A mathematical formalism to quantify drug-target residence time

Despite drug-target residence time (RT) is a key topic in binding kinetics, little information exists on its theoretical quantification. The two most frequent mathematical expressions found in the literature correspond to two particular and simple pharmacological cases: the binary ligand-receptor complex and the induction-fit model. In this article, we propose a mathematical formalism to obtain an expression of RT that can be of general applicability. RT is calculated from the system of ordinary differential equations (ODE) obtained by applying the Law of Mass Action to the selected chemical process. Then, a subsystem is constructed by defining which chemical species are of interest and omitting their global formation processes. RT maintains its accepted definition of 1/k_off, where k_off is here defined as the absolute value of the smallest-modulus eigenvalue of the subsystem. The proposed procedure is successfully used to derive RT for a wide variety of pharmacological cases. In particular, the theoretical expressions of RT obtained for binary ligand-receptor binding and induction-fit coincide with those previously found in the literature. An extension of the RT pharmacological framework is proposed by including the concept of relaxation time (RXT), which involves pharmacological conditions associated with receptor activation rather than receptor binding. To conclude, the herein presented formalism for RT and RXT provides a mathematical framework that can be of general applicability in many pharmacological systems. It is expected that the procedure may be helpful in different pharmacological areas such as binding kinetics, PK/PD and enzymology.