Regulatory feedback effects on tissue growth dynamics in a two-stage cell lineage model

Identifying the mechanism of intercellular feedback regulation is critical for the basic understanding of tissue growth control in organisms. In this paper, we analyze a tissue growth model consisting of a single lineage of two cell types regulated by negative feedback signaling molecules that undergo spatial diffusion. By deriving the fixed points for the uniform steady states and carrying out linear stability analysis, phase diagrams are obtained analytically for arbitrary parameters of the model. Two different generic growth modes are found: blow-up growth and final-state controlled growth which are governed by the nontrivial fixed point and the trivial fixed point, respectively, and can be sensitively switched by varying the negative feedback regulation on the proliferation of the stem cells. Analytic expressions for the characteristic timescales for these two growth modes are also derived. Remarkably, the trivial and nontrivial uniform steady states can coexist and a sharp transition occurs in the bistable regime as the relevant parameters are varied. Furthermore, the bistable growth properties allows for the external control to switch between these two growth modes. In addition, the condition for an early accelerated growth followed by a retarded growth can be derived. These analytical results are further verified by numerical simulations and provide insights on the growth behavior of the tissue. Our results are also discussed in the light of possible realistic biological experiments and tissue growth control strategy. Furthermore, by external feedback control of the concentration of regulatory molecules, it is possible to achieve a desired growth mode, as demonstrated with an analysis of boosted growth, catch-up growth and the design for the target of a linear growth dynamic.

Figure 1

Schematic picture showing a two-stage cell lineage model with negative feedback. $c_0$ and $c_1$ are the concentration of SC and TD cell, respectively. (a) SC replicates with a rate of $\nu$ and has a probability $\mathcal{P}$ are of duplicating itself and $1-\mathcal{P}$ of differentiating into two TD cells. (b) The molecule $A$, which negative regulates $\mathcal{P}$, is produced by the TD cells at a rate $\mu$, and decays with a rate $d$.

Figure 2

The trivial (green curve) and nontrivial (blue curve) fixed points $\gamma A$ as a function of $\tilde{\mu }\equiv \frac{\gamma \mu }{d}$ for $m=2$. The solid lines show the stable states and the dashed lines show the unstable states. The corresponding physical ($c_0⩾0$) and stable $c_0$ states are also shown by the red curve. (a) $\tilde{\nu }\equiv \frac{\nu }{d}=0.5, p=0.6$; (b) $\tilde{\nu }=0.5, p=0.74$; and (c) $\tilde{\nu }=0.5, p=0.9$. The red vertical dotted lines shows the hysteresis behavior and the bistable phase region lies between them. (d) $\tilde{\nu }=2, p=0.52$; (e) $\tilde{\nu }=2, p=0.6$; and (f) $\tilde{\nu }=2, p=0.9$.

Figure 3

The phase diagram of $\tilde{\mu }\equiv \frac{\gamma \mu }{d}$ vs $p$ with $m=2$. The phase boundaries for the number of roots from equation (11), ${\tilde{\mu }}_t^{+}$ and ${\tilde{\mu }}_t^{-}$ are denoted by blue dashed and red dot-dashed curves, respectively. The $\tilde{\mu }={\tilde{\mu }}_0\left(p\right)\equiv {(2p-1)}^{\frac{1}{m}}$ curve is denoted by a green dotted curve separating the blow-up growth (region I) and final-state growth (region II). The bistable regime (hatched region III) in which the $c_0=0$ (trivial state) and $c_0>0$ (nontrivial state) uniform steady states coexist is marked by the shaded region between the ${\tilde{\mu }}_t^{-}$ and ${\tilde{\mu }}_0$ curves. (a) $\tilde{\nu }\equiv \frac{\nu }{d}=0.5$. The critical $p_t\left(\nu \right)$ given by (A23) and $p_c\left(\nu \right)$ given by (13) are shown by the vertical dot-dashed arrow and solid arrow, respectively. (b) Similar phase diagram for $\tilde{\nu }=2$.

Figure 4

(a) The critical value at which the coexistence regime vanishes, $p_c$ [given by (13) as a function of $\tilde{\nu }$ for different values of the Hill coefficient], $m=1$ (dashed black), $m=2$ (solid blue), and $m=4$ (dot-dashed red). (b) The size of the coexistence regime $\mathrm{\Delta }\tilde{\mu }\equiv {\tilde{\mu }}_t^{-}-{\tilde{\mu }}_0$ [given by (11) and (9)] as a function of $p$ for different values of $m$ for $\tilde{\nu }=0.5$ and (c) for $\tilde{\nu }=2$.

Figure 5

Numerical solution of the lineage model with $m=2, \tilde{\nu }=0.5$, and $p=0.6$ for $\tilde{\mu }=0.2$ inside region I of the phase diagram in Fig. 3 corresponding to the case of blow-up tissue growth. Time evolution of the (a) SC distributions $c_0\left(z\right)$ and (b) feedback molecule concentrations $A\left(z\right)$. The nontrivial USS value $c_0^{*}$ is marked by the horizontal dashed line in (a). Panels (c) and (d) are similar to (a) and (b) for $\tilde{\mu }=2$ inside region II of the phase diagram in Fig. 3 corresponding to the case of saturated tissue growth to the final state. Time and space are in units of $1/d$ and $\sqrt{D/d}$, respectively. Distribution curves in (a) and (b) as well as (c) and (d) are separated by a time of 10 and 5 units, respectively.

Figure 6

Numerical solution of the lineage model with $m=2, \tilde{\nu }=2$ and $p=0.6$ for $\tilde{\mu }=0.2$ inside region I of the phase diagram in Fig. 3 corresponding to the case of blow-up tissue growth. Time evolution of the (a) SC distributions $c_0\left(z\right)$, (b) feedback molecule concentrations $A\left(z\right)$. Panels (c) and (d) are similar to panels (a) and (b) for $\tilde{\mu }=2$ inside region II of the phase diagram in Fig. 3 corresponding to the case of saturated tissue growth to the final state. Time and space are in units of $1/d$ and $\sqrt{D/d}$, respectively. Distribution curves in (a) and (b) as well as (c) and (d) are separated by a time of 3 and 0.5 units, respectively.

Figure 7

Numerical solution of the lineage model with $m=2, \tilde{\nu }=0.5$, and $p=0.9$ for $\tilde{\mu }=1$ inside the bistable region of the phase diagram in Fig. 3. Time evolution of the (a) SC distributions $c_0\left(z\right)$, (b) feedback molecule concentrations $A\left(z\right)$, for initial concentration of $c_0=0.1$. Panels (c) and (d) are similar to panels (a) and (b) but for initial concentration of $c_0=0.5$. Distribution curves in (a) and (b) as well as (c) and (d) are separated by time intervals of 10 and 2, respectively. Time and space are in units of $1/d$ and $\sqrt{D/d}$, respectively.

Figure 8

The $A$ vs $c_0$ phase plane portrait obtained from (7) and (8) showing (a) the case of final state with $p=0.6, \tilde{\mu }=0.8$; (b) the case of blow-up state with $p=0.6, \tilde{\mu }=0.2$; and (c) in the bistable regime with $p=0.9, \tilde{\mu }=1$. In all cases, $\tilde{\nu }=0.5$ and $\gamma =1$. The stable and unstable fixed points are denoted by a filled and open circles, respectively.

Figure 9

Numerical results on the tissue size dynamics of the lineage model with $m=2$ for (a) $\tilde{\mu }=2$ corresponding to the case of the tissue size is governed by the final state for $\tilde{\nu }=0.5$ and 2. The solid curves are fitting of results of the form $L_m\left(t\right)=a_0+a_1{e}^{-\frac{t}{{\tau }_s}}$. (b) Semilog plot of the tissue size as a function of time for $\mu =0.2$ corresponds to the case of the blow-up growth. The straight lines in the large time regimes are fitting using an exponential form from which the exponential growth timescales $\tau$ are obtained. Time and $L_m$ are in units of $1/d$ and $\sqrt{D/d}$, respectively.

Figure 10

Different tissue growth modes in the bistable region. (a) The tissue size $L_m\left(t\right)$ for the system in Fig. 7 under three different initial values of $c_0$. Time and $L_m$ are in units of $1/d$ and $\sqrt{D/d}$, respectively. (b) The growth speed of the leading edge of the tissue as a function time in (a) for the initial $c_0=0.2$, showing a peak at a time ${\tau }_{\mathrm{sw}}$ (marked by vertical dashed line) the accelerated growth is switched to retarded growth.

Figure 11

Characteristic timescales (in unit of $1/d$) for the controlled growth and blow-up growth, for $m=2$. Symbols are timescales obtained from fitting of the tissue size from the numerical solutions. Curves are the corresponding theoretical predictions. (a) $\mu =2$ for the case of controlled growth to the final state. Theoretical predictions are from (29). (b) $\mu =0.2$ for the case of blow-up growth. Theoretical predictions are from (22).

Figure 12

Numerical results on the tissue size dynamics of the lineage model showing the $\mathsf{S}$-shape growth curve, with $m=2$ for (a) leading edge tissue size as a function of $t$ for $\tilde{\mu }=0.48, p=0.6$, and $\tilde{\nu }=0.5$ and 2. (b) The instantaneous growth speed of the leading edge of the tissue in (a) for $\tilde{\nu }=0.5$ and initial SC cell density $c_0\left(0\right)=0.1$ and 0.2. The maximum of the speed is marked by a vertical dashed line, which occurs at $t={\tau }_{\mathrm{sw}}$ with the corresponding SC cell density decay to $c_0^{\mathrm{sw}}\simeq 0.05$ (marked by dashed horizontal arrow), which agree reasonably well with the theoretical prediction [from (30)] of 0.047 and is independent of the values of $\tilde{\nu }$. Time and $L_m$ are in units of $1/d$ and $\sqrt{D/d}$, respectively.

Figure 13

(a) Tissue size time evolution to the final state from numerical solutions for $\tilde{\nu }=0.5$ and 2 (symbols) with $c_0\left(0\right)=0.1$ and $p=0.4$. The theoretical approximations from (27) (with $f=0.5$) for $\tilde{\nu }=0.5$ (solid curve) $\tilde{\nu }=2$ (dashed curve) are also shown. Time and $L_m$ are in units of $1/d$ and $\sqrt{D/d}$, respectively. (b) Normalized $L_m\left(\infty \right)$ as a function of $\tilde{\mu }$ measured from the numerical solutions at long times. The theoretical results from (28) are also shown.

Figure 14

Switching of growth modes by pulse control of the regulatory molecular concentration. (a) Tissue size growth dynamics with the initial growth condition as in Fig. 7. Pulse control of $A$ is applied for $5⩽t⩽15$ (between the vertical dotted lines) by keeping $A=0.2$ in this period. The original final-state growth is boosted to the blow-up mode. (b) Tissue size growth dynamics with the initial growth condition as in Fig. 7. Pulse control of $A$ is applied for $5⩽t⩽15$ by keeping $A=1$ in this period. The original blow-up growth is suppressed to the final-state mode. Time and $L_m$ are in units of $1/d$ and $\sqrt{D/d}$, respectively.

Figure 15

Tissue designed to grow linearly by feedback control of the concentration of the regulatory molecules. $m=2$. (a) Time evolution of the $c_0\left(z\right)$ profiles for $\tilde{\nu }=0.5$, each curve is separated by time intervals of 60 units. Time and space are in units of $1/d$ and $\sqrt{D/d}$, respectively. (b) Tissue size as a function of time showing the linear growth behavior for $\tilde{\nu }=0.5$ and 2. Time and $L_m$ are in units of $1/d$ and $\sqrt{D/d}$, respectively.

Figure 16

(a) Catch-up growth modeled by a short duration of increase in the progenitor cell proliferation rate parameter $\tilde{\nu }$. The original final state grows with $p=0.8, \tilde{\mu }=1, c_0=0.3$, and $\tilde{\nu }=0.5$ (dashed curve). A catch-up growth occurs at an early stage (solid curve) in $5⩽t⩽7$ (between two vertical solid lines) during which $\tilde{\nu }=2$. Another similar catch-up growth occurs at a later stage (dot-dashed curve) in $10⩽t⩽12$ (between two vertical dot-dashed lines) during which $\tilde{\nu }=2$. (b) Boosted growth modeled by a pulse control of the secretion rate parameter of the regulatory molecules, $\tilde{\mu }$. The original growth of the tissue is with parameters $m=2, \tilde{\nu }=0.5, \tilde{\mu }=2$, and $p=0.6$. The tissue size growth curves are shown for the original (dashed curve), early boosted (solid curve), and late boosted (dot-dashed curve) growths. An early pulse control of $\tilde{\mu }$ is applied for $5⩽t⩽15$ with the change $\tilde{\mu }=0.1$ in this period. Another similar boosted pulse control is also applied but at a later time in $10⩽t⩽20$. The vertical lines show the corresponding boosted periods during which $\tilde{\mu }$ is changed. Time and $L_m$ are in units of $1/d$ and $\sqrt{D/d}$, respectively.