  In host-pathogen interactions, often the host (attacked organism) defends
itself by some toxic compound and the parasite, in turn, responds by producing
an enzyme that inactivates that compound. In some cases, the host can respond
by producing an inhibitor of that enzyme, which can be considered as a
counter-counter defence. An example is provided by cephalosporins,
beta-lactamases and clavulanic acid (an inhibitor of beta-lactamases). Here, we
tackle the question under which conditions it pays, during evolution, to
establish a counter-counter defence rather than to intensify or widen the
defence mechanisms. We establish a mathematical model describing this
phenomenon, based on enzyme kinetics for competitive inhibition. We use an
objective function based on Haber's rule, which says that the toxic effect is
proportional to the time integral of toxin concentration. The optimal
allocation of defence and counter-counter defence can be calculated in an
analytical way despite the nonlinearity in the underlying differential
equation. The calculation provides a threshold value for the dissociation
constant of the inhibitor. Only if the inhibition constant is below that
threshold, that is, in the case of strong binding of the inhibitor, it pays to
have a counter-counter defence. This theoretical prediction accounts for the
observation that not for all defence mechanisms, a counter-counter defence
exists. Our results should be of interest for computing optimal mixtures of
beta-lactam antibiotics and beta-lactamase inhibitors such as sulbactam, as
well as for plant-herbivore and other molecular-ecological interactions and to
fight antibiotic resistance in general.
