Transient Turing patterns in a morphogenetic model

Abstract

One of the most surprising mechanisms to explain the symmetry breaking phenomenon linked to pattern formation is known as Turing instabilities. These patterns are self-organising spatial structures resulting from the interaction of at least two diffusive species in specific conditions. The ideas of Turing have been used extensively in the specialised literature both to explain developmental patterns, as well as synthetic biology design. In the present work we study a previously proposed morphogenetic synthetic circuit consisting of two genes controlled by the same regulatory system. The spatial homogeneous version of this simple model presents a rich phase diagram, since it has a saddle-node bifurcation, spirals and limit cycle. Linear stability analysis and numerical simulations of the complete model allow us to determine the conditions for the development of Turing patterns, as well as transient patterns. We found that the parameter region where Turing patterns are found is much smaller than the region where transient patterns occur. We observed that the temporal evolution towards Turing patterns can present one or two different length scales, depending on the initial conditions. Further, we found a parameter region where the persistence time of the transient patterns depends on the distance between the parameters values on which the system is operating and the boundary of Turing patterns. This persistence time has a singularity at a critical distance that gives place to metastable patterns. To the best of our knowledge, transient and metastable patterns associated with Turing instabilities have not been previously reported in morphogenetic models.