Statistical dynamics of spatial-order formation by communicating cells

Communicating cells can coordinate their gene expressions to form spatial patterns. 'Secrete-and-sense cells' secrete and sense the same molecule to do so and are ubiquitous. Here we address why and how these cells, from disordered beginnings, can form spatial order through a statistical mechanics-type framework for cellular communication. Classifying cellular lattices by 'macrostate' variables - 'spatial order paramete' and average gene-expression level - reveals a conceptual picture: cellular lattices act as particles rolling down on 'pseudo-energy landscapes' shaped by a 'Hamiltonian' for cellular communication. Particles rolling down represent cells' spatial order increasing. Particles trapped on the landscapes represent metastable spatial configurations. The gradient of the Hamiltonian and a 'trapping probability' determine the particle's equation of motion. This framework is extendable to more complex forms of cellular communication

Nonautonomous fractional problems with exponential growth

We study a class of nonlinear non-autonomous nonlocal equations with subcritical and critical exponential nonlinearity. The involved potential can vanish at infinity.Comment: 12 page