Physical Aspects of Axonemal Beating and Swimming

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provided by CERN Document Server S´ebastien Camalet and Frank Julic¨ her PhysicoChimie Curie, UMR CNRS/IC 168, 26 rue d’Ulm, 75248 Paris Cedex 05, France

In addition, it contains a large number of other proteins We discuss a two-dimensional model for the dynamics of such as nexin which provide elastic links between micro- axonemal deformations driven by internally generated forces tubule doublets, see Fig. 1. The axoneme is inherently of molecular motors. Our model consists of an elastic filament pair connected by active elements. We derive the dynamic active. A large number of dynein molecular motors are equations for this system in presence of internal forces. In located in two rows between neighboring microtubules the limit of small deformations, a perturbative approach al- and can induce forces and local displacements between lows us to calculate filament shapes and the tension profile. adjacent microtubules (Albert et al. 1994). We demonstrate that periodic filament motion can be gener- Axonemal flagella can generate periodic waving or ated via a self-organization of elastic filaments and molecular beating patterns of motion. In the case of sperm for ex- motors. Oscillatory motion and the propagation of bending ample, a bending wave of the flagellum propagates from waves can occur for an initially non-moving state via an in- the head which contains the chromosomes towards the stability termed Hopf bifurcation. Close to this instability, tail. In a viscous environment, the surrounding fluid is the behavior of the system is shown to be independent of set in motion and hydrodynamic forces act on the fila- microscopic details of the axoneme and the force-generating ment. For typical values of frequencies and length scales mechanism. The oscillation frequency however does depend on properties of the molecular motors. We calculate the oscil- given by the size of a flagellum, the Reynolds number lation frequency at the bifurcation point and show that a large is small and inertia terms in the fluid hydrodynamics frequency range is accessible by varying the axonemal length can be neglected (Taylor 1951). Therefore, only friction between 1 and 50µm. We calculate the velocity of swimming forces resulting from solvent viscosity can contribute to of a flagellum and discuss the effects of boundary conditions propulsion. Under such conditions self-propulsion is pos- and externally applied forces on the axonemal oscillations. sible if a wave propagates towards one end, a situation which breaks time-reversal invariance of the sequence of deformations of the flagellum (Purcell 1977).

M D I. INTRODUCTION

N Many small organisms and cells swim in a viscous environment using the active motion of cilia and flagella. These are hair-like appendages of the cell which can undergo periodic motion and use hydrodynamic friction to induce cellular self-propulsion (Bray 1992). In this paper, we are interested in those flagella and cilia which con- tain force generating elements integrated along the whole length of the elastic filamentous structure. They represent rod-like elastic structures which move and bend as a result of internal stresses. Examples for these systems are paramecium which has a large number of cilia on its surface; sperm, which use a single flagellum to swim; and FIG. 1. Schematic representation of the cross-section of the chlamydomonas which uses two flagella to swim (Bray axoneme. Nine doublets of microtubules (M) are arranged in 1992). Cilia also occur in very different situations. An a cylindrical fashion, two microtubules are located in the cen- example is the kinocilium which exists in many hair bun- ter. Dynein motors (D) are attached to microtubule doublets dles of mechanosensitive cells and has the ability to beat and interact with a neighboring doublet. Elastic elements (N) periodically (Rusc¨ h and Thurm 1990). such as nexin are indicated as springs. The common structural theme of cilia and flagella is the axoneme, a characteristic structure which occurs in How can bending waves be generated by axone- a large number of very different organisms and cells and mal dyneins which act internally within the axoneme? which appeared early in evolution. The axoneme consists Dynein molecular motors induce relative forces between of a cylindrical arrangement of 9 doublets of parallel mi- parallel elastic filaments, which are microtubule doublets. crotubules and one pair of microtubules in the center. As a result of these forces, the filaments have the tendency to slide with respect to each other. If such a slid-

1 ing is permitted globally, the filaments simply separate by the facts that flagellar dyneins are capable of gener- but no bending occurs. Bending results if global sliding ating oscillatory motion (Shingyoji et al. 1998) and that is suppressed by rigidly connecting the filament pair in experiments suggest the existence of dynamic transitions the region close to one of the two ends. In this situation, in many-motor systems (Riveline et al. 1998, Fujita and sliding is still possible locally, however only if the fila- Ishiwata 1998). Patterns of motion of cilia and flag- ments undergo a bending deformation. This coupling of ella can be complex and embedded in three dimensional axonemal bending to local microtubule sliding has been space. Many examples of propagating bending waves, demonstrated experimentally. In situations where the ax- however, are planar and can thus be considered as con- oneme is cut at its basal end, filaments slide and in the fined to two-dimensions (Brokaw 1991). presence of ATP separate without bending (Warner and In the following sections, we present a systematic study Mitchell 1981). Small gold-beads specifically attached to of a simplified model for the axoneme which has been in- microtubules in fully functioning flagella can be used to troduced recently (Camalet et al. 1999) and which cap- directly visualize the local relative sliding during beating tures the basic physical properties that are relevant for (Brokaw 1991). its dynamics. In Section II, we present a thorough anal- The theoretical problem of oscillatory axonemal bend- ysis of the dynamics of flexible filaments driven by in- ing and wave patterns have been addressed by several ternal forces. This work is inspired by recent studies authors. One can distinguish two principally different of the dynamics of semiflexible filaments subject to ex- mechanisms to generate oscillatory forces within the ax- ternal forces (Goldstein and Langer 1995, Wiggins and oneme: (i) Deterministic forcing: a chemical oscillator Goldstein 1998, Wiggins et al. 1998). We show how the could regulate the dynein motors which are activated dynamic equations can be solved perturbatively and we and deactivated periodically. In this case, the regula- calculate the shapes of bending waves, the velocity of tory system defines a dynamical force-pattern along the swimming and the tension profile along the flagellum. In axoneme and drives the system in a deterministic way Section III, we briefly review a simple two-state model (Sugino and Naitoh 1982). (ii) Self-organized beating: for a large number of coupled molecular motors. This the axoneme oscillates spontaneously as a result of the model is well suited to represent the dynein molecular interplay of force-generating elements and the elastic fil- motors which act within the axoneme. Self-organized aments (Machin 1963, Brokaw 1975, Brokaw 1985, Lin- bending waves and oscillations via a Hopf bifurcation of demann and Kanous 1995). In particular, Brokaw has the coupled motor-filament system are studied in section studied thoroughly self-organized patterns of beating IV. We calculate the wave-patterns close to a Hopf bi- using numerical simulations of simple models (Brokaw furcation and determine the frequencies selected by the 1985, Brokaw 1999). Such models are based on the bend- system. Finally, we discuss the relevance of our simple ing elasticity of the flagellum and an assumption on the model to real axonemal cilia and flagella and propose ex- coupling of motor activity to the flagellar deformations. periments which could be performed to test predictions In the present work, we are mostly interested in self- that follow from our work. organized beating. Our approach is conceptually different from other works as we focus on an oscillating instability of the motor-filament system. In general, sponta- II. A SIMPLE MODEL FOR AXONEMAL neous oscillations occur via a so-called Hopf-bifurcation DYNAMICS where an initially stable quiescent state becomes unstable and starts to oscillate. There is evidence for the The cylindrical arrangement of microtubule doublets existence of such a dynamic instability in the axoneme. within the axoneme can be modeled effectively as an Demembranated flagella show a behavior which depends elastic rod. Deformations of this rod lead to local slid- on the ATP-concentration CAT P in the solution. For ing displacements of neighboring microtubules. Here, we small CATP, the flagellum is straight and not moving. If consider planar deformations. In this case, the geomet- CATP is increased, oscillatory motion sets in at a critical ric coupling of bending and sliding can be captured by value of the ATP concentration and persists for larger considering two elastic filaments (corresponding to two concentrations (Gibbons 1975). This implies an instabil- microtubule doublets) arranged in parallel with constant ity of the initial straight state with respect to a wave-like separation a along the whole length of the rod. At one mode. Recently, it has been demonstrated using a sim- end, which corresponds to the basal end of an axoneme ple model for molecular motors that a large number of and which we call “head”, the two filaments are rigidly motors working against an elastic element can generate attached and not permitted to slide with respect to each oscillations via a Hopf bifurcation by a generic mecha- other. Everywhere else, sliding is possible, see Fig. 2. nism (Julic¨ her and Prost 1995, Julic¨ her and Prost 1997). The configurations of the system are described by the This suggests that a system consisting only of molecu- shape of the filament pair given by the position of the lar motors and semiflexible filaments can in general un- neutral line r(s) as a function of the arclength s, where dergo self-organized oscillations. This idea is supported

2 r is a point in two dimensional space. The shapes of the Therefore, two filaments are then given by s ∆ = a ds0C (7) r (s) = r(s) − a n(s)/2 1 Z0 r (s) = r(s) + a n(s)/2 , (1) 2 is given by the integrated curvature along the filament. where n with n2 = 1 is the filament normal. The local geometry along the filament pair is characterized by the B. Enthalpy functional relations

r˙ = t (2) The bending elasticity of filaments (microtubules) characterizes the energetics of the filament pair. In ad- t˙ = Cn (3) dition to filament bending, we also have to take into ac- n t ˙ = −C , (4) count the large number of passive and active elements (e.g. nexin links and dynein molecular motors) which where t denotes the normalized tangent vector and C = give rise to relative forces between neighboring micro- t˙ · n is the local curvature. Throughout this paper dots tubule doublets. These internal forces can be charac- denote derivatives with respect to s, i.e. r˙ ≡ ∂ r ≡ s terized by a coarse-grained description defining the force ∂r/∂s. per unit length f(s) acting at position s in opposite directions on the two microtubules, see Fig. 2 (internal forces ∆ r must balance). This force density is assumed to arise as 2 r the sum of active and passive forces generated by a large r 1 number of proteins. This internal force density corresponds to a shear stress within the flagellum which tends f(s) to slide the two filaments with respect to each other. A static configuration of a filament pair of length L can -f(s) thus be characterized by the enthalpy functional

s L κ Λ G ≡ C2 + f∆ + r˙ 2 ds . (8) 2 2 Z0 a Here, κ denotes the total bending rigidity of the filaments. The incompressibility of the system is taken into FIG. 2. Two filaments (solid lines) r1 and r2 at constant account by the Lagrange multiplier function Λ(s) which is separation a are rigidly connected at the bottom end with used to enforce the constraint r˙ 2 = 1 which ensures that s = 0, where s is the arclength of the neutral line r (dashed). s is the arclength (Goldstein and Langer 1995). The in- Internal forces f(s) are exerted in opposite directions, tan- ternal force density f couples to the sliding displacement gential to the filaments. The sliding displacement ∆ at the ∆ as described by the contribution dsf∆. The varia- tail is indicated. tion of this term under small deformations of the filament shape represents the work performedRby internal stresses. A. Bending and sliding of a filament pair Using Eq. (7) leads after a partial integration to L κ Λ The two filaments are assumed to be incompressible G = ds C2 − aCF + r˙ 2 , (9) 2 2 and rigidly attached to each other at one end where s = 0. Z0 Bending of the filament pair and local sliding displace- where ments are then coupled by a geometric constraint. The sliding displacement at position s along the neutral line L F (s) ≡ − ds0f (10) s Zs ∆(s) ≡ ds0(|r˙ | − |r˙ |) (5) 1 2 is the force density integrated to the tail. From Eq.(9) Z0 it follows that if the internal stresses or F are imposed, is defined to be the difference of the total arclengths along G is minimized for a filament curvature C = C0 where the two filaments up to the points r1(s) and r2(s) which C0(s) = aF (s)/κ is a local spontaneous curvature. The face each other along the neutral line. From Eqns. (1) internal forces therefore induce filament bending. In or- r t and (4) follows that ˙ 1,2 = (1 aC/2) and thus der to derive the filament dynamics, we determine the variation δG with respect to variations δr. Details of |r˙ 1,2| = 1 aC/2 . (6)

3 this calculation are given in Appendix A. As a result, we Eqns. (16) and (17) determine the filament dynamics. find The filament shape follows from

δG ˙ s = ∂s[ (κC − af) n − τt ] . (11) r r 0 δr (s, t) = (0, t) + (cos ψ, sin ψ)ds , (18) Z0 Here, where r(0, t) can be obtained from Eq. (15) evaluated at τ = Λ + κC2 − aCF , (12) s = 0. plays the role of the physical tension. This becomes apparent since from Eq. (11) it follows that D. Boundary Conditions

L 0 τ(s) = t(s) · ds δG/δr + Fext (L) (13) The filament dynamics depends on the imposed bound- s ary conditions. The variation δG has contributions at Z ! the boundaries which can be interpreted as externally Fext where (L) is the external force applied at the end applied forces Fext and torques Text acting at the ends, which satisfies Eq. (20). Therefore, τ is the tangent see Appendix A. At the head with s = 0, component of the integrated forces acting on the filament. Fext = (κC˙ − af)n − τt L 0 C. Dynamic equations Text = −κC − a ds f . (19) Z0 We derive the dynamic equation with the simplifying Similarly, at the tail for s = L, assumption that the hydrodynamics of the surrounding fluid can be described by two local friction coefficients ξk Fext = (−κC˙ + af)n + τt and ξ⊥ for tangential and normal motion, respectively. Text = κC . (20) The equations of motion in this case are given by (Wig- gins and Goldstein 1998, Wiggins et al. 1998) If constraints on the positions and/or angles are imposed at the ends, forces and torques have to be applied to 1 1 δG ∂ r = − nn + tt · . (14) satisfy these constraints. In the following, we will discuss t ξ ξ δr ⊥ k different boundary conditions as specified in Table I: For the following, it is useful to introduce a coordinate Case A is the situation of a filament with clamped system r = (X, Y ) and the angle ψ between the tangent head, i.e. both the tangent and the position at the head t = (cos ψ, sin ψ) and the X-axis which satisfies C = ψ˙. are fixed, the tail is free. Case B is a filament with fixed We find with Eq. (11) head, i.e. the tangent at the head can vary. The situation of a swimming sperm corresponds to case C where the 1 ... friction force of a viscous load attached at the head is ∂tr = n (−κ ψ +af˙ + ψ˙τ) ξ⊥ taken into account, the head is otherwise free. As an example of a situation with an external force F (L) 1 t ˙ ¨ ˙ ext + (κψψ − aψf + τ˙) . (15) applied at the tail we consider Case D. For simplicity, we ξk assume a force parallel to the (fixed) tangent at the head. Noting that ∂tr˙ = n∂tψ, we obtain an equation of motion for ψ(s) alone: boundary head s = 0 tail s = L .... condition 1 ¨ ˙ ¨ ∂tψ = (−κ ψ +af + ψτ˙ + τψ) A ∂tr = 0 ∂tt = 0 Fext = 0 Text = 0 ξ⊥ r 0 ext Fext 0 ext 1 B ∂t = T = 0 = T = 0 + ψ˙(κψ˙ψ¨ − afψ˙ + τ˙) . (16) C Fext = −ζ∂tr Text = 0 Fext = 0 Text = 0 ξk D ∂tr = 0 ∂tt = 0 Fext =6 0 Text = 0 The tension τ is determined by the constraint of in- TABLE I. Different boundary conditions studied. (A) 2 compressibility ∂tr˙ = 2t · ∂tr˙ = 0. This condition and clamped head, free tail; (B) fixed head, free tail; (C) swim- Eq. (15) leads to a differential equation for the tension ming flagellum with viscous load ζ; (D) clamped head, exter- profile: nal force applied at the tail. ... ξk 2 ξk τ¨ − ψ˙ τ = a∂s(ψ˙f) − κ∂s(ψ˙ψ¨) + ψ˙(af˙ − κ ψ) . ξ⊥ ξ⊥ (17)

4 E. Small deformations The boundary conditions for h(s) are given in table II.

In the absence of internal forces f, the filament relaxes y Y passively to a straight rod with ψ˙ = 0, i.e. ψ constant. Without loss of generality, we choose ψ = 0 in this state which implies that straight filament is parallel to the X- axis. Internal stresses f(s) induce deformations of this _ n straight conformation. For small internal stresses, we can v ψ perform a systematic expansion of the filament dynamics X,x in powers of the stress amplitude. We introduce a dimensionless parameter which scales t the amplitudes of the internal stresses, f(s, t) = f1(s, t), where f1 is an arbitrary stress distribution. We can now solve the dynamic equations (16) and (17) perturbatively FIG. 3. Resting frame (X, Y ) and frame (x, y) moving with by writing the filament at velocity v¯. The tangent t and the normal n are 2 3 indicated, the angle between the X, x-axis and t is denoted ψ = ψ1 + ψ2 + O( ) ψ. 2 3 τ = τ0 + τ1 + τ2 + O( ) , (21) which allows us to determine the coefficients ψn(s, t) and F. Swimming τn(s, t). This procedure is described in Appendix B. We find that ψ2 and τ1 always vanish and τ0 = σ is a con- The velocity of swimming v¯ can be calculated pertur- stant tension which is equal to the X-component of the batively in . We consider for simplicity periodic internal external force applied at the end. Note, that for bound- stresses with frequency ω given by1 ary conditions A-C, σ = 0. The filament shape is thus iωt −iωt characterized by the behavior of ψ1(s, t) which obeys f(s, t) = f˜(s) + f˜∗(s)e (28) .... ξ⊥∂tψ1 = −κ ψ1 +σψ¨1 + af¨1 . (22) which after long time leads to periodic filament motion

In order to discuss the filament motion in space, we define h(s, t) = h˜(s)eiωt + h˜∗(s)e−iωt (29) the average velocity of swimming where h˜ satisfies according to Eq. (26) t v 1 0 r ¯ = lim dt ∂t (23) 4 2 t→∞ t κ∂ h˜ − σ∂ h˜ + ξ iωh˜ = a∂ f˜ . (30) Z0 s s ⊥ s which is independent of s. This velocity is different from zero only in the case of boundary condition C. We choose v boundary head s = 0 tail s = L the X-axis parallel to ¯ and introduce a coordinate sys- L 2 condition F = − 0 dsf ∂s h = 0 tem (x, y) which moves with the filament 3 A h = 0 ∂sh = 0 κ∂s h = af R 2 3 (x, y) = (X − v¯t, Y ) , (24) B h = 0 κ∂s h = aF κ∂s h = af 3 2 3 C ζ∂th = af − κ∂s h κ∂s h = aF κ∂s h = af 3 where v¯ = |v¯| and the sign depends on whether motion D h = 0 ∂sh = 0 κ∂s h is towards the positive or negative X-direction, see Fig. −σ∂sh = af 3. It is useful to introduce the transverse and longitudinal TABLE II. Boundary conditions for small amplitude mo- deformations h and u, respectively, which satisfy tion. (A) clamped head, free tail; (B) fixed head, free tail; (C) swimming flagellum with viscous load ζ; (D) clamped head, (x, y) = (s + u(s), h(s)) (25) external force applied at the tail. and which vanish for = 0. The quantities h, u and v¯ can be calculated perturbatively in , see Appendix B. To second order in transverse motion satisfies

ξ ∂ h = −κ∂4h + σ∂2h + a∂ f (26) ⊥ t s s s 1 ˜ inωt Note, that more general periodic stresses f = n fne Longitudinal displacements satisfy can also be considered. Since the equation of motion (26) is linear, diﬀerent modes superimpose linearly andP we can s 1 without loss of generality restrict our discussion to a single u(s) = u(0) − (∂ h)2ds0 (27) 2 s mode n = 1. Z0

5 1.0 H 0.8 A B 0.6 C

0.4 A B

0.2 (a) 0.0 1.0 0.0 0.2 0.4 s 0.6 0.8 C 20.0 Φ A 10.0 FIG. 5. Filament shapes in the (x, y) plane that correspond to the modes displayed in Fig. 4. Snapshots taken at differ- 0.0 ent times to illustrate bending waves are shown for boundary conditions A-C. The arrows indicate the direction of wave propagation. -10.0 C B (b) 0.4 0.6 0.8 1.0 0.0 0.2 s III. INTERNAL FORCES GENERATED BY MOLECULAR MOTORS FIG. 4. Fundamental modes of filament motion −iφ h˜1(s) = He close to a Hopf bifurcation are displayed for The filament dynamics is driven by internal stresses an oscillation frequency of about 50Hz and boundary condi- −8 f(s, t) generated by a large number of dynein motors and tions A-C. In case C, a viscous load ζ ' 5.10 Ns/m acts at passive elements. We describe the properties of a force- the head. (a) Amplitude H in arbitrary units as a function generating system using a simplified two-state model. of arclength s. (b) Same plot for the gradient of the phase φ˙.

˜ −iφ Writing h = He where H and φ denote the am- A. Two state model for molecular motors plitude and phase, respectively, filament motion can be expressed as We briefly review the two-state model for a large number of motor molecules attached to a filament which slides h(s, t) = H(s) cos(ωt − φ(s)) , (31) with respect to a second filament (Julic¨ her and Prost 1995, Julic¨ her et al. 1997). Each motor is assumed to which represents bending waves for which vp = ω/φ˙ can be interpreted as the local wave propagation velocity. Ex- have two different chemical states, a strongly bound state amples of bending waves for self-organized beating dis- 1 and a weakly bound state or detached state 2. The cussed in section IV are displayed in Figs. 4 and 5. For interaction between a motor and a filament in states 1 a freely oscillating filament with a viscous load of fric- and 2 is characterized by energy landscapes W1(x) and tion coefficient ζ attached (case C), we find an average W2(x), where x denotes the position of a motor along the filament. The potentials reflect the symmetry of the fila- propulsion velocity v¯ = V0/(1 + ζ/ξkL) where ments: they are periodic with period l, Wi(x) = Wi(x+l) L and are spatially asymmetric, Wi(x) 6= Wi(−x). In the ξ⊥ ω 2 ˙ V0 = − − 1 dsH φ , (32) presence of ATP which is the chemical fuel that drives ξk 2L Z0 the system, the motors are assumed to undergo transi- is the velocity for ζ = 0. Eq. (32) which is correct tions between states. The corresponding transition rates to second order in reveals that motion is only possible are denoted ω1 for detachments and ω2 for attachments. if the filament friction is anisotropic ξk/ξ⊥ 6= 1 and if a We introduce the relative position ξ of a motor with re- wave is propagating, φ˙ 6= 0. This is consistent with earlier spect to the potential period where x = ξ + nl, 0 ≤ ξ < l work on swimming (Taylor 1951, Purcell 1977, Stone and and n is an integer. The probability to find a motor in state i at position ξ at time t is denoted Pi(ξ, t), P1 + P2 Samuel 1996). For a rod-like filament with ξk < ξ⊥, swimming motion is opposite to the direction of wave is normalized within one period. The dynamic equations propagation. of the system are given by

6 ∂tP1 + v∂ξP1 = −ω1P1 + ω2P2 Here, the summation over common indices is implied. The coeﬃcients F (n) are calculated for the two-state ∂tP2 + v∂ξP2 = ω1P1 − ω2P2 . (33) model in Appendix C. However, Eq. (38) is more gen- The sliding velocity v = ∂tx is determined by the force- eral and characterizes a whole class of active nonlinear balance mechanical systems.

l f = λv + ρ (P ∂ W + P ∂ W )dξ + Kx . (34) 1 ξ 1 2 ξ 2 C. Symmetry considerations Z0 Here, the coefficient λ describes the total friction per unit The interaction of the motors with the two filaments length in the system. The number density of motors is asymmetric. Motors are rigidly connected to one of along the filament is denoted by ρ and f is the force the filaments and slide along the second. In addition, per unit length generated by the system. The elastic the filaments are polar and filament sliding is induced by modulus per unit length K occurs in presence of elastic motors towards one particular end of the two filaments. elements such as nexins. If motors have an incommensu- As a consequence, the system has a natural tendency to rate arrangement compared to the filament periodicity, create bending deformations with one particular sign of P + P = 1/l and the equations simplify and can be 1 2 the curvature. Exchanging the role of the two filaments expressed by P alone 1 reverses this sign of bending deformations. This broken symmetry does however not reflect the symmetry of the ∂tP1 + v∂ξP1 = −(ω1 + ω2)P1 + ω2/l l axoneme which is symmetric with respect to all microtubule doublets. Each of the nine microtubule doublet f = λv + ρ P1∂ξ∆W + Kx (35) Z0 within the cylindrical arrangement of the axoneme play identical roles and interact with rigidly attached motors where ∆W = W − W . 1 2 as well as with motors which slide along their surfaces. If all motors generate a constant force, there is consequently B. Molecular motors coupled to a filament pair no resulting bending deformation since all bending mo- ments cancel by symmetry. The two state model for a large number of motors is well suited to represent the internal forces acting within (a) - the axoneme. We assume that at any position s an independent two-state model described by Eq. (35) is located - which generates the internal force density f(s). The filament sliding displacement is identified with the motor displacement, ∆ ≡ x. Note, that here we neglect for simplicity fluctuations which arise from the chemical activity of a finite number of force generators and we assume ho- (b) mogeneity of all properties along the axonemal length. - The energy source of the active system is the chemical activity characterized by the transition rates ω1 and ω2, - the generated forces induce bending deformations of the filaments. FIG. 6. Asymmetric (a) and symmetric (b) mo- The dynamic equations (35) represent a nonlinear ac- tors/filament pair. The arrows indicate the polarity of the tive system which generates time-dependent forces. Since filaments. In case (a), spontaneous bending occurs. In case we will study periodic motion, we can express forces and (b) both filaments play identical roles, no spontaneous bend- displacements by a Fourier expansion ing occurs.

inωt In order to introduce this axonemal symmetry in our f(t) = fne (36) n two-dimensional model, we assume that both filaments X have motors which are rigidly attached and which slide ∆(t) = ∆ einωt . (37) n on the second filament, see Fig. 6. The consequence n X of this symmetry is that the filament polarity becomes The relation between force amplitudes fn and amplitudes unimportant. Exchanging the role of the two filaments of sliding displacements ∆n can in general be expressed is equivalent to replacing the sliding displacement ∆ of in powers of the amplitudes ∆n (Julic¨ her and Prost 1997) motors by −∆. Therefore, in the context of the two-state model this symmetrization is equivalent to the require- (1) (2) 3 fn = Fnk ∆k + Fnkl∆k∆l + O(∆k) . (38) ment that energy landscapes Wi(x) and transition rates

7 ωi(x) are symmetric functions with respect to x → −x. chemical rates of the ATP hydrolysis cycle. In an exper- As a consequence of this symmetry, there is no preferred imental situation, it could be varied e.g. by changing the direction of bending. In the following, we adopt this ATP concentration. If the molecular motors are regu- choice of a symmetric motor/ﬁlament pair. If we use the lated by some other ion concentration such as e.g. Ca2+, expansion given in Eq. (38), this symmetry implies that this concentration could also play the role of the control all even coeﬃcients F (2n) = 0. parameter.

D. Linear response function A. Generic aspects

The perturbative treatment to study small filament de- For oscillations in the vicinity of a Hopf-bifurcation, formations introduced in section II can be naturally ex- filament deformations are small and we can thus use Eq. tended to the coupled motor-filament situation using the (26) to describe the filament dynamics. The force ampli- expansion (38). Up to second order in , only the lin- tude f˜ can be obtained from the expansion ear response coefficient of the active system plays a role (1) f˜ = χ(Ω, ω)∆˜ + O(∆˜ 3) (42) which for the two-state model is given by Fnk = χδnk with in the amplitude of sliding displacements. In addition, l ∂ξR this amplitude is related to the filament shape: χ = K + λiω − ρ dξ ∂ξ∆W iω . (39) 0 α + iω 3 Z ∆(s) = a(∂sh(s) − ∂sh(0)) + O(h ) . (43)

Here, we have introduced R ≡ ω2/αl and α = ω1 + ω2. Higher order terms F (2n+1) have to be taken into account With Eq. (30), we find that spontaneously oscillating modes h˜(s) at frequency ω are to linear order near the if the third or higher order in is considered. The linear bifurcation solutions to response function χ as well as nonlinear coefficients can be calculated most easily for a simple choice of symmetric 4 2 2 2 κ∂ h˜ − σ∂ h˜ + ξ⊥iωh˜ = a χ∂ h˜ (44) potential and transition rates s s s Note that this equation is general and its structure does ∆W (ξ) = U cos(2πξ/l) not depend on the specific model chosen for the force ω1(ξ) = β − β cos(2πξ/l) generating elements. Boundary conditions corresponding ˜ ω2(ξ) = α − β + β cos(2πξ/l) (40) to cases A-D follow from Table II by replacing h → h, ∂th → iωh˜ and f → f˜ = aχ(∂sh˜ − ∂sh˜(0)). where α and β are ξ-independent rate constants. For this As discussed in Appendix D, Eq. (44) has the struc- convenient choice ture of an eigenvalue problem, with χ playing the role of a complex eigenvalue. For every choice of parameters, iω/α + (ω/α)2 there exists an infinite set of nontrivial solutions hñ to χ(Ω, ω) = K + iλω − ρkΩ 2 , (41) 1 + (ω/α) Eq. (44) if χ is equal to the corresponding eigenvalue χ , n = 1, 2, ... We order these values according to their where k ≡ U/l2 is the cross-bridge elasticity of the mo- n amplitude: |χ | ≤ |χ |. The functional dependence tors. We have introduced the dimensionless parameter n n+1 of the Eigenvalues χ on the model parameters is com- Ω = 2π2β/α with 0 < Ω < π2 which plays the role of n pletely determined by dimensionless functions χ¯ (σ¯, ω¯) a control parameter of the motor-filament system, α is a n according to characteristic ATP cycling rate. κ χ = χ¯ (σ¯, ω¯) , (45) n L2a2 n IV. SELF-ORGANIZED BEATING VIA A HOPF 2 4 BIFURCATION where σ¯ ≡ σL /κ and ω¯ ≡ ωξ⊥L /κ are a dimensionless tension and frequency, respectively. A Hopf-bifurcation is an oscillating instability of an ini- The existence of discrete eigenvalues reveals that only tial non-oscillating state which occurs for a critical value particular values of the linear response function of the active material are consistent with being in the vicinity Ωc of a control parameter. For Ω < Ωc, the system is of a Hopf-bifurcation. If the actual value of χ(Ω, ω) of passive and not moving, while for Ω > Ωc it exhibits spontaneous oscillations. As we demonstrate below, self- the system differs from one of the eigenvalues χn, the organized oscillations of the driven filament pair are a system is either not oscillating, or it oscillates with large natural consequence of its physical properties. The con- amplitude for which higher order terms in cannot be trol parameter Ω is in our two-state model a ratio of neglected. An important consequence of this observa- tion is that the modes hñ of filament beating close to a

8 Hopf-bifurcation can be calculated without knowledge of which can be seen by the fact that all eigenvalues χ¯n the properties of the active elements. Not even knowl- have negative real and imaginary parts which cannot be edge of the linear response coefficient is required since it matched by the linear response of a passive system. If Ω follows as an eigenvalue at the bifurcation. Therefore, is increased, we are interested in a critical point Ω = Ωc filament motion close to a Hopf bifurcation has generic for which the straight filament configuration becomes un- or universal features which do not depend on details of stable. This happens as soon as the linear response func- the problem such as the structural complexity within the tion χ matches for a particular frequency ω = ωc one of axoneme. The motion is given by one of the modes hñ the complex eigenvalues, which only depend on σ¯ and ω¯. κ χ¯ (σ¯, ω¯ ) = χ(Ω , ω ) , (46) a2L2 n c c c 0 _ 5 _ 4 ω=10 χ where ω¯c = ξ⊥L ωc/κ. Since χ is complex, both the real _ 1 and the imaginary part of Eq. (46) represent indepen- Imχ _ dent conditions. Therefore, Eq. (46) determines both χ the critical point Ω as well as the selected frequency ω . 3 _ c c -30 χ The selected mode n is the first mode (beginning from 2 Ω = 0) to satisfy this condition. Since |χ(Ω, ω)| typi- cally increases with increasing Ω, the instability occurs almost exclusively for n = 1 since |χ¯1| has been defined _ 3 _ ω=5.10 3 to have the smallest value. For Ω > Ωc, but close to the ω =1,7.10 transition, the system oscillates spontaneously with fre- -60 -60 _ -30 0 quency ωc. The shapes of filament beating are character- Reχ ized by the corresponding mode hñ(s) whose amplitude is determined by nonlinear terms. For the case of a con- FIG. 7. (a) Eigenvalues χ¯n for boundary conditions B. tinuous transition, the deformation amplitude increases Each eigenvalue is represented by a line which traces the loca- 1/2 as |hñ| ∼ (Ω − Ωc) . tions of χ¯n in the complex plane for varying reduced frequency ω¯. The lines begin for ω¯ = 0 on the real axis, the value of ω¯ at the other ends are indicated. C. Axonemal vibrations for different lengths

Fig. 4 displays the amplitude H and the gradient of We choose the parameters of our model to correspond the phase φ˙ of the fundamental mode h˜ = He−iφ. Note, 1 to the axonemal structure. We estimate the bending that we use arbitrary units for H since the amplitude of rigidity κ ' 4 · 10−22Nm2 which is the rigidity of about h˜(s) is not determined by Eq. (44). Corresponding fil- 20 microtubules. Furthermore, we choose a microtubule ament shapes represented according to Eq. (25) in the separation a ' 20nm, motor density ρ ' 5 · 108m−1, (x, y) plane, are displayed in Fig. 5 for boundary con- friction per unit length λ ' 1kg/ms, rate constant ditions A-C as snapshots taken at different times. The α ' 103s−1, cross-bridge elasticity k ' 10−3N/m and oscillation amplitudes are chosen in such a way that the a perpendicular friction ξ ' 10−3kg/ms which is of maximal value of ψ ' π/2. The direction of wave propa- ⊥ the order of the viscosity of water. A rough estimate gation depends on the boundary conditions. For clamped of the elastic modulus per unit length associated with head (case A), waves propagate towards the head whereas filament sliding can be obtained by comparing the num- for fixed or free head (cases B and C), they propagate ber of dynein heads to the number of nexin links within towards the tail. The positions of the lowest eigenvalues the axoneme. This suggests that K is relatively small, χ¯ in the complex plane for boundary conditions B are n K < kρ/10. displayed in Fig. 7 for σ¯ = 0 and varying ω¯. ∼ The selected frequency ωc at the bifurcation point for case B is shown in Fig. 8 as a function of the axoneme B. Selection of eigenmodes and frequency length for K = 0. For small lengths, the oscillation frequency is large and increases as L decreases. In this high-frequency regime, which occurs for L < 10µm, the In order to determine which of the modes h˜ is selected ∼ n system vibrates in a mode with no apparent wave prop- and at what frequency ω the system oscillates at the bi- agation. For L > 10µm the frequency is only weakly L- furcation point, explicit knowledge of the linear response ∼ dependent and the system propagates bending waves of function χ(Ω, ω) is necessary. As a simple example, we a wave-length shorter than the filament length. use χ as given by Eq. (41) for a two-state model. For The limit of small lengths corresponds to small ω¯ and Ω = 0, χ = K + iλω which is a passive viscoelastic re- can be studied analytically. For σ¯ = 0, the eigenvalue sponse. In this case, no spontaneous motion is possible 2 χ1 is given to linear order in ω¯ by χ¯1 ' −π /4 − iγω¯.

9 The coefficient γ depends on boundary conditions but values χ¯n depend on σ¯, therefore the tension affects shape not on any model parameters. For a clamped head (A), and frequency of an unstable mode. Inspection of Eq. γ ' 0.184, for a fixed head (B), γ ' 0.008, see Appendix (44) suggests that the tension σ can be interpreted as an D. The criterion for a Hopf bifurcation for small L is additive contribution to the linear response coefficient 2 2 2 χ → χ + σ/a . Taking into account the boundary condi- π κ ξ⊥ωcL χ(Ω , ω ) ' − − iγ (47) tions complicates the situation, the σ¯-dependence of the c c 4a2L2 a2 eigenvalues is in general nontrivial. However, for a fila- Together with Eq.(41), we find the critical frequency ment with clamped head and a force applied at the tail 1/2 2 1/2 parallel to the tangent at the head (case D) the effect of π κα 1 + (L/LK) ω ' , (48) tension can be easily studied. In this particular case, the c 2L2 γξ 1 + (L /L)2 ⊥ λ tension leads to the same contribution to χ both in Eq. 2 2 1/2 2 1/2 where LK ≡ (π κ/4Ka ) and Lλ ≡ (λa /γξ⊥) are (44) and in the boundary conditions and the eigenvalues two characteristic lengths. For K and λ small, LK thus depend linearly on σ: 2 L Lλ and the critical frequency behaves as ωc ∼ 1/L . χ¯n(σ¯, ω¯) = χ¯n(0, ω¯) − σ¯ . (50) 10 Since the tension corresponds to a change of the real part 5 of χ, its presence has the same effect on filament motion ω /2π as an increase of the elastic modulus K per unit length in c 2 (kHz) Eq. (41) by σ/a . Therefore, we can include the effects of tension on the critical frequency in Eq. (48) by replac- 1 ing K → K + σ/a2 which reveals that in those regimes 0.5 where the frequency depends on K, it will increase for increasing tension. The critical value of the control parameter Ωc is a function of the tension, with Eq. (49), we find 1 5 10 50 L (µm) 1 Ω (σ) = Ω (0) + σ . (51) c c ρka2

FIG. 8. Oscillation frequency ωc/2π at the bifurcation For our choice of parameters, ρka2 ' 200pN which in- point as a function of the axoneme length L for boundary dicates that forces of the order of 102pN should have a conditions B and model parameters as given in the text. The insets show characteristic patterns of motion for small and significant effect on the bifurcation. Furthermore, Eq. large lengths. (51) indicates that the tension σ can play the role of a second control parameter for the bifurcation. Consider The critical value of the control-parameter for small L a tensionless filament oscillating close to the bifurcation is given by for Ω > Ωc(0). If a tension is applied, the critical value Ω (σ) increases until the system reaches for a critical 2 2 c κπ K + λα αγξ⊥L tension σ a bifurcation point with Ω (σ ) = Ω and the Ωc ' + + . (49) c c c 4a2ρkL2 ρk ρka2 system stops oscillating. For axoneme lengths below a characteristic value, L < 2 Lmin, the condition Ωc ≤ π is violated. Therefore, os- V. DISCUSSION cillations exist only for L > Lmin. If (K + λα)/ρk 1, 2 Lmin ' κ/4a ρk ' 1µm for our choice of the parameters. The maximal frequency of oscillations occurs for In the previous sections, we have shown that many p L = Lmin. Within the range of lengths between 1µm and aspects of axonemal eating can be described by a simple 50µm, we find frequencies ωc/2π which vary between tens model based on the idea of local sliding of microtubule of Hz and about 20kHz. The parameters chosen in Fig. doublets driven by molecular motors inside the axoneme. 8 lead to frequencies above 150Hz. Lower frequencies This model represents a class of physical systems termed are obtained for larger values of the friction λ or smaller internally driven filaments (Camalet et al. 1999) which chemical rate α. have characteristic properties that are closely linked to the geometric constraint that couples global bending and local filament sliding. D. Effect of external forces applied at the tail Our work shows that axonemal beating can be studied both numerically and analytically using a coarse-grained In the presence of an external force applied at the end description which ignores many details of the proteins in- σ¯ 6= 0 and the filament pair is under tension. The eigen- volved and is based on effective material properties such

10 as bending rigidities, internal friction coefficient and elas- sional dynamics has to be accounted for in the dynamic tic modulus per unit length as well as the frequency de- equations. Finally, we have restricted most calculations pendent linear response function of the active elements. to the limit of small deformations which corresponds to Our main results are: (i) The filament pair introduces a filament shapes that are almost straight. This regime has geometric coupling of the active elements at different po- several important features. The filament dynamics can sition along the filaments. This coupling is suited for the be studied by an analytic approach by a systematic ex- generation of periodic beating motion by a general self- pansion in the deformation amplitude. This allows us to organization mechanism of the system. (ii) The genera- characterize linear and nonlinear terms both for the fila- tion of oscillations and propagating bending waves occurs ment dynamics as well as for the properties of the active via a Hopf bifurcation for a critical value of a control- elements. Patterns of motion close to a Hopf bifurcation parameter such as the ATP concentration. The patterns are fully characterized by the linear terms of this expan- of motion generated close to this bifurcation can be cal- sion. These linear terms are given by the structure of the culated without any knowledge of the internal force gen- problem and their form does not depend on molecular erating mechanism if the oscillation frequency is known. details of the system. For example, most details of the (iii) The frequency depends on chemical rates of the mo- operation of the molecular motors are unimportant for tors, coefficients of internal elasticities and frictions, the the shape of the filament oscillations at the bifurcation, solvent viscosity and the microtubule rigidity. Our cal- only the linear response function of the active material culations using a simple two-state model for molecular plays a role. motors suggest significant variations of the oscillation fre- If filament beating with larger amplitude is of interest, quency if the axonemal length is varied. Short axonemes nonlinearities become relevant. Nonlinearities arise due are predicted to be able to oscillate at frequencies of sev- to nonlinear geometric terms in the bending energy or via eral kHz. This fact is particularly interesting in the case non-Hookian corrections of the elasticity. Furthermore, of kinocilia which are located in the hair bundles of many nonlinearities in the force-generation process of molecular mechanosensitive cells that are involved in the detection motors exist. All these nonlinearities determine the large of sounds. If cilia are able to vibrate at high frequencies, amplitude motion and could give rise to new types of be- this suggests that the kinocilium could play as active role havior such as additional dynamic instabilities. However, in sound detection (Camalet et al. 2000). (iv) Forces ap- in contrast to the linear terms, the form of the dominant plied to the axoneme can be an important tool to study nonlinearities does depend on structural details of the ax- the mechanism of force generation. An external force oneme. Therefore, an analysis of large amplitude motion can play the role of a second control parameter for the is difficult and knowledge of the nature of the dominating bifurcation which has a stabilizing effect for increasing nonlinearities is required. However, if no new instabili- forces. ties after the initial Hopf-bifurcation occur, the princi- These results have been obtained using several simpli- pal effect of nonlinear terms is to fix the amplitude of fying assumptions. Our model represents the solvent hy- propagating waves. We therefore expect that propagat- drodynamics by an anisotropic local friction acting on the ing waves with larger amplitude are in many cases well rod-like filament. This approach ignores hydrodynamic approximated by our calculation to linear order. interactions between different parts of the filament which Our work shows that propagating bending waves and lead to logarithmic corrections. These hydrodynamic ef- oscillatory motion of internally driven filaments can oc- fects do not change the basic physics but can lead to cor- cur naturally by a simple physical mechanism. Complex rections to the numerical results. It is straightforward to biochemical networks to control the system are not re- generalize our model to incorporate the effects of hydro- quired for wave propagation to occur. This suggests that dynamic interaction, but this does not change our results the basic axonemal structure intrinsically has the ability qualitatively. A more serious simplification of our model to oscillate. Biochemical regulation systems are thus ex- is the restriction to a two-dimensional system and to fila- pected to control this activity on a higher level but are ment configurations which are planar. This choice is mo- not responsible for oscillations. Our work shows that ex- tivated by the fact that observed bending waves of flagella perimental studies of axonemal beating close to a Hopf are planar in many cases (Brokaw 1991). If filament con- bifurcation would be very valuable. Such experiments figurations are planar then our model applies and is com- could be e.g. performed by using demembranated flag- plete. However, this model cannot explain why motion ella and ATP concentrations not far from the level where is confined to a plane and it misses all non-planar modes beating sets in. The observed behavior close to the bifur- of beating. In order to address such questions, a three cation would give insight in the self-organization at work. dimensional generalizations have to be used. Such gen- Furthermore, externally applied forces and manipulation eralizations introduce additional aspects. In particular, of boundary conditions could be sensitive tools to test torsional deformations become relevant. The local sliding some of the predictions of our work. displacements depend in the three dimensional case both on bending and torsional deformations and the full tor-

11 ACKNOWLEDGMENTS ψ2 = 0. The transverse and longitudinal displacements h, u and the velocity v¯ are obtained perturbatively as

We thank A. Ajdari, M. Bornens, H. Delacroix, T. 2 3 Duke, R. Everaers, K. Kruse, A. Maggs, A. Parmeggiani, h = h1 + h2 + O( ) 2 3 M. Piel, J. Prost and C. Wiggins for useful discussions. u = u1 + u2 + O( ) 2 3 v¯ = v¯1 + v¯2 + O( ) . (B3)

APPENDIX A: FUNCTIONAL DERIVATIVE OF Using Eqns. (21) and (18) we ﬁnd u1(s) = u1(0), h2(s) = THE ENTHALPY h2(0) and

s The variation δG of the enthalpy G given by Eq. (8) 0 0 h1(s) = h1(0) + ψ1(s )ds (B4) under variations δr of a shape r(s) is 0 Z s L 1 0 2 0 u2(s) = u2(0) − ψ1(s ) ds . (B5) δG = [(κC − aF ) δC + Λr˙ · δr˙ ] ds . (A1) 2 Z0 Z0 The dynamics of h(s) and u(s) to second order in is Note, that r˙ 2 = 1 but (r˙+δr˙)2 6= 1 in general. Under such therefore determined by ψ1(s, t) and the motion ∂tr for a variation, δC = δ(n·¨r) = n·δ¨r since ¨r·δn = Cn·δn = 0. s = 0. For the latter, we ﬁnd from Eq. (15) v¯1 = 0, Therefore, ∂tu1(0) = 0, ∂th2(0) = 0 and L n r t r .... δG = [(κC − aF ) · δ¨ + Λ · δ ˙ ] ds . (A2) ξ⊥∂th1(0) = −κ h1 (0) + τ0h¨1(0) + af˙1(0) (B6) Z0 ˙ ¨ ξk(v¯2 + ∂tu2(0)) = −ξkψ1(0)∂th1(0) + κψ1(0)ψ1(0) Two subsequent partial integrations lead to − aψ˙1(0)f1(0) + τ˙2(0) . (B7) n r L δG = [(κC − aF ) · δ ˙]0 (A3) L For the lateral motion, we obtain with Eq. (B2) and 2 + −(κC˙ − af)n + (κC − aCF + Λ)t · δr (B4), 0 hL i .... s 2 ¨ ˙ + ∂s (κC˙ − af)n − (κC − aCF + Λ)t · δr ds . ξ⊥(∂th1(s) − ∂th1(0)) = −κ h1 +σh1 + af1 (B8) 0 0 Z h i The functional derivative δG/δr given by Eq. (11) can be Using Eq. (B6), we obtain read oﬀ from the integrand, the boundary terms provide .... expressions for forces and torques at the ends given by ξ⊥∂th1 = −κ h1 +σh¨1 + af˙1 , (B9) Eqs. (19) and (20). which is the result given in Eq. (26). In order to calculate τ2(s) and v¯2 we integrate Eq. (B1). Together with Eq. APPENDIX B: SMALL DEFORMATIONS (B9),

s ˙ ¨ Inserting the expansions (21) in the differential Eq. τ˙2(s) = τ˙2(0) + ψ1(af1 − κψ1) 0 (17) for the tension profile leads at each order in to a s h i ˙ 0 separate equation. Up to second order we find + ξk ψ1∂th1ds . (B10) Z0 τ¨0 = 0 (B1) After elimination of τ˙2(0) via Eq. (B7) we find the τ¨1 = 0 tension profile ξk ... τ¨2 = ∂s(ψ˙1(af1 − κψ¨1)) + ψ˙1(−κ ψ1 +af˙1 + τ0ψ˙1) . ξ⊥ τ2(s) = τ2(0) + sξk(v¯2 + ∂tu2(0)) s 0 ˙ ¨ From the boundary conditions, it follows that τ0 = σ is + ds ψ1(af1 − κψ1) + ξkψ1∂th1 constant and τ1 = 0. Repeating the same procedure for 0 Z 0 s h s i the dynamic Eq. (16) using τ1 = 0 and τ0 = σ, we obtain 0 00 − ξk ds ds ψ1∂tψ1 . (B11) .... 0 0 ξ⊥∂tψ1 = −κ ψ1 +af¨1 + σψ¨1 Z Z .... The tension at the head τ2(0), as well as v¯2 + ∂tu2(0) are ξ⊥∂tψ2 = −κ ψ2 (B2) fixed using the boundary conditions. In cases A,B and The Equation for ψ2 is independent of f1 and only de- D where the head is fixed v¯2 + ∂tu2(0) = 0. In this case, scribes transient behavior which depends on initial con- τ2(0) follows from Eq. (B11) at s = L. For boundary ditions. After long times and for limit cycle motion, condition C, the velocity of the head is determined from

12 the condition τ (L) = 0 together with the boundary con- 4 2 qis/L h˜ = Aie . (D2) ditions τ2(0) = −ψ1(0)(κψ¨1(0)−af1(0))+ζ(v¯2 +∂tu2(0)) ... i=1 and κ h1 (L) = af1(L) which leads to X In order to determine the amplitudes Ai the four bound- L ξ⊥ − ξk ary conditions have to be used. Since Eq. (44) is linear v¯2 + ∂tu2(0) = h˙ 1∂th1ds ζ + ξkL 0 and homogeneous, the equations for the coeﬃcients Ai Z have the form ξ L s k 0 1 ˙ 2 + ds ds ∂t(h1) . (B12) 4 ζ + ξkL 0 0 2 Z Z AiMij = 0 , (D3) i=1 Averaging this equation for periodic motion h1(s, t) = X H(s) cos(ωt − φ(s)) leads to Eq. (32). where Mij is a 4×4 matrix which depends on χ¯, σ¯ and ω¯. Nontrivial solutions exist only for those values χ¯ = χ¯n for which det M = 0. The corresponding eigenmode APPENDIX C: LINEAR AND NONLINEAR ij RESPONSE FUNCTIONS OF THE TWO-STATE is the solution for Ai of Eq. (D3). These eigenvalues MODEL and eigenfunctions can be determined by numerically ob- taining solutions for det Mij = 0. In the limit of small ω¯, analytic expressions can be obtained by inserting the We introduce the Fourier modes of the distribution ansatz χ¯ = χ¯(0) + χ¯(1)ω¯ + O(ω¯2) for the eigenvalue which function P (ξ, t) = P (ξ, k)eikωt. The dynamic equa- 1 k 1 leads to an expansion of det M in powers of ω¯1/2. This tion (35) of the two-state model can then be expressed ij (0) 2 as P procedure leads for σ¯ = 0 to χ¯n = −[(n − 1)π + π/2] independent of boundary conditions. The linear coeﬃ- lδk,l+m cient χ¯(1) = −iγ depends on boundary conditions. For P1(ξ, k) = δk,0R(ξ) − iω ∆l∂ξP1(ξ, m) (C1) 1 α + iωk boundary conditions A Xlm 12π3 − 32π2 + 1728π − 4608 Inserting the Ansatz γ = ' 0.184 , (D4) π5 + 144π3 P (ξ, k) = Rδ + P (1)∆ + P (2) ∆ ∆ + O(∆3) (C2) 1 k,0 kl l klm l m while for case B, into Eq. (C1), one obtains a recursion relation 192 π5 π4 γ = + − π3 + 2π2 − 2π ' 0.008 . (D5) π7 32 12 (n) lδk,kn+l (n−1) P (ξ) 1 n = −iω ∂ P . k,k ,..,k α + iωk ξ l,k1,..,kn−1 Xl (C3)

(n) the knowledge of the functions Pk,k1,..,kn allows one to calculate the coefficients of the expansion (38). In par- (0) ticular, with P = R, one obtains the result (39) and B. Alberts, D. Bray, J. Lewis, M. Raff, K. Roberts and J.D. Watson, The molecular biology of the cell, (Garland, New l (n) (n) York) (1994). Fk,k1,..,kn = ρ dξPk,k1 ,..,kn (ξ)∂ξ∆W (C4) 0 D. Bray, Cell Movements, (Garland, New York) (1992). Z C.J. Brokaw, Molecular mechanism for oscillation in flagella and muscle, Proc. Natl. Acad. Sci. USA 72, 3102-3106 (1975). APPENDIX D: EIGENVALUES AND C.J. Brokaw, Computer simulation of flagellar movement VI. EIGENMODES NEAR A HOPF BIFURCATION Biophys. J. 48, 633-642 (1985). C.J. Brokaw, Microtubule sliding in swimming sperm flagella: Nontrivial solutions to Eq. (44) can be found using the direct and indirect measurement on sea urchin and tunicate Ansatz h˜ = Aeqs/L with q being solution to spermatozoa, J. Cell Biol. 114, 1201-1215 (1991). C.J. Brokaw, Computer simulation of flagellar movement VII. q4 − (σ¯ + χ¯)q2 + iω¯ = 0 (D1) Cell motility and the cytoskeleton 42, 134-148 (1999). S. Camalet, F. Julic¨ her and J. Prost, Self-organized beating Here, σ¯ ≡ σL2/κ, ω¯ ≡ ωξ L4/κ, and χ¯ ≡ χa2L2/κ are and swimming of internally driven filaments, Phys. Rev. Lett. ⊥ 82, 1590-1593 (1999). a dimensionless tension, frequency and linear response S. Camalet, T. Duke, F. Julic¨ her and J. Prost, Auditory sensi- coefficient, respectively. Eq. (D1) has four complex so- tivity provided by self-tuned critical oscillations of hair cells, lutions qi, i = 1, .., 4. Therefore, Proc. Natl. Acad. Sci. USA, 97, 3183-3188 (2000).

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