Field theory of enzyme-substrate systems with restricted long-range interactions

Enzyme-substrate kinetics form the basis of many biomolecular processes. The interplay between substrate binding and substrate geometry can give rise to long-range interactions between enzyme binding events. Here, we study a general model of enzyme-substrate kinetics with restricted long-range interactions described by an exponent $-\lambda$. We employ a coherent-state path integral and renormalization group approach to calculate the first moment and two-point correlation function of the enzyme-binding profile. We show that starting from an empty substrate the average occupancy follows a power law with an exponent $1/(1-\lambda)$ over time. The correlation function decays algebraically with two distinct spatial regimes characterized by exponents $-\lambda$ on short distances and $-(2/3)(2-\lambda)$ on long distances. The crossover between both regimes scales inversely with the average substrate occupancy. Our work allows to associate experimental measurements of bound enzyme locations with their binding kinetics and the spatial confirmation of the substrate.