Cross-diffusion induced patterns for a single-step enzymatic reaction

Several different enzymes display an apparent diffusion coefficient that increases with the concentration of their substrate. Moreover, their motion becomes directed in substrate gradients. Currently, there are several competing models for these transport dynamics. Here, we use mathematical modeling and numerical simulations to analyze whether the enzymatic reactions can generate a significant feedback from enzyme transport onto the substrate profile. We find that this feedback can generate spontaneous spatial patterns in the enzyme distribution, with just a single-step catalytic reaction. However, patterns are formed only for a subclass of transport models. For such models, nonspecific repulsive interactions between the enzyme and the substrate, or attractive interactions between the enzyme and the product, cause the enzyme to accumulate in regions of low substrate concentration. Reactions then amplify local substrate and product fluctuations, causing enzymes to further accumulate where substrate is low. Experimental analysis of this pattern formation process could discriminate between different transport models.