Quasi-steady state approximation for PDE

Multiscale systems are of the most encountered problems in natural as well as social sciences. They are systems in which different components contribute to the behaviour of the whole system in a different time scale, i.e. they might happen very fast/slow relatively to the other components. In many situations, one can utilise these different time scales to reduce the complexity of the system and consequently study the reduced problem efficiently. This idea is commonly called quasi-steady-state-approximation (QSSA) and has been founded at the beginning of the twentieth century for the case of enzyme reactions in biology. While the reduction procedure is mostly intuitive, rigorous mathematical proofs are significantly important to ensure its accuracy as well as to estimate the error of approximation. This is particularly challenging when spatial heterogeneity is taken into account. The main goal of this project is to rigorously investigate the QSSA for spatially heterogeneous systems which are modelled by partial differential equations. One novelty is the combination of two approaches: functional-analytic technique and geometric technique for fast-slow systems. We are going to start with rigorous derivation of the Michaelis-Menten kinetic, one of the most frequently used kinetics when it comes to enzyme reactions, from the law of mass action in a heterogeneous medium. The next step is to investigate structural assumptions for the validity of QSSA for general models, that is to answer the question under which conditions we can apply the QSSA to a problem without obtaining an inaccurate reduced system?. The potential success of this project will, in our expectation, significantly advance the theory of QSSA for PDE as well as provide more insights into its applications to practical problems.

Systems with multiple timescales-where different processes occur at (very) different rates-appear frequently in many realistic problems. This phenomenon has been investigated in the context of fast-slow systems for decades, yielding numerous geometrical properties. Recently, these problems have been also studied in the context of fast reaction limits for PDEs, revealing intriguing mathematical structures. In this project, we combine these two directions and provide an adequate geometrical theory for fast reaction limit problems in PDEs. One of the interesting results of this project is the derivation of Michaelis-Menten kinetics for enzyme reactions in the presence of diffusion. More precisely, it is proven that the modelling system contains a cross-diffusion effect, where the diffusion flux of the enzyme-complex mixture is influenced by the substrate concentration. This provides a meaningful model for enzymatic, or generally catalytic, reactions in the presence of diffusion.