Flexible search for single-axon morphology during neuronal spontaneous polarization

Abstract

Polarization, a disruption of symmetry in cellular morphology, occurs spontaneously, even in symmetrical extracellular conditions. This process is regulated by intracellular chemical reactions and the active transport of proteins and it is accompanied by cellular morphological changes. To elucidate the general principles underlying polarization, we focused on developing neurons. Neuronal polarity is stably established; a neuron initially has several neurites of similar length, but only one elongates and is selected to develop into an axon. Polarization is flexibly controlled; when multiple neurites are selected, the selection is eventually reduced to yield a single axon. What is the system by which morphological information is decoded differently based on the presence of a single or multiple axons? How are stability and flexibility achieved? To answer these questions, we constructed a biophysical model with the active transport of proteins that regulate neurite growth. Our mathematical analysis and computer simulation revealed that, as neurites elongate, transported factors accumulate in the growth cone but are degraded during retrograde diffusion to the soma. Such a system effectively works as local activation-global inhibition mechanism, resulting in both stability and flexibility. Our model shows good accordance with a number of experimental observations.

Figure 1. Flexible search for single-axon morphology.

(A) In stage 2, a neuron has several short neurites of similar length. (B) A developing neuron enters stage 3 if one neurite spontaneously elongates and develops to an axon. This established polarity is stably maintained. (C) Sometimes multiple neurites accidentally elongate, but this unfavorable pattern is flexibly reversed, leading to the single-axon pattern (B).

Figure 2. Model of neuronal polarization.

(A) The model neuron consists of one soma, a well-mixed compartment, and several neurites, each of which is represented as a continuous cable compartment. Each neurite has a growth cone at the tip. Factor X is produced by gene expression in the soma and is actively transported to each growth cone. Factor X is degraded throughout the cell and diffuses along the neurite shafts. denotes the concentration of factor X in the soma at time t, and denotes the concentration of factor X at x µm from the neck of neurite i at time t. is the length of neurite i. (B) The axon is specified according to the activity of factor Y at the growth cone. Factor Y activity depends on the concentration of factor X but also has hysteresial behavior. Between the threshold concentrations of η and θ, the system exhibits bistability with one unstable and two stable states (on and off states). When factor Y is in the on state, the neurite elongates, and when factor Y is in the off state, the neurite shrinks.

Figure 3. Local activation-global inhibition.

(A) Relationship between neurite length and factor X concentration at the growth cone at steady state. This curve is mathematically obtained by Equation (4). (B) Relationship between neurite length and the proportion of factor X diffusing back to the soma without being degraded. This curve is mathematically obtained by Equation (5).

Figure 4. Logic of Neuronal Polarization.

(A) Expected time course of the somatic concentration of factor X during the establishment of polarization. The black line shows the time course when no neurites are elongated, mathematically obtained as solutions to Equation (9). The expected time courses with one and two elongated neurites are depicted by the blue and red dashed curves, respectively. The solid blue, red, cyan and purple lines depict the equilibrium concentrations when one, two, three and four neurites, respectively, elongate (100 µm), as obtained from Equation (6). The green line indicates the threshold at which factor Y is activated to the on state in the growth cone of a neurite of length L_o. (B) Activity of factor Y (upper panel) and concentration of factor X (lower panel) in the growth cone. The blue, red, cyan and purple lines in the lower panel indicate the concentration of factor X given as a function of neurite length when the somatic concentration is in equilibrium, as depicted by the solid blue, red, cyan and purple lines in (A), respectively. Points (a – f) indicate states when the concentration of factor X in growth cones is above (a) or below (b) the threshold θ, when the somatic concentration is in equilibrium as depicted by the solid blue (c, d) and red (e, f) lines in (A) and when the neurite length is infinite (c, e) and L_o (d, f). The green arrows in (A) and (B) indicate the time that elapses until the concentration of factor X in one growth cone reaches the threshold θ. The parameters used in this analysis were , , , , , , , , and . A neurite was assumed to elongate to 100 µm if factor Y switches to the on state in the corresponding growth cone.

Figure 5. Computer simulations.

The biophysical model is simulated under single-axon conditions, i.e., the threshold η is between the points indicated by (c) and (e) in Figure 4 (). The rates of neurite growth and shrinkage are set to and , respectively. (A–D) Typical simulation results showing a direct transition from an initial state to a single-axon state, depicted as a transition from state (A) to state (B) in Figure 1. (E–H) Typical simulation result showing an indirect transition from an initial state to a single-axon state via a two-axon state, depicted as a transition from state (A) to state (B) via state (C) in Figure 1. Time courses of the concentration of factor X in the soma (A, E), at growth cones (B, F), state of factor Y (C, G) and neurite lengths (D, H) are shown. Note that four curves with different colors are plotted in (B–D, F–H) to correspond to four neurites, although distinguishing them in some cases is difficult. The other parameters used in this simulation were the same as those in Figure 4.