Critical Point in Self-Organized Tissue Growth

We present a theory of pattern formation in growing domains inspired by biological examples of tissue development. Gradients of signaling molecules regulate growth, while growth changes these graded chemical patterns by dilution and advection. We identify a critical point of this feedback dynamics, which is characterized by spatially homogeneous growth and proportional scaling of patterns with tissue length. We apply this theory to the biological model system of the developing wing of the fruit fly Drosophila melanogaster and quantitatively identify signatures of the critical point.

Figure 1

Minimal model for growth control in biological tissues by scaling morphogen gradients. (a) The wing imaginal disk of the fruit fly is a two-dimensional epithelial sheet with a source releasing Dpp molecules (red) at the anterior-posterior compartment boundary (dark red). We consider a simplified morphogen system with a source at the left boundary. Panels (b)–(f) show numeric solutions of Eqs. (1) and (2) for $k=0$. Color code defined in (b) applies to all panels [in (c) and (e) most lines overlap]. (b) Spatial profiles for morphogen concentration $C$ for different tissue lengths. (c) Rescaled concentration profiles from (b) collapse on a master curve, thus showing scaling (inset: log-normal plot). (d) Amplitude $C_0$ of the concentration profile obeys a power-law relationship with tissue length $\ell$. (e) Self-consistently regulated growth becomes spatially homogeneous after an initial transient period. (f) Growth slows down inversely with time [solid line: Eq. (10)]. Parameters are $D=0.1\mu {\mathrm{m}}^2/\mathrm{s}$ [12], $\nu =1\text{conc}/\mathrm{s}$, $w=0.1\ell$, $ϵ=0.83$ [2], $\beta =2/(1+ϵ)$.

Figure 2

Critical point and growth dynamics for $k\sim {\ell }^{-2}$. (a) Concentration profiles as a function of relative position $r=x/\ell$ for different tissue length (color code) and different values of $\beta$. Scaling of the concentration profiles at the critical point with $\beta ={\beta }_c$ results in a collapse of the normalized concentration profiles for different tissue lengths. Above and below the critical point (here, $0.8{\beta }_c$, $1.2{\beta }_c$) deviations from scaling occur. (b) Growth rate profiles as a function of $r$ for different tissue length. At the critical point growth becomes homogeneous. (c) Growth rate as a function of time. For $\beta >{\beta }_c$, the growth rate decreases exponentially with time, while for $\beta \le {\beta }_c$, a power-law behavior leads to unbounded growth [solid line: Eq. (10)]. (d) Growth behaviors for super- and subcritical $\beta$ for different degradation scenarios. (e) Different growth regimes as a function of the source scaling exponent $\gamma$ for $k\sim {\ell }^{-2}$. Regimes of unbounded growth (light red) and growth arrest (light blue) are separated by the line $\gamma =2\beta /{\beta }_c-1$ for $\gamma <1$. Numerical results (dots; see Ref. [46]), critical point with $\gamma =1$ (red dot), parameters corresponding to fit to experimental data shown in Fig. 3 (blue dot). A constant source width corresponds to $\gamma =0$. Parameters are $D=0.1\mu {\mathrm{m}}^2/\mathrm{s}$ [12], $\nu =1\text{conc}/\mathrm{s}$, $w=0.1\ell$ [$w=0.3\mu \mathrm{m}(\ell /30\mu \mathrm{m}{)}^{\gamma }$ in (e)], $ϵ=0.83$ [2], $k{\ell }^2=9\mu {\mathrm{m}}^2/\mathrm{s}$. The color code defined in (a) also applies to (b) and (c).

Figure 3

Growth and gradient scaling in the fly wing. (a) Schematic illustration of time-dependent morphogen profiles $C(x)$ in a growing posterior compartment of size ${\ell }_p$ regulated by an expander mechanism. The morphogen is produced in a source region of width $w$ that increases with tissue length $\ell$ as $w=w_0{\ell }^{\gamma }$. The expander is produced in a source of constant width $w_E$, located at the posterior end; see Ref. [46]. (b) Numerical solutions for morphogen profiles $C(x)$. (c) Position and time dependence of local growth rates $g$. Inset: Average growth rate in the posterior compartment as a function of time. The growth rate relaxation time, 48.2 h, is consistent with experiments [2, 50]. (d) Collapse of relative concentration profiles $C/C_0$ as function of relative position $x/{\ell }_p$ at different times. (e)–(h) Comparison of experimental data (dots) [2] and numerical solutions (solid lines). (e) Morphogen profile amplitude $C_0$ as a function of posterior tissue size ${\ell }_p$. (f) Posterior tissue area $A$ as a function of time. (g) Decay length $\lambda$ of the morphogen profile in the posterior compartment as a function of ${\ell }_p$. Boxed data points in (e) are excluded from the fits. Initial conditions are steady state of Eq. (1). Parameters estimated from experimental measurements are $D=0.1\mu {\mathrm{m}}^2{\mathrm{s}}^{-1}$ [12], $\beta =0.7$, $ϵ=0.83$ [2]. Parameters estimated by a fit to the data are $\gamma =0.3$, $w_0=5.75\mu {\mathrm{m}}^{1-\gamma }$, $w_E=2.5\mu \mathrm{m}$, $\nu /{\nu }_E=0.21$, $k_E=5\times {10}^{-6}{\mathrm{s}}^{-1}$, $D_E=10\mu {\mathrm{m}}^2{\mathrm{s}}^{-1}$, $\eta {\nu }_E^2=2.56\times {10}^{-11}{\mathrm{s}}^{-3}$.