Differentiation and Replication of Spots in a Reaction Diffusion System with Many Chemicals

The replication and differentiation of spots in reaction diffusion equations are studied by extending the Gray-Scott model with self-replicating spots to include many degrees of freedom needed to model systems with many chemicals. By examining many possible reaction networks, the behavior of this model is categorized into three types: replication of homogeneous fixed spots, replication of oscillatory spots, and differentiation from ‘multipotent spots’. These multipotent spots either replicate or differentiate into other types of spots with different fixed-point dynamics, and as a result, an inhomogeneous pattern of spots is formed. This differentiation process of spots is analyzed in terms of the loss of chemical diversity and decrease of the local Kolmogorov-Sinai entropy. The relevance of the results to developmental cell biology and stem cells is also discussed. A simple chemical reaction-diffusion system, the Gray-Scott (GS) model[1], has recently been studied extensively. This model exhibits self-replicating spots within some range of parameter values[2]. Similar replicating spots have been found in experiment[3]. In each model system studied to this time, only a single type of spots appear, and there is no differentiation to create different types of spots. To search for possibility in complex pattern formation as in biological systems, it is interesting to study a reaction-diffusion system displaying not only the division but also differentiation of such cell-like spots. There it is more necessary to consider chemical reaction dynamics within each spot which are more complex than those adopted in the GS model. Also, in order for spots to continue replication and differentiation, their division process and their internal complex chemical dynamics must somehow organize a proper relationship. We consider a model with such sufficiently complex dynamics and study how this relationship is organized. In so doing, we find the connection between self-replicating spots and cell differentiation in dynamical systems models assuming the cell itself[4, 5, 6].