Emergent dynamics of coordinated cells with time delays in a tissue

In this article, we investigate the emergence of tissue dynamics with time delays of diffusion. Such emergent dynamics, describing the tissue homeostasis, usually correspond to particular tissue functions, which are attracting a tremendous amount of attention from both communities of mathematical modeling and systems biology. Specifically, in addition to the within-cell genome dynamics and the diffusion among the cells, we consider several types of time delays of diffusion present in the coordinated cells. We establish several generalized versions of the "monotonicity condition" (MC), whose traditional version [I. Rajapakse and S. Smale, Proc. Natl. Acad. Sci. U.S.A. 114, 1462-1467 (2017)] guaranteed the stability of the equilibrium in a system of coordinated cells without time delay. Indeed, we find that one generalized MC we establish still guarantees the stability of the time-delayed system's equilibrium, which corresponds to a formation of tissue functions depending primarily on individual genome dynamics but less on interacting structures and time delays of diffusion. We also find that, when the generalized MC is further relaxed, the system is able to sustain periodic oscillations, whose periods are verified to have delicate dependence with the selected time delays. These produced oscillations usually represent realistic behaviors of "alive" cells. We use several representative examples to demonstrate the usefulness of the established analytical conditions to the understanding of the emergent dynamics observed in computational models and in real systems as well.