In antagonistic interactions among organisms, often one organism “defends” itself by some toxic compound and the counterpart, in turn, responds by producing an enzyme that inactivates that compound [1]. In some cases, the former organism can respond by producing an inhibitor of that enzyme (counter-counter defence) [2,3]. An example is provided by cephalosporins, β-lactamases and clavulanic acid (an inhibitor of β-lactamases).

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 [3]. 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 predicts that only if the inhibition constant is below a threshold, it pays to have a counter-counter defence. This 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 β-lactam antibiotics and β-lactamase inhibitors, as well as for plant-herbivore and other molecular-ecological interactions.

References

[1] S. Dühring, …, S. Schuster: Host-pathogen interactions between the human innate immune system and Candida albicans - Understanding and modeling defense and evasion strategies. Front. Microbiol. 6 (2015) 625.

[2] J. Ewald, …, S. Schuster, B. Ibrahim: Trends in mathematical modeling of host-pathogen interactions. Cell. Mol. Life Sci. 77 (2020) 467–480.

[3] S. Schuster, …, S. Dühring: Optimizing defence, counter-defence and counter-counter defence in parasitic and trophic interactions - A modelling study. arXiv: 1907.04820 (2019)

Biosketch

Stefan Schuster graduated in biophysics from Humboldt University, Berlin in 1986 and obtained his Ph.D in 1988 from the same university (supervisor: Reinhart Heinrich). He did postdocs in Bordeaux and Amsterdam, returned to Humboldt University in 1993 and worked at Max Delbrück Center, Berlin-Buch, funded by a Heisenberg grant from 1998-2003. Since 2003, he holds a chair in Bioinformatics at the University of Jena. His research interests include the modelling of metabolic networks and host-pathogen interactions, evolutionary game theory and other topics in Theoretical Systems Biology.