# American Institute of Mathematical Sciences

## Stochastic dynamics of cell lineage in tissue homeostasis

 1 Department of Mathematics, University of California, Irvine, Irvine, CA 92697, USA 2 Department of Mathematics, University of California, Riverside, Riverside, CA 92507, USA 3 Department of Mathematics, Department of Developmental and Cell Biology, University of California, Irvine, Irvine, CA 92697, USA

* Corresponding author: Qing Nie

Contributed in honor of Peter Kloeden on the occasion of his 70th birthday

Received  June 2018 Revised  August 2018 Published  January 2019

Fund Project: This work is supported by the NIH grants U01AR073159, R01GM107264, and R01NS095355; a grant from the Simons Foundation (594598, QN), and the NSF grant DMS1763272, DMS1562176, and DMS1762063

During epithelium tissue maintenance, lineages of cells differentiate and proliferate in a coordinated way to provide the desirable size and spatial organization of different types of cells. While mathematical models through deterministic description have been used to dissect role of feedback regulations on tissue layer size and stratification, how the stochastic effects influence tissue maintenance remains largely unknown. Here we present a stochastic continuum model for cell lineages to investigate how both layer thickness and layer stratification are affected by noise. We find that the cell-intrinsic noise often causes reduction and oscillation of layer size whereas the cell-extrinsic noise increases the thickness, and sometimes, leads to uncontrollable growth of the tissue layer. The layer stratification usually deteriorates as the noise level increases in the cell lineage systems. Interestingly, the morphogen noise, which mixes both cell-intrinsic noise and cell-extrinsic noise, can lead to larger size of layer with little impact on the layer stratification. By investigating different combinations of the three types of noise, we find the layer thickness variability is reduced when cell-extrinsic noise level is high or morphogen noise level is low. Interestingly, there exists a tradeoff between low thickness variability and strong layer stratification due to competition among the three types of noise, suggesting robust layer homeostasis requires balanced levels of different types of noise in the cell lineage systems.

Citation: Yuchi Qiu, Weitao Chen, Qing Nie. Stochastic dynamics of cell lineage in tissue homeostasis. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2018339
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A schematic diagram of a main cell lineage in epithelium. Stem cells and TA cells proliferate with probabilities $p_0$ and $p_1$ and differentiate with probabilities $1-p_0$ and $1-p_1$. TD cells undergo cell death with rate $d_2$. All three types of cells can secrete molecule A that inhibits self-renewal probability $p_0$. TD and TA cells secrete molecule G that inhibits self-renewal probability $p_1$. Molecules A and G are diffusive in the epithelium. The apical surface is moving with the dynamic position $z_{\max}$ and no-flux boundary condition is imposed. On the other hand, leaky boundary condition is imposed at the basal lamina with its position fixed.
A baseline simulation for the system containing all three kinds of noise. The spatial distribution of three types of cells and different mophogens at four different time points: A. t = 0; B. t = 330; C. t = 860; D. t = 1200. E. Layer thickness in one particular stochastic simulation. F. Stratification factor of stem cells ($sf(C_0)$). G. Stratification factor of TA cells ($sf(C_1)$). In E-G, the black dash line is the steady-state value for corresponding quantities in the deterministic system. The noise levels used are $\varepsilon_0 = \varepsilon_1 = 0.6$, $\sigma_0 = \sigma_1 = 10^{-4}$, and $\omega_0 = \omega_1 = 0.58$.
Simulations with only cell-intrinsic noise. Dash lines represent the corresponding quantities at homeostasis. A. Layer thickness in three simulations with $\varepsilon = 0.2$, $0.6$ and $1$. B. The mean $TH$. The error bars show the standard deviation. C. The mean $CV$. The error bars show the standard deviation of $CV$. The mean $SF$ of D. stem cells and E. TA cells. The error bars show the standard deviation. F. Distribution of cells and morphogens in a specific simulation with $\varepsilon = 0.6$ at time $t = 400$. In (B-E), all statistical quantities are captured based on $20$ simulations, and the standard deviations (error bars) are negligible compared to the means.
Simulations with only cell-extrinsic noise. Dash lines represent the corresponding quantities at homeostasis. A. Layer thickness in three simulations with $\sigma = 1\times 10^{-3}$, $2\times 10^{-3}$ and $4\times 10^{-3}$. B. The mean $TH$. The error bars show the standard deviation. C. The mean $CV$. The error bars show the standard deviation of $CV$. The mean $SF$ of D. stem cells and E. TA cells. The error bars show the standard deviation. F. Distribution of cells and morphogens in a specific simulation with $\sigma = 3\times 10^{-3}$ at time $t = 400$. In (B-E), all statistical quantities are captured based on $20$ simulations, and the standard deviations (error bars) are negligible compared to the means.
Simulations with only morphogens noise. Dash lines represent the corresponding quantities at homeostasis. A. Layer thickness in three simulations with $\omega = 0.4$, $0.6$ and $1$. B. The mean $TH$. The error bars show the standard deviation. C. The mean $CV$. The error bars show the standard deviation of $CV$. The mean $SF$ of D. stem cells and E. TA cells. The error bars show the standard deviation. F. Distribution of cells and morphogens in a specific simulation with $\omega = 0.6$ at time $t = 400$. In (B-E), all statistical quantities are captured based on $20$ simulations, and the standard deviations (error bars) are negligible compared to the means.
Simulations with both cell-intrinsic noise and cell-extrinsic noise. Simulations with different noise levels are shown in (A-I). In each subfigure, the panel on the top shows the dynamics of layer thickness, the panel on the bottom shows the dynamics of layer stratification of stem cells ($sf(C_0)$). The dash line represents for the corresponding quantity at homeostasis. Three different levels are chosen for each type of noise. For cell-intrinsic noise level $\varepsilon$: $0.2$ (Low), $0.6$ (Medium), $1$ (High). For cell-extrinsic noise level $\sigma$: $5\times 10^{-4}$ (Low), $1\times 10^{-3}$ (Medium), $2\times 10^{-3}$ (High).
Simulations with both cell-intrinsic noise and morphogen noise. Simulations with different noise levels are shown in (A-I). In each subfigure, the panel on the top shows the dynamics of layer thickness, the panel on the bottom shows the dynamics of layer stratification of stem cells ($sf(C_0)$). The dash line represents for the corresponding quantity at homeostasis. Three different levels are chosen for each type of noise. For cell-intrinsic noise level $\varepsilon$: $0.2$ (Low), $0.6$ (Medium), $1$ (High). For morphogen noise level $\omega$: $0.2$ (Low), $0.6$ (Medium), $1$ (High).
Simulations for maintaining homeostasis ($SS$ = 0.49mm) with different combinations of three types of noise. Points with the same color and the same marker represent for simulations with the same cell-intrinsic noise level $\varepsilon$, where $\varepsilon = 0.2$, $0.4$, $0.6$, $0.8$ and $1$ respectively. The strips, filled with color gradient, roughly divide the plane into several regions. Data points located in the region next to dark/light color of an individual strip have more/less desirable properties. A. The relation between the cell-extrinsic noise level $\sigma$ and the morphogen noise level $\omega$. The blue strip sketches the green points with maximal cell-intrinsic noise level $\varepsilon = 1$. It divides this plane into stabilized region (region Ⅰ-Ⅳ) and non-stabilized region (region Ⅴ). The stabilized region is divided into four parts (region Ⅰ-Ⅳ) by a red strip and a green strip. These regions will be introduced next. B. The relation between layer thickness variability ($CV$) and layer stratification factor of stem cells ($SF(C_0)$). The red strip with $CV = 20%$ divides this plane into two regions with low $CV$ or high $CV$. Also the green strip with $SF(C_0) = 0.4$ divides the plane into two regions with high $SF$ or low $SF$. The red and the green strips together divide the stabilized region into four regions (Region Ⅰ: low $CV$ and high $SF$; Region Ⅱ: high $CV$ and high $SF$; Region Ⅲ: low $CV$ and low $SF$; Region Ⅳ: high $CV$ and low $SF$). C. The relation between $\sigma$ and $CV$. D. The relation between $\omega$ and $CV$. E. The relation between $\sigma$ and $SF(C_0)$. F. The relation between $\omega$ and $SF(C_0)$.
The statistics of $TH$, $CV$ and $SF(C_0)$ with combined cell-intrinsic ($\varepsilon$) and cell-extrinsic ($\sigma$) noise. All quantities are captured based on $20$ simulations.
 $0$ $5\times10^{-4}$ $1\times10^{-3}$ $2\times10^{-3}$ $0$ $TH$ $0.49$mm $0.53$mm $0.58$mm $0.75$mm $CV$ $0%$ $1%$ $3%$ $7%$ $SF$ $0.91$ $0.90$ $0.88$ $0.40$ $0.2$ $TH$ $0.45$mm $0.49$mm $0.54$mm $0.69$mm $CV$ $7%$ $7%$ $7%$ $8%$ $SF$ $0.91$ $0.90$ $0.82$ $0.39$ $0.6$ $TH$ $0.24$mm $0.28$mm $0.30$mm $0.44$mm $CV$ $30%$ $25%$ $23%$ $17%$ $SF$ $0.84$ $0.69$ $0.52$ $0.27$ $1$ $TH$ $0.12$mm $0.19$mm $0.22$mm $0.38$mm $CV$ $89%$ $36%$ $32%$ $19%$ $SF$ $0.88$ $0.42$ $0.32$ $0.18$
 $0$ $5\times10^{-4}$ $1\times10^{-3}$ $2\times10^{-3}$ $0$ $TH$ $0.49$mm $0.53$mm $0.58$mm $0.75$mm $CV$ $0%$ $1%$ $3%$ $7%$ $SF$ $0.91$ $0.90$ $0.88$ $0.40$ $0.2$ $TH$ $0.45$mm $0.49$mm $0.54$mm $0.69$mm $CV$ $7%$ $7%$ $7%$ $8%$ $SF$ $0.91$ $0.90$ $0.82$ $0.39$ $0.6$ $TH$ $0.24$mm $0.28$mm $0.30$mm $0.44$mm $CV$ $30%$ $25%$ $23%$ $17%$ $SF$ $0.84$ $0.69$ $0.52$ $0.27$ $1$ $TH$ $0.12$mm $0.19$mm $0.22$mm $0.38$mm $CV$ $89%$ $36%$ $32%$ $19%$ $SF$ $0.88$ $0.42$ $0.32$ $0.18$
The statistics of $TH$, $CV$ and $SF(C_0)$ with combined cell-intrinsic ($\varepsilon$) and morphogen ($\omega$) noise. All quantities are captured based on $20$ simulations.
 $0$ $0.2$ $0.6$ $1$ $0$ $TH$ $0.49$mm $0.58$mm $1.06$mm $1.33$mm $CV$ $0%$ $3%$ $9%$ $11%$ $SF$ $0.91$ $0.92$ $0.92$ $0.91$ $0.2$ $TH$ $0.45$mm $0.54$mm $0.98$mm $1.23$mm $CV$ $7%$ $7%$ $11%$ $13%$ $SF$ $0.91$ $0.91$ $0.92$ $0.90$ $0.6$ $TH$ $0.24$mm $0.26$mm $0.43$mm $0.52$mm $CV$ $30%$ $29%$ $30%$ $33%$ $SF$ $0.84$ $0.83$ $0.84$ $0.81$ $1$ $TH$ $0.12$mm $0.13$mm $0.15$mm $0.16$mm $CV$ $89%$ $87%$ $97%$ $108%$ $SF$ $0.88$ $0.88$ $0.87$ $0.84$
 $0$ $0.2$ $0.6$ $1$ $0$ $TH$ $0.49$mm $0.58$mm $1.06$mm $1.33$mm $CV$ $0%$ $3%$ $9%$ $11%$ $SF$ $0.91$ $0.92$ $0.92$ $0.91$ $0.2$ $TH$ $0.45$mm $0.54$mm $0.98$mm $1.23$mm $CV$ $7%$ $7%$ $11%$ $13%$ $SF$ $0.91$ $0.91$ $0.92$ $0.90$ $0.6$ $TH$ $0.24$mm $0.26$mm $0.43$mm $0.52$mm $CV$ $30%$ $29%$ $30%$ $33%$ $SF$ $0.84$ $0.83$ $0.84$ $0.81$ $1$ $TH$ $0.12$mm $0.13$mm $0.15$mm $0.16$mm $CV$ $89%$ $87%$ $97%$ $108%$ $SF$ $0.88$ $0.88$ $0.87$ $0.84$
Parameters used in Eq. (2) to Eq. (7).
 Parameters Values Units $\nu_0$, $\nu_1$ $1$ $\ln 2*$(cell cycle)$^{-1}$ $d_2$ $0.01$ $\ln 2*$(cell cycle)$^{-1}$ $D_A$, $D_G$ $10^{-5}$ mm$^2$s$^{-1}$ $\mu_0$, $\mu_1$, $\mu_2$, $\eta_1$, $\eta_2$ $10^{-3}$ s$^{-1}\mu M$ $a_{\deg}$, $g_{\deg}$ $10^{-3}$ s$^{-1}$ $\alpha_A$, $\alpha_G$ $10$ mm$^{-1}$ $\bar{p}_0$ $0.95$ - $\bar{p}_1$ $0.5$ - $\gamma_A$ $1.6$ $\mu M^{-1}$ $\gamma_G$ $2$ $\mu M^{-1}$
 Parameters Values Units $\nu_0$, $\nu_1$ $1$ $\ln 2*$(cell cycle)$^{-1}$ $d_2$ $0.01$ $\ln 2*$(cell cycle)$^{-1}$ $D_A$, $D_G$ $10^{-5}$ mm$^2$s$^{-1}$ $\mu_0$, $\mu_1$, $\mu_2$, $\eta_1$, $\eta_2$ $10^{-3}$ s$^{-1}\mu M$ $a_{\deg}$, $g_{\deg}$ $10^{-3}$ s$^{-1}$ $\alpha_A$, $\alpha_G$ $10$ mm$^{-1}$ $\bar{p}_0$ $0.95$ - $\bar{p}_1$ $0.5$ - $\gamma_A$ $1.6$ $\mu M^{-1}$ $\gamma_G$ $2$ $\mu M^{-1}$
Noise levels used in Eq. (7) and (8) in different figures.
 $\varepsilon_0$, $\varepsilon_1$ $\sigma_0$, $\sigma_1$ $\omega_0$, $\omega_1$ Figure 2 $0.6$ $10^{-4}$ 0.58 Figure 3F $0.6$ $0$ $0$ Figure 4F $0$ $3\times10^{-3}$ $0$ Figure 5F $0$ $0$ $0.6$ Figure 6 Low; $0.2$ Low: $5\times 10^{-4}$ $0$ Medium: $0.6$ Medium: $1\times10^{-3}$ High: $1$ High: $2\times10^{-3}$ Figure 7 Low: $0.2$ $0$ Low: $0.2$ Medium: $0.6$ Medium: $0.6$ High: $1$ High: $1$
 $\varepsilon_0$, $\varepsilon_1$ $\sigma_0$, $\sigma_1$ $\omega_0$, $\omega_1$ Figure 2 $0.6$ $10^{-4}$ 0.58 Figure 3F $0.6$ $0$ $0$ Figure 4F $0$ $3\times10^{-3}$ $0$ Figure 5F $0$ $0$ $0.6$ Figure 6 Low; $0.2$ Low: $5\times 10^{-4}$ $0$ Medium: $0.6$ Medium: $1\times10^{-3}$ High: $1$ High: $2\times10^{-3}$ Figure 7 Low: $0.2$ $0$ Low: $0.2$ Medium: $0.6$ Medium: $0.6$ High: $1$ High: $1$
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