Mathematical modelling of axonal cortex contractility

The axonal cortex is composed of a regular structure of F-actin and spectrin able to contract thanks to myosin II motors. Such an active tension is of fundamental importance in controlling the physiological shape of axons. Recent experiments show that axons modulate the contraction of the cortex when subject to mechanical deformations, exhibiting a non-trivial coupling between the hoop and the axial active tension. However, the underlying mechanisms are still poorly understood. In this paper, we propose a continuum model of the axon based on the active strain theory. Exploiting the Coleman-Noll procedure, we shed light on the coupling between the hoop and the axial active strain through the Mandel stress tensor. We propose a qualitative analysis of the system under the simplifying assumption of incompressibility, showing the existence of a stable equilibrium solution. In particular, our results show that the axon regulates the active contraction to maintain a homeostatic stress state. Finally, we propose numerical simulations of the model, using a more suitable compressible constitutive law. The outcomes are compared with experimental results, showing an excellent quantitative agreement. Statement of Significance. The mechanics of cortical contractility in axons is still poorly understood. Unraveling the mechanisms underlying axial and hoop stress generation in the cortex will give insight on the active regulation of axon diameter. The understanding of this phenomenon may shed new light on the physical causes of axonal morphological degeneration as a consequence of neurodegenerative diseases, viral infections, and traumatic brain injuries.