Diffusion-controlled reaction

Diffusion-controlled (or diffusion-limited) reactions are reactions that occur so quickly that the reaction rate is the rate of transport of the reactants through the reaction medium (usually a solution).^[1] As quickly as the reactants encounter each other, they react. The process of chemical reaction can be considered as involving the diffusion of reactants until they encounter each other in the right stoichiometry and form an activated complex which can form the product species. The observed rate of chemical reactions is, generally speaking, the rate of the slowest or "rate determining" step. In diffusion controlled reactions the formation of products from the activated complex is much faster than the diffusion of reactants and thus the rate is governed by diffusion.

Diffusion control is rare in the gas phase, where rates of diffusion of molecules are generally very high. Diffusion control is more likely in solution where diffusion of reactants is slower due to the greater number of collisions with solvent molecules. Reactions where the activated complex forms easily and the products form rapidly are most likely to be limited by diffusion control. Examples are those involving catalysis and enzymatic reactions. Heterogeneous reactions where reactants are in different phases are also candidates for diffusion control.

One classical test for diffusion control is to observe whether the rate of reaction is affected by stirring or agitation; if so then the reaction is almost certainly diffusion controlled under those conditions.

Applications in biology

The theory of diffusion-controlled reaction was originally utilized by R.A. Alberty, G.G. Hammes, and Manfred Eigen to estimate the upper limit of enzyme-substrate reaction ^{[2] [3]} According to their estimation ^[2][3], the upper limit of enzyme-substrate reaction was 10⁹ / Msec.

In 1972, it was observed that in the dehydration of H₂CO₃ catalyzed by carbonic anhydrase, the second-order rate constant obtained experimentally was about $1.5 \times 10^{10}/\text{Msec}$, ^[4] which was one order of magnitude higher than the upper limit estimated by Alberty, Hammes, and Eigen based on a simplified model^[2][3]. However, after taking into account the spatial factor and force field factor between the enzyme and its substrate, Kuo-Chen Chou and co-workers found that the upper limit could reach 10¹⁰ / Msec, ^{[5] [6] [7]} and can be used to explain some surprisingly high reaction rates in molecular biology. ^{[4] [8] [9]}

The new upper limit found by Chou et al. for enzyme-substrate reaction was further confirmed by a series of follow-up studies (see, e.g., ^{[10] [11]}). A detailed comparison between the simplified Alberty-Hammes-Eigen’s model and the Chou’s model was elaborated in the paper. ^[12]

References

1. ^ Atkins, Peter (1998), Physical Chemistry (6th ed.), New York: Freeman, pp. 825–828 
2. ^ ^{a b c} Alberty R. A., Hammes G. G. (1958) Application of the theory of diffusion-controlled reactions to enzyme kinetics. J. Phys. Chem. 62, 154-159.
3. ^ ^{a b c} Eigen M., Hammes G. G. (1963) Elementary steps in enzyme reactions (as studies by relaxation spectrometry). Advances In Enzymology and Related Subjects of Biochemistry 25, 1-38.
4. ^ ^{a b} Koening S. H., Brown R. D. (1972) H₂CO₃ as substrate for carbonic anhydrase in the dehydration of H2CO3-. Proc Natl Acad Sci USA 69, 2422-2425.
5. ^ Chou K. C., Jiang S. P. (1974) Studies on the rate of diffusion-controlled reactions of enzymes. Scientia Sinica 17, 664-680.
6. ^ Kuo-Chen Chou (1976) The kinetics of the combination reaction between enzyme and substrate. Scientia Sinica 19, 505-528.
7. ^ Li T. T., Chou K. C. (1976) The quantitative relations between diffusion-controlled reaction rate and characteristic parameters in enzyme-substrate reaction system: 1. Neutral substrate. Scientia Sinica 19, 117-136.
8. ^ Riggs A. D., Bourgeois S., Cohn M. (1970) The lac repressor-operator interaction. 3. Kinetic studies. Journal of Molecular Biology 53, 401-17.
9. ^ Kirschner K., Gallego E., Schuster I., Goodall D. (1971) Co-operative binding of nicotinamide-adenine dinucleotide to yeast glyceraldehyde-3-phosphate dehydrogenase. I. Equilibrium and temperature-jump studies at pH 8-5 and 40 degrees C. Journal of Molecular Biology 58, 29-50.
10. ^ Chou K. C., Zhou G. P. (1982) Role of the protein outside active site on the diffusion-controlled reaction of enzyme. Journal of American Chemical Society 104, 1409-1413.
11. ^ Zhou G. Z., Wong M. T., Zhou G. Q. (1983) Diffusion-controlled reactions of enzymes. An approximate analytic solution of Chou's model. Biophys Chem 18, 125-32.
12. ^ Zhou G. Q., Zhong W. Z. (1982) Diffusion-controlled reactions of enzymes. A comparison between Chou's model and Alberty-Hammes-Eigen's model. Eur J Biochem 128, 383-7.