Michaelis-Menten kinetics in shear flow: Similarity solutions for multi-step reactions

Models for chemical reaction kinetics typically assume well-mixed conditions, in which chemical compositions change in time but are uniform in space. In contrast, many biological and microfluidic systems of interest involve non-uniform flows where gradients in flow velocity dynamically alter the effective reaction volume. Here, we present a theoretical framework for characterizing multi-step reactions that occur when an enzyme or enzymatic substrate is released from a flat solid surface into a linear shear flow. Similarity solutions are developed for situations where the reactions are sufficiently slow compared to a convective time scale, allowing a regular perturbation approach to be employed. For the specific case of Michaelis- Menten reactions, we establish that the transversally averaged concentration of product scales with the distance x downstream as x^5/3. We generalize the analysis to n-step reactions, and we discuss the implications for designing new microfluidic kinetic assays to probe the effect of flow on biochemical processes.