TThe sound of an axon’s growth
Frederic Folz, Lukas Wettmann, and Giovanna Morigi
Theoretische Physik, Universit¨at des Saarlandes, 66041 Saarbr¨ucken, Germany
Karsten Kruse
NCCR Chemical Biology, Departments of Biochemistry and Theoretical Physics,University of Geneva, 1211 Geneva, Switzerland (Dated: April 18, 2019)Axons are linear structures of nerve cells that can range from a few tens of micrometers up tometers in length. In addition to external cues, the length of an axon is also regulated by unknowninternal 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 presenta mechanism for determining the axon length that couples the mechanical properties of an axon tothe spectral decomposition of the oscillatory signal.
In order to assure proper function, the size of a biolog-ical system typically needs to be regulated. There is cur-rently no general understanding of the underlying mech-anisms. Best studied are processes based on gradientsfor setting the extension of linear structures. Prominentexamples are provided by the length of cytoskeletal fila-ments [1–8] and the extension of antiparallel microtubuleoverlaps [9–11]. In addition, length-dependent mechani-cal forces can play a role as has been suggested for set-ting the size of stereocilia [12]. Finally, oscillations areinvolved in positioning the division plane of some bacte-ria [13, 14] and thereby determine the size of the daugh-ter cells. They may be viewed as a specific case of cavityresonances of chemical waves that have been proposedto provide a general mechanism for size determination ofbiological structures [15].Axons are linear structures along which electrical sig-nals emanating from the body (soma) of a nerve cellare transported to other cells. The length of an axoncan vary from a few micrometers up to meters. It isset in part by extrinsic mechanisms, for example, stretchgrowth: axons that have connected to other cells are ex-periencing mechanical tension as the organism is grow-ing, which induces axonal extension [16]. Prior to makingcontact with other cells and driven by a structure calledthe growth cone, axons extend at their tip. Growth conesare guided in part by external physical [17] and chemicalcues [18]. In addition, there are intrinsic mechanisms toset the axon length that are notably used in early stagesof organismal development [19].It has been shown that transport by molecular mo-tors – with kinesins moving from the soma to the axontip and cytoplasmic dynein in the opposite direction –is essential for intrinsic axonal length regulation [20–22]. However, the precise role of the motors in this pro-cess is currently unknown. Similar to motor-dependentlength-regulation of filaments [1–4, 6], one possibilityis that they generate a gradient along the axon. Yet,a gradient-based mechanism is unlikely to operate overmore than a few micrometers in cells, whereas in devel-
I IO O v v R J soma growthconeaxon L FIG. 1. (color online) Illustration of axon length regulation.Motors transport signals I and O along microtubules at veloc-ity v , respectively, from the soma to the growth cone and viceversa . Activation of O -transport in the growth cone by I andsuppression of I at the soma by O generate oscillations with afrequency depending on the axon length L . The total influx of I comprises both a constant J corresponding to the maximalinflux of motors and the inhibitory effect of O . Furthermore,the signal I also stimulates actin polymerization leading togrowth cone extension, as well as inhibition of the response R . In turn, R induces actin network contraction leading togrowth cone retraction. The extending and contracting actinnetworks are drawn separately for clarity, but can be colocal-ized. Lines with arrow heads indicate activation/stimulation,lines with blunt ends inhibition. oping embryos, axons up to a few hundred micrometerscan emerge [19]. Furthermore, decreased motor concen-trations would imply shortened axons. However, the op-posite is observed [20, 22]. These considerations have ledto the proposal that interactions between motors mov-ing in opposite directions along the axon generate an os-cillating signal with a length-dependent frequency [20].Indeed, if kinesins transported some signaling moleculeto the tip, where it initiated dynein-mediated transportof another signalling molecule that in turn stopped theoriginal chemical signal at the soma, then the concen-trations of these factors at the growth cone and at thesoma can oscillate [23], see Fig. 1. For motors with aconstant velocity the associated period would increaseproportional to the axon length. However, it is not clear a r X i v : . [ q - b i o . S C ] A p r how this frequency-dependence could be used for lengthregulation. One proposition is that there is a networkgenerating a frequency-dependent average of some sig-naling molecule coupled to a switch turning off furtheraxonal growth as a certain concentration threshold andtherefore length is reached [24].The actual growth dynamics of an axon regulated bylength-dependent oscillations has not yet been studied.Here, we propose a mechanism for this regulation thatcouples the oscillations to the axon’s mechanical proper-ties. Axons are known to be under mechanical tension:on one hand, the growth cone pulls on the axon [25], onthe other hand, the axon itself generates contractile me-chanical stress by the actin cytoskeleton [26, 27]. Theregulation of cytoskeletal stresses has been suggested togenerate bouts of elongation and retraction [28]. The os-cillations can be analyzed through differential equationswith delayed feedback [23].We start from these equations and explicitly includethe axon growth dynamics, obtaining a set of equationswith a state-dependent delay. We show that our mecha-nism, rather than reading out the oscillation frequency,exploits the information contained in the signal’s spec-tral composition for regulating axon growth and length.Notably, we find regions in parameter space where a re-duction of the motor concentration leads to an increasein the final axon length, consistent with experimentalfindings [20, 22].Let us begin our discussion by noting that, typically,the cellular response to a chemical signal shows a sig-moidal dose-response curve [29]. It is often given in termsof a Hill function f κ ( c ) = c n κ n + c n (1)with Hill coefficient n and half-concentration κ . For anoscillating chemical signal, the average response is in-dependent of the signal’s period T : Let c I denote theconcentration of an incoming signaling molecule with c I ( t + T ) = c I ( t ). The response c R is then given by˙ c R = J R (1 − f κ ( c I )) − γ R c R , (2)where J R is the coupling constant between c I and c R and γ R is the response’s decay rate. In the following we willscale the concentrations by κ and denote them by thesame symbols as before. The average response is then (cid:104) c R (cid:105) ≡ T (cid:82) T c R ( t ) d t = ¯ J R (1 − (cid:104) f ( c I ) (cid:105) ), which is scaledby the dimensionless parameter ¯ J R ≡ J R / ( γ R κ ). It holdsthat T (cid:82) T f ( c I ( t )) d t = βT (cid:82) T/β f ( c I ( βt )) d t for any β > (cid:104) c R (cid:105) is independent of the oscillationperiod.This general result can be illustrated in the limit n → ∞ , when the Hill function turns into a Heavisidefunction, and one can determine the time dependence of c R explicitly. Let the input signal be some oscillatory FIG. 2. (color online) Two-step transformation of a pe-riodic signal into a frequency-dependent average response,Eqs. (2) and (4). a) Dynamics of the response c R as afunction of time and corresponding mean value (cid:104) c R (cid:105) fortwo different frequencies of a sinusoidal oscillatory signal O , c O ( t ) = ¯ J R (1 + sin(2 πt/T )) /
2. b) Response average value (cid:104) c R (cid:105) as a function of the oscillatory signal frequency for twodifferent Hill coefficients. Full line represents the analytic ex-pression Eq. (3) with t × replaced by t > . Parameter valuesare J R = J I = 55 × − µ m − s − , κ = 2 × − µ m − , γ = 10 − s − as well as n = 4 (a) and n = 4 (dotted line withcircles) and n = 50 (full line with circles) in panel (b). function with exactly one minimum and one maximumper period T and let t = 0 and t = t × be the times, whenthe input signal equals the threshold, such that c I ( t ) > ≤ t < t × and c I ( t ) < t × ≤ t < T . Then theaverage is given by (cid:104) c R (cid:105) = ¯ J R (cid:18) − t × T (cid:19) . (3)Apart from the coupling constant ¯ J R , (cid:104) c R (cid:105) only dependson the fraction of the period during which the incomingsignal c I is larger than 1, but is independent of the period.Whereas (cid:104) c R (cid:105) does not depend on the frequency of c I ,the form, i.e., the spectrum of c R does. In turn, (cid:104) c R (cid:105) typically depends on the spectrum of c I . This is eas-ily seen if f κ is a Heaviside function as in the previousparagraph: As long as min c I < κ < max c I , a change inthe spectrum of c I typically changes the fraction of time c I > κ and hence (cid:104) c R (cid:105) . Consequently, by first transform-ing the frequency variation of a signal O into a variationof the shape of the incoming signal I via a process simi-lar to Eq. (2) and then reading out its shape via Eq. (2),variations in frequency can be ultimately turned into avariation of the average value of the response R .To give a specific illustration of this mechanism, con-sider an oscillatory signal O , which feeds into I through˙ c I = J I (1 − f κ ( c O )) − γ I c I , (4)where J I is the coupling constant between O and I and γ I is the decay rate of the incoming signal. Furthermore,we used the same sigmoidal function f κ in Eqs. (2) and(4), but our general results do not rely on these specificchoices. We will also choose J I = J R and γ I = γ R ≡ γ to not blur the general mechanisms by a multitudeof parameters. As above we will scale the densities by κ . As anticipated, the incoming signal I responds tofrequency changes in O by variations of its shape, seeFig. 2a. Furthermore, the average value of the eventualresponse R varies with the frequency of O , see Fig. 2b.In case f κ is a Heaviside function, the average of c R as afunction of the period T of c O takes the same form as Eq.(3) with the replacement t × → t > . Here, t > is the lengthof the time interval during which c I >
1, where t > > J I >
1. The value of t > is determined by atranscendent equation, which we omit here. The averageresponse increases with T and (cid:104) c R (cid:105) → ¯ J R (1 − t > /T ) for n → ∞ .Having established a mechanism for reading out thefrequency of the incoming signal, we now return to axonlength regulation. As mentioned in the introduction, mo-tors moving in opposite directions and carrying motor-transport activating or inhibiting signals can generateaxon-length dependent oscillations. Specifically, let I de-note a signal released from the soma and transported intothe growth cone. Its concentration at the growth cone isdenoted by c I ( t ). In addition to eliciting the response R ,it also triggers transport of an outgoing signal O fromthe growth cone to the soma. There, O suppresses fur-ther transport of I . The concentration of O at the somais denoted by c O ( t ). This dynamics can be captured asin Ref. [23] by the following delay-differential equations˙ c I ( t ) = J − J I f κ ( c O ( t − τ )) − γ I c I ( t ) (5)˙ c O ( t ) = J O f κ ( c I ( t − τ )) − γ O c O ( t ) . (6)The delay τ = L/v accounts for the time motors need totransport cargo along an axon of length L at a velocity v ,which we assume for simplicity to be the same for bothkinds of motors. Note that for τ = 0, Eq. (5) is thesame as Eq. (4) if J = J I . The incoming signal decaysat rate γ I at the growth cone and the outgoing signalat rate γ O at the soma. J is the maximal incomingflux of signal I and J I ≤ J as well as J O denote therespective coupling constants between O and I . To avoidunnecessary complications, we consider again the samesigmoidal function f κ as in Eqs. (2) and (4) and focus onthe case γ I = γ O ≡ γ and J = J I = J O ≡ J .For fixed length L , a linear stability analysis of thestationary state c I, and c O, of Eqs. (5) and (6) showsthat the system undergoes a Hopf-bifurcation when theparameters fulfil ¯ J sin (cid:16) γτ (cid:112) ¯ J − (cid:17) = 1 , (7)where ¯ J = √ αJ/ ( κγ ) and α = (cid:104) f (cid:48) ( c O ) (cid:105)(cid:104) f (cid:48) ( c I ) (cid:105) [23]. Inparticular, f (cid:48) ( x ) = df ( x ) /dx and the mean values aretaken at the instability point c I = c ∗ I , c O = c ∗ O , cf. Sup-plementary Material (S.M.). This equation can only befulfilled if ¯ J >
1. The linear stability analysis also showsthat there is a minimal axon length L min = vτ min , belowwhich the system does not oscillate, which is in agree-ment with numerical solutions of Eqs. (5) and (6), see FIG. 3. (color online) Motor-induced oscillations for fixedaxon length. a) Oscillation of the incoming and outgoingsignals I and O , respectively, obtained by solving Eqs. (5) and(6). b) Frequency of the oscillations generated by Eqs. (5) and(6). The full line is obtained from Eq. (9). Parameter valuesas in Fig. 2 and J = J O = J I , γ O = γ I = γ as well as n = 4(a) and n = 4 (squares) and n = 50 (circles) in panel (b). Fig. 3a. The frequency ω min at this critical axon lengthis finite and fulfilscot ( ω min τ min ) = ω min γ . (8)Remarkably, this relation between the axon length andthe oscillation frequency also determines the frequency ofthe full nonlinear oscillations, see Fig. 3b. As a functionof the axon length L it is approximately given by ω ∼ (cid:113) Lv (cid:0) L v + γ − (cid:1) . (9)For Lγ (cid:29) v , we have ω ≈ √ v/L , which is the solutionof a wave equation with a rescaled sound velocity. In theopposite limit, the frequency scales as ω ≈ (cid:112) vγ/L .Whereas the previous analysis was for a fixed axonlength, we will now consider L to be a dynamic variable.The axon length is regulated by two processes: an exten-sion of the growth cone and a shortening of the axon dueto contractile stresses [26, 27]. The growth cone movesforward by a process similar to mesenchymal cell migra-tion on a flat substrate: Extension of the leading edge isdriven by the polymerization of actin, which is anchoredto a large actin network and thus able to exert protrudingforces on the membrane. The protrusion velocity is regu-lated by various processes. We assume here that chemicalregulation through signal I dominates and write for theprotrusion velocity v g c I . We notably neglect an effect ofmembrane tension on the protrusion velocity [30].The contractile stresses that are generated by molecu-lar motors in the axon and the growth cone can be cap-tured phenomenologically by a term ζ ∆ µ [31]. Here,∆ µ ≡ µ ATP − µ ADP − µ P , where µ ATP is the chem-ical potential of Adenosine-triphosphate, µ ADP that ofAdenosine-diphosphate, and µ P that of inorganic phos-phate, such that ∆ µ is the chemical energy liberated dur-ing an event of ATP-hydrolysis. The phenomenologicalcoefficient ζ describes the coupling of the liberated chem-ical energy to the mechanical stress generated. For con-tractile stresses ζ <
0. The phenomenological coefficient ζ depends on regulatory signals [32]. In particular, weassume ζ ≡ ζ ( c R ). Contractile stresses tend to reducethe distance between the cell body and the growth cone.We assume the cell body to be anchored to the substrate,such that contractile stresses are balanced by dissipativeforces as the growth cone retracts. The latter can be writ-ten as ξ ˙ x gc = ζ ∆ µ , where x gc denotes the position of thegrowth cone and which we identify with the axon length L . Assuming a linear dependence of ζ on c R , ζ = ζ c R ,and adding the growth cone protrusion velocity to thevelocity due to contraction, we arrive at˙ L = v g c I − v s c R , (10)where v s = − ζ ∆ µ/ξ >
0. The response R still dependson I through Eq. (2). We will scale the length by L min and concentrations by κ , while keeping the same nota-tion for the axon length as well as for the growth andshrinkage velocities. Furthermore, we take γ R = γ .In Figure 4a, we present an example of the solutionto the dynamic equations (2), (5), (6), and (10). Start-ing from length zero, the length increases and eventuallyoscillates with an amplitude that is less than 1% of theaverage length. The final average length decreases withan increasing coupling parameter J R , Fig. 4b. This isbecause an increase of the coupling between the incom-ing signal and actomyosin contractility will increase thelatter, which opposes the extension of the axon. A sim-ilar effect is observed when reducing the actomyosin ac-tivity, which is generally achieved by decreasing ζ ∆ µ .Importantly, in Fig. 4c it is visible that for sufficientlysmall values of J the stationary length increases withdecreasing J , that is with decreasing kinesin motor con-centration. This is consistent with experimental find-ings [20, 22]. Only beyond a certain critical value of J the average final length increases with an increasing mo-tor concentration [33]. An increase of the axon lengthshas been also observed when the dynein concentration isreduced [20]. Accordingly, our model shows an increasein the stationary length, when the parameter J O is re-duced, see S.M.We have checked that the solutions are stable againstfluctuations of the delay time up to 10%, see S.M. Wealso found that lossy transport of the signals along theaxon does not qualitatively change the system’s behavior,as we show in the S.M.From a more general point of view, our system achieveslength regulation through an adaptive delay, which is de-termined by the axon length. Adaptive delays have alsobeen proposed as a mechanism to retrieve informationfrom chaotic neural networks [34]. It is also interest-ing to compare our mechanism to Laughlin’s proposalof regulating lengths by resonant chemical waves [15],where he exploits a formal analogy of excitable systems FIG. 4. (color online) Axon length dynamics. a) Solutionto the dynamic equations (2), (5), (6), and (10). Parametervalues are J R = 26 × − µ m − s − , J = 55 × − µ m − s − , κ = 2 × − µ m − , κ R = 5 × − µ m − , γ = 10 − s − , v g = 0 . µ m s − , v s = 0 . µ m s − and n = 4. b) Depen-dence of the mean length on J R for different values of theHill coefficient n . c) Dependence of the mean length on J for different values of the coupling constant J R and J I = J .The other parameters are as in (a), L min is calculated fromEq. (7) with J I = 55 × − µ m − s − . with an effective amplifying and saturable medium, likein lasers or electronic circuits. Even though a mappingfrom the present model to reaction-diffusion equations isnot evident, let us exploit the similarity of the term oflength growth and shrinkage in Eq. (10) with gain andloss terms of a laser. As the system tunes its length, iteventually reaches a state in which the gains on averageequal the losses. 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