Motivated by the operation of myogenic (self-oscillatory) insect flight
muscle, we study a model consisting of a large number of identical oscillatory
contractile elements joined in a chain, whose end is attached to a damped
mass-spring oscillator. When the inertial load is small, the serial coupling
favors an antisynchronous state in which the extension of one oscillator is
compensated by the contraction of another, in order to preserve the total
length. However, a sufficiently massive load can sychronize the oscillators and
can even induce oscillation in situations where isolated elements would be
stable. The system has a complex phase diagram displaying quiescent,
synchronous and antisynchrononous phases, as well as an unsual asynchronous
phase in which the total length of the chain oscillates at a different
frequency from the individual active elements.Comment: 5 pages, 4 figures, To appear in Phys. Rev. Let

A. Pikovsky

A. Vilfan

B. Alberts

D. A. Smith

D. Riveline

F. Jülicher

H. Fujita

J. Howard

J. W. S. Pringle

K. A. P. Edman

K. E. Machin

K. Yasuda

N. Okamura

R. K. Josephson

T. Anazawa

Y. Kuramoto

English

arXiv

Motivated by the operation of myogenic (self-oscillatory) insect flight muscle, we study a model consisting of a large number of identical oscillatory contractile elements joined in a chain, whose end is attached to a damped mass-spring oscillator. When the inertial load is small, the serial coupling favors an antisynchronous state in which the extension of one oscillator is compensated by the contraction of another, in order to preserve the total length. However, a sufficiently massive load can synchronize the oscillators and can even induce oscillation in situations where isolated elements would be stable. The system has a complex phase diagram displaying quiescent, synchronous and antisynchronous phases, as well as an unusual asynchronous phase in which the total length of the chain oscillates at a different frequency from the individual active elements

Vilfan, A.

Duke, T.

UCL Discovery

arXiv:physics/0307155v1  [physics.bio-ph]  31 Jul 2003Synchronization of active mechanical oscillators by an inertial loadAndrej Vilfan1,2, ∗ and Thomas Duke1, †1Cavendish Laboratory, Madingley Road, Cambridge CB3 0HE, UK2J. Stefan Instutute, Jamova 39, 1000 Ljubljana, Slovenia(Dated: February 2, 2008)Motivated by the operation of myogenic (self-oscillatory) insect ﬂight muscle, we study a modelconsisting of a large number of identical oscillatory contractile elements joined in a chain, whoseend is attached to a damped mass-spring oscillator. When the inertial load is small, the serialcoupling favors an antisynchronous state in which the extension of one oscillator is compensated bythe contraction of another, in order to preserve the total length. However, a suﬃciently massive loadcan sychronize the oscillators and can even induce oscillation in situations where isolated elementswould be stable. The system has a complex phase diagram displaying quiescent, synchronous andantisynchrononous phases, as well as an unsual asynchronous phase in which the total length of thechain oscillates at a diﬀerent frequency from the individual active elements.PACS numbers: 05.45.Xt,87.19.Ff,87.16.NnThe origin of movement in biological systems can fre-quently be traced to molecular motors — specialized pro-teins that convert chemical energy to mechanical work.Kinesin and dynein, which travel along microtubules, andmyosin, which pulls on actin ﬁlaments, are typical exam-ples [1, 2]. In many cases, such as muscle contraction orintracellular transport, molecular motors generate uni-directional motion. But there are also a number of phys-iological systems which incorporate motor proteins thatdisplay oscillatory dynamics. These include eucaryoticﬂagella and cilia whose undulation is driven by dyneinmolecules, and the ﬂight muscles of many insects whichcontract rythmically at a frequency that is out of stepwith the excitory neural impulses [3, 4, 5, 6, 7, 8]. Appar-ently these systems are self-oscillatory, and the dynamicalinstability that leads to vibration is directly generated bythe action of the motor proteins; it is usually attributedto delayed stretch activation, which can be caused bya variety of diﬀerent microscopic mechanisms (reviewedin Refs. [6] & [7]). Oscillations have also been observedin the sarcomeres of skeletal muscle in non-physiologicalconditions [9, 10, 11, 12] and in experiments that probethe interaction between individual dynein molecules andmicrotubules [13]. Even normal muscle ﬁbers can displaya damped oscillatory response to sudden changes in load[14].Theoretical analysis has demonstrated that a single ﬁl-ament interacting with an ensemble of motors can havean anomalous force-velocity relation [15, 16], wherebytwo diﬀerent sliding speeds, one positive and one neg-ative, can occur at a given load. Experimental conﬁr-mation of this phenomenon has been obtained in glid-ing motility assays for both actin- [17] and microtubule-based systems [18]. In such a situation the motors cancollectively generate oscillations when the ﬁlament is con-∗Andrej.Vilfan@ijs.si†td18@cam.ac.uk￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿￿Hopf Hopf Hopf M HopfFIG. 1: Model system: A chain of serially coupled active Hopfoscillators is attached to a damped inertial oscillator.nected in series with an elastic element [19, 20, 21], be-cause the solution for the sliding speed then switches pe-riodically between the two stable branches. However,direct application of this model to muscle ﬁbers, com-posed of hundreds of sarcomeres (contractile units) inseries, omits a crucial point: the oscillations will only bemacroscopically observable if there is at least some de-gree of synchrony between the oscillations of individualsarcomeres. How might this occur? One possibility isthat the activity of myosin motors in diﬀerent sarcom-eres is coordinated by some chemical signaling. An al-ternative suggestion is that torsion of actin ﬁlaments isinvolved [11]. In this Letter, we propose a mechanismof synchronization that does not rely on any such spe-ciﬁc molecular process. We investigate the dynamics of achain of active mechanical elements each of which, whenisolated, can undergo a dynamical instability from a qui-escent (stable) to an oscillatory (unstable) regime. Inthe case where each element is individually stable, weshow that the entire chain can be set into synchronizedvibration by the application of a suﬃciently massive in-ertial load at its end. In the alternative case where eachcontractile element is unstable, we demonstrate the ex-istence of a variety of dynamical regimes, including anunusual asynchronous state in which individual elementsoscillate at a faster frequency than the mass to whichthey are connected.The model system that we investigate is shown inFig. 1. Each contractile element in the chain is an ac-tive mechanical system whose displacement xi is couplednonlinearly to an internal variable yi (which might, forexample, be the fraction of bound motors, or the concen-2tration of a regulator). We suppose that an element mayeither stay still or generate self-sustained oscillations, de-pending on the value of a control parameter. In the vicin-ity of the critical point where the quiescent state becomesunstable, the dynamics may be described by the canoni-cal equation for a Hopf bifurcation. Writing the complexvariable zi = xi + iyi, we have˙ zi = (iω + ǫ)zi − B |zi|2 zi + F/ζ , (1)where ǫ is the control parameter and ω is the character-istic frequency of each active oscillator. F is the forceacting on each of the oscillators in the chain, due tothe coupled mass-spring system and is determined by itsequation of motion:F = −MN Xi=1Re￿¨ zi + Ω2zi + γ ˙ zi￿, (2)where M is the mass of the load, γ is a measure of thedamping and Ω is the natural frequency of the mass-spring system. In order to proceed further with the anal-ysis and obtain numerical solutions, we replace Eq. (2)with a ﬁrst-order diﬀerential equation by inserting ¨ zifrom the ﬁrst derivative of Eq. (1)˙ F = −ζNMF +ζNN Xi=1￿−ǫRe ˙ zi + ωIm ˙ zi+ B Red(z2i z∗i )dt− Ω2 Rezi − γ Re ˙ zi￿, (3)where the terms containing ˙ zi can be substituted fromEq. (1).These equations constitute a system of globally cou-pled oscillators [22, 23, 24, 25] (the oscillators interactwith each other via the single variable F). However, ourmodel diﬀers in a crucial way from classical models ofsynchronization, such as the Kuramoto model [26], inthat the coupling variable F is determined by a ﬁrst-orderdiﬀerential equation (Eq. 3), rather than as a function ofthe variables zi. By expressing all frequencies in termsof ω and amplitudes in terms ofpω/B, and assumingN ≫ 1, the number of model parameters is reduced tofour: ǫ/ω, Ω/ω, γ/ω and ζ/NMω.We start our analysis by considering the situation inwhich the active elements would be stable if isolated,ǫ < 0. With ǫ ﬁxed, the frequency of the inertial oscil-lator Ω acts as a control parameter for the system as awhole. We can identify a transition between a quiescentphase (for Ω < Ωc) in which the entire system is at rest,and a synchronized phase (for Ω > Ωc) in which all ofthe contractile elements and the massive load oscillatetogether (see Fig. 2). At the critical value Ω = Ωc thesystem undergoes a Hopf bifurcation. In its vicinity wecan use linear stability analysis, because the amplitudetends to zero there. In the synchronized phase, all oscil-lators have identical displacement zi ≡ z = x + iy andEqs. (1) & (3) form a system of three coupled diﬀerential-0.4 -0.2 0 0.2 0.4 0.6ε/ω0.00.20.40.60.81.0Ω/ωSQAntiAa)0 0.2 0.4 0.6ε/ω0.70.80.9Ω/ωSS+AntiAntiA S+Ab)FIG. 2: Phase diagram (Q quiescent; S synchronous; Antiantisynchronous; A asynchronous) as a function of the con-trol parameter ǫ of the active elements and the frequency Ωof the inertial oscillator. Other parameters have the valuesζ/NMω = 0.3125 and γ/ω = 0.2. The diagram was estab-lished by numerical solution of Eqs. (1) & (3). The phaseboundary of the synchronous state is given by Eq. (4) for ǫ < 0and can be determined by perturbation theory for ǫ > 0.equations for x, y and F. The characteristic equation forthe eigenvalues of the Jacobian reads λ3 −(ǫ −ζ/NM −γ)λ2+(Ω2−ǫγ−2ǫζ/NM)λ−Ω2ǫ+(ǫ2+ω2)ζ/NM = 0.The transition occurs when the real part of the complexeigenvalue pair changes sign, givingΩc =sω2/￿1 +NMγζ￿+ ǫ￿γ − ǫ + 2ζNM￿. (4)The frequency f of synchronized oscillations at the bi-furcation is given by the imaginary part of the complexeigenvalues:f =pω2/(1 + γNM/ζ) − ǫ2 . (5)Note that it is always the case that Ωc < f < ω. Awayfrom the bifurcation, Ω < Ωc, the synchronized oscilla-tions have a frequency lower than f, thus f representsthe maximal frequency of the system.3FIG. 3: The asynchronous state. (a) Phase φi = arg(zi) ofeach of the N = 128 active elements as a function of time.The phase velocity ˙ φi of a particular element is slow for anumber of periods of oscillation, then speeds up over a fewperiods, then slows again. The average frequency of the ac-tive elements is therefore faster than the speed of propaga-tion of the the high-density area in the phase space, whichis the frequency of the inertial load. (b) Total extensionx = Σi Rezi (in units Npω/B) as a function of time, il-lustrating the lower frequency of the inertial oscillator. Pa-rameters: ζ/NMω = 0.3125, γ/ω = 0.2, ǫ/ω = 0.6 andΩ/ω = 0.83.To interpret this result in terms of the underlying phys-ical model, we can consider two diﬀerent ways of adjust-ing the load. In the ﬁrst, illustrated in Fig. 2a, we ﬁxthe mass and the damping of the inertial oscillator (thusM = const, γ = const) and change the stiﬀness of thespring (a stiﬀer spring implies a higher frequency Ω andvice versa). Then a suﬃciently strong spring, such thatΩ > Ωc, will always maintain the stability of the system.But decreasing the stiﬀness can provoke a transition tosynchronized oscillations, provided that the control pa-rameter ǫ lies above some threshold value, determined bythe solution Ωc = 0 of Eq. (4) (i.e. the active elementsmust be suﬃciently close to their dynamical instability).We also note that the chain can oscillate synchronouslyeven when the inertial oscillator itself is overdamped(i.e. Q = Ω/γ < 1). In the second situation, we vary themass M of the inertial oscillator while keeping the springstiﬀness MΩ2 and the damping Mγ constant. Now weﬁnd that the system will remain quiescent if the mass issuﬃciently small, but that a larger mass can make thesystem oscillate provided that ǫ > −ω/p1 + NMγ/ζ.We continue the analysis by considering the situa-tion where the active elements are individually unstable,ǫ > 0, and would oscillate spontaneously if isolated. Inthis case the system displays a greater variety of phases,as indicated in Fig. 2. For small values of Ω, there is asynchronous phase as described above. For large valuesof Ω there is an antisynchronous phase: All of the active0 0.2 0.4 0.6 0.8 1Ω/ω0.00.20.40.60.81.01.2S AAntiFIG. 4: Frequency and amplitude of oscillations as a func-tion of Ω for a ﬁxed, positive value of the control param-eter (ǫ/ω = 0.6). These parameters correspond to movingup the right vertical axis of Fig. 2a. The system transitsfrom the synchronous (S), to the asynchronous (A), to theantisynchronous (Anti) phase. Black curves show the fre-quencies of the inertial load (continuous) and of the activeelements (dashed). Gray curves show the RMS amplitudeof the mass (continuous) in units Npω/B and that of eachactive element (dashed) in unitspω/B. Other parameters:ζ/NMω = 0.3125, γ/ω = 0.2.elements oscillate with the same frequency ω, but witha distribution of phases φi = arg(zi) such that the sumof their extensions is always zero; thus the massive loadremains stationary. When the inertial frequency Ω quiteclosely matches the characteristic frequency ω of the ac-tive oscillators, there is a remarkable phase in which theindividual elements oscillate at one frequency, while thetotal extension oscillates at a diﬀerent, lower frequency.We label this phase asynchronous. As shown in Fig. 3,active oscillators form clusters with diﬀerent phases φiand their phase velocities ˙ φi periodically slow down andspeed up. The number of oscillators in each cluster gen-erally decreases with increasing ǫ/ω and can be as smallas one, which is the case in the example shown in Fig. 3.The ensemble of oscillators resembles a set of vehicles ona congested ring road, which repeatedly enter a traﬃcjam. Because vehicles enter the jam at its rear end andleave at the front, the jam progresses more slowly thanthe average speed of an individual vehicle. In our model,the total extension depends on the phase of the major-ity of active oscillators, and the oscillation frequency ofthe inertial load therefore corresponds to the speed ofpropagation of the traﬃc jam.The dependencies on Ω of the frequencies of individ-ual elements, and of the total displacement, are shownin Fig. 4. Also shown are the corresponding amplitudesof motion. In the synchronous phase, both frequenciesare equal and the amplitude of the inertial load is Ntimes greater than that of each active element. In theasynchronous phase, the amplitude of the inertial oscil-4lator falls towards zero, while that of the active elementsapproachespǫ/B — the amplitude of spontaneous os-cillations for an isolated element. At the same time, thefrequency of the active oscillators approaches their char-acteristic frequency ω. In the antisynchronous phase thefrequency and amplitude of the active elements remainunchanging at these limiting values, and the inertial massdoes not move. For large values of ǫ all transitions aresecond order, and the system passes directly from thesynchronized, to the asynchronous, to the antisychronousphase as Ω increases. For small, positive values of ǫ thetransitions become ﬁrst order and there is a region inwhich the synchronous phase and the asynchronousphasecoexist as two metastable states. At intermediate valuesof ǫ, a complex asynchronous phase arises in which threeor more clusters of oscillators are synchronized withineach cluster, but the clusters are out of phase with eachother (see Fig. 2b).One experimental situation which appears to corre-spond to the regime ǫ > 0 is the spontaneous oscilla-tory contraction (SPOC) of skeletal muscle in conditionsof high ADP/ATP ratio. Oscillations of the length ofa myoﬁlament have been observed when its end was at-tached to a ﬂexible microneedle [11], and oscillations inthe length of individual sarcomeres have been seen whenthe total length was held ﬁxed [9]. These correspondto the synchronous and antisynchronous phases of ourmodel. Additionally a ‘metachronal’ phase, in which con-tractile waves propagate along a myoﬁlament, has beenobserved [10]. This cannot be explained by our model. Itwould require either chemical signaling between adjacentsarcomeres, or a gradient in one or more of the sarcomereproperties (e.g. the number of myosin molecules inter-acting with the thin ﬁlament) as suggested by Smith andStephenson [27].The mechanics of insect ﬂight muscle has been inves-tigated in detail by Machin and Pringle [4, 5]. Theyfound that a sudden increase in ﬁber length caused asubsequent rise in tension — the phenomenon known as‘delayed stretch activation’. Consequently a muscle sub-jected to a sinusoidal change of length produced net work[5]. They also observed that a muscle could be made tooscillate by attaching it to an inertial load [4], providedthat the damping was not too great. They suggested thatthe load must be resonant (i.e. Q = Ω/γ > 1) for oscilla-tions to occur, and found that the frequency of vibrationwas primarily determined by the inertia and elasticityof the load. All of these results are consistent with ourmodel in the regime ǫ < 0 and ζ/NMω ≪ 1. But wepredict that it is not in general necessary for the loadto be resonant; oscillations should be observable when-ever Ω is smaller than Ωc, given by Eq. (4). And wenote that the oscillation frequency generally depends onthe characteristic frequency ω of the sarcomeres, as wellas on the nature of the inertial load. It is often statedthat the wings and thorax of insects provide a resonantload whose oscillation is maintained by energy suppliedby the ﬂight muscles [7, 8]. In the light of our investi-gation, it would be interesting to conduct experimentsin which the mass and damping of this load is modiﬁed,to verify whether the muscles can generate oscillations inthe absence of resonance.AcknowledgmentsWe have beneﬁted from discussions with Martin Fal-cke and thank Karsten Kruse for a critical reading ofthe manuscript. A.V. would like to acknowledge supportfrom the European Union through a Marie Curie Fellow-ship (No. HPMFCT-2000-00522) and from the SlovenianOﬃce of Science (Grant No. Z1-4509-0106-02).[1] B. Alberts, D. Bray, J. Lewis, M. Raﬀ, K. Roberts, andJ. D. Watson, Molecular Biology of the cell (GarlandPubl., New York, 1994), 3rd ed.[2] J. Howard, Mechanics of Motor Proteins and the Cy-toskeleton (Sinauer, Sunderland, MA, 2001).[3] These ﬂight muscles are variously called ‘myogenic’, ‘ﬁb-rillar’, or ‘asynchronous’; this last term refers to the factthat contraction is not directly related to the nerve im-pulses and has nothing to do with the synchronization ofsarcomere oscillations discussed in this Letter.[4] K. E. Machin and J. W. S. Pringle, Proc. R. Soc. Lond.B 151, 204 (1959).[5] K. E. Machin and J. W. S. Pringle, Proc. R. Soc. Lond.B 152, 311 (1960).[6] J. W. S. Pringle, Proc. R. Soc. Lond. B 201, 107 (1978).[7] M. H. Dickinson and M. S. Tu, Comp. Biochem. Physiol.116A, 223 (1997).[8] R. K. Josephson, J. G. Malamud, and D. R. Stokes, J.Exp. Biol. 203, 2713 (2000).[9] N. Okamura and S. Ishiwata, J. Muscle Res. Cell Motil.9, 111 (1988).[10] T. Anazawa, K. Yasuda, and S. Ishiwata, Biophys. J. 61,1099 (1992).[11] K. Yasuda, Y. Shindo, and S. Ishiwata, Biophys. J. 70,1823 (1996).[12] H. Fujita and S. Ishiwata, Biophys. J. 75, 1439 (1998).[13] C. Shingyoji, H. Higuchi, and T. Yanagida, Nature 393(1998).[14] K. A. P. Edman and N. A. Curtin, J. Physiol. 534, 553(2001).[15] F. J¨ ulicher and J. Prost, Phys. Rev. Lett. 75, 2618(1995).[16] A. Vilfan, E. Frey, and F. Schwabl, Europhys. Lett. 45,283 (1999).[17] D. Riveline, et al., Eur. Biophys. J. 27, 403 (1998).[18] S. A. Endow and H. Higuchi, Nature 406, 913 (2000).[19] F. J¨ ulicher and J. Prost, Phys. Rev. Lett. 78, 4510(1997).5[20] N. Thomas and R. A. Thornhill, J. Phys. D. 31, 253(1998).[21] F. J¨ ulicher, C.R. Acad. Sci. (Paris) IV pp. 849–860(2001).[22] A. Pikovsky, M. Rosenblum, and J. Kurths, Synchroniza-tion, A Universal Concept in Nonlinear Sciences (Cam-bridge Univeristy Press, Cambridge, 2001).[23] D. Golomb, D. Hansel, B. Shraiman, and H. Sompolin-sky, Phys. Rev. A 45, 3516 (1992).[24] V. Hakim and W.-J. Rappel, Phys. Rev. A 46, R7347(1992).[25] N. Nakagawa and Y. Kuramoto, Physica D 75, 74 (1994).[26] Y. Kuramoto, Chemical Oscillations, Waves, and Turbu-lence (Springer, Berlin, 1984).[27] D. A. Smith and D. G. Stephenson, J. Muscle Res. CellMotil. 15, 369 (1994).

Synchronization of active mechanical oscillators by an inertial load

Vilfan, A

Duke, T

Synchronization of Active Mechanical Oscillators by an Inertial LoadAndrej Vilfan1,2,* and Thomas Duke1,*1Cavendish Laboratory, Madingley Road, Cambridge CB3 0HE, United Kingdom2J. Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia(Received 20 December 2002; published 9 September 2003)Motivated by the operation of myogenic (self-oscillatory) insect ﬂight muscle, we study a modelconsisting of a large number of identical oscillatory contractile elements joined in a chain, whose end isattached to a damped mass-spring oscillator.When the inertial load is small, the serial coupling favorsan antisynchronous state in which the extension of one oscillator is compensated by the contraction ofanother, in order to preserve the total length. However, a sufﬁciently massive load can synchronize theoscillators and can even induce oscillation in situations where isolated elements would be stable. Thesystem has a complex phase diagram displaying quiescent, synchronous and antisynchronous phases, aswell as an unusual asynchronous phase in which the total length of the chain oscillates at a differentfrequency from the individual active elements.DOI: 10.1103/PhysRevLett.91.114101 PACS numbers: 05.45.Xt, 87.16.Nn, 87.19.FfThe origin of movement in biological systems canfrequently be traced to molecular motors—specializedproteins that convert chemical energy to mechanicalwork. Kinesin and dynein, which travel along microtu-bules, and myosin, which pulls on actin ﬁlaments, aretypical examples [1,2]. In many cases, such as musclecontraction or intracellular transport, molecular motorsgenerate unidirectional motion. But there are also a num-ber of physiological systems which incorporate motorproteins that display oscillatory dynamics. These in-clude eucaryotic ﬂagella and cilia whose undulation isdriven by dynein molecules, and the ﬂight muscles ofmany insects which contract rhythmically at a frequencythat is out of step with the excitatory neural impulses [3–8]. Apparently these systems are self-oscillatory, and thedynamical instability that leads to vibration is directlygenerated by the action of the motor proteins; it is usuallyattributed to delayed stretch activation, which can becaused by a variety of different microscopic mechanisms(reviewed in Refs. [6,7]). Oscillations have also beenobserved in the sarcomeres of skeletal muscle in non-physiological conditions [9–12] and in experiments thatprobe the interaction between individual dynein mole-cules and microtubules [13]. Even normal muscle ﬁberscan display a damped oscillatory response to suddenchanges in load [14].Theoreticalanalysishasdemonstratedthat a single ﬁla-ment interacting with an ensemble of motors can have ananomalous force-velocity relation [15,16], whereby twodifferent sliding speeds, one positive and one negative,can occur at a given load. Experimental conﬁrmation ofthis phenomenon has been obtained in gliding motilityassaysfor both actin- [17] and microtubule-based systems[18]. In such a situation the motors can collectively gen-erate oscillations when the ﬁlament is connected in serieswith an elastic element [19–21], because the solution forthe sliding speed then switches periodically between thetwo stable branches. However, direct application of thismodel to muscle ﬁbers, composed of hundreds of sarco-meres (contractile units) in series, omits a crucial point:the oscillations will be macroscopically observable onlyif there is at least some degree of synchrony between theoscillations of individual sarcomeres. How might thisoccur? One possibility is that the activity of myosinmotors in different sarcomeres is coordinated by somechemical signaling. An alternative suggestion is that tor-sion of actin ﬁlaments is involved [11]. In this Letter, wepropose a mechanism of synchronization that does notrely on any such speciﬁc molecular process. We in-vestigate the dynamics of a chain of active mechanicalelements each of which, when isolated, can undergo adynamical instability from a quiescent (stable) to an os-cillatory (unstable) regime. In the case where each ele-ment is individually stable, we show that the entire chaincan be set into synchronized vibration by the applicationof a sufﬁciently massive inertial load at its end. In thealternative case where each contractile element is un-stable, we demonstrate the existence of a variety of dy-namical regimes, including an unusual asynchronousstate in which individual elements oscillate at a fasterfrequency than the mass to which they are connected.Themodelsystemthat weinvestigateisshown in Fig.1.Each contractile element in the chain is an active me-chanical system whose displacement xi is coupled non-linearly to an internal variable yi (which might, forexample, be the fraction of bound motors, or the concen-tration of a regulator). We suppose that an element mayeither stay still or generate self-sustained oscillations,depending on the value of a control parameter. In thevicinity of the critical point where the quiescent stateHopf Hopf Hopf M HopfFIG. 1. Model system: A chain of serially coupled activeHopf oscillators is attached to a damped inertial oscillator.PHYSICAL REVIEW LETTERS week ending12 SEPTEMBER 2003 VOLUME 91, NUMBER 11114101-1 0031-9007=03=91(11)=114101(4)$20.00  2003 The American Physical Society 114101-1becomes unstable, the dynamics may be described by thecanonical equation for a Hopf bifurcation. Writing thecomplex variable zi   xi   iyi, we have_ z z i    i!     zi   Bjzij2zi   F= ; (1)where   is the control parameter and ! is the character-istic frequency of each active oscillator. F is the forceacting on each of the oscillators in the chain, due to thecoupled mass-spring system and is determined by itsequation of motion:F    MX Ni 1Re   z zi    2zi     _ z zi ; (2)where M is the mass of the load,   is a measure of thedamping, and   is the natural frequency of the mass-spring system. In order to proceed further with the analy-sis and obtain numerical solutions, we replace Eq. (2)with a ﬁrst-order differential equation by inserting   z zifrom the ﬁrst derivative of Eq. (1)_ F F    NMF   NX Ni 1    Re _ z zi   !Im _ z zi   BRed z2iz i dt   2 Rezi    Re _ z zi ; (3)where the terms containing _ z zi can be substituted fromEq. (1).These equations constitute a system of globallycoupled oscillators [22–25] (the oscillators interact witheach other via the single variable F). However, our modeldiffers in a crucial way from classical models of synchro-nization, such as the Kuramoto model [26], in that thecoupling variable F is determined by a ﬁrst-order differ-ential equation [Eq. (3)], rather than as a function of thevariables zi. By expressing all frequencies in terms of !and amplitudes in terms of          !=Bp, and assuming N   1,the number of model parameters is reduced to four:  =!, =!,  =!,a n d =NM!.We start our analysis by considering the situation inwhich the active elements would be stable if isolated, <0. With   ﬁxed, the frequency of the inertial oscil-lator   acts as a control parameter for the system as awhole. We can identify a transition between a quiescentphase (for   <  c) in which the entire system is at rest,and a synchronized phase (for   >  c) in which all ofthe contractile elements and the massive load oscillatetogether (see Fig. 2). At the critical value      c thesystem undergoes a Hopf bifurcation. In its vicinity wecan use linear stability analysis, because the amplitudetends to zero there. In the synchronized phase, all oscil-lators have identical displacement zi   z   x   iy,a n dEqs. (1) and (3) form a system of three coupled differ-ential equations for x, y,a n dF. The characteristic equa-tion for the eigenvalues of the Jacobian reads  3       =NM     2     2        2  =NM      2     2   !2  =NM   0. The transition occurs when the realpart of the complex eigenvalue pair changes sign, giving c                                                                                   !2  1  NM              2 NM  s: (4)The frequency f of synchronized oscillations at the bi-furcation is given by the imaginary part of the complexeigenvalues:f                                                   !2= 1    NM=    2q: (5)Note that it is always the case that  c <f<! . Awayfrom the bifurcation,   <  c, the synchronized oscilla-tions have a frequency lower than f, thus f represents themaximal frequency of the system.To interpret this result in terms of the underlyingphysical model, we can consider two different ways ofadjusting the load. In the ﬁrst, illustrated in Fig. 2(a), weﬁx the mass and the damping of the inertial oscillator-0.4 -0.2 0 0.2 0.4 0.6ε/ω0.00.20.40.60.81.0Ω/ωSQAntiAa)0 0.2 0.4 0.6ε/ω0.70.80.9Ω/ωSS+AntiAntiA S+Ab)FIG. 2. (a) Phase diagram (Q, quiescent; S, synchronous;Anti, antisynchronous; A, asynchronous) as a function of thecontrol parameter   of the active elements and the frequency  of the inertial oscillator. Other parameters have the values =NM!  0:3125 and  =!   0:2. The diagram was estab-lished by numerical solution of Eqs. (1) and (3). The phaseboundary of the synchronous state is given by Eq. (4) for  <0and can be determined by perturbation theory for  >0.(b) Magniﬁcation of the boxed region in (a).PHYSICAL REVIEW LETTERS week ending12 SEPTEMBER 2003 VOLUME 91, NUMBER 11114101-2 114101-2(thus M   const,     const) and change the stiffness ofthe spring (a stiffer spring implies a higher frequency  and vice versa). Then a sufﬁciently strong spring, suchthat  > c, will always maintainthe stability of the sys-tem. But decreasing the stiffness can provoke a transitionto synchronized oscillations, provided that the controlparameter   lies above some threshold value, determinedby the solution  c   0 of Eq. (4) (i.e., the active elementsmust be sufﬁciently close to their dynamical instability).We also note that the chain can oscillate synchronouslyeven when the inertial oscillator itself is overdamped (i.e.,Q    =  < 1). In the second situation, we vary the massM of the inertial oscillator while keeping the springstiffness M 2 and the damping M  constant. Now weﬁnd that the system will remain quiescent if the mass issufﬁciently small, but that a larger mass can make thesystem oscillate provided that  > !=                          1   NM = p.We continue the analysis by considering the situ-ation where the active elements are individually un-stable,  >0, and would oscillate spontaneously ifisolated. In this case the system displays a greater varietyof phases, as indicated in Fig. 2. For small values of  ,there is a synchronous phase as described above. For largevalues of   there is an antisynchronous phase: All of theactive elements oscillate with the same frequency !,b u twith a distribution of phases  i   arg zi  such that thesum of their extensions is always zero; thus the massiveload remains stationary. When the inertial frequency  quite closely matches the characteristic frequency ! ofthe active oscillators, there is a remarkable phase inwhich the individual elements oscillate at one frequency,while the total extension oscillates at a different, lowerfrequency.We label this phase asynchronous. As shown inFig. 3, active oscillators form clusters with differentphases  i and their phase velocities _    i periodicallyslow down and speed up. The number of oscillators ineach cluster generally decreases with increasing  =! andcan be as small as one, which is the case in the exampleshown in Fig. 3. The ensemble of oscillators resembles aset of vehicles on a congested ring road, which repeatedlyenter a trafﬁc jam. Because vehicles enter the jam at itsrear end and leave at the front, the jam progresses moreslowly than the average speed of an individual vehicle. Inour model, the total extension depends on the phase of themajority of active oscillators, and the oscillation fre-quency of the inertial load therefore corresponds to thespeed of propagation of the trafﬁcj a m .The dependencies on of the frequencies of individualelements, and of the total displacement, are shown inFig. 4. Also shown are the corresponding amplitudes ofmotion. In the synchronous phase, both frequencies areequal and the amplitude of the inertial load is N timesgreater than that of each active element. In the asynchro-nous phase, the amplitude of the inertial oscillator fallstowards zero, while that of the active elements approaches           =Bp—the amplitude of spontaneous oscillations for anisolated element. At the same time, the frequency ofthe active oscillators approaches their characteristic fre-quency !. In the antisynchronous phase the frequencyand amplitude of the active elements remain unchang-ing at these limiting values, and the inertial mass doesnot move. For large values of   all transitions aresecond order, and the system passes directly from the0 5 10 15 20 0π1π2πφ(zi)0 5 10 15 20ωt-0.4-0.200.20.4x(t)a)b)FIG. 3. The asynchronous state. (a) Phase  i   arg zi  ofeach of the N   128 active elements as a function of time.The phase velocity _    i of a particular element is slow for anumber of periods of oscillation, then speeds up over a fewperiods, then slows again. The average frequency of the activeelements is therefore faster than the speed of propagation of thehigh-density area in the phase space, which is the frequency ofthe inertial load. (b) Total extension x    i Rezi (in unitsN          !=Bp) as a function of time, illustrating the lower frequencyof the inertial oscillator. Parameters:  =NM!  0:3125, =!   0:2,  =!   0:6,a n d =!   0:83.0 0.2 0.4 0.6 0.8 1Ω/ω0.00.20.40.60.81.01.2S AAntiFIG. 4. Frequency and amplitude of oscillations as a functionof   for a ﬁxed, positive value of the control parameter ( =!  0:6). These parameters correspond to moving up the rightvertical axis of Fig. 2(a). The system transits from the syn-chronous (S), to the asynchronous (A), to the antisynchronous(Anti) phase. Black curves show the frequencies of the inertialload (continuous) and of the active elements (dashed). Graycurves show the rms amplitude of the mass (continuous) inunits N          !=Bpand that of each active element (dashed) in units           !=Bp. Other parameters:  =NM!  0:3125,  =!   0:2.PHYSICAL REVIEW LETTERS week ending12 SEPTEMBER 2003 VOLUME 91, NUMBER 11114101-3 114101-3synchronized, to the asynchronous, to the antisynchro-nous phase as   increases. For small, positive values of  the transitions become ﬁrst order and there is a region inwhich the synchronous phase and the asynchronous phasecoexist as two metastable states. At intermediate valuesof  , a complex asynchronous phase arises in which threeor more clusters of oscillators are synchronized withineach cluster, but the clusters are out of phase with eachother [see Fig. 2(b)].One experimental situation which appears to corre-spond to the regime  >0 is the spontaneous oscilla-tory contraction of skeletal muscle in nonphysiologicalconditions. Oscillations of the length of a myoﬁlamenthave been observed when its end was attached to a ﬂex-ible microneedle [11], and oscillations in the length ofindividual sarcomeres have been seen when the totallength was held ﬁxed [9]. These correspond to the syn-chronous and antisynchronous phases of our model.Additionally a ‘‘metachronal’’ phase, in which contractilewaves propagate along a myoﬁlament, has been observed[10]. This cannot be explained by our model. It wouldrequire either chemical signaling between adjacent sarco-meres or a gradient in one or more of the sarcomereproperties (e.g., the number of myosin molecules interact-ing with the thin ﬁlament) as suggested by Smith andStephenson [27].The mechanics of insect ﬂight muscle has been inves-tigated in detail by Machin and Pringle [4,5].They foundthat a sudden increase in ﬁber length caused a subsequentrise in tension—the phenomenon known as ‘‘delayedstretch activation.’’ Consequently, a muscle subjected toa sinusoidal change of length produced net work [5].Theyalso observed that a muscle could be made to oscillate byattaching it to an inertial load [4], provided that thedamping was not too great. They suggested that the loadmust be resonant (i.e., Q    =  > 1) for oscillations tooccur, and found that the frequency of vibration wasprimarily determined by the inertia and elasticity of theload. All of these results are consistent with our model intheregime <0 and =NM!  1. But we predict that itis not in general necessary for the load to be resonant;oscillations should be observable whenever   is smallerthan c, given by Eq. (4). And we note that the oscillationfrequency generally depends on the characteristic fre-quency ! of the sarcomeres, as well as on the nature ofthe inertial load. It is often stated that the wings andthorax of insects provide a resonant load whose oscilla-tion is maintained by energy supplied by the ﬂightmuscles [7,8]. In the light of our investigation, it wouldbe interesting to conduct experiments in which the massand damping of this load are modiﬁed, to verify whetherthe muscles can generate oscillations in the absence ofresonance.We have beneﬁted from discussions with MartinFalcke and thank Karsten Kruse for a critical reading ofthe manuscript. A.V. acknowledges support from theEuropean Union through a Marie Curie Fellowship(No. HPMFCT-2000-00522) and from the SlovenianOfﬁce of Science (Grant No. Z1-4509-0106-02).*Electronic address: andrej.vilfan@ijs.si†Electronic address: td18@cam.ac.uk[1] B. Alberts et al., Molecular Biology of the Cell (GarlandPublishing, New York, 1994), 3rd ed.[2] J. Howard, Mechanics of Motor Proteins and theCytoskeleton (Sinauer, Sunderland, MA, 2001).[3] These ﬂight muscles are variously called ‘‘myogenic,’’‘‘ﬁbrillar,’’ or ‘‘asynchronous’’; this last term refers tothe fact that contraction is not directly related to thenerve impulses and has nothing to do with the synchroni-zation of sarcomere oscillations discussed in this Letter.[4] K. E. Machin and J.W.S. Pringle, Proc. R. Soc. London,Ser. B 151, 204 (1959).[5] K. E. Machin and J.W.S. Pringle, Proc. R. Soc. London,Ser. B 152, 311 (1960).[6] J.W.S. Pringle, Proc. R. Soc. London, Ser. B 201,1 0 7(1978).[7] M.H. Dickinson and M.S. Tu, Comp. Biochem. Physiol.,Part A, Physiol. 116A, 223 (1997).[8] R.K. Josephson, J.G. Malamud, and D.R. Stokes, J. Exp.Biol. 203, 2713 (2000).[9] N. Okamura and S. Ishiwata, J. Muscle Res. Cell Motil. 9,111 (1988).[10] T. Anazawa, K. Yasuda, and S. Ishiwata, Biophys. J. 61,1099 (1992).[11] K. Yasuda, Y. Shindo, and S. Ishiwata, Biophys. J. 70,1823 (1996).[12] H. Fujita and S. Ishiwata, Biophys. J. 75, 1439 (1998).[13] C. Shingyoji et al., Nature (London) 393, 711 (1998).[14] K. A. P. Edman and N. A. Curtin, J. Physiol. 534, 553(2001).[15] F. Ju ¨licher and J. Prost, Phys. Rev. Lett. 75, 2618 (1995).[16] A. Vilfan, E. Frey, and F. Schwabl, Europhys. Lett. 45,283 (1999).[17] D. Riveline et al., Eur. Biophys. J. 27, 403 (1998).[18] S. A. Endow and H. Higuchi, Nature (London) 406,9 1 3(2000).[19] F. Ju ¨licher and J. Prost, Phys. Rev. Lett. 78, 4510 (1997).[20] N.Thomas and R. A.Thornhill, J. Phys. D 31, 253 (1998).[21] F. Ju ¨licher, C.R. Acad. Sci. (Paris) IV 2, 849 (2001).[22] A. Pikovsky, M. Rosenblum, and J. Kurths, Synchro-nization, A Universal Concept in Nonlinear Sciences(Cambridge University Press, Cambridge, 2001).[23] D. Golomb et al., Phys. Rev. A 45, 3516 (1992).[24] V. Hakim and W.-J. Rappel, Phys. Rev. A 46, R7347(1992).[25] N. Nakagawa and Y. Kuramoto, Physica (Amsterdam)75D, 74 (1994).[26] Y. Kuramoto, Chemical Oscillations, Waves, andTurbulence (Springer, Berlin, 1984).[27] D. A. Smith and D.G. Stephenson, J. Muscle Res. CellMotil. 15, 369 (1994).PHYSICAL REVIEW LETTERS week ending12 SEPTEMBER 2003 VOLUME 91, NUMBER 11114101-4 114101-4

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Synchronization of Active Mechanical Oscillators by an Inertial Load

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