The Emergence of Complexity from a Simple Model for Tissue Growth

Abstract

The growth of living tissue is known to be modulated by mechanical as well as biochemical signals. We study a simple numerical model where the tissue growth rate depends on a chemical potential describing biochemical and mechanical driving forces in the material. In addition, the growing tissue is able to adhere to a three-dimensional surface and is subjected to surface tension where not adhering. We first show that this model belongs to a wider class of models describing particle growth during phase separation. We then analyse the predicted tissue shapes growing on a solid support corresponding to a cut hollow cylinder, which could be imagined as an idealized description of a broken long bone. We demonstrate the appearance of complex shapes described by Delauney surfaces and reminiscent of the shapes of callus appearing during bone healing. This complexity of shapes arises despite the extreme simplicity of the growth model, as a consequence of the threedimensional boundary conditions imposed by the solid support.

Emergent collective organization of bone cells in complex curvature fields

“…Alternatively, one could consider hyperbolic sheet-based scaffolds, such as those based on triply periodic minimal surfaces (TPMS), which have ≤ 0 at every point. However, developing geometrically-optimized scaffold designs requires further investigation into the intricacies of cell-geometry interaction, likely involving computational studies that take geometry explicitly into consideration 48 . Nevertheless, fuelled by rapid advances in high-resolution free-form fabrication, we anticipate exciting avenues for geometric control of cells and tissues, relying on surface curvature as the language of shape.…”

Microenvironment-mediated cancer dormancy: Insights from metastability theory

“…2 B and C ). This simple model shows that a size-dependent parameter, named the critical radius ( ), exists, which, if exceeded, will result in the sustained growth of the droplet; otherwise, it will shrink and disappear ( 24 ) ( Fig. 2 B ).…”

“…In this respect, some parallels can be drawn with critical states [also known as the emergence of multistability ( 8 )], such as their reduced sensitivity to the details of the system, while acquiring increased susceptibility to environmental perturbations ( 8 , 9 , 11 ). Building on the fact that, at least for spherical shapes, tissue growth models fall under the same category as phase separation models of nodular structures ( 24 , 29 , 30 ) (e.g., growth of a particle within a solution, or diffusion along dislocation lines or grain boundaries), here we propose that cancer dormancy shares several traits of thermodynamic metastability. The duration (or stability) of this “dormant” state will then depend on the activation energy barrier, which, in thermodynamic terms, can be described as the amount of energy that needs to be overcome in order to transition from particle nucleation to its critical size and eventual growth.…”

“…A crossing of the barrier may result from various perturbations, such as fluctuations in the driving force for growth or a decrease in the barrier height through a modification of the microenvironment. In this sense, metastability is the resilience of the system against small mechanobiological perturbations ( 24 – 28 ) that would ultimately lead to unrestricted growth (beyond the unstable state; see the sketch in Fig. 2 C ).…”

“…In particular, we assume the growing tissue “droplet” to behave like a fluid, in that the relaxation of local stress concentrations is much faster than growth itself. Under these conditions, the Young–Laplace law leads to droplet shapes with constant curvature and kinetic metastability (see details in SI Appendix ) that results from the fact that a small increase in droplet volume would lead to a negative change in growth rate ( 24 , 29 ).…”