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). 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 collision frequency.
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 Robert A. Alberty, Gordon Hammes, and Manfred Eigen to estimate the upper limit of enzyme-substrate reaction. According to their estimation, the upper limit of enzyme-substrate reaction was 109 M−1sec−1.
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 × 1010 M−1sec−1, which was one order of magnitude higher than the upper limit estimated by Alberty, Hammes, and Eigen based on a simplified model.
To address such a paradox, Prof. Kuo-Chen Chou (周国城) and his co-workers proposed a model by taking into account the spatial factor and force field factor between the enzyme and its substrate and found that the upper limit could reach 1010 M−1sec−1, and can be used to explain some surprisingly high reaction rates in molecular biology.  The new upper limit found by Chou et al. for enzyme-substrate reaction was further discussed and analyzed by a series of follow-up studies.  
A detailed comparison between the simplified Alberty-Hammes-Eigen’s model (Figure 1a) and the Chou’s model (Figure 1b) in calculating the diffusion-controlled reaction rate of enzyme with its substrate, or the upper limit of enzyme-substrate reaction, was elaborated in the paper.
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