Influence of survival, promotion, and growth on pattern formation in zebrafish skin

Linking discrete and continuous models of cell birth and migration

Self-organization of individuals within large collectives occurs throughout biology. Mathematical models can help elucidate the individual-level mechanisms behind these dynamics, but analytical tractability often comes at the cost of biological intuition. Discrete models provide straightforward interpretations by tracking each individual yet can be computationally expensive. Alternatively, continuous models supply a large-scale perspective by representing the 'effective' dynamics of infinite agents, but their results are often difficult to translate into experimentally relevant insights. We address this challenge by quantitatively linking spatio-temporal dynamics of continuous models and individual-based data in settings with biologically realistic, time-varying cell numbers. Specifically, we introduce and fit scaling parameters in continuous models to account for discrepancies that can arise from low cell numbers and localized interactions. We illustrate our approach on an example motivated by zebrafish-skin pattern formation, in which we create a continuous framework describing the movement and proliferation of a single cell population by upscaling rules from a discrete model. Our resulting continuous models accurately depict ensemble average agent-based solutions when migration or proliferation act alone. Interestingly, the same parameters are not optimal when both processes act simultaneously, highlighting a rich difference in how combining migration and proliferation affects discrete and continuous dynamics.

Turing patterns by supramolecular self-assembly of a single salphen building block

In the 1950s, Alan Turing showed that concerted reactions and diffusion of activating and inhibiting chemical species can autonomously generate patterns without previous positional information, thus providing a chemical basis for morphogenesis in Nature. However, access to these patterns from only one molecular component that contained all the necessary information to execute agonistic and antagonistic signaling is so far an elusive goal, since two or more participants with different diffusivities are a must. Here, we report on a single-molecule system that generates Turing patterns arrested in the solid state, where supramolecular interactions are used instead of chemical reactions, whereas diffusional differences arise from heterogeneously populated self-assembled products. We employ a family of hydroxylated organic salphen building blocks based on a bis-Schiff-base scaffold with portions responsible for either activation or inhibition of assemblies at different hierarchies through purely supramolecular reactions, only depending upon the solvent dielectric constant and evaporation as fuel.

Effect of obstructions on growing Turing patterns

We study how Turing pattern formation on a growing domain is affected by discrete domain discontinuities. We use the Lengyel-Epstein reaction-diffusion model to numerically simulate Turing pattern formation on radially expanding circular domains containing a variety of obstruction geometries, including obstructions spanning the length of the domain, such as walls and slits, and local obstructions, such as small blocks. The pattern formation is significantly affected by the obstructions, leading to novel pattern morphologies. We show that obstructions can induce growth mode switching and disrupt local pattern formation and that these effects depend on the shape and placement of the objects as well as the domain growth rate. This work provides a customizable framework to perform numerical simulations on different types of obstructions and other heterogeneous domains, which may guide future numerical and experimental studies. These results may also provide new insights into biological pattern growth and formation, especially in non-idealized domains containing noise or discontinuities.

Recombinant COL6 α2 as a Self-Organization Factor That Triggers Orderly Nerve Regeneration Without Guidance Cues

Collagen VI (COL6) in the microenvironment was recently identified as an extracellular signal that bears the function of promoting orderly axon bundle formation. However, the large molecular weight of COL6 (≈2,000 kDa) limits its production and clinical application. It remains unclear whether the smaller subunit α chains of COL6 can exert axon bundling and ordering effects independently. Herein, based on a dorsal root ganglion (DRG) ex vivo model, the contributions of three main COL6 α chains on orderly nerve bundle formation were analyzed, and COL6 α2 showed the largest contribution weight. A recombinant COL6 α2 chain was produced and demonstrated to promote the formation of orderly axon bundles through the NCAM1-mediated pathway. The addition of COL6 α2 in conventional hydrogel triggered orderly nerve regeneration in a rat sciatic nerve defect model. Immunogenicity assessment showed weaker immunogenicity of COL6 α2 compared to that of the COL6 complex. These findings suggest that recombinant COL6 α2 is a promising material for orderly nerve regeneration.