Synchronization dynamics and diffusion-induced spatiotemporal patterns in cells with activator-inhibitor pathways.

Abstract

The cellular transport through the plasma membrane determines cells’ fate, function and phenotype. It controls what enters or leaves the cells through both active and passive transport processes such as diffusion and osmosis, while relying on several important signaling schemes involving both short and long range interactions. In this thesis, we contribute to the understanding, on one part, of the impact of these two types of coupling on the synchronization and control of the cellular rhythms crucial for their coordinated collective behavior, in order to preserve homeostasis and enhance cellular communication. On the other part, to highlight the emergence of both stable and unstable patterns which are fundamental in cellular exchange, organisms morphology, and specialization amid living beings. In this framework, cells with activator-inhibitor pathways stand as our core biochemical models for these investigations. Firstly, synchronization - wise, while bearing in mind the adaptive character of the extracellular medium, we start by analyzing the effects of chemical, electrical and environment-mediated coupling types on the exchange of metabolites across concentrations of cells, in order to study the onset of phase and complete synchronization in the chaotic biochemical system. In these contexts, the stability of the synchronization manifold is studied by means of both mathematical and numerical tools such as the master stability function approach, the conditional Lyapunov exponents, the rate of change of the Lyapunov function, the construction of localized sets via Poincare sections, the phase difference, the Kuramoto parameter for phase entrainment, and the phase space conditional observation spreading parameter for stroboscopic maps. All these indicators continuously advocate for the prevalence of a phase synchronized dynamics among the interacting cells in the phase space. Yet, the results indicate that the system cannot completely synchronize under the sole action of the chemical coupling. Both the combined effects of the chemical and electrical couplings on one side, and the adaptive environment-mediated coupling scheme with feedbacks and control mechanisms on the other side, are found to be of capital importance in the onset of complete synchronization and high quality synchronization in the system. Secondly, we ascertain the emergence of spatiotemporal structures in 1 - D and 2 - D spatial lattices of the biochemical system. In 1 - D ensembles, we report several phenomena of patterns formation with and without suppression of chaos in the heterogeneous lattices, with bubbling scenarios. Next, we also construct a normalized formalism emulating the dynamics of the morphogens concentrations in two-dimensional spatial arrays of such pathways. The stability analysis of the spatially homogeneous equilibrium state of our biochemical lattice is performed both analytically and numerically. Some suitable parameter conditions are highlighted as conducive for the occurrence of diffusion-driven instabilities, crucial for the emergence of spatiotemporal patterns. The patterning events include inhomogeneous stationary (Turing) spatial patterns and oscillatory spatial patterns encompassing travelling wave phenomena and spatiotemporal chaos. Their relevance in biology stems from their staple prevalence in the communicative and developmental processes of cells.