Abstract
We investigate how the size, the number, and the spatial arrangement of identical nonoverlapping reactive patches on a sphere influence the overall reaction kinetics of bimolecular diffusion-limited (or diffusion-controlled) reactions that occur between the patches and the reactants diffusing around the sphere. First, in the arrangement of two patches, it is known that the overall rate constant increases as the two patches become more separated from each other but decreases when they become closer to each other. In this work, we further study the dependence of the patch arrangement on the kinetics with three and four patches using the finite element method (FEM). In addition to the patch arrangement, the kinetics is also dependent on the number and size of the patches. Therefore, we study such dependences by calculating the overall rate constants using the FEM for various cases, especially for large-sized patches, and this study is complementary to the kinetic studies that were performed by Brownian dynamics (BD) simulation methods for small-sized patches. The numerical FEM and BD simulation results are compared with the results from various kinetic theories to evaluate the accuracies of the theories. Remarkably, this comparison indicates that our theory, which was recently developed based on the curvature-dependent kinetic theory, shows good agreement with the FEM and BD numerical results. From this validation, we use our theory to further study the variation of the overall rate constant when the patches are arbitrarily arranged on a sphere. Our theory also confirms that to maximize the overall rate constant, we need to break large-sized patches into smaller-sized patches and arrange them to be maximally separated to reduce their competition.
Highlights
In reaction kinetics, competition effects have been a central theme in understanding the details of reaction processes and in calculating accurate physical quantities, such as rate coefficients [1]
Overall, our formula and the boundary homogenization (BH) formula give better results than others in a comparison of all the formulas except the LBW formula in the Šolc-Stockmayer, curved reactive surface (CRS), and BP models, the Zwanzig formula only gives the best agreement with the Brownian dynamics (BD) simulation performed by Northrup for the BP model [27]
We investigate various cases, including those with large-sized patches and large total reactive area fractions, using a numerical method called the finite element method (FEM), which has been recently used for the study of a single reactive patch on a sphere [44]
Summary
Competition effects have been a central theme in understanding the details of reaction processes and in calculating accurate physical quantities, such as rate coefficients [1]. In a confined space or on a restricted surface area, we can speculate that the key to achieving the maximal overall rate constant is to arrange the reactants or the reactive patches in a way that they are maximally separated under a given constraint The latter case of a restricted surface area arises in a model of N reactive circular or curved disk-like patches on a sphere, which can be considered a simple model for ligand binding on a cell surface [4]. A theoretical study on this subject was initially carried out by Berg and Purcell [4] with their famous Berg-Purcell model They assumed that the patches were evenly distributed over a sphere and derived a simple formula for the overall rate constant as a function of the patch size, the number of patches, and the radius of the sphere.
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