Equations of Interdoublet Separation During Flagella Motion Reveal Mechanisms of Wave Propagation and Instability

View metadata, citation and similar papers at core.ac.uk brought to you by CORE

provided by Elsevier - Publisher Connector

1756 Biophysical Journal Volume 107 October 2014 1756–1772 Article

Equations of Interdoublet Separation during Flagella Motion Reveal Mechanisms of Wave Propagation and Instability

Philip V. Bayly1,* and Kate S. Wilson1 1Washington University, St. Louis, Missouri

ABSTRACT The motion of flagella and cilia arises from the coordinated activity of dynein motor protein molecules arrayed along microtubule doublets that span the length of axoneme (the flagellar cytoskeleton). Dynein activity causes relative sliding between the doublets, which generates propulsive bending of the flagellum. The mechanism of dynein coordination remains incompletely understood, although it has been the focus of many studies, both theoretical and experimental. In one leading hypothesis, known as the geometric clutch (GC) model, local dynein activity is thought to be controlled by interdoublet separation. The GC model has been implemented as a numerical simulation in which the behavior of a discrete set of rigid links in viscous fluid, driven by active elements, was approximated using a simplified time-marching scheme. A continuum mechanical model and associated partial differential equations of the GC model have remained lacking. Such equations would provide insight into the underlying biophysics, enable mathematical analysis of the behavior, and facilitate rigorous comparison to other models. In this article, the equations of motion for the flagellum and its doublets are derived from mechanical equilibrium principles and simple constitutive models. These equations are analyzed to reveal mechanisms of wave propagation and instability in the GC model. With parameter values in the range expected for Chlamydomonas flagella, solutions to the fully nonlinear equations closely resemble observed waveforms. These results support the ability of the GC hypothesis to explain dynein coordination in flagella and provide a mathematical foundation for comparison to other leading models.

INTRODUCTION Cilia and flagella are thin, active organelles that propel tral pair (7–9) or with interdoublet sliding (10–14). Re- cells or move fluid. The axoneme (the cytoskeleton of view articles by Brokaw (15), Lindemann and Lesich the flagellum or cilium) consists of nine microtubule (16), and Woolley (17) summarize and assess the leading doublets surrounding a central pair of singlet microtubules theories of dynein regulation in light of today’s (1). The axoneme undergoes large bending deformations knowledge. under the action of dynein, a motor protein powered Mathematical modeling and simulation of flagellar mo- by ATP hydrolysis. Dynein molecules form an array of tion have contributed greatly to our current understanding cross-bridges between pairs of outer doublets, and exert of flagella motion. Brokaw (18) originally confirmed in forces that cause sliding of one doublet relative to the computer simulations that the sliding filament model of other. These active dynein forces interact with passive flagella with delayed curvature control of dynein activity structural elements (doublets, nexin links, and radial could generate autonomous, oscillatory waveforms. Brokaw spokes) to produce coordinated propagation of bends along and Rintala (10,19) and Brokaw (20–23) have continued to the flagellum (2). investigate different models of dynein regulation in a well- The timing of bend formation defines the recovery and known series of articles. Hines and Blum (24) contributed power strokes, and is critical for propulsive effectiveness. a rigorous and complete derivation of the nonlinear, contin- However, the mechanisms that coordinate dynein activity uum equations of the flagellum in two dimensions. The are not well understood (3). Imposed mechanical bending model, which includes delayed curvature feedback to regu- can initiate or change flagella motion (4,5), but the spe- late dynein activity, succeeds in generating propagating cific mechanisms by which bending or sliding affects waves qualitatively similar to those seen in beating flagella. dynein activity are not clear. Bending does induce changes Several other authors have used similar continuum models in interdoublet spacing, which may affect the ability of to investigate dynein regulation (12–14) or to study the dynein to form cross-bridges (6). Bending may also affect motion of flagella in nonlinear and viscoelastic fluids dynein through coupling with torsion of the twisted cen- (25–27). Continuum models and the associated partial differential equations (PDEs) have been widely used to explore Submitted May 15, 2014, and accepted for publication July 22, 2014. concepts of dynein regulation. In the original Hines- *Correspondence: [email protected] Blum model (24) the biophysical mechanism behind their Editor: Charles Wolgemuth. Ó 2014 by the Biophysical Society 0006-3495/14/10/1756/17 $2.00 http://dx.doi.org/10.1016/j.bpj.2014.07.064 Propagation and Instability in Flagella Motion 1757 mathematical model of curvature feedback was not speci- global transverse force, which results from tension differ- fied, but Cibert (28) proposed that bending may change ences between doublets in curved regions.) Thus, local sta- the pitch of the tubulin-dimer spiral pattern in microtu- bility and bifurcations in this model have not been bules, which could regulate the attachment of dynein characterized, nor have the mechanisms of dynein propa- arms. The excitable dynein concept, in which sliding gation been thoroughly analyzed. This study focuses on beyond a specific threshold stimulates dynein activity, such objectives. The sections below describe the was proposed in Murase (11,31), Murase and Shimadzu following: (29), and Murase et al. (30). Sliding-controlled collective 1. Derivation of the basic differential equations of flagella behavior has been explored further by Camalet and Ju- ¨ mechanics in the GC model, including a partial differen- licher (12), Riedel-Kruse et al. (13), and Hilfinger et al. tial equation describing interdoublet separation; (14) to explain dynein-driven oscillations. These studies 2. Solution approaches for these equations (phase plane postulate that local sliding enhances local dynein activity analysis, local linearization, eigenanalysis, and time- by decreasing the force per dynein head, which allows domain simulation); and more dynein to be recruited, thus increasing net shear 3. The results from achieving objectives 1 and 2, using force (13). This mechanism is incorporated as a complex parameters representing the flagella of Chlamydomonas impedance with negative stiffness and friction compo- reinhardtii. nents. Sliding-controlled models with this negative impedance can exhibit periodic solutions very similar to These analyses confirm the ability of the GC hypothesis to observed flagellar waveforms (13,14). Agreement is explain key aspects of flagella motion, and place the theory particularly close if the model includes relative motion be- on a firm mathematical foundation. tween doublets at the base (13). Sliding-controlled models (12–14) may also exhibit retrograde (distal-proximal) propagation. EQUATIONS OF MOTION Lindemann (32–34) has argued convincingly for a model Flagellar bending in which dynein activity depends on the separation between adjacent doublets. In this model, if doublets are close Hines and Blum (24) derived the equations of motion for a enough, dynein arms form cross-bridges and produce force flagellum in two dimensions, modeling it as a slender both tangential and normal to the doublets. If the doublets elastic beam in viscous fluid, subject to both active and are too far apart, fewer cross-bridges are formed and less passive internal shear forces (Fig. 1). The derivation below force is produced. Lindemann (32,33) showed that axial ten- follows the approach of Hines and Blum (24), but con- sion or compression in curved doublets, as well as local siders the bending of independent doublets as well as dynein activity and elastic stretching of nexin links, leads the entire flagellum. The model is herein simplified, con- to transverse forces that either bring doublets together sisting of two active pairs of doublets (Fig. 1, d and e), (increasing dynein cross-bridging and activity) or push dou- with activity in one pair producing the principal (P) blets apart (decreasing dynein activity). Lindemann (32,33) bend, and activity in the other producing a reverse (R) postulates a relationship between the activities of dynein in bend (32,33,36). adjacent sections (spatial coupling) as one that encourages We first consider a single doublet pair (Fig. 2). The propagation. Computer models based on this geometric equations of force equilibrium at any point along each clutch (GC) hypothesis (32–34) have produced simulated of two doublets (labeled 1 and 2) are written in terms behavior that closely resembles observed waveforms of cilia of the tangent angles j1 and j2, the net internal tangen- and flagella. In these models, the flagellum was represented tial and normal force components in each doublet (T1, by a set of discrete links, subject to forces from the viscous N1, T2, N2), and the external viscous force components fluid environment, elastic components of the axoneme, and per unit length (qT and qN). The baseline separation be- dynein arms. In a recent study, Brokaw (35) provides more tween doublets in this model is the effective diameter, theoretical support for the GC hypothesis; he has simulated a. The net interdoublet shear (tangential) force compo- cyclic splitting and reattachment of two doublets using a nent, fT, is due to distributed active (dynein arms) and detailed mechanochemical model of the dynein cross-bridge passive elements (nexin links e.g.). In addition, these el- cycle. ements also provide a transverse interdoublet force The GC hypothesis of flagellar beating has not yet been component, fN. cast in terms of a continuum model with associated PDEs The key variables and parameters of this model of overall that can be analyzed mathematically, like those found in flagella mechanics are summarized in Table 1. the literature (12–14,24–27). (Note that in Riedel-Kruse The equilibrium equations, separated into tangential et al. (13), a simplified version of the GC model was and normal components for each doublet, and the analyzed but the key feedback mechanism was absent. In moment balance for any element on each doublet, are particular, the equations did not include the dominant as follows:

Biophysical Journal 107(7) 1756–1772 1758 Bayly and Wilson a

b cd

1 9 2

8 3

7 4 FIGURE 1 (a) Diagram of flagellum with shape 6 5 defined by tangent angle, j(s,t). (b–d) Cross-section, electron micrograph, and functional schematic (rotated 180). (e) The simplified model is e based on two pairs of doublets driving flagellar Acve bending. Side views of P and R doublet pairs show sliding displacement, u(s,t); effective diam- Passive eter a; active (green); and passive (red) shear forces

(fTP(s,t) and fTR(s,t)) and internal doublet tension (T , T , T , and T ). Schematic cross-sectional T T 1P 2P 1R 2R 1P 1R views illustrate activity of P and R doublet pairs f (s,t) f (s,t) (note: distortion is exaggerated). P activity drives TP T TR T 2P 2R doublet 4 tipward relative to doublet 2; R activity P R drives doublet 9 tipward relative to doublet 7. The dashed line segment, formed by connecting a a doublets 3 and 8, is normal to the beat plane. To s s see this ﬁgure in color, go online.

5-6 5-6 4 7 4 7 P R 2 9 2 9 1 1

Doublet 1 Doublet 2

vT1 vj1 vT2 vj2 eT : N1 þ fT þ qT1 ¼ 0; N2 fT þ qT2 ¼ 0; (1) vs vs vs vs

Doublet 1 Doublet 2

vN1 vj1 vN2 vj2 eN : þ T1 fN þ qN1 ¼ 0; þ T2 þ fN þ qN2 ¼ 0: (2) vs vs vs vs

Doublet 1 Doublet 2

vMB1 a vMB2 a þ fT þ N1 ¼ 0; þ fT þ N2 ¼ 0: (3) vs 2 vs 2

We define the net tangential force in the element T ¼ T1 þ j ¼ (j1 þ j2)/2 defines the shape of the flagellum. By add- T2, the net normal force N ¼ N1 þ N2, and the total moment ing Doublet 1 to Doublet 2 in Eq. 1, Doublet 1 to Doublet 2 due to elastic bending MB ¼ MB1 þ MB2. The mean angle in Eq. 2, and Doublet 1 to Doublet 2 in Eq. 3, and by

Biophysical Journal 107(7) 1756–1772 Propagation and Instability in Flagella Motion 1759

modeled as a set of slender elastic beams with combined ﬂexural rigidity, EI: vj MB ¼ EI : (7) vs

The ﬂuid at low Reynolds number is assumed to provide resistive force proportional to velocity (24). The velocity of any point along the ﬂagellum can be written as v ¼ vNeN þ vTeT. The spatial derivatives of the tangential and normal components of velocity are

vvT vj ¼ vN ; (8) vs vs

vvN vj vj ¼ vT ; (9) vs vt vs

and using resistive force theory to model viscous drag, as in Hines and Blum (24), the corresponding tangential and FIGURE 2 Free-body diagram of a differential axial element of one doublet normal force components are pair. Interdoublet components (dynein, nexin links, radial spokes) are repre- qT ¼cT vT ; (10) sented as a single lumped element contributing net forces fTds and fNds and net moment afTds/2 on each doublet. To see this figure in color, go online. qN ¼cNvN: (11) neglecting the small differences in curvature between the doublets The equilibrium, kinematic, and constitutive equations can be combined to form three equations describing the motion ð Þ v j1 j2 vj ; of a slender elastic beam with internal shear moving in vs vs viscous fluid, which again match equations in Hilfinger and using the variables defined above, we obtain the equa- et al. (14) and Hines and Blum (24): tions below, which match analogous equations in Hilfinger cT cT 2 et al. (14) and Hines and Blum (24): T;ss Nj;ss 1 þ N;sj;s Tj;s ¼ 0; (12) cN cN vT vj N þ qT ¼ 0; (4) vs vs c c N þ 1 þ N T þ T N N 2 ¼ c ; (13) ;ss c ;sj;s j;ss c j;s Nj;t vN vj T T þ T þ qN ¼ 0; (5) vs vs EIj;ss þ afT þ N ¼ 0; (14)

vMB þ afT þ N ¼ 0: (6) , ¼ v , v vs where ( ),z ( )/ z. Boundary conditions are also required to specify the solu- Next, constitutive models are needed for the elastic bending tion. We focus on the case in which the base is fixed and the moment and viscous forces. The flagellum (doublet pair) is distal end is free of external moments and force: a reasonable approximation to the actual situation (37). The TABLE 1 Key variables and parameters of flagellum following boundary conditions specify the fixed-free case: equations Zero angle at base Variable Definition j(s,t) Tangent angle of the flagellum at location s and time t jð0; tÞ¼0: (15) T(s,t), N(s,t) Net internal tangential and normal force in the doublets (pN) cN, cT Viscous drag coefficients (normal and tangential) (pN-s/mm2) Zero normal velocity at base fT(s,t) Net interdoublet tangential (shear) force per unit length (pN/mm) F (s,t) Net interdoublet normal force per unit length (pN/mm) N N a Effective diameter of flagellum (beam) (nm) ðv ð0; tÞ¼0Þ : v þ T vj ¼ 0: (16) 2 N EI Flexural rigidity of flagellum (pN-mm ) vs vs s ¼ 0

Biophysical Journal 107(7) 1756–1772 1760 Bayly and Wilson

Zero tangential velocity at base Denoting the change in separation between the doublets by h, so the interdoublet distance is a h (Fig. 3), we can write 1 h vT vj v ðv ð0; tÞ¼0Þ : N ¼ 0: (17) j1 ¼ j þ ; T 2 vs vs vs s ¼ 0 1 vh j ¼ j þ ; and 2 2 vs Zero bending moment at distal end vh j j ¼ : 1 2 vs EIj;sðL; tÞ¼0: (18) We note that h << a << L and vh/vs << j. We assume the distributed viscous forces act approximately equally on both Zero transverse force at distal end doublets (qT1 – qT2) z (qN1 – qN2) z 0, and that the constitutive relationship for each doublet is NðL; tÞ¼0: (19) M ¼ EI vj1; B1 d s v (24) Zero tangential force at distal end vj2 MB2 ¼ EId ; vs

TðL; tÞ¼0: (20) where EId is the flexural modulus of an individual doublet. Combining Eqs. 21–24, defining S ¼ T1 – T2, leads to v4h vj Interdoublet separation EId ¼ 2fN S ; (25) vs4 vs Models of dynein regulation are expressed mathematically by equations that relate the interdoublet shear force fT(s,t) Z L to mechanical variables such as the curvature or the inter- S ¼ 2 fT dz: (26) doublet sliding velocity. In the GC hypothesis (32–34), s dynein is regulated by the distance between doublets. In this section, we derive continuum mechanical equations Equations 25 and 26 are the key equations that relate the in- h f that govern interdoublet separation. terdoublet separation to the transverse internal force ( N) vj vs Subtracting Doublet 2 from Doublet 1 in Eq. 1, and like- and to the curvature of the flagellum ( / )(Fig. 3). (Small nonlinear terms have been dropped for simplicity; their in- wise for Eqs. 2 and 3, we obtain clusion in the full model is straightforward, and does not vðT1 T2Þ vj1 vj2 perceptibly affect results.) The dependence on curvature is N1 N2 þ 2fT þðqT1 qT2Þ¼0; vs vs vs modulated by the difference in internal doublet tension, (21) S ¼ T1 – T2, which arises from the integration of the interdoublet shear force, fT. Boundary conditions for h are in Ap- pendix A. vðN1 N2Þ vj1 vj2 þ T1 T2 2fN þðqN1 qN2Þ¼0; vs vs vs (22) Cross-bridge attachment, dynein activity, and interdoublet forces vðMB1 MB2Þ Interdoublet forces are the sum of active and passive contri- þðN1 N2Þ¼0: (23) vs butions. The active force is modeled by a constant ab FIGURE 3 Schematic diagrams of interdoublet separation. (a) Doublet spacing with zero mean curvature. Interdoublet spacing is a smooth function of axial position, s, modulated by doublet flexural modulus and active and passive axoneme components. Shapes and sizes are exaggerated for illustration. (b) Effect of curvature and doublet tension on interdoublet force. (Blue arrows) Resul- tants. To see this figure in color, go online.

Biophysical Journal 107(7) 1756–1772 Propagation and Instability in Flagella Motion 1761 maximum active force (f or f ) multiplied by the aggregate K T N K ðhÞ¼ 0 : (32) probability of dynein cross-bridge attachment, p. Passive on 1 þ expð20ðhon hÞ=hmaxÞ shear and transverse forces oppose the active forces, and are assumed to be proportional to the corresponding compo- Incorporation of the effects of interdoublet separation on nents of displacement and velocity. The tangential force dynein activity completes the basic model of flagella mo- (k u b vu/vt) and transverse force (k h b vh/vt) T T N N tion. Summarizing the continuum version of the GC model, are the passive forces that oppose axoneme deformation in the coupled equations for j, N, T, h, A, and S are written the absence of dynein activity. A local transverse interdoub- together in Table 2. let force due to link stretching by shear displacement was included in the discrete GC models (32,33,36); this term is less significant than other transverse force terms Principal and reverse bending via opposing (32,33,36) and is left out of this model for simplicity. The doublet pairs interdoublet sliding displacement is u(s,t) ¼ aj(s,t) when the base is fixed (j(0,t) ¼ 0 and u(0,t) ¼ 0) (13,24). Thus, The equations summarized in Table 2 describe the me- the tangential and normal interdoublet forces are chanics by which dynein activity might lead to global flagella bending and interdoublet separation in a flagellum vj fT ¼ f T p kT aj bT a ; (27) with a single pair of doublets. However, as given above, vt these equations will only produce active bending in one direction, and will not lead to switching or oscillations. vh f ¼ f p k h b : (28) To explain this behavior, we invoke the widely accepted N N N N t v notion that two subsystems exist (Fig. 1 e), each consist- ing of a set of doublets and associated dynein motors In the GC model, the probability of attachment is governed (32,33) and each governed by a set of equations like those by the interdoublet distance, following the literature in Table 2. The principal (P) bend subsystem comprises (32,33,36). One way to model this is to assume that the var- doublets 2–4 and the reverse (R) subsystem comprises iable p increases with a specified rate constant toward a doublets 7–9. The resulting equations for the two sides maximum value p when h exceeds a threshold, h , and de- 1 on are given in Table 3. creases to a baseline value p0 when h is less than a different threshold, hoff. This behavior is captured by the following equations and illustrated in Fig. 4. The variable A is an acti- TABLE 2 Summary of the equations of flagella motion and vation variable that modulates the probability of attachment. interdoublet separation Deflections are defined relative to h ¼ p1f =kN, Equations of global flagella motion max N c c p ¼ p þ Aðp p Þ; (29) T N 1 þ T N T T 2 ¼ 0 0 1 0 Tangential force ;ss j;ss c ;sj;s c j;s balance (Eq. 12) N N cN cN 2 vA Normal force N;ss þ 1 þ T;sj;s þ Tj;ss Nj;s ¼ cNj;t ¼ K ð1 AÞK A; (30) cT cT vt on off balance (Eq. 13) Moment balance EIj,ss þ afT þ N ¼ 0 (Eq. 14) K0 Interdoublet separation EI v4h/vs4 ¼ 2f – Svj/vs KoffðhÞ¼ ; (31) d N 1 þ exp 20 h hoff hmax (Eq. 15) R L Tension difference S ¼ 2 s fT dz in doublets (Eq. 16) Tangential interdoublet fT ¼ f T p kT aj bT avj=vt force density (Eq. 27) Normal interdoublet fN ¼ f Np kN h bN vh=vt force density (Eq. 28) Dynein activity

Cross-bridge p ¼ p0 þ A(p1 – p0) attachment probability (Eq. 29)

Cross-bridge vA/vt ¼ Kon(1 – A)–KoffA attachment FIGURE 4 Simplified model of the effect of interdoublet separation on dynamics (Eq. 30) the rate of cross-bridge attachment or detachment (Eqs. 31 and 32). K0 Detachment rate Koff h ¼ When the doublets become sufficiently close (h > hon), attachment proba- 1 þ expð20ðh h Þ=h Þ (Eq. 31) off max bility increases at a characteristic rate, k0. When h drops below a different K0 Attachment rate KonðhÞ¼ threshold (h < hoff). the probability of attachment decreases at a rate that 1 þ expð20ðh hÞ=h Þ (Eq. 32) on max approaches k0. To see this figure in color, go online.

Biophysical Journal 107(7) 1756–1772 1762 Bayly and Wilson

TABLE 3 Equations for principal (P) and reverse doublet (R) pairs Parameters PR 4 4 v hP vj v hR vj Interdoublet separation EId ¼ 2fNP SP EId ¼ 2fNR SR vs4 vs vs4 vs R L R L Tension difference SP ¼ 2 s fTPdz SR ¼ 2 s fTRdz

Tangential force density fTP ¼ f T pP kT aj bT avj=vtfTR ¼f T pR kT aj bT avj=vt

Normal force density fNP ¼ f NpP kNhP bNvhp=vtfNR ¼ f NpR kN hR bN vhR=vt

Attachment probability PP ¼ p0 þ AP(p1 – p0) PR ¼ p0 þ AR(p1 – p0)

Attachment dynamics vAp/vt ¼ Kon(1 – Ap)–KoffAp vAR/vt ¼ Kon(1 – AR)–KoffAR

K0 K0 Attachment rate Koff ðhPÞ¼ Koff ðhRÞ¼ 1 þ expð20ðhP hoff Þ=hmaxÞ 1 þ expð20ðhR hoff Þ=hmaxÞ K0 K0 Detachment rate Kon hP ¼ KonðhRÞ¼ 1 þ expð20ðhon hPÞ=hmaxÞ 1 þ expð20ðhon hRÞ=hmaxÞ

The equations for the R side are identical to those of the P the fact that differential tension in doublets either pulls the side, after replacing P with R, except that the active force doublets together or pushes them apart depending on their (f TpR) is negative, inasmuch as it bends the flagella in average curvature (Fig. 3 b). Inserting the expression for the opposite direction. The equations of global flagella mo- fN (Eq. 28) into the equation for interdoublet separation tion (Eqs. 12–14 and Table 2) remain the same, except that (Eq. 25) gives activity on both the P and R sides contributes to the active 4h h bending moment: a ¼ a(f þ f ). To model asymmetric EI v þ 2k h þ 2b v ¼ 2f ½p þðp p ÞAS vj: fT TP TR d s4 N N t N 0 1 0 s beating, values of parameters (such as h or h ) may differ v v v on off (33) between sides P and R. For simplicity, mechanical coupling between opposing doublets (36) is not included. Doublets on opposing sides are in fact connected mechanically by a ring The activation variable A is governed by Eq. 30. The local of links and the system of radial spokes, so that some dynamics of Eqs. 30 and 33 may be examined by ignoring coupling is likely. In the discrete GC model, this coupling spatial variations in h and A and uncoupling the resulting v improved agreement with observed waveforms; however, equations from the global flagellar motion (letting S j/ v ¼ it did not change the fundamental behavior (36). Accord- s 0. This leads to a pair of ordinary differential equations ingly we postpone, for now, the inclusion of P-R coupling. dh f Nðp1 p0Þ kN f N ¼ A h þ p0; (34) dt bN bN bN

ANALYSIS AND RESULTS dA ¼ K ð1 AÞK A: (35) dt on off Excitability and propagation of interdoublet separation The null clines of this system (where dh/dt ¼ 0ordA/dt ¼ 0) In intact flagella, dynein activity appears to propagate from are shown in Fig. 5 a. For illustrative purposes, we consider base to tip during flagella beating, and activity can also the case p0 ¼ 0; other parameters are as in Table 4. The in- propagate in pairs of isolated doublets (38). To explain the tersections of the null clines define the equilibria (fixed mechanism of propagation we consider the simplified model points) of the local system. of interdoublet separation in Eq. 25 with interdoublet forces For these parameter values, the behavior of this local two- described by Eqs. 28–32. Equation 25 is coupled to overall dimensional system is closely approximated by a one- motion of the flagellum by the term Svj/vs, which describes dimensional subsystem, because the cross-bridge variable

FIGURE 5 (a) Null clines and vector ﬁeld corresponding to the local dynamics of interdoublet separation: Eqs. 34 and 35. (b) Trajectories in the A h plane rapidly approach a curve representing the quasi-equilibrium value of A for a given value of

h: Aeq(h). (c) The system approaches the behavior of a particle on a curved surface with two stable equilibria separated by an unstable equilibrium. To see this ﬁgure in color, go online.

Biophysical Journal 107(7) 1756–1772 Propagation and Instability in Flagella Motion 1763

TABLE 4 Typical parameter values of the continuum GC Examining the plots of dh/dt versus h and dA/dt versus A model (see Appendix B) (Fig. 6) in this local system, it is apparent that the right- Name Value Description hand side of the equation may be approximated by a cubic 3 2 CN 2.50 10 pN-s/mm Resistive (viscous) force coefficient, polynomial function, as in many excitable systems (e.g., normal (13,33,51) (B,C) Fitzhugh-Nagumo). Thus, local dynamics of h follow 3 2 CT 1.25 10 pN-s/mm Resistive (viscous) force coefficient, tangential (51) (C) dh 2 zC hðh h Þðh hÞ; (37) EI 500 pN-mm Flexural modulus of flagellum dt h th max (59–61). (M,U) 2 EId 50 pN-mm Flexural modulus of doublet (59) (M) L 12 mm Length of flagellum (49,50) (C) where hmax ¼ f Nðp1 p0Þ=kN. The other parameters of the k 125 pN/mm2 Interdoublet shear stiffness T cubic analogy, Ch and hth, depend on kN, bN, hon, and hoff. coefficient (61) (U) 4 Similarly, the local dynamics of the attachment variable A k3T 250 pN/mm Nonlinear (cubic) shear stiffness coefficient (estimate) may be approximated by (Fig. 6) 2 bT 2.5 pN-s/mm Interdoublet shear viscous dA coefficient (estimate) zEðAÞ¼CAAðA AthÞð1 AÞ: (38) 2 kN 5000 pN/mm Interdoublet normal stiffness dt coefficient (estimate z 40 kT) 8 4 k3N 1 10 pN/mm Nonlinear (cubic) transverse stiffness If we make the corresponding approximation in the PDE coefficient (estimate) 2 (Eq. 33), we obtain bN 100 pN-s/mm Interdoublet normal viscous coefficient (estimate z 40 b ) T h EI 4h S a 200 nm Effective diameter of flagellum v ¼ d v þ C hðh h Þðh hÞ vj: (39) 4 h th max (13,61) (B,U) vt 2bN vs 2bN vs k0 2000/s Maximum dynein cross-bridge attachment rate (estimate) honP 8 nm [4 nm] Attachment threshold, principal (P) Excluding the last term, Eq. 39 is a form of the parabolic side (estimate) nonlinear fourth-order PDE with bistable nonlinearity hoffP 0nm[ 4 nm] Detachment threshold, principal (P) known as the extended Fisher-Kolmorogov (EFK) equation side (estimate) (39). The term Svj/vs couples this system to global flagella honR 8 nm Attachment threshold, reverse (R) v v ¼ side (estimate) motion. This equation (with S j/ s 0) is analogous to the hoffR 0 nm Detachment threshold, reverse (R) classic reaction-diffusion system, and is known to exhibit side (estimate) traveling wave-front solutions (40,41). Fig. 7 shows exam- f T 2000 pN/mm Maximum tangential dynein force ples of propagation of an initial disturbance in h at the left density (32) (B,C,U) end of the domain, using: f N 400 pN/mm Maximum normal dynein force zf =5 density (estimate T ) 1. The original model with cross-bridge kinetics (Eqs. 33 p0P 0.10 [0.05] Baseline probability of dynein and 30), and cross-bridge attachment (P) (33) 2. The model with cubic approximation (Eq. 39). p0R 0.10 [0.01] Baseline probability of dynein cross-bridge attachment (R) (33) Thus, the equations of interdoublet separation and dynein p1P 0.30 [0.30] Maximum probability of dynein cross-bridge attachment (P) (estimate) activity, uncoupled from the global motion of the flagellum, p1R 0.30 [0.26] Maximum probability of dynein clearly support propagation of disturbances. Results from cross-bridge attachment (R) (estimate) theoretical studies of the EFK equation (39–44) can illuminate the relationship between flagella properties and behavior. (A) approaches a quasi-equilibrium for each value of the spacing (h). Trajectories from many initial conditions are shown in Fig. 5 b and Fig. 6; trajectories rapidly approach Stability analysis and small-amplitude flagellar a curve that comprises the unstable manifold of the unstable oscillations fixed point and the stable manifolds of the two locally stable To produce flagellar motion, dynein activity not only prop- fixed points. Trajectories then more slowly approach one of agates but also changes flagellar curvature, and periodically the stable fixed points. The asymptotic behavior resembles and autonomously switches the direction of bending. This that of a damped particle on the energy surface of Fig. 5 c. behavior is exhibited by the discrete GC model (32,33,36) After the very short initial transient, the interdoublet sep- and is replicated by our set of PDEs. We will first consider aration is well described by the simplified model small-amplitude (linear) motion. Chlamydomonas flagella exhibit small-amplitude, symmetric oscillations that propel dh f Nðp1 p0Þ kN z AeqðhÞ h: (36) the cells backward under conditions of increased cytosolic dt bN bN calcium concentration (45).

Biophysical Journal 107(7) 1756–1772 1764 Bayly and Wilson

FIGURE 6 Trajectories in (a) dh/dt h plane and (b) dA/dt A plane rapidly approach one- dimensional curves as in Fig. 5. These curves resemble cubic polynomial functions, with three zeros representing two stable equilibria and one unstable (threshold) equilibrium. (c) The cubic polynomial analogy to local attachment dynamics:

vA/vt z E(A) ¼ CAA(A – Ath)(1 – A). Derivatives are normalized by tN ¼ bN/kN. To see this ﬁgure in color, go online.

Linearized equations of motion The time constant tN ¼ bN/kN describes the local (linear- To describe small-amplitude motion about a straight, equi- ized) behavior of A; the new variable S0 is the baseline dif- librium configuration, nonlinear terms in Eqs. 12–14 may ference in tension in the doublets. This resting difference in be neglected, leading to a single equation (13,14,26) that tension provides a baseline level of coupling between curva- can be written: ture and dynein activity even when the flagellum is almost straight. The parameter CS ¼ (dAeq/dh)/2bN controls the EI þ af þ c ¼ 0: (40) j;ssss T;ss Nj;t magnitude of the coupling. The stability of a straight flagellum is significantly affected when S s 0. Solutions to Eq. 40 must also satisfy appropriate boundary 0 Generalizing the approach of Camalet and Ju¨licher (12), conditions (see Appendix A). Riedel-Kruse et al. (13), and Hilfinger et al. (14), we seek A model of dynein regulation is needed to express fT in solutions of the forms terms of j and system parameters. In the GC model, fT ¼ f þ f , where f and f are given in Table 3. The vari- TP TR TP TR ðs; tÞ¼ ð tÞ~ðsÞ; able A ¼ A – A represents net dynein activity in the prin- j exp s j P R f ðs; tÞ¼ ð tÞ~f ðsÞ; (45) cipal bend direction. The net shear force is thus expressed as T exp s T Aðs; tÞ¼expðstÞA~ðsÞ; vj fT ¼ f ðp1 p0ÞA 2kT aj 2bT a : (41) T vt with s ¼ a þ iu (a and u are real). Each such solution that Using the results above (Figs. 5 and 6 and Eqs. 36–39), satisfies the equation of motion and all boundary conditions dynein dynamics on each side are governed by an equation is a solution mode. If a > 0, the mode grows exponentially. such as If a number M such modes are found with exponents s and ðmÞ m shape j~ ðsÞ, then a solution can also be formed from any vAP vhP dAeq z , : (42) linear combination of these modes: vt vt dh

XM ðmÞ Dynein activity is thus governed by the propagation dy- smt ~ jðs; tÞ¼ ame j ðsÞ: namics of Eq. 39 coupled to global flagella motion by ten- m ¼ 1 sion and curvature (SPvj/vs or SRvj/vs). After linearization, we obtain the expressions In general, for arbitrary initial conditions, the least stable 4 mode will dominate the response. vA 1 vj EId v A ¼ A CSS0 þ ; (43) To illuminate the mechanisms of mechanical feedback t t s 2b s4 v N v N v from flagellar shape and facilitate comparison to other models, we neglect for the moment the relatively small ZL fourth-order spatial derivative term in Eq. 43. The resulting S ¼ 2 f p d ¼ 2f p ðL sÞ: (44) 0 T 0 z T 0 expressions can be combined and simplified to obtain an or- s dinary differential equation in j~,

FIGURE 7 Propagation of interdoublet separation. (a) The decrease in separation h(s,t)is computed from the original PDE of interdoublet separation (Eqs. 33 and 30), and plotted versus s/L at discrete times t ¼ 0.025 n, with n ¼ 1, 2, ...20. (b) Solutions of the simpliﬁed excitable system (Eq. 39) plotted versus s/L at the same discrete times. See Movie S1 and Movie S2 in the Support- ing Material for corresponding animations. Note the domain is extended to s ¼ 2/L to better visualize propagation. To see this ﬁgure in color, go online.

Biophysical Journal 107(7) 1756–1772 Propagation and Instability in Flagella Motion 1765 4 ~ 2 ~ 2 ~ d j d dj d j ~ can these effects become significant and lead to instability? EI c1ðsÞ ð1s=LÞ c2ðsÞ þscNj ¼ 0; ds4 ds2 ds ds2 Both bifurcation conditions are favored when c1(s) becomes (46) relatively large. For a flagellum with fixed physical parameters (e.g., a, f T, EI, and kN,) this can occur if the baseline attachment probability p0 and the maximal attachment prob- with coefficients ability p1 increase. If these probabilities increase with cyto- 2 solic calcium concentration, it could explain the propensity c1ðsÞ¼2p0ðp1 p0Þaf T CStNL ðtNs þ 1Þ; (47) for fast, low amplitude, oscillations under high calcium conditions. 2 c2ðsÞ¼2a kT þ sbT ; (48) One notable characteristic of the curvature-feedback term in the GC model is that it is proportional to 1 s/L.The feedback is thus strongest at the proximal end, which will or, equivalently, encourage switching at the base and proximal-to-distal d4 ~ d3 ~ propagation. This spatial dependence also complicates the EI j ½c ð Þð1 s=LÞ j 4 1 s 3 solution of the eigenvalue problem, so that numerical ds ds (49) 2 ~ methods (e.g., weighted-residual or finite-element calcula- d j ~ ½c2ðsÞ2c1ðsÞ þ scNj ¼ 0: tions) are required to find even the natural modes and fre- ds2 quencies of oscillation. Numerical eigenanalysis and simulation Comparison to sliding-controlled and curvature-controlled models Stability analysis of the linearized equations of motion (Eq. 49 and boundary conditions Appendix A) was performed by By comparing Eqs. 46 and 49 to analogous equations (below using the method of weighted residuals (47) with up to N ¼ and in Appendix C) describing sliding-controlled models 12 test functions to obtain a matrix form of the eigenvalue (12–14) and curvature-controlled models (15,21,24,46), it problem. The resulting matrix eigenvalue problem was is apparent that the GC model includes feedback from solved using the software MATLAB (The MathWorks, Na- both curvature and shear deformation. For example, in the tick, MA). The free vibration modes of a uniform, fixed-free model of sliding-controlled dynein regulation proposed in beam were used as trial and test functions. Riedel-Kruse et al. (13), the equation analogous to Eq. 46 is To check the stability analysis and characterize the subse- d4 ~ d2 ~ quent behavior, solutions to the equations of motion were j 2 j ~ EI a cðsÞ þ scNj ¼ 0; (50) ds4 ds2 also found using the PDE modeling capability of a commer- cial finite-element software package (COMSOL, Ver. 4.3a; COMSOL, Burlington, MA). The one-dimensional domain ð Þ¼~f ðsÞ=u~ðsÞ¼~k ð Þþ b~ ð Þ where c s T T s s T s is the dynamic was discretized into 50 elements with quartic interpolation. shear stiffness, which is complex. To account for active Eigenvalue/eigenfunction calculations (300 maximum itera- forces that depend on sliding, c can have both negative tions, relative tolerance 1 106) and time-marching sim- real or imaginary components (negative stiffness or damp- ulations (backward differentiation formula, variable time ing) at a given value of s. In the classic curvature-controlled step, relative tolerance 1 104) were performed. Repre- model of Hines and Blum (24), the analogous equation is sentative results were confirmed at finer spatial resolution 4 ~ h i 3 ~ and smaller tolerance values. d j m0 d j ~ EI þ scNj ¼ 0: (51) ds4 st þ 1 ds3 Results of stability analysis By analogy to other models, parameter changes that in- Both the sliding-controlled and curvature-controlled models crease the magnitude of the delayed curvature feedback above exhibit dynamic instability at critical values of the pa- and decrease the value of the shear feedback were expected rameters c and m0, respectively. For the sliding-controlled to lead to instability. Increasing the baseline probability of model, sperm-flagella-like oscillations occur when both cross-bridge attachment, p0, does both. Fig. 8 illustrates real and imaginary parts of c are negative due to load- the effects of p0, L, and EI on the stability of the straight dependence of dynein activity (13). In the curvature- equilibrium position. A dynamic instability occurs when controlled model, instability is associated with relatively complex eigenvalues cross into the right half-plane with large values of the ratio m0L/EI (24). nonzero imaginary part: Re(s) > 0; Im(s) s 0. Notably, the GC hypothesis can explain both negative Physically, the oscillations in Fig. 8 can be interpreted as apparent shear stiffness (the coefficient of d2j~=ds2 in Eq. the result of a switching mechanism (16,36). Active shear in 49) and large-magnitude, time-delayed curvature feedback one doublet pair (e.g., the P side) induces a tension differ- (the term proportional to d3j~=ds3). Under what conditions ence in the doublets on that side, which combined with

Biophysical Journal 107(7) 1756–1772 1766 Bayly and Wilson

FIGURE 8 (a) Frequency (imaginary part of eigenvalue: u ¼ Im(s)) as a function of baseline

attachment probability p0 and ﬂexural modulus, EI.(b) Region in which the straight position is unstable (R(s) > 0). At a given value of EI, a Hopf bifurcation occurs (eigenvalues of the GC model cross into the right half-plane) as the baseline probability of attachment is increased. (c and d) Anal-

ogous plots of Im(s) and Re(s) > 0 in the p0L plane. (e and f) The least stable mode of the linear-

ized GC model (either Eq. 46 or 49) with p0 ¼ 0.10, p1 ¼ 0.30, CS ¼ 0.5 mm/pN-s; other parameters are as in Table 4. The real (solid) and imaginary (dashed) part of the mode is shown; the frequency u ¼ 310 rad/s (49.3 Hz). See Movie S3 and Movie S4 for corresponding animations. To see this ﬁgure in color, go online.

3 curvature, produces a transverse force (SP vj/vs) that pushes is –kNh – bN vh/vt – kN3h (see Eqs. 27–28 and Appendix D). the doublets apart and eventually terminates the active shear. Simulations were started from an initial straight configura- This transverse force corresponds to the ‘‘global transverse tion. Bending is initiated by asymmetry in the baseline prob- force’’ described by Lindemann (32,33,36). At the same abilities of attachment. time, the corresponding passive shear force on the opposite For a wide range of parameter values, the system exhibited doublet pair produces a transverse force (SR vj/vs), which large-amplitude oscillations (Fig. 9). The solutions are char- pulls the doublets on the passive side together and initiates acterized by proximal-distal propagation of dynein activity, active shear on that side. sustained by the local positive feedback between dynein activity and interdoublet separation. As predicted by Linde- mann (32,33,36), when the curvature and doublet tension Large-amplitude, nonlinear flagella oscillations become large enough, the global transverse force (S vj/vs) To explore large-amplitude, nonlinear behavior of the promotes doublet separation at the proximal end of the active model, direct numerical simulations (i.e., time-marching) side. This leads to switching of dynein activity and reversal of the full flagellum equations (Eqs. 12–14) with fixed- of the direction of motion. For reasonable parameter values free boundary conditions (Eqs. 15–20) together with the (Table 4), the frequency, amplitude, and waveform resemble equations of interdoublet separation (Eqs. 25 and 26) and those of Chlamydomonas in forward swimming. dynein activity (Eqs. 27–32), were performed in the soft- Thus, local positive feedback between dynein activity and ware COMSOL as described above. Time-marching simu- interdoublet separation, combined with global negative lation allows exploration of large-amplitude, transient, feedback between flagella curvature (modulated by doublet and nonperiodic behavior. In simulations, an additional tension) and interdoublet separation, can explain the large- nonlinear elastic restoring force, proportional to the cube amplitude oscillations of flagella. Notably, when the base- of the displacement, was added to the passive shear line probability of attachment is symmetric and relatively and transverse interdoublet forces to account for physical large, small-amplitude oscillations arise via a classical dy- nonlinearities that limit motion (46). With only linear namic instability. Such higher levels of baseline dynein ac- elastic elements, unstable oscillations tend to grow expo- tivity increase the steady-state difference in tension in the nentially. The passive shear force in the full equations is doublets, which amplifies the destabilizing effects of curva- 3 thus –kTu – bT vu/vt – kT3u , and the passive transverse force ture feedback.

Biophysical Journal 107(7) 1756–1772 Propagation and Instability in Flagella Motion 1767

FIGURE 9 Results of time-marching simulation. (a) Asymmetric waveform with baseline

probabilities of dynein attachment p0P ¼ 0.05 and p0R ¼ 0.01. (b) Symmetric waveform with baseline probabilities of dynein attachment p0P ¼ p0R ¼ 0.10. (c–h) Corresponding time series of tangent angle, attachment probability, and deﬂec- tion at s ¼ L/2. (c, e, and g) Correspondence to the waveform in panel a.(d, f, and h) Correspon- dence to the waveform in panel b. Except as noted, parameter values are in Table 4. See Movie S5 and Movie S6 for animations.

DISCUSSION AND CONCLUSIONS turn, dynein activity is modulated by the distance between doublets. PDEs corresponding to a continuum version of the GC hy- This model includes both local positive feedback, as pothesis of flagella motion were derived and analyzed. dynein activity pulls doublets together and increases the The GC hypothesis has been supported previously by a likelihood of attachment, and global negative feedback, in discrete computer model (32,33,36).The solutions to the equations derived here exhibit characteristic features of which accumulated tension difference and increasing curva- the behavior of Chlamydomonas flagella. Three key behav- ture tend to pull the doublets apart. This negative feedback iors are reproduced by these equations: is greater proximally than distally, favoring proximal to distal propagation. 1. Propagation of dynein activity after a suprathreshold Our analysis provides theoretical justification for the GC stimulus; hypothesis in several ways. First, the equations of interdoub- 2. Instability and emergence of small-amplitude, symmet- let separation reduce to a form of the EFK equations ric oscillations under some conditions; and (40,41,48), which is an example of an excitable system 3. The occurrence of large-amplitude, asymmetric oscilla- known to exhibit propagating wavefronts. This system is tions under other conditions. closely related to classical reaction-diffusion models (40). The PDEs were derived from a simplified but rigorous Direct simulation of the original equations of interdoublet mechanical analysis of doublet pairs on opposite sides of separation (Eqs. 33 or 30) or a simplified excitable model the flagellum. The effects of dynein activity on interdoublet (Eq. 39) led to patterns of dynein activity that propagated separation, both directly, and indirectly through the effects from the location of a suprathreshold stimulus. Propagation of accumulated tension and curvature, are included. In speed depends on elastic and viscous properties of the

Biophysical Journal 107(7) 1756–1772 1768 Bayly and Wilson

flagellum, as well as parameters that describe the effects of emphasize key mechanisms; in each such case, the accuracy interdoublet spacing on cross-bridge formation. of the approximation was verified by simulation of the full The emergence of oscillations via dynamic instability is equations. confirmed in a linearized version of the system of equations. The modes and frequencies predicted by this linearized analysis correspond to the small-amplitude solutions of CONCLUSIONS the full nonlinear model under certain conditions. These The partial differential equations of motion for a flagellum modes resemble the symmetric motion of Chlamydomonas in viscous fluid, accounting for changes in interdoublet sep- flagella during backward swimming (49). aration, reproduce propagating dynein activity, dynamic Direct simulations of the full GC model (Eqs. 12 and 13), instability, small-amplitude symmetric oscillations, and with unequal parameter values governing dynein activity on large-amplitude asymmetric oscillations. The derivation the P and R sides (Table 4), lead to large-amplitude, asym- and analysis of these equations represent useful contribu- metric oscillations similar to those exhibited by Chlamydo- tions to the development of the GC hypothesis, which in monas in forward swimming (49–51). In the absence of this form can be directly compared to other PDE models active dynein forces, these equations predict the expected of flagellar oscillation. behavior of an elastic beam in viscous fluid: the flagellum returns asymptotically to its straight position. The equations derived here clearly represent a simplified APPENDIX A: BOUNDARY CONDITIONS description of a real flagellum. Only two doublet pairs are Interdoublet separation modeled, one on each side of the axoneme; their mechanical properties are described by idealized linear elastic and In the fixed-free flagellum, the boundary conditions for h associated with viscous constitutive equations. The model of dynein regula- Eqs. 25 and 26 are the following: tion is simplistic, although it captures the key postulate of Zero deflection at base the GC hypothesis—namely, that cross-bridge attachment and detachment are governed by interdoublet separation. hð0; tÞ¼0: (52) For simplicity, and to clarify the role of doublet tension and curvature, we have not included two mechanical forces Zero angle of deflection at base that may modulate interdoublet separation: h 1. The local transverse force due to stretching of nexin links v ð0; tÞ¼0: (53) by tangential sliding (32,33,36), and vs 2. The mechanical coupling force between doublets on one side (e.g., R) due to dynein activity on the other (P) side Zero bending moment at distal end (32,33,36). 2h The local transverse forces due to sliding are thought to be v ðL; tÞ¼0: (54) small relative to other interdoublet forces (32,36). vs2 There is substantial evidence that the central pair and radial spokes are important in dynein regulation, inasmuch Zero transverse force at distal end as mutants lacking these components are immotile (52– 55) and dynein activity in specific doublets correlates with 3h v ðL; tÞ¼0: (55) central pair orientation (54,56). Structurally, the spokes vs3 and central pair are critical to axoneme mechanics (Fig. 1); their passive mechanical properties may be their primary contribution and underlie the correlation of orientation to dynein activity (9). In the context of GC hypothesis, Linearized equations of flagella motion the central apparatus may also modulate parameters such as In the linearized model (Eq. 40), if the flagellum is fixed at its proximal end baseline probability of cross-bridge formation, or attach- (s ¼ 0) and free at its distal end (s ¼ L), solutions satisfy the following: ment/detachment rates. Note that our equations do not Zero angle at base take into account the variations in fluid force due to motion of the body and neighboring flagella, which were recently jð0; tÞ¼0: (56) measured in Guasto et al. (57) and Drescher et al. (58). Despite its limitations, the continuum equations of the GC model derived here reproduce many features of flagellar Zero normal velocity at base motion. In the derivations described above, various approx- ðv ð0; tÞ¼0Þ : EI ð0; tÞþaf 0; t ¼ 0: (57) imations are made and terms are neglected in order to N j;sss T;s

Biophysical Journal 107(7) 1756–1772 Propagation and Instability in Flagella Motion 1769 ~ ~ ~ f ðsÞ¼cðsÞu~ðsÞ; u~ðsÞ¼u~0 þ a jðsÞjð0Þ ; Zero bending moment at distal end (62) cðsÞ¼K þ sl: EIj;sðL; tÞ¼0: (58)

Substitution into the equation of motion Eq. 11 as in Riedel-Kruse et al. Zero transverse force at distal end (13) leads to Eq. 50, rewritten below as ðNðL; tÞ¼0Þ : EI ðL; tÞþaf L; t ¼ 0: (59) d4 ~ d2 ~ j;ss T j 2 j ~ EI a c þ scNj ¼ 0: (63) ds4 ds2

APPENDIX B: PARAMETER VALUES Following Riedel-Kruse et al. (13), a nondimensional version of this equation is Table 4 contains typical parameter values for the equations of flagella mo- 0000 00 tion and interdoublet separation, summarized in Tables 2 and 3. To model j~ cj~ þ sj~ ¼ 0 (64) an asymmetric beat, values of parameters such as hon and hoff may differ between sides P and R (see values expressed in parentheses below). These parameter values are either from prior studies in the literature (as cited in Ta- with nondimensional parameters defined analogy to those in Riedel-Kruse ble 5) or estimated by comparison to such values. Results of this study are et al. (13), generally robust to variations in parameter values of 550%. The parameter EI falls within a range bounded by estimates derived from s ¼ s=L; 2 the flexural rigidity of multiple individual doublets (EI z 500 pN-mm )(59), 4 z s ¼ scNL EI; estimates of flexural modulus from intact sperm flagella doublets (EI 900 pN-mm2)(60), and estimates (61) in which the authors accounted for the ef- ¼ a2L2 EI; 2 c c (65) fects of shear stiffness, kT to obtain a value for flexural EI z 100 pN-mm . Parameter were either measured in Chlamydomonas reinhardtii flagella and (C), or estimated from results in bull sperm (B), sea urchin sperm (U), or dð , Þ isolated microtubules (M). ð , Þ0 ¼ : ds

APPENDIX C: COMPARISON OF STABILITY Boundary conditions for the ﬁxed-free case of Eq. 21 are also written in ANALYSES nondimensional form (13), as follows:

Nondimensional, linearized versions of sliding-controlled dynein regula- Zero angle at base tion (12–14), curvature-controlled regulation (24), and the current continuum version of the GC model are compared below. j~ð0Þ¼0: (66)

Model of sliding-controlled dynein regulation Zero normal velocity at base

Riedel-Kruse et al. (13), Camalet and Julicher (12), and Hilfinger et al. (14) ¨ 000 0 have suggested that interdoublet sliding provides the feedback necessary to j~ ð0Þcj~ ð0Þ¼0: (67) produce sustained, propulsive flagellar oscillations. In these studies the effects of the feedback-controlled dynein motors, plus the shear stiffness and friction due to passive axoneme components, are combined into effective Zero bending moment at distal end stiffness (K) and friction (l) coefficients as 0 vu j~ ð1Þ¼0: (68) fT ðs; tÞ¼ Ku þ l : (60) vt Zero transverse force at distal end Interdoublet sliding displacement, u(s,t), is related to bending by the ki- 00 nematic relationship (13,24)as ~ ~ ~ j ð1Þc u0 þ jð1Þjð0Þ ¼ 0; (69)

uðs; tÞ¼u0 þ aðjðs; tÞjð0; tÞÞ; (61) where u0 ¼ u~0=a (13). where u0 ¼ u(0,t) is the sliding permitted at the base of the ﬂagellum. In the sliding-controlled feedback mechanism, local interdoublet sliding reduces the load per dynein motor, leading to recruitment of more dyneins Model of curvature-controlled dynein regulation and greater net force (13). To represent this hypothesis mathematically, the effective stiffness and friction coefﬁcients are allowed to become negative Hines and Blum (24) proposed a model for curvature control in a continuum (12–14). beam model of the form of Eq. 51. The total shear force is expressed as To investigate stability, the sliding-controlled model can be expressed using Eqs. 60 and 61 in Eqs. 40 and 45 to express the shear force in terms af ðs; tÞ¼S ðs; tÞS ðs; tÞ; (70) of a complex transfer function c(s): T d r

Biophysical Journal 107(7) 1756–1772 1770 Bayly and Wilson

dð , Þ where Sr is the shear force due to deformation of passive components and Sd ð , Þ0 ¼ ; s ¼ s L; is the active dynein force. Local dynein shear force is dynamically regulated ds by curvature, according to the equation vSd 1 vj 4 ¼ m0 Sd ; (71) and again s ¼ scNL =EI. The dimensionless coefficients are vt t vs dG where m0 (N-m) controls the dynein response to curvature. The passive g1ðsÞ¼ ; (79) component of shear is related to interdoublet sliding displacement u(s,t) 1 þ hGs by the nonlinear relationship .pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2L2 a2 2 ð Þ¼2 k þ b ð Þ ; (80) Sr ¼ k1u 1 1 1 þ k2u ; (72) g2 s T s 2 T g1 s EI L cN where k1 and k2 are nonlinear stiffness parameters. By direct analogy to the 2 2 4 analysis of the sliding-controlled model, the nondimensional equation and where dG ¼ 2p0ðp1 p0Þaf T CStNL =EI, and hG ¼ tN EI=cNL . It is clear boundary conditions for the curvature-controlled model of Hines and Blum that the GC model includes feedback analogous to classical delayed curva- (24) are ture control, as well as the possibility of negative shear impedance (as in the

0000 000 sliding-controlled model). These destabilizing effects increase with dynein ~ d ~ ~ f j j þ sj ¼ 0: (73) force ( T ), attachment probability (p0), and length (L), and decrease with 1 þ hs ﬂexural rigidity (EI).

APPENDIX D: NONDIMENSIONAL FORM OF THE Zero angle at base FULL EQUATIONS OF THE CONTINUUM GC j~ð0Þ¼0: (74) MODEL In the main text, the equations of the continuum GC model are derived and Zero normal velocity at base analyzed in dimensional form. This is done for simplicity and to keep the physics as clear as possible. The equations may also be written in nondimensional form if desired. To do so, we specify a characteristic force, char- 000 d 00 j~ ð0Þ j~ 0 ¼ 0: (75) acteristic force density, and characteristic time, as well as aspect ratio and 1 þ hs ﬂexural rigidity ratio: 2 3 4 F0 ¼ EI L ; f0 ¼ EI L ; T0 ¼ cNL EI; Zero bending moment at distal end (81) ε ¼ a L; G ¼ EId EI: 0 j ð1Þ¼0: (76) Next, we deﬁne the nondimensional independent variables:

Zero transverse force at distal end 0 ð , Þ 1 ð , Þ s ¼ s L; ð , Þ ¼ v ¼ v ; (82) vs L vs 00 d 0 j~ ð1Þ j~ 1 ¼ 0: (77) 1 þ hs _ vð , Þ 1 vð , Þ t ¼ t T0; ð , Þ¼ ¼ : (83) vt T0 vt 4 4 where s ¼ scNL =EI. The new nondimensional variables are h ¼ tEI=cN L (the ratio of the dynein time constant to the viscoelastic time constant of the flagellum), and d ¼ m0L=EI (the ratio of the characteristic dynein moment Finally, we define the nondimensional dependent variables: m0 to the moment required to bend the flagellum to curvature 1/L). Note that passive shear forces may be included due to elastic or dissipative ele- h ¼ h a; (84) ments, but in the original model (24) the linearized shear restoring force (the linear approximation to Eq. 72) vanishes. Others, notably Brokaw T~ ¼ T F ; (21,22) have also explored the potential role of curvature control. 0 N~ ¼ N F ; (85) 0 and S~ ¼ S F ; The continuum GC model 0 The linearized, continuum GC model (Eq. 46 or 49) becomes, in nondimen- ~f ¼ f f ; T T 0 sional form, ~ (86) f N ¼ fN f0; 0000 000 00 j~ g ðsÞð1 sÞj~ g ðsÞj~ þ sj~ ¼ 0; (78) 1 2 K~ ¼ K T ; off off 0 (87) K~ ¼ K T ; where on on 0

Biophysical Journal 107(7) 1756–1772 Propagation and Instability in Flagella Motion 1771

TABLE 5 Nondimensional equations of flagella motion and sperm flagella at very low ATP concentrations. Cell Struct. Funct. interdoublet separation 32:17–27. Equations of global flagella motion 5. Hayashibe, K., C. Shingyoji, and R. Kamiya. 1997. Induction of tem- porary beating in paralyzed flagella of Chlamydomonas mutants by 00 00 0 0 0 ~ ~ cT ~ cT ~ 2 application of external force. Cell Motil. Cytoskeleton. 37:232–239. Tangential force balance T Nj 1 þ c N j c Tðj Þ ¼ 0 N N 6. Lindemann, C. B., and D. R. Mitchell. 2007. Evidence for axonemal 00 0 0 00 0 ~ cN ~ ~ cN ~ 2 _ Normal force balance N þ 1 þ c T j þ Tj c Nðj Þ ¼ j distortion during the flagellar beat of Chlamydomonas. Cell Motil. T T Cytoskeleton. 64:580–589. j00 þ ε~f þ N~ ¼ 0 Moment balance T 7. Omoto, C. K., and G. B. Witman. 1981. Functionally significant cen- ~0000 ~ ~ 0 Interdoublet separation εGh ¼ 2f N Sj tral-pair rotation in a primitive eukaryotic flagellum. Nature. 290: R 1 708–710. Tension difference S~ ¼ 2 ~f dz s T 8. Omoto, C. K., I. R. Gibbons, ., G. B. Witman. 1999. Rotation of the in doublets Mol. Biol. Cell. ~ e ~ ~ _ ~ 3 central pair microtubules in eukaryotic flagella. 10:1–4. Tangential interdoublet f ¼ f p kT j bT j k3T j T T 9. Mitchell, D. R. 2003. Orientation of the central pair complex during force density 3 flagellar bend formation in Chlamydomonas. Cell Motil. Cytoskeleton. ~f ¼ fe p ~k h~ b~ h~_ ~k h~ Normal interdoublet N N N N 3N 56:120–129. force density 10. Brokaw, C. J., and D. R. Rintala. 1975. Computer simulation of Dynein activity ¼ þ flagellar movement. III. Models incorporating cross-bridge kinetics. Cross-bridge attachment p p0 A(p1 – p0) J. Mechanochem. Cell Motil. 3:77–86. probability 11. Murase, M. 1990. Simulation of ciliary beating by an excitable dynein Cross-bridge attachment vA=vt ¼ K~ ð1 AÞK~ A on off model: oscillations, quiescence and mechano-sensitivity. J. Theor. Biol. dynamics 146:209–231. 12. Camalet, S., and F. Ju¨licher. 2000. Generic aspects of axonemal beating. New J. Phys. 2:241–243. ~k ¼ ak f ; T T 0 13. Riedel-Kruse, I. H., A. Hilfinger, .,F.Ju¨licher. 2007. How molecular b~ ¼ ab ðT f Þ; (88) T T 0 0 and motors shape the flagellar beat. HFSP J. 1:192–208. ~ 3 k3T ¼ a k3T f0; 14. Hilfinger, A., A. K. Chattopadhyay, and F. Ju¨licher. 2009. Nonlinear dynamics of cilia and flagella. Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79:051918. ~k ¼ ak f ; N N 0 15. Brokaw, C. J. 2009. Thinking about flagellar oscillation. Cell Motil. b~ ¼ ab ðT f Þ; Cytoskeleton. 66:425–436. N N 0 0 (89) ~k ¼ a3k f ; 16. Lindemann, C. B., and K. A. Lesich. 2010. Flagellar and ciliary 3N 3N 0 and beating: the proven and the possible. J. Cell Sci. 123:519–528. ~c ¼ ac =ðf T Þ: N N 0 0 17. Woolley, D. M. 2010. Flagellar oscillation: a commentary on proposed mechanisms. Biol. Rev. Camb. Philos. Soc. 85:453–470. Using these quantities, a full set of equations in nondimensional form are 18. Brokaw, C. J. 1971. Bend propagation by a sliding filament model for given in Table 5, corresponding to the dimensional versions in Table 2. flagella. J. Exp. Biol. 55:289–304. ~ ~ In simulations, cubic nonlinear stiffness terms involving k3T and k3N 19. Brokaw, C. J., and D. Rintala. 1977. Computer simulation of flagellar were added to the interdoublet forces to reflect the limits of extensibility movement. V. Oscillation of cross-bridge models with an ATP-concen- of elastic elements. tration-dependent rate function. J. Mechanochem. Cell Motil. 4: 205–232. 20. Brokaw, C. J. 1999. Computer simulation of flagellar movement. VII. SUPPORTING MATERIAL Conventional but functionally different cross-bridge models for inner and outer arm dyneins can explain the effects of outer arm dynein Six movies are available at http://www.biophysj.org/biophysj/supplemental/ removal. Cell Motil. Cytoskeleton. 42:134–148. S0006-3495(14)00813-3. 21. Brokaw, C. J. 1985. Computer simulation of flagellar movement. VI. Simple curvature-controlled models are incompletely specified. The authors thank Anders Carlsson, Susan Dutcher, and Ruth Okamoto for Biophys. J. 48:633–642. helpful discussions. 22. Brokaw, C. J. 2002. Computer simulation of flagellar movement. VIII. Support from National Science Foundation grant No. CMMI-1265447 and Coordination of dynein by local curvature control can generate helical bending waves. Cell Motil. Cytoskeleton. 53:103–124. the Children’s Discovery Institute is gratefully acknowledged. 23. Brokaw, C. J. 2005. Computer simulation of flagellar movement. IX. Oscillation and symmetry breaking in a model for short flagella and nodal cilia. Cell Motil. Cytoskeleton. 60:35–47. REFERENCES 24. Hines, M., and J. J. Blum. 1978. Bend propagation in flagella. I. Deri- vation of equations of motion and their simulation. Biophys. J. 23: . 1. Nicastro, D., C. Schwartz, , J. R. McIntosh. 2006. The molecular ar- 41–57. chitecture of axonemes revealed by cryoelectron tomography. Science. 313:944–948. 25. Fu, H. C., T. R. Powers, and C. W. Wolgemuth. 2007. Theory of swimming filaments in viscoelastic media. Phys. Rev. Lett. 99:258101. 2. Satir, P. 1968. Studies on cilia. 3. Further studies on the cilium tip and a ‘‘sliding filament’’ model of ciliary motility. J. Cell Biol. 39:77–94. 26. Fu, H. C., C. W. Wolgemuth, and T. R. Powers. 2008. Beating patterns of filaments in viscoelastic fluids. Phys. Rev. E Stat. Nonlin. Soft Matter 3. Mitchison, T. J., and H. M. Mitchison. 2010. Cell biology: how cilia Phys. 78:041913. beat. Nature. 463:308–309. 27. Fu, H. C., C. W. Wolgemuth, and T. R. Powers. 2009. Swimming 4. Ishikawa, R., and C. Shingyoji. 2007. Induction of beating by imposed speeds of filaments in nonlinearly viscoelastic fluids. Phys Fluids bending or mechanical pulse in demembranated, motionless sea urchin (1994). 21:33102.

Biophysical Journal 107(7) 1756–1772 1772 Bayly and Wilson

28. Cibert, C. 2008. Are the local adjustments of the relative spatial fre- 44. Peletier, L. A., A. I. Rotariu-Bruma, and W. C. Troy. 1998. Pulse-like quencies of the dynein arms and the b-tubulin monomers involved in spatial patterns described by higher-order model equations. J. Differ. the regulation of the ‘‘9þ2’’ axoneme? J. Theor. Biol. 253:74–89. Equ. 150:124–187. 29. Murase, M., and H. Shimizu. 1986. A model of flagellar movement 45. Bray, D. 2001. Cell Movements: from Molecules to Motility. Garland based on cooperative dynamics of dynein-tubulin cross-bridges. Publishing, New York. J. Theor. Biol. 119:409–433. 46. Brokaw, C. J. 1972. Computer simulation of flagellar movement. I. 30. Murase, M., M. Hines, and J. J. Blum. 1989. Properties of an excitable Demonstration of stable bend propagation and bend initiation by the dynein model for bend propagation in cilia and flagella. J. Theor. Biol. sliding filament model. Biophys. J. 12:564–586. 139:413–430. 47. Meirovitch, L. 1997. Principles and Techniques of Vibrations. Prentice 31. Murase, M. 1991. Excitable dynein model with multiple active sites for Hall, Upper Saddle River, NJ. large-amplitude oscillations and bend propagation in flagella. J. Theor. 48. Bonheure, D. 2004. Multitransition kinks and pulses for fourth order Biol. 149:181–202. equations with a bistable nonlinearity. Ann. Inst. Henri Poincare, Anal. Non Lineaire. 21:319–340. 32. Lindemann, C. B. 1994. A model of flagellar and ciliary functioning which uses the forces transverse to the axoneme as the regulator of 49. Brokaw, C. J., and D. J. Luck. 1983. Bending patterns of Chlamydomo- dynein activation. Cell Motil. Cytoskeleton. 29:141–154. nas flagella. I. Wild-type bending patterns. Cell Motil. 3:131–150. 33. Lindemann, C. B. 2002. Geometric clutch model version 3: the role of 50. Bayly, P. V., B. L. Lewis, ., S. K. Dutcher. 2010. Efficient spatiotem- the inner and outer arm dyneins in the ciliary beat. Cell Motil. Cytoskel- poral analysis of the flagellar waveform of Chlamydomonas reinhard- eton. 52:242–254. tii. Cytoskeleton (Hoboken). 67:56–69. . 34. Lindemann, C. B. 2007. The geometric clutch as a working hypothesis 51. Bayly, P. V., B. L. Lewis, , S. K. Dutcher. 2011. Propulsive forces on for future research on cilia and flagella. Ann. N. Y. Acad. Sci. 1101: the flagellum during locomotion of Chlamydomonas reinhardtii. 477–493. Biophys. J. 100:2716–2725. 35. Brokaw, C. J. 2014. Computer simulation of flagellar movement X: 52. Smith, E. F., and W. S. Sale. 1992. Regulation of dynein-driven micro- Science. doublet pair splitting and bend propagation modeled using stochastic tubule sliding by the radial spokes in flagella. 257:1557–1559. dynein kinetics. Cytoskeleton (Hoboken). 71:273–284. 53. Smith, E. F. 2002. Regulation of flagellar dynein by the axonemal central apparatus. Cell Motil. Cytoskeleton. 52:33–42. 36. Lindemann, C. B. 1994. A geometric clutch hypothesis to explain oscillations of the axoneme of cilia and flagella. J. Theor. Biol. 168: 54. Wargo, M. J., and E. F. Smith. 2003. Asymmetry of the central appa- 175–189. ratus defines the location of active microtubule sliding in Chlamydomo- nas flagella. Proc. Natl. Acad. Sci. USA. 100:137–142. 37. Brokaw, C. J. 1991. Microtubule sliding in swimming sperm flagella: direct and indirect measurements on sea urchin and tunicate spermato- 55. Smith, E. F., and P. Yang. 2004. The radial spokes and central appa- zoa. J. Cell Biol. 114:1201–1215. ratus: mechano-chemical transducers that regulate flagellar motility. Cell Motil. Cytoskeleton. 57:8–17. 38. Vernon, G. G., and D. M. Woolley. 1995. The propagation of a zone of activation along groups of flagellar doublet microtubules. Exp. Cell 56. Wargo, M. J., M. A. McPeek, and E. F. Smith. 2004. Analysis of micro- Res. 220:482–494. tubule sliding patterns in Chlamydomonas flagellar axonemes reveals dynein activity on specific doublet microtubules. J. Cell Sci. 117: 39. Peletier, L. A., and W. C. Troy. 1996. Chaotic spatial patterns described 2533–2544. by the extended Fisher-Kolmogorov equation. J. Differ. Equ. 129: 57. Guasto, J. S., K. A. Johnson, and J. P. Gollub. 2010. Oscillatory flows 458–508. induced by microorganisms swimming in two dimensions. Phys. Rev. 40. Akveld, M. E., and J. Hulshof. 1998. Travelling wave solutions of a Lett. 105:168102. fourth-order semilinear diffusion equation. Appl. Math. Lett. 11: 58. Drescher, K., R. E. Goldstein, ., I. Tuval. 2010. Direct measurement 115–120. of the flow field around swimming microorganisms. Phys. Rev. Lett. 41. van den Berg, J. B., J. Hulshof, and R. C. Vandervorst. 2001. Traveling 105:168101. waves for fourth-order parabolic equations. SIAM J. Math. Anal. 59. Gittes, F., B. Mickey, ., J. Howard. 1993. Flexural rigidity of micro- 32:1342–1374. tubules and actin filaments measured from thermal fluctuations in 42. Peletier, L. A., W. C. Troy, and R. C. A. M. Vandervorst. 1995. Station- shape. J. Cell Biol. 120:923–934. ary solutions of a 4th-order nonlinear diffusion equation. Differ. Equ. 60. Okuno, M. 1980. Inhibition and relaxation of sea urchin sperm flagella 31:301–314. by vanadate. J. Cell Biol. 85:712–725. 43. Peletier, L. A., and W. C. Troy. 1997. Spatial patterns described by the 61. Pelle, D. W., C. J. Brokaw, ., C. B. Lindemann. 2009. Mechanical extended Fisher-Kolmogorov equation: periodic solutions. SIAM J. properties of the passive sea urchin sperm flagellum. Cell Motil. Cyto- Math. Anal. 28:1317–1353. skeleton. 66:721–735.

Biophysical Journal 107(7) 1756–1772