Contractile Stress and Morphogen Diffusion in Developing Cell Assemblies

A key feature of growth and pattern formation of cell assemblies in embryos is that the steady-state diffusion profile of the signaling morphogen molecules scales with the size of the embryo1. Recent research2 has suggested biochemical mechanisms used by the embryo to achieve such scaling. However many of the processes involved in development are mechanical3 in nature. In particular, the elastic forces due to cytoskeletal contractility are by their very nature, long-ranged4, and could facilitate global effects in the pattern forming processes. As a first-step, we study a model that couples the morphogens to the "mechanical state" of the cells. In our model, the contractility profile decreases as a power law, instead of decaying exponentially (as expected for systems with local interactions) and is thus sensitive to the system boundaries. The effect of the elastic interactions on the diffusive behavior of morphogens is more subtle. We consider several possible models and boundary conditions for the effects of elasticity on diffusion in a finite system. Specific boundary conditions, such as stress-free boundary with a concentration fixed by biochemical feedback at one end, can lead to morphogen profiles that indeed scale with the size of the embryo.1 De Robertis, E. M., Nature Rev. Mol. Cell Biol.7, 296-302 (2006).2 Ben-Zvi, D., Shilo, B. Z., Fainsod, A., & Barkai, N., Nature453, 1205-1211 (2008).3 Forgacs, G and Newman A., Biological Physics of the Developing Embryo, (Cambridge University Press, 2005).4 Schwarz, U. S. & Safran, S. A., Phys. Rev. Lett.88, 048102 (2002).