.35 0.49 0.09
 Parameter S B L S B L $\gamma$ 0.95 0.7 0.65 S 0.58 0.04 0.01 $\kappa$ 0.02 0.03 0 B 0.07 0.47 0 $\mu$ 0.1 0.1 0.1 L 0.35 0.49 0.09
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