Slow-fast dynamics in non-linear enzyme cascades gives rise to spatial multiscaling.

Slow-fast dynamics in a well-mixed biological system is known to be associated with functional modularity with temporal hierarchy of processes. Yet, the significance of slow-fast dynamics in spatially distributed reaction- diffusion systems remains to be clarified. We assessed such systems with slow-fast dynamics, such as blood coagulation and predator-prey models to identify spatial patterns with independent parametric regulation, and performed their reaction-rate based sensitivity analysis with the help of computational one-dimensional reaction-diffusion models. In blood coagulation, spatial distribution of fibrin had two scales: a larger region with a high and almost constant fibrin concentration (corresponding to solid blood clot) and a narrow transition zone, where fibrin concentration decayed to zero (semi-liquid clot). Slow variables (e.g., the concentration of the coagulation factor IXa) regulated the clot size (large scale), while fast variables (e.g., concentrations of thrombin and factor Xa) regulated the thickness of the gel-to-liquid transition zone (small scale). Analysis of the simplified model of blood coagulation, composed only of factor X and IX activation, and of the predator-prey model confirmed this finding to be sufficiently general, and revealed that the limitation of fast enzyme production (due to precursor depletion or to kinetics with saturation) was crucial for this multi-scaling. Independent regulation of spatial scales also required species with slow kinetics to be upstream from species with fast kinetics. The shared slow-fast dynamics of the examined systems endow them with the ability to form spatial patterns, representing a new mechanism complementing the Turing-type and the Positional Information type pattern formation.