Force-Regulated State Transitions of Growing Axons

Growing axons are one-dimensional active structures that are important for wiring the brain and repairing nerves. However, the biophysical mechanisms underlying the complex kinetics of growing axons remain elusive. Here, we develop a theoretical framework to recapitulate force-regulated states and their transitions in growing axons. We demonstrate a unique negative feedback mechanism that defines four distinct kinetic states in a growing axon, whose transitional boundaries depend on the interplay between cytoskeletal dynamics and axon-substrate adhesion. A phase diagram for axonal growth is formulated based on two dimensionless numbers.

Figure 1

(a) Schematics of an axon extending from soma depicting cytoplasmic regions and cytoskeletal structures, including axoplasm (AP) and cortical membrane (CM) of actin filaments (F-actin). (b) The rod model for a growing axon. (c) The mechanical response of the axon is depicted by a growth dashpot (blue element) connected to an actomyosin contractile element (red element) in series. The axon-substrate adhesion is simplified as a friction dashpot (black element). (d) The elongation rate $\dot{\delta }(s)$ depends on the tension $f(s)$, where $T_{\mathrm{c}}$ is the characteristic tension of actomyosin motors.

Figure 2

Growth kinetics of the axon. (a) Schematics of two distinct tension regimes of growth kinetics. (b) Our theoretical results agree with velocities $v$ of the normal (control) and ciliobrevin D (CilD)-treated axons of chick sensory neurons [28, 41]. (c) Tip velocity $v_t$ as a function of the towing force $T$ in two cases: axon-substrate adhesion and debonding. Inset: details for the case of adhesion. (d) Phase diagram of the state transition defined by ${\tilde{c}}_0$ and $\tilde{G}$. Heat map: dimensionless rest tension ${\tilde{T}}_r$. The black dashed line represents the stalling state. Two regions are distinguished to represent collapse and growth states.

Figure 3

Kinetic features of the growing axon. Spatiotemporal kinetics of (a) tension $f$ and (b) velocity $v$ under a reference towing force, 1.4 nN, during the axonal growth. (c) Stress contour at different ages. (d) Tip velocity $v_t$ vs length $L$.

Figure 4

Effects of axon-substrate adhesion and cytoskeletal properties on the kinetic behaviors. (a)–(c) Velocity $v$ for different $\tilde{\zeta }$, $\tilde{G}$, and ${\tilde{c}}_0$, respectively. (d)–(f) Rest tension $T_r$ and kinetic characteristic length $\ell$ versus $\tilde{\zeta }$, $\tilde{G}$, and ${\tilde{c}}_0$, respectively. Other parameters: $L=200\mu \mathrm{m}$, $\mathrm{T}=1.4\mathrm{nN}$.

Figure 5

Four kinetic states with transitional boundaries governed by two dimensionless numbers. (a) Phase diagram of the growth kinetics, obtained by examining the dimensionless velocity $\tilde{v}$ and tension $\tilde{f}$ relation. Parameter space $(\tilde{G},{\tilde{c}}_0,\tilde{\zeta })$ is split by two transitional boundaries [Eqs. (10) and (11)]. (b) $\tilde{v}\text{−}\tilde{f}$ relations of four kinetic states: (I) linear growth, (II) nonlinear growth, (III) linear collapse, and (IV) nonlinear collapse. Parameters used in the calculations are included in Table S1 of the Supplemental Material [40].