Patch-distribution effect on diffusion-limited process in dilute suspension of partially active spheres

Abstract

The normalized overall rate constant, kp/kf for diffusion-limited processes in a dilute suspension of spheres, partially covered with active patches of varying distribution states, is studied with sped-up Brownian dynamic simulations. A dimensionless separation index Is is defined to quantify the characteristics of patch distribution on the sphere surfaces, with values of 0 and 1 corresponding to the states of the most compact and loosest patch distributions, respectively. Remarkably, the normalized overall rate constant is found to strongly correlate with the dimensionless separation index at fixed patch coverage fcover exhibiting a positive, linear relationship. In addition, the slope of the kp/kf vs Is line, a measure of sensitivity of kp/kf to variation in the separation state of the distributed patches, is found to depend on patch coverage and patch size. This sensitivity exhibits a maximum value with respect to an increase in patch coverage, and the magnitude of the maximum sensitivity decreases with increasing patch size. The patch coverage, at which the maximum sensitivity occurs, increases with increasing patch size.

Effects of the Size, the Number, and the Spatial Arrangement of Reactive Patches on a Sphere on Diffusion-Limited Reaction Kinetics: A Comprehensive Study

“…Specifically, in this work, we numerically solve the Smoluchowski equation using the FEM [47,48]. Alternatively, we may use a BD simulation method [26,29,30,33], another numerical method based on a stochastic equation equivalent to the Smoluchowski equation. However, since the FEM allows us to directly solve the Smoluchowski equation even with complicated geometric boundary conditions, we use the FEM.…”

“…In particular, 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]. This model, called the Berg-Purcell (BP) model, and its variant models have been studied with a various number of N patches [4,[26][27][28][29][30][31][32][33][34][35][36]. In fact, the competition-minimization problem in this N-patch model is similar to the one in finding the spatial arrangement of N electrons (or electron pairs) on a sphere in the Thomson problem [37,38] (or valence shell electron pair repulsion (VSEPR) theory [39]) giving the minimum potential energy.…”

“…One interesting finding from the study is that, as the patch size decreases, the rate constant increases and eventually can reach the Smoluchowski rate constant, which is obtained when the whole surface of the reactant is fully reactive [1,43]. Additionally, Wu and Lu investigated how the spatial distribution of patches, characterized by a quantity called the separation index, affects the overall reaction kinetics [30]. They defined the separation index as a measure of relative patch-patch distance.…”

“…However, since the previous comparison was made using BD simulations with small-sized patches and small reactive area fractions, and new theories and methods were recently proposed, it is necessary to extend the comparison to include cases with large-sized patches and new results from recently developed theories and other numerical methods. Specifically, in previous studies, relatively small-sized patches and small reactive area fractions σ were used: a/R = 0.063 with σ ≤ 0.25 in the work of Northrup [26]; a/R = 0.025, 0.05, 0.1, and 0.2 with σ ≤ 0.5 in the work of Berezkowskii et al [33]; a/R = 0.063 and 0.314 with σ ≤ 0.4 in the work of Wu and Lu [30], where a and R are the radii of a curved disk and sphere; and σ = 0.125 and 0.25 in the work of Lu [29]. In fact, in addition to BD simulations, recently sophisticated numerical methods, such as the spectral boundary element method with Zernike polynomials [36], have been developed for the BP model.…”

“…Berg and Purcell also discussed that the diffusive current (flux) to reactive patches for a random patch arrangement would be only slightly smaller than that for the uniform (symmetric) patch arrangement if the number of patches is sufficiently large [4]. Later, Wu and Lu also investigated the distributions of rate constants in BD simulations of random patch arrangements with two fixed patch sizes, and they found that the distributions of rate constants are very narrow for both cases [30], which is consistent with Northrup's results [26].…”