title: Effective reaction rates for diffusion-limited reaction cycles.
abstract: Biological signals in cells are transmitted with the use of reaction cycles, such as the phosphorylation-dephosphorylation cycle, in which substrate is modified by antagonistic enzymes. An appreciable share of such reactions takes place in crowded environments of two-dimensional structures, such as plasma membrane or intracellular membranes, and is expected to be diffusion-controlled. In this work, starting from the microscopic bimolecular reaction rate constants and using estimates of the mean first-passage time for an enzyme-substrate encounter, we derive diffusion-dependent effective macroscopic reaction rate coefficients (EMRRC) for a generic reaction cycle. Each EMRRC was found to be half of the harmonic average of the microscopic rate constant (phosphorylation c or dephosphorylation d), and the effective (crowding-dependent) motility divided by a slowly decreasing logarithmic function of the sum of the enzyme concentrations. This implies that when c and d differ, the two EMRRCs scale differently with the motility, rendering the steady-state fraction of phosphorylated substrate molecules diffusion-dependent. Analytical predictions are verified using kinetic Monte Carlo simulations on the two-dimensional triangular lattice at the single-molecule resolution. It is demonstrated that the proposed formulas estimate the steady-state concentrations and effective reaction rates for different sets of microscopic reaction rates and concentrations of reactants, including a non-trivial example where with increasing diffusivity the fraction of phosphorylated substrate molecules changes from 10% to 90%.
authors.affiliation: College of Inter-Faculty Individual Studies in Mathematics and Natural Sciences, University of Warsaw, Warsaw, Poland.
authors.forename: Paweł
authors.initials: P
authors.lastname: Nałęcz-Jawecki
authors.affiliation: College of Inter-Faculty Individual Studies in Mathematics and Natural Sciences, University of Warsaw, Warsaw, Poland.
authors.forename: Paulina
authors.initials: P
authors.lastname: Szymańska
authors.affiliation: Institute of Fundamental Technological Research, Polish Academy of Sciences, Warsaw, Poland.
authors.forename: Marek
authors.initials: M
authors.lastname: Kochańczyk
authors.affiliation: Institute of Applied Mathematics and Mechanics, University of Warsaw, Warsaw, Poland.
authors.forename: Jacek
authors.initials: J
authors.lastname: Miękisz
authors.affiliation: Institute of Fundamental Technological Research, Polish Academy of Sciences, Warsaw, Poland.
authors.forename: Tomasz
authors.initials: T
authors.lastname: Lipniacki
mesh_terms: D000255:Adenosine Triphosphate / Q000378:metabolism; D000818:Animals; D055162:Biocatalysis; D003198:Computer Simulation; D004058:Diffusion; D006801:Humans; D007700:Kinetics; D008954:Models, Biological; D009010:Monte Carlo Method; D010744:Phosphoric Monoester Hydrolases / Q000378:metabolism*; D010766:Phosphorylation; D011494:Protein Kinases / Q000378:metabolism*
chemical_list: D000255:Adenosine Triphosphate; D011494:Protein Kinases; D010744:Phosphoric Monoester Hydrolases