Fractional logarithmic susceptible-infected model. Definition and applications to the study of COVID-19 main protease

Abadias, Luciano; Estrada-Rodriguez, Gissell; Estrada, Ernesto

We propose a model for the transmission of perturbations across the amino acids of a protein represented as an interaction network. The dynamics consists of a Susceptible-Infected (SI) model based on the logarithmic Caputo fractional derivative. We find an approximate analytical solution for this model which represents an upper bound for the fractional SI dynamics on a network. This upper bound is expressed in terms of the Mittag-Leffler function of the adjacency matrix of the network of inter-amino acids interactions. We consider some network descriptors based on these Mittaf-Leffler matrix functions which account for the"circulability"and"transmissibility"of the perturbations across the residues of a protein. We then apply this model and descriptors to the analysis of the communication effects produced by inhibitors of the main protease of COVID-19. We find that the perturbation produced by strong inhibitors of the protease are propagated up to 60{\AA} away from the binding site, confirming the long-range nature of intra-protein communication. These findings may help to the design of drug candidates against this new coronavirus.