Protein pattern formation induced by the joint effect of noise and delay in a multi-cellular system

We explore the combined effect of the intrinsic noise and time delay on the spatial pattern formation within the framework of a multi-scale mobile lattice model mimicking two-dimensional epithelium tissues. Every cell is represented by an elastic polygon changing its form and size under pressure from the surrounding cells. The model includes the procedure of minimization of the potential energy of tissue. The protein fluctuations in the tissue are driven by transcription/translation processes in cells exchanging chemical and mechanical signals. Network architecture includes an autorepressor model with time-delayed negative feedback with the only gene defining oscillations. Simultaneously, the expressed protein of the autorepressor acts as a positive regulator of the signaling protein by activating its transcription. The signaling species is assumed to spread from one cell to the other by the diffusion mechanism. We provide both deterministic and stochastic descriptions. The numerical simulation of spatially-extended stochastic oscillations is performed using a generalized Gillespie algorithm. We developed this method earlier to account for the non-Markovian properties of random biochemical events with delay. Finally, we demonstrate that time delay, intrinsic noise, and spatial signaling can cause a system to develop the protein pattern even when its deterministic counterpart exhibits no pattern formation.