Effective reaction rates for diffusion-limited reaction cycles.

Biological signaling often relies on reaction cycles involving opposing enzymes. These cycles operate in two-dimensional environments like cell membranes. Prior research has shown that such environments can limit diffusion. However, the impact of diffusion on effective reaction rates remains unclear. This paper addresses how diffusion affects macroscopic reaction rates in crowded membranes. The study focuses on a generic cycle involving phosphorylation and dephosphorylation. Existing models do not account for the influence of enzyme concentration on motility. No prior work had resolved how motility and enzyme density interact. This gap motivated the development of a new framework for estimating effective reaction rates. The approach combines microscopic rate constants with motility estimates.

The goal is to derive effective macroscopic reaction rate coefficients for a reaction cycle in two-dimensional environments. The study aims to quantify how diffusion and crowding influence these rates. The researchers focus on the phosphorylation-dephosphorylation cycle as a model system. The approach integrates microscopic reaction rates with motility estimates. The study tests whether the derived formulas can predict steady-state concentrations. The motivation stems from the need to understand how diffusion affects signaling in membranes. The work builds on prior knowledge of enzyme-substrate interactions. The study fills a gap in modeling diffusion-limited reaction cycles.

The researchers begin with microscopic bimolecular reaction rate constants. They use estimates of the mean first-passage time for enzyme-substrate encounters. The method derives effective macroscopic reaction rate coefficients (EMRRC). The approach incorporates motility as a function of enzyme concentration. The study applies a logarithmic function to account for crowding effects. The researchers validate their formulas using kinetic Monte Carlo simulations. The simulations take place on a two-dimensional triangular lattice. The method allows for single-molecule resolution in the model.

Each EMRRC is half of the harmonic average of the microscopic rate constant and motility. The motility term is divided by a logarithmic function of enzyme concentration. The study shows that EMRRCs scale differently when phosphorylation and dephosphorylation rates differ. The steady-state fraction of phosphorylated substrate becomes diffusion-dependent. The simulations confirm the analytical predictions. The model predicts a 10% to 90% shift in phosphorylated substrate with changing diffusivity. The results apply to various microscopic reaction rates and concentrations. The formulas estimate steady-state concentrations accurately.

The authors demonstrate that diffusion affects the effective reaction rates in two-dimensional environments. The derived formulas account for both microscopic rates and motility. The study shows that the steady-state fraction of phosphorylated substrate depends on diffusion. The results suggest that motility and enzyme concentration interact in a non-linear way. The researchers validate their approach with kinetic Monte Carlo simulations. The study provides a framework for modeling diffusion-limited reaction cycles. The findings may help explain how signaling is regulated in crowded membranes. The authors propose that their method can be applied to other reaction cycles.