Hill equation and Hatze's muscle activation dynamics complement each other: enhanced pharmacological and physiological interpretability of modelled activity-pCa curves

- Steady-states of the muscle activation dynamics from Hatze are connected to the famous Hill equation with benefits for physiology, pharmacology and biomechanics.

In pharmacology, particularly receptor theory, the drug dose-effect relation of bio-active substances is frequently described by a sigmoidal function formulated by A.V. Hill. In biomechanics and muscle physiology then again, H. Hatze had elaborated a mathematical model for the stimulation- and length-dependent dynamics of the calcium-induced activation of mammalian skeletal muscle. Here, we prove that muscular activity-pCa curves described by the Hill equation and the equilibrium state predicted by Hatze's activation dynamics are equivalent. Thus, the exponent introduced by Hatze can be directly identified with its counterpart in the Hill equation, by which the former model gains further physiological interpretability. Conversely, the Hill constant can now be interpreted as a function of the fibre length, generally allowing for advanced Hill plots based on model ideas. We derive and examine the complementary relation of both model approaches, highlight the benefits of mutually viewing one approach from the perspective of the other, and address the physiology behind sigmoidal curves.