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In host-pathogen interactions, often the host (attacked organism) defendsitself by some toxic compound and the parasite, in turn, responds by producingan enzyme that inactivates that compound. In some cases, the host can respondby producing an inhibitor of that enzyme, which can be considered as acounter-counter defence. An example is provided by cephalosporins,beta-lactamases and clavulanic acid (an inhibitor of beta-lactamases). Here, wetackle the question under which conditions it pays, during evolution, toestablish a counter-counter defence rather than to intensify or widen thedefence mechanisms. We establish a mathematical model describing thisphenomenon, based on enzyme kinetics for competitive inhibition. We use anobjective function based on Haber's rule, which says that the toxic effect isproportional to the time integral of toxin concentration. The optimalallocation of defence and counter-counter defence can be calculated in ananalytical way despite the nonlinearity in the underlying differentialequation. The calculation provides a threshold value for the dissociationconstant of the inhibitor. Only if the inhibition constant is below thatthreshold, that is, in the case of strong binding of the inhibitor, it pays tohave a counter-counter defence. This theoretical prediction accounts for theobservation that not for all defence mechanisms, a counter-counter defenceexists. Our results should be of interest for computing optimal mixtures ofbeta-lactam antibiotics and beta-lactamase inhibitors such as sulbactam, aswell as for plant-herbivore and other molecular-ecological interactions and tofight antibiotic resistance in general.

Optimizing defence, counter-defence and counter-counter defence in parasitic and trophic interactions -- A modelling study