Sound of an axon's growth

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Sound of an axon's growth

Frederic Folz, Lukas Wettmann, Giovanna Morigi, and Karsten Kruse

Phys. Rev. E 99, 050401(R) – Published 1 May 2019

Abstract

Axons are linear structures of nerve cells that can range from a few tens of micrometers up to meters in length. In addition to external cues, the length of an axon is also regulated by unknown internal mechanisms. Molecular motors have been suggested to generate oscillations with an axon-length-dependent frequency that could be used to measure an axon's extension. Here, we present a mechanism for determining the axon length that couples the mechanical properties of an axon to the spectral decomposition of the oscillatory signal.

Received 12 July 2018
Revised 2 November 2018

DOI:https://doi.org/10.1103/PhysRevE.99.050401

Physics Subject Headings (PhySH)

Cell growth Cell signaling

Physics of Living Systems

Authors & Affiliations

Frederic Folz, Lukas Wettmann, and Giovanna Morigi

Theoretische Physik, Universität des Saarlandes, 66041 Saarbrücken, Germany

Karsten Kruse

NCCR Chemical Biology, Departments of Biochemistry and Theoretical Physics, University of Geneva, 1211 Geneva, Switzerland

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Issue

Vol. 99, Iss. 5 — May 2019

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Images

Figure 1
Illustration of axon length regulation. Motors transport signals $I$ and $O$ along microtubules at velocity $v$ , respectively, from the soma to the growth cone and vice versa. Activation of $O$ transport in the growth cone by $I$ and suppression of $I$ at the soma by $O$ generate oscillations with a frequency depending on the axon length $L$ . The total influx of $I$ comprises both a constant $J_{0}$ corresponding to the maximal influx of motors and the inhibitory effect of $O$ . Furthermore, the signal $I$ also stimulates actin polymerization leading to growth cone extension, as well as inhibition of the response $R$ . In turn, $R$ induces actin network contraction leading to growth cone retraction. The extending and contracting actin networks are drawn separately for clarity, but can be colocalized. Lines with arrowheads indicate activation, and lines with blunt ends inhibition.
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Figure 2
Two-step transformation of a periodic signal into a frequency-dependent average response, Eqs. (2) and (4). (a) Dynamics of the response $c_{R}$ as a function of time and corresponding mean value $〈 c_{R} 〉$ for two different frequencies of a sinusoidal oscillatory signal $O$ , $c_{O} (t) = {\bar{J}}_{R} (1 + sin (2 π t / T)) / 2$ . (b) Response average value $〈 c_{R} 〉$ as a function of the oscillatory signal frequency for two different Hill coefficients. The solid line represents the analytic expression Eq. (3) with $t_{\times}$ replaced by $t_{>}$ . Parameter values are $J_{R} = J_{I} = 55 \times 10^{- 5} μ m^{- 1} s^{- 1}$ , $κ = 2 \times 10^{- 2} μ m^{- 1}$ , $γ = 10^{- 2} s^{- 1}$ , as well as $n = 4$ (a) and $n = 4$ (dotted line with circles) and $n = 50$ (solid line with circles) in (b).
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Figure 3
Motor-induced oscillations for fixed axon length. (a) Oscillation of the incoming and outgoing signals $I$ and $O$ , respectively, obtained by solving Eqs. (5) and (6). (b) Frequency of the oscillations generated by Eqs. (5) and (6). The solid line is obtained from Eq. (9). Parameter values as in Fig. 2 and $J_{0} = J_{O} = J_{I}$ , $γ_{O} = γ_{I} = γ$ as well as $n = 4$ (a) and $n = 4$ (squares) and $n = 50$ (circles) in (b).
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Figure 4
Axon length dynamics. (a) Solution to the dynamic equations (2), (5), (6), and (10). Parameter values are $J_{R} = 26 \times 10^{- 5} μ m^{- 1} s^{- 1}$ , $J = 55 \times 10^{- 5} μ m^{- 1} s^{- 1}$ , $κ = 2 \times 10^{- 2} μ m^{- 1}$ , $κ_{R} = 5 \times 10^{- 3} μ m^{- 1}$ , $γ = 10^{- 2} s^{- 1}$ , $v_{g} = 0.1 μ m^{2} s^{- 1}$ , $v_{s} = 0.5 μ m^{2} s^{- 1}$ , and $n = 4$ . (b) Dependence of the mean length on $J_{R}$ for different values of the Hill coefficient $n$ . (c) Dependence of the mean length on $J_{0}$ for different values of the coupling constant $J_{R}$ and $J_{I} = J_{0}$ . The other parameters are as in (a), and $L_{min}$ is calculated from Eq. (7) with $J_{I} = 55 \times 10^{- 5} μ m^{- 1} s^{- 1}$ .
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