# Diffusion-controlled reaction rates for two active sites on a sphere

- David E Shoup
^{1}Email author

**7**:3

https://doi.org/10.1186/2046-1682-7-3

© Shoup; licensee BioMed Central Ltd. 2014

**Received: **29 January 2014

**Accepted: **27 May 2014

**Published: **4 June 2014

## Abstract

### Background

The diffusion-limited reaction rate of a uniform spherical reactant is generalized to anisotropic reactivity. Previous work has shown that the protein model of a uniform sphere is unsatisfactory in many cases. Competition of ligands binding to two active sites, on a spherical enzyme or cell is studied analytically.

### Results

The reaction rate constant is given for two sites at opposite ends of the species of interest. This is compared with twice the reaction rate for a single site. It is found that the competition between sites lowers the reaction rate over what is expected for two sites individually. Competition between sites does not show up, until the site half angle is greater than 30 degrees.

### Conclusions

Competition between sites is negligible until the site size becomes large. The competitive effect grows as theta becomes large. The maximum effect is given for theta = pi/2.

## Background

## Methods

Where D is the diffusion coefficient of the ligands and R is the radius of the sphere. The accuracy of the constant flux boundary condition may be seen, for small θ˳ (e.g. small binding sites), by considering a reactive disk in an insulating plane. For this problem, the exact solution is known [6]. It is k_{DC} = 4 Da, where a = the disk radius. The constant flux method yields [6] k_{DC} = 3.7 Da. Thus the constant flux boundary condition is accurate for small reactive sites.

*P*

_{ m }(x) are Legendre polynomials of order m. Application of the boundary condition given by equation (2) yields

The coefficients a_{2m} in equation (10) are found using equations (5) and (6). Details, parallel a previous derivation [6].

## Results and discussion

In this section we present the results and discuss how the method represents a step forward in the field of diffusion-controlled kinetics and biophysics in general.

The results are given by the solution of the model in the preceding section, and by comparing it to twice the rate limited constant for the one site problem [6]. The difference between the two problems, gives a measure of competition between the two sites for reaction with ligands.

*k2*(θ˳) (the rate constant for 2 sites), twok1 (θ˳) (twice the rate constant for 1 site)

*and k*1 (θ˳) (the rate constant for one site [6]) versus θ

_{°}. For small θ˳, the two site rate constant and 2 × the single site rate constant are in agreement with each other, as would be expected(both behaving as two active sites on a large sphere). As θ˳ grows above 30 degrees, the curves grow apart. The two site curve, being less than 2 × the one site curve.

The difference between the curves is a measure of the competition between the two sites. The competition effect does not show up till around 30 degrees. For $\mathrm{\theta}\circ =\frac{\mathit{\pi}}{2},\mathrm{k}2=1$, which is the exact result. Thus, the constant flux boundary condition, is good for large θ˳.

This paper represents progress in the field, by presenting a new model for the interpretation of experimental data. This is for macromolecule-ligand binding reactions that fall in the diffusion-controlled regime. Previously, only one site models were available for modeling proteins. Proteins with multiple binding sites can now be studied.

## Conclusions

The analytical expression for the diffusion-limited rate constant to two active sites on a sphere has been given. The result was used to study the competitive effects between the two sites. The effect doesn’t show up until the site half-angles is greater than 30 degrees. The competitive effect grows until its maximum value is reached at θ˳ = π/2.

## Declarations

### Acknowledgements

I would like to thank Attila Szabo, of the NIH, for suggesting this problem.

## Authors’ Affiliations

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