Diffusion-controlled reaction rates for two active sites on a sphere
© Shoup; licensee BioMed Central Ltd. 2014
Received: 29 January 2014
Accepted: 27 May 2014
Published: 4 June 2014
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.
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.
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.
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 . It is kDC = 4 Da, where a = the disk radius. The constant flux method yields  kDC = 3.7 Da. Thus the constant flux boundary condition is accurate for small reactive sites.
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 . The difference between the two problems, gives a measure of competition between the two sites for reaction with ligands.
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 , 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.
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.
I would like to thank Attila Szabo, of the NIH, for suggesting this problem.
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