Continuum mechanical modeling of axonal growth

Abstract

Axonal growth is a complex phenomenon in which many intra-and extra-cellular signals collaborate simultaneously. Two different compartments can be identified in the growing axon: the growth cone, the leading tip that guides and steers the axon, and the axonal shaft, connecting the soma to the growth cone. The complex relations between both compartments and how their interaction leads the axon to its final synaptic target remain a topic of intense scrutiny. Here, we present a continuum and computational model for the development of the axonal shaft. Two different regions are considered: the axoplasm, filled with microtubules, and the surrounding cortical membrane, consisting mainly of F-actin, Myosin II motor proteins and the membrane. Based on the theory of morphoelasticity, the deformation gradient is decomposed into anelastic and viscoelastic parts. The former corresponds to either a growth tensor for the axoplasm, or a composition of growth and contractile tensors for the cortical membrane. The biophysical evolution for the anelastic parts is obtained at the constitutive level, in which the polymerization and depolymerization of microtubules and F-actin drive the growth, while the contractility is due to the pulling exerted by the Myosin II on the F-actin and depends on the stress. The coupling between cytoskeletal dynamics and mechanics is naturally derived from the equilibrium equations. The framework is exploited in two representative scenarios in which an external force is applied to the axonal shaft either along the axis or off the axis. In the first case three states are found: growth, collapse and stall. In the second case, axonal turning is observed. This framework is suitable to investigate the complex relationship between the local mechanical state, the cytoskeletal polymerization/depolymerization rates, and the contractility of the cortical membrane in axonal guidance.

Modeling molecular mechanisms in the axon

“…The axonal cytoskeleton consists of longitudinally aligned microtubules and neurofilaments that are connected by a variety of different crosslinks [20]. The cytoskeleton is encapsulated by an actin cortex [27, 28] that is held together by spectrin [29,38,76]. At the tip of the axon, the growth cone is leading axonal growth and is responsible for path finding and axonal steering [48].…”

“…Myosin, in constrast, is a molecular motor that is located primarily in the actin cortex and in the growth cone where it generates contractile forces [14, 53, 80]. Force equilibrium and axonal elongation is a competition between the extensile forces of dynein and the compressive forces of myosin [28, 33, 66]. In addition to these actively force-generating motors, molecules like tau that passively crosslink the axonal cytoskeleton also play a critical role in network mechanics [82].…”

SoftFEM: The Soft Finite Element Method

“…The test cases are chosen to represent: a bioengineering problem (chosen here to be a simplified axonal growth) with a strong continuity of the outputs with respect to the inputs, ie, small variations of input values lead to small variations of output values; a plant sciences problem (chosen here to be a simplified plant cell wall shape study) with a “noise” behavior of the outputs; a materials engineering problem (chosen here to be a crystal plasticity problem) with a computing‐intensive complex interactions between inputs. …”

“…The constitutive model is linear elastic with arbitrary material properties. This model is coupled to a growth model where the growth deformation gradient tensor is defined by the following: where t is the time, G c is the growth rate parameter, and n 0 is the growth direction vector (see the work of Garcıa‐Grajales et al for more details). The rod is clamped at its base and pulled dynamically in the z‐direction on its top face at 0.2 m/s for 1 s. The top face is not allowed to move laterally in the y‐direction but is free to move in the x‐direction.…”

“…If the rod is pulled slower than it grows, the top face area is also expected to change: either increasing if the rod “fattens” (Δ A top > 0), decreasing because of intense buckling (Δ A top < 0), or an alternance between both (this behavior is a priori dependent on the aspect ratio of the rod). There should thus be an optimum G c allowing the top surface area to remain the same (or as close to as possible) after 1 s. The problem is run on OxFEMM (see Figure ).…”