We present a simple theory that explains how surface curvature affects the reaction kinetics of diffusion-limited reactions on spherically curved surfaces. In this theory, we derive a quadratic equation under the conditions that the rate constant satisfies the Hill and Smoluchowski rate constants at the lowest and highest curvatures, respectively, and that at a certain intermediate curvature, there should be a maximum value of the rate constant, which was recently found in our previous work. We find that the result obtained from our theory is in good agreement with the corresponding one obtained from numerical calculation. In addition, we show that our theory can be directly applied to the Šolc-Stockmayer model of axially symmetric reactants, which can be considered as a spherical reactant with a single reaction site. Furthermore, we discuss using our theory to improve the formula for the rate constant in the Berg-Purcell ligand-binding model of a cell membrane covered by multiple receptors. Our simple theory yields insight into the effect of curvature on diffusion-influenced reactions and provides a useful formula for easily and quantitatively evaluating the curvature effect.

中文翻译：

---

扩散受限反应的曲率依赖性动力学理论及其在配体与具有多个受体的球体结合中的应用

我们提出了一个简单的理论，该理论解释了表面曲率如何影响球形曲面上扩散受限反应的反应动力学。在该理论中，我们得出一个二次方程，其条件是速率常数分别满足最低曲率和最高曲率下的Hill和Smoluchowski速率常数，并且在一定的中间曲率下，速率常数应为最大值，这是在我们之前的工作中最近发现的。我们发现，从我们的理论中获得的结果与从数值计算中获得的相应结果非常吻合。此外，我们证明了我们的理论可直接应用于轴对称反应物的Šolc-Stockmayer模型，该模型可被视为具有单个反应位点的球形反应物。此外，我们讨论了使用我们的理论来改进被多个受体覆盖的细胞膜的Berg-Purcell配体结合模型中速率常数的公式。我们的简单理论可以洞察曲率对扩散影响的反应的影响，并为轻松，定量地评估曲率效应提供有用的公式。