Statistics of noisy growth with mechanical feedback in elastic tissues

Abstract

Tissue growth is a fundamental aspect of development and is intrinsically noisy. Stochasticity has important implications for morphogenesis, precise control of organ size, and regulation of tissue composition and heterogeneity. However, the basic statistical properties of growing tissues, particularly when growth induces mechanical stresses that can in turn affect growth rates, have received little attention. Here, we study the noisy growth of elastic sheets subject to mechanical feedback. Considering both isotropic and anisotropic growth, we find that the density-density correlation function shows power law scaling. We also consider the dynamics of marked, neutral clones of cells. We find that the areas (but not the shapes) of two clones are always statistically independent, even when they are adjacent. For anisotropic growth, we show that clone size variance scales like the average area squared and that the mode amplitudes characterizing clone shape show a slow [Formula: see text] decay, where n is the mode index. This is in stark contrast to the isotropic case, where relative variations in clone size and shape vanish at long times. The high variability in clone statistics observed in anisotropic growth is due to the presence of two soft modes-growth modes that generate no stress. Our results lay the groundwork for more in-depth explorations of the properties of noisy tissue growth in specific biological contexts.

Keywords: clonal dynamics; mechanical feedback; noisy growth; tissue mechanics.

Fig. 1.

Noisy growth leads to tissue deformation. A shaded square of side $L$ in the initial state has grown at time $t$ into a larger region with deformed boundaries (red); dashed grid lines indicate the average, uniform tissue dilation. A material point (green circle) initially at $\mathbf{R}$ is displaced to $\mathbf{R}{e}^{{\gamma }_{0}t}+\mathbf{w}$.

Fig. 2.

(A) Independence of nearby clone areas. A localized region of growth (marked by the red star) at the center of the left clone (light blue) leads to an increase in the area of that clone, but it leaves the area of the adjacent clone (dark blue) unchanged even while distorting its shape. (B) Longitudinal soft mode. Sinusoidal growth (color scale gives $\tilde{G}$) leads to a deformation field $\mathbf{w}$ (arrows) that exactly compensates for the growth, leaving the density unchanged.