Anti-cooperativity in diffusion-controlled reactions with pairs of anisotropic domains: a model for the antigen-antibody encounter

The encounter between anisotropic agents in diffusion-controlled reactions is a topic of very general relevance in chemistry and biology. Here we introduce a simplified model of encounter of an isotropic molecule with a pair of partially reacting agents and apply it to the encounter reaction between an antibody and its antigen. We reduce the problem to the solution of dual series relations, which can be solved iteratively, yielding the exact solution for the encounter rate constant at any desired order of accuracy. We quantify the encounter effectiveness by means of a simple indicator and show that the two binding centers systematically behave in an anti-cooperative fashion. However, we demonstrate that a reduction of the binding active sites allows the composite molecule to recover binding effectiveness, in spite of the overall reduction of the rate constant. In addition, we provide a simple formula that enables one to calculate the anti-cooperativity as a function of the size of the binding site for any values of the separation between the two active lobes and of the antigen size. Finally, some biological implications of our results are discussed.