Peripheral doublet microtubules and wave generation in eukaryotic flagella

J. theor. Biol. (1981) 93,769-784

Peripheral

Doublet Microtubules and Wave Generation in Eukaryotic Flagella

PHILIP H. CROWLEY, CRAIG J. BENHAM, SUZANNE M. LENHART AND JANET L. MORGAN Thomas Hunt Morgan School of Biological Sciences and the Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506, U.S.A. (Received

19 March

1981)

The shape and propagation of waves produced by eukaryotic flagella depend on the three-dimensional arrangement and physical-chemical properties of peripheral substructures. The modeling analysis presented here, which assumes force-moment equilibrium and neglects the viscous resistances of the medium, shows how substructural arrangements characteristic of 9 + 0, 9 + 1, and 9 + 2 axonemes can yield their characteristic wave patterns. When flexural stiffnesses are equal along all axonemal radii, any non-uniform doublet shearing pattern propagated distally at constant rate, with successive pairs 4 cycle out of phase, should generate helical waves. When stiffnesses differ greatly on different radii, but the stiffness pattern is the same for all cross-sections, any such shearing pattern should yield planar waves resembling sine-generated curves. Propagated axonemal bending results from the active bending moment produced by local shearing of doublet pairs. Uniformly twisting the doublets about the axonemai axis cannot directly alter the magnitude of the active bending moment. If dynein cross-bridges are activated by shear displacement between peripheral doublets, then the resulting distribution of the active bending moment will be appropriate for balancing the elastic moment in a propagated bending wave. 1. Introduction It is now widely accepted that motile eukaryotic flagella generate traveling waves by the active sliding displacement of peripheral doublet microtubules-the sliding microtubule hypothesis (Satir, 1965, 1968). The basic relations between flagellar forces and moments on one hand and wave shapes on the other have been investigated using analytical models and computer simulation (e.g. Machin, 1958; Brokaw, 1971; Rikmenspoel, 1971; Lubliner & Blum, 1971; Hines & Blum, 1979; Morgan et al., 1980). But such studies have generally not focused on how the three-dimensional shape and physical-chemical properties of peripheral substructures are 769

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responsible for wave shape and propagation, the question of primary concern here. We begin by citing some observations and making several straightforward assumptions about axonemal structure and function. Then from equations for the moments along axonemes having the appropriate physical properties, we obtain solutions for the curvature in two or three orthogonal planes with respect to fixed axes. Additional assumptions about the force distribution near a region of maximal doublet shearing yield plausible distributions of shearing along the length and around a cross-section. This approach should ultimately provide a means of obtaining quantitative predictions of the shape under various conditions. 2. Methods,

Assumptions

and Derivation

An isolated demembranated flagellum in reactivation buffer generates essentially normal traveling waves (Gibbons & Gibbons, 1972). Any region along the length of the axoneme appears capable of active bending (Brokaw & Gibbons, 1973; Shingyoji, Murakami & Takahashi, 1977), and even short segments sectioned from a flagellum can exhibit sustained beating (Douglas & Holwill, 1972; Gibbons, 1973). Thus the basic machinery for translating microtubule sliding into bending apparently consists of the structures present along the entire length of the axoneme: the nine peripheral doublet microtubules, inner and outer dynein arms, nexin linksand, for flagella that contain them, radial spokes, central sheath, and singlet microtubule (Afzelius, 1959; Warner & Satir, 1974; Summers & Gibbons, 1971; Gibbons & Gibbons, 1973; Witman, 1978) (see Fig. 1). In the interest of clarity and tractability, we therefore focus here on a flagellum unattached to any structure capable of damping or otherwise modifying the waves generated by the flagellum itself (see also Lubliner & Blum, 1971). The inner and outer arms, containing the dynein ATPase, are known to supply most or all of the shear force responsible for the axial displacement (sliding) of adjacent microtubules (see Gibbons & Gibbons, 1972). We assume that attachment of these dynein arms between adjacent tubules and hydrolysis of ATP (i.e. “firing”) are associated with maximal intertubule shear (see Hines & Blum, 1979). But the dynein arms appear incapable of also providing the shear resistance necessary for bending (see Summers & Gibbons, 197 1,1973); shearing is resisted instead by the elastic nexin links connecting each pair of adjacent peripheral doublets (Brokaw & Simonick, 1977; Blum & Hines, 1979; see Fig. 1). Activity of the dynein ATPase may be facilitated mechanically (Brokaw & Rintala, 1977), perhaps by

MICROTUBULES Outer membrane

AND CWllU3l singlet microtubule

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Peripheral doubler microtubules

FIG. 1. Diagrammatic representation of a short segment of a helical 9 + 1 flagellum, oblique view. Structures of known or possible importance in wave generation are identified. The firing locus (a region of maximal shear) on a pair of peripheral doublet microtubules is viewed through the outer membrane, with arrows indicating the directions of sliding brought about by the dynein arms (not shown). Notice the sigmoid shapes of the firing pair and the stretching and tilting of the nexin links. The radial spokes and nexin links are actually arranged helically around the axoneme (Warner, 1974), but for clarity we represent them in this and the next figure as co-planar within evenly spaced cross-sections. Local firing may tend to twist the peripheral doublets as indicated by the curved arrows.

adjacent shearing on the same doublet pair or by nexin-induced shifts in the distance between tubules (see below). We therefore assume that with adequate ATP present, firing (maximal shearing) travels at constant speed down the length of each peripheral doublet pair. Local firing brings the adjacent region on each pair to a “firing threshold”, like an action potential transmitted along a neural axon. In contrast, firing at a locus along one doublet pair should inhibit firing in the same cross-section by adjacent pairs (see Fig. 2); this is probably a facet of the poorly understood process of co-ordinating firing among doublet pairs and implies that at any instant such firing is likely out of phase along the length, at least between adjacent doublets. We assume that firing in any cross-section shifts at a constant rate around the peripheral doublet assemblage once per flagellar beating cycle (Bradfield, 1955; Costello, 1973; Rikmenspoel, 1977; Hiramoto & Baba, 1978). Such a pattern is consistent with firing distribution along adjacent pairs of doublets that are out of phase along the length by one-ninth cycle; it

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FIG. 2. A simplified view of an unrolled segment of peripheral doublets (vertical lines l-9) joined by shear-distorted nexin links (other lines a-i). If the figure were rolled up such that the lines indicating doublet 1 were superimposed, the contiguous nexin links would trace out the edges of oblique planar slices through the cylinder or peripheral doublets. Note that firing (maximal shear) loci shown with heavy diagonal lines are longitudinally offset between adjacent pairs, perhaps partly because firing is longitudinally facilitated but laterally inhibited. If pair 5-6 fires at e, and all other pairs are relaxed, elastic forces should arrange the other e-links according to the broken line; thus the 5-6 firing shears the other pairs opposire to the direction that presumably facilitates firing.

also agrees with the hypothesis for the 9 + 2 axoneme that the central pair rotates and acts as a distributor to co-ordinate firing around the periphery (Omoto & Kung, 1979). Though the central elements may thus participate in co-ordination or help determine axonemal stiffness, the beating documented in IZ+ 0 flagella (e.g. Golstein etal., 1979; Presier etal., 1980) and in clusters of several doublets excised from 9 +2 flagella (Nakamura & Kamiya, 1978) indicates that these central structures are not fundamental to flagellar wave generation (Blum & Hines, 1979). A key assumption of this analysis is that at any instant the axoneme can be considered to be at force-moment equilibrium: No net forces or moments act on the axoneme, so that during beating it simply assumes a sequence of equilibrium (or near-equilibrium) configurations (see also Blum & Hines, 1979). One justification for this approach is the striking similarity between rigor waves of ATP-starved flagella and waves produced by actively swimming flagella (Gibbons & Gibbons, 1974). Moreover, beating flagella predominantly assume the two- and three-dimensianal shapes (Hiramoto & Baba, 1978) that minimize potential energy (sine-generated waves and helices respectively-see below), as expected at or near equilibrium. Also, by taking beat frequency and wavelength to be model parameters rather than variables to be predicted, we can neglect the relatively small external viscous forces and moments (Brokaw, 1966; Lubliner & Blum, 1971) that

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influence those factors much more than axonemal shape. We conceptually partition internal bending moments into active (M,, imposed on the axoneme by the sliding filaments) and elastic (M,, the passive response of the axoneme to curvature) Lubliner & Blum, 1971; Lubliner, 1973). At equilibrium kf,+hf,=o. (1) Equations describing the relations among M,, and M,, the axonemal curvature K, and the active shear force per unit length f have been obtained (see Machin, 1958; 1963; Brokaw, 1966, 1971; Lubliner & Blum, 1971; Lubliner, 1973), subject to the following assumptions. (a) The flagellum is a set of parallel filaments (microtubules) that are thin, flexible, and maintain constant length. (b) The filaments have linearly elastic connections (nexin links) that offer passive elastic resistance only. (c) An additional set of transient connections (dynein bridges) contributes an active shear force per unit length f The resulting equations are: (2)

and

(3) where s is the contour distance along the axonemal axis, K is the coefficient relating inter-doublet shearing to shear force, E is Young’s modulus for a doublet, I,, is the second moment of area of the axonemal cross-section, and I is the inertial moment of all doublets. The derivatives are partial derivatives because the moments M, and M,, curvature K, and active shear force f are also functions of time t. Equations (l)-(3) combined yield [:I$-[I+;-K=[+]$

(4)

Equations (2)-(4), derived for bending in a single plane, are strictly applicable to bending in two orthogonal planes (i.e. three-dimensional bending) only with uniform flexural stiffness in all directions around the axonemal cross-section. This is because unequal stiffnesses imply a preferred (but not exclusive) bending plane toward which the axoneme will twist, considerably complicating the algebra. But we will be concerned here only with substructural configurations that can be assumed to fit the special cases to which equations (2)-(4) apply.

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f(s, t) is obtained as follows. Pairwise shearing of adjacent doublets is responsible for the active bending moments directed radially away from and perpendicular to the axonemal axis (see Fig. 3). If firing moves around Central sheath

mIcrotubule--*

FIG. 3. Diagrammatic partial cross-section of a 9 + 1 or 9 + 2 flagellum showing the net active bending moment on a firing pair and the axonemal substructures that may participate in producing and damping this moment. As the right-hand doublet moves up (toward the base) and the left-hand doublet moves down (toward the tip; Sale & Satir, 1977), elastic restoring forces from doublets and nexin pull the pair slightly closer together and nearer to the central sheath (heavy arrows; Warner, 1978); this may stimulate the dynein ATPase between the doublets (Summers, 1975) and allow the radial spokes to transmit torque to the central sheath. During firing, the plane of the cross-section tilts up on the right and down on the left relative to the axonemal axis.

a cross-section and along a doublet pair at constant rates (as argued above), then the entire pattern of active bending moments, including the net active bending moment per unit length m, must be similarly translated on the axoneme. For a given cross-section, projecting m on an axis x orthogonal to the axonemal axis (z) and fixed with respect to the peripheral doublets then yields m as a sine function of contour distance S, so that at some later time the projection of m on the axis x in the same cross-section will be consistent with the moment moving into the cross-section from elsewhere along the doublet pair. But since that active bending moment is proportional in magnitude to the active shear force (i.e. m, afx), we have fx(s, t) =f* sin [27r(~t +s/h)]

(5)

for the boundary condition f,(O, 0) = 0, where fX is the magnitude of the active shear force per unit length along the x axis, f* is the magnitude of

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the active shear force per unit length on a cross-section, w is the flagellar beat frequency, and A is the wavelength of the flagellum measured along the axonemal (z) axis. (Note that this particular boundary condition is chosen simply for convenience: the cross-section at s = 0 must have fV increasing to zero at some time within the “initial” cycle; this time is arbitrarily taken to be t = 0.) Similarly, fY(S' t,=f"

cos [27T(wt+s/h)],

(6)

where f, is the magnitude of the active shear force per unit length along the y axis, mutually orthogonal to the x and z axes. Because the active bending moments are perpendicular to the z axis-and because the flexural stiffnesses along the x and y axes are assumed equal--f,(s, t) = 0. At force-moment equilibrium, equation (4) can be solved for K,(s, t) and Kv(S, t) as an ordinary differential equation in s by substituting from equations (5) and (6) as if t were constant. Since f is a sinusoidal function of contour distance s, a glance at the form of equation (4) suggests that K is also sinusoidal. In fact, 27rAI,f* cos [2r(wt + s/n 11 K,(S,

t)=-

47r”ElJ+KA*(lo+I)

’

(7)

and sin [27r(wt+s/A)] 47r%IJ+KA’(I,,+1)

2?rAI,f*

Kv(S’ ‘)=

’

(8)

Note also that because fZ(s, t) = 0, no twist can be imposed on the axoneme by the active shearing considered here. Nine + 1 and n + 0 flagella seem consistently to exhibit radial symmetry of structure about the axonemal axis in electron micrographs (Phillips, 1974), suggesting equal stiffness on x and y as in equations (7) and (8) (see Figs 4a and b). Nine + 2 flagella, on the other hand, are not radially symmetrical and are believed commonly to retain permanent rigid linkages between the central singlets and between the peripheral doublets conventionally designated as numbers 5 and 6 (Warner & Satir. 1974; see Fig. 4~). To date, the 5-6 linkage has been found only in metazoa (C. K. Omoto, personal communication). Rough calculations taking the peripheral doublets and central singlets to be hollow rods, with dimensions estimated from electron micrographs (e.g. Warner & Satir, 1974), suggest that the sum of inertial moments is increased by a factor of about three by the permanent linkages (Hayden et al., in preparation). But if these linkages were rigid enough to increase Young’s modulus disproportionately in the plane of the linkages, then the overall impact on relative flexural stiffnesses could

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Outer membrane

(b) 9*1

(a) 9+0

Cc) 9+2

FIG. 4. Diagrammatic comparison of axonemal substructures in 9 + 0, 9 + 1, and 9 + 2 flagella. Radially symmetric 9 + 0 (and other n i- 0) and 9 + 1 flagella are reported to generate helical waves, whereas 9 + 2 flagella exhibit several different wave patterns. When the 9 + 2 flagellum features “permanently” linked central and 5-6 pairs, these linkages are generally parallel to each other and perpendicular to the plane of bending (Tamm & Horridge, 1970).

be considerably greater. If stiffness were much greater in one plane, say y-z, than in the other (x-z), then fV and K~ could be negligibly small, a case considered further below. We determine the axonemal shape implied in equations (7) and (8) for any particular time by the method of Euler angles: (Hayden et al., in preparation). Three simultaneous, first-order differential equations are solved numerically for the angular orientation of arbitrarily many points on the axoneme relative to fixed “laboratory axes”, the angles are transformed into Cartesian co-ordinates, and the results are displayed graphically. (A)

THE

ACTIVE

SHEAR

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ON

A

CROSS-SECTION

Consider the distribution about a firing point of active shear forces along an isolated pair of peripheral doublets at a particular time (Fig. 5a). The force per unit length f(s) should be maximal (fO) at the firing site so, declining distally at an initially accelerating rate in both directions, and finally declining more gradually toward asymptotes at zero. This kind of pattern is well described by a Gaussian (normal) curve: IS~-s,,P/?d* f(s)

=fo

e

(9)

where d is the standard deviation of the distribution. (Note that the magnitudes of fO and d probably reflect doublet stiff nesses and the elasticity of the nexin links.) Suppose now that several firing points spaced A units apart are located along the isolated doublet pair, each contributing an identical (but longitudinally displaced) Gaussian distribution of forces. Summing the forces yields an undulating pattern along the length, symmetrical about its maxima and minima, which can very closely resemble the oscillations of a cosine function (see Fig. 5b).

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(b)

5, Contour

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firmg locus

distance

FIG. 5. (a) Pattern of shearing and nexin-link orientation in an isolated pair of peripheral doublets firing at the locus indicated by the arrows. (b) Distribution of shear force along the length of an isolated pair of peripheral doublets. Arrows distinguish the pattern for single-locus firing (as in Fig. 5a; roughly a Gaussian distribution) from that for multiple-locus tiring (overlapping Gaussian curves successively displaced by the wavelength A ; the summed force is approximately sinusoidal along the length).

Now connecting this doublet pair to the others via nexin links requires at equilibrium that the sum of shear forces around any cross-section be zero (Lubliner, 1973); this constraint also applies for each wavelength along a doublet pair, because cross-sectional and lengthwise bending patterns must match, as argued above. Ideally, the resulting shape should also (a) minimize changes per unit length in the magnitude of shear force, smoothing out the pattern, and (b) minimize the total amount of stretching by nexin links, thus minimizing the elastic forces along the links. Since a sine-generated wave best satisfies (a) (Langbein & Leopold, 1966), and a saw-tooth wave best satisfies (b), the actual pattern probably represents a compromise between these wave forms. The rule of thumb in mechanics that any cross-section of an object subjected to bending remains planar (e.g. Popov, 1968) implies a sinusoidal distribution of shears around the cross-section. A sine (or cosine) function does yield absolute magnitudes that are intermediate between those of sine-generated and saw-tooth functions of the same amplitude and wavelength. We therefore assume that the distribution of shear forces (and of shear displacements, with linear elasticity of the nexin) is at least closely approximated by cosine functions around the cross-section and along the length of a single doublet pair. This result will now permit us to express the net active shear force f” in terms of the shear force at a firing site on an isolated doublet pair, fo. Let r be the distance from the center of an unsheared peripheral doublet to the center of the axoneme. We choose a cross-section containing one firing pair, of which the right-hand doublet pushes up through the plane of the cross-section as the left-hand doublet pushes down, creating active

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bending moments on the axoneme. The components of these moments along the axis parallel to the link joining the firing tubules cancel, but the components outward along the axis bisecting the 40” angle between the radial spokes sum to 2rfo sin (7r/9), where 7~/9 is half of the spoke angle (i.e. half of 40” in radians; see Fig. 3). The above analysis of the force distribution implies that the bending moments between adjacent doublets will follow the function [2rf0 sin (7r/9)] cos (2~i/9), where i designates a particular pair of adjacent doublet microtubules (1 d i 5 9; i = 9 indicates the firing pair with the maximal bending moments, specified by the bracketed term). Thus the entire flagellar cross-section rotates about the axis of the net bending moment bisecting the firing pair (Fig. 3). Projecting the resultant bending moment of each pair on the resultant for the firing pair introduces another multiplicative cosine term; summing over all pairs yields m = 2rfo sin (7r/9) i

co? (27ri/9) = 9rfo sin (r/9),

(10)

i=l

where m is the net active bending moment in the direction of the firing pair. (By symmetry, all compoents orthogonal to this axis cancel.) The magnitude of m can now be readily decomposed into two multiplicative terms: the axonemal diameter 2r, and the active shear force on a cross section f* = 9fo sin (r/9)/2.

(11

This can be substituted into equations (5)-(g). (B)

LOCALIZING

FIRING

Experimental results from the local reactivation of detergent-extracted flagella by iontophoretic application of ATP (Shingyoji eta/., 1977) provide a test of the analysis presented here. Removing the outer flagellar membrane from a sea urchin spermatozoan and diluting out the ATP from the medium yields a straight, relaxed axoneme. Then adding a small amount of ATP at one point on the axoneme produces a sigmoid bend, with the inflection point at the site of ATP application. We argue that at the very low ATP levels for which this was observed, the pairs in the cross-section of application compete for sufficient ATP to achieve firing, and the first pair to fire tends to inhibit the others (see Fig. 2). Because of physical priming of dynein of the firing pair, the firing region should be extended on that pair rather than moving to any of the others as additional ATP is added. Alternatively, locally high levels of ATP stimulate a small segment

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of the axoneme to exhibit the entire beating sequence, though damped by the adjacent ATP-starved regions, as observed in the experiments. We assume that the active bending moment of the firing pair follows equation (9), being maximal (fO) at the firing point. So that shears around the firing cross-section sum to zero with minimal stretching of nexin links, each of the other eight pairs contributes shear forces of magnitude -f,,/8. Expressing these as moments projected on an axis bisecting the firing pair [cf. equations (10) and (ll)] gives mr = 9rfo sin (7rj9 114,

(12)

f? = 9fo sin (r/9)/8,

(131

and

where subscripts I specify firing at one point on the axoneme (via iontophoresis). Now f? replaces fO in equation is), since fO applies to the maximal shear force on an isolated doublet pair; but with f? substituted, equation (9) refers to the entire axoneme. For this case, equation (4) must be solved numerically to find K~, where x is taken to be the axis bisecting the firing pair. Clearly, K, = K, = 0.

3. Results When the flexural stiffnesses are the same on both x and y axes, and sufficient ATP is present to permit active beating, the flagellum assumes a left-handed helical shape having contour wavelength A (Fig. 6a); through time, the wave progresses along the length at the frequency w. When the flexural stiffness is greatly increased on one of the axes, as in Figs 6b, c, and d, bending is restricted to the plane of the page, with zero curvature on orthogonal axes. Here the flagellum has assumed the distinctive sinegenerated wave form (cf. a sine wave), most noticeable in the low-stiffness example Fig. 6d. And finally, when firing is restricted to a single point on the axoneme, it becomes sigmoid, with the inflection point at the firing point (Figs 6e and f). Two different stiffnesses in the plane of bending are illustrated. If the firing point is stretched out as a firing region along a single pair, this region remains straight, and the axoneme retains the same shape beyond the firing region as for point firing (Fig. 6g). The latter example may apply best to the experiments of Shingyoj: et al. (1977), since it is probably very difficult to restrict the zone of ATP application to what could be considered a single point.

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FIG. 6. Wave shapes obtained by computer simulation. Magnitudes of lumped parameters, chosen arbitrarily, are indicated below. Total length is 1; for ad, A = f. (a) A helix, with an arrow showing the orientation of its axis relative to the plane of illustration; K, = (2n/9) sin (27rs/A), ~~ = (2~/9) cos (27rs/h), and K, = 0. (b-d) Sine-generated waves, with bending restricted to the plane of observation. (b) K” = K~ = 0. (d) K, = K, = (r/9) sin (2rr$/A ), q = K; = 0. (c) K, = (2n/9) sin (2m/A), (47r/9) sin (27rs/A), Q = K* = 0. (e-g) Single-locus and extended-locus firing. (e) Single-locus pattern; firing at the midpoint sc = 0.5 (heavy dot), d = s0/2, the curvature coefficient is 7r/9 (as for Fig. 6b), K,, = K, = 0. (f) Single-locus pattern as for e, except that the curvature coefficient is 2~r/9. (g) Extendedlocus firing (heavy line) with parameters as for f.

4. Discussions and Conclusions

We emphasize several significant implications of the analysis and results. (1) Established properties of axonemal substructures, and a minimal set of defensible assumptions, suffice to account for common two- and three-dimensional wave shapes in space and through time. Substructural arrangements characteristic of 9 + 0 and 9 + 1 axonemes yield helical beating patterns, and arrangements typical of many 9+ 2

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axonemes yield shapes approaching planar sine-generated waves; these are the wave forms and propagation patterns generally observed for the respective flagella (Hiramoto & Baba, 1978; Phillips, 1974; Schrevel & Besse, 1975; Rikmenspoel, 1978; Baccetti et al., 1979). (2) With negligible viscous moments, active bending moments normal to the plane of observation are applied only minimally or not at all along an axonemal bend; dynein-induced maximal shear forces and bending moments occur in straight regions or at axonemal inflection points, as implied by equations (5)-(6) (see Brokaw, 1971; also Shingyoji et al., 1977). This has not been generally appreciated in the interpretation of longitudinal transmission electron micrographs of flagella and cilia, for which maximal bending and maximal shear forces are often assumed to coincide (e.g. Warner & Satir, 1974; Warner, 1978). (3) Propagated axonemal bending depends on local shearing of doublet pairs. Simultaneous uniform shearing along the entire lengths of the doublet pairs cannot produce bending. Unless the active shear force or the shear resistance varies along the length of a doublet, no active bending moment is imposed and thus no curvature results (see equations (5)-(6); compare swimming and straight flagella in Goldstein, 1979). (4) Since twist is orthogonal to the two components of the active bending moment, twisting the doublets about the axonemal axis cannot directly alter the magnitude of the active bending moment. (5) When flexural stiffnesses are equal along all axonemal radii, an) non-uniform doublet shearing pattern propagated distally at constant rate, with successive pairs around a cross-section $ cycle out of phase, should generate helical waves. (6) If dynein cross-bridges are activated by shear displacement between peripheral doublets, then the resulting distributions of the active bending moment will be appropriate for balancing the elastic moment in a propagated bending wave. As indicated in the derivation, this conclusion does not depend on restrictive boundary conditions. From the considerable diversity of axonemal architecture (see Phillips, 1974; Blum & Hines, 1979), we have focused on the peripheral elements of 9 + n axonemes. These structures seem currently to offer the best combination of parsimony (i.e. all functional axonemes in this class feature basically the same peripheral elements) and available information (i.e. most electron micrography, experimental analyses, and mutant studies are of 9 + n) for understanding flagellar swimming. Nevertheless, the mechanisms considered here may be generally applicable to all axonemes capable of

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producing waves and having parallel doublets equally spaced around the periphery of a circular cross-section. If so, then what is the function of the central structures in flagella that contain them, and what peripheral structures perform that function in flagella that lack them? Despite a heavy bias in total research effort favoring flagella having central structures, which may be partly responsible for the apparent diversity within this category, it seems clear that such flagella do in fact exhibit a much broader repertoire of wave forms than do those lacking central structures. For 9 +2 axonemes, this includes both sinegenerated (often considered to be arc-1ine-e.g. Johnson, Silvester & Holwill, 1979) and helical (Brokaw, 1966) waves, the “breast-stroke” patterns of Chlamydomonas (Hyams & Borisy, 1978), and several other patterns and pattern reversals (e.g. Holwill, 1966); it also includes the standard ciliary rowing pattern (e.g. Baba, 1979) and some more complex movements by cilia (e.g. Kuznicki, John & Fonseca, 1970). Though some of this variety may simply be attributable to differences in axonemal length (see Costello, 1973) or in the presence or absence of a basal body (Douglas & Holwill, 1972), much of it may depend on regulation of dynein firing, doublet sliding, or the plane(s) of bending via the central structures (Omoto, 1979: Omoto & Kung, 1979; Satir, 1980; Blum & Hines, 1979). In that case, for n +0 flagella, an adequate but perhaps less flexible kind of co-ordination must emerge from the various inhibitions, stimulations, and other interactions among the peripheral structures alone. In addition to a possible regulatory role in II + 1 and n + 2 flagella, the central structures certainly help stiffen the axoneme; besides a possible influence on the plane of bending emphasized in the foregoing analysis, greater axonemal stiffness may permit much higher beat frequencies (compare n + 0 in Baccetti et al., 1979; Goldstein, Besse & Schrevel, 1979; Prensier et al., 1980; with 9+2 in Brokaw, 1966), thus greatly increasing swimming speed. Finally, the central structures may also function as a skeleton, helping to avoid the collapse or entanglement of the peripheral assemblage (Blum & Hines, 1979). The two most commonly observed flagellar wave patterns, the planar sine-generated wave and the helix, apparently share some intriguing mathematical properties. A sine-generated wave is a very close approximation to the most frequent random path for a continuous planar curve of a given length between two points, where the probability distribution of path-deviation angles is Gaussian Won Schelling, 1951, 1964). This curve also minimizes the variance of the change in path direction per unit length (Langbein & Leopold, 1966), which may minimize the potential energy of a curved axoneme as it does for the meander path of a river (Leopold & Langbein, 1966). Since a helix clearly minimizes the variance of the change

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in path direction per unit length in three dimensions (in fact this variance is zero for a helix, as implied by the interchangeability of all finite segments of equal length), a helix may represent the curved shape of minimal potential energy in three dimensions. Though there certainly are other constraints involved in determining the optimal shape of an axoneme, the least energetically expensive from the set of functionally equivalent shapes will be the most stable, and this may generally resemble the shape of minimal potential energy. We thank W. Dentler, S. F. Goldstein, M. Hines, and D. C. Leigh for helpful suggestions in the early stages of this study, and T. L. Hayden for challenging us to write this paper. We are especially grateful to C. J. Brokaw for encouragement and thorough critiques of several drafts, and to C. K. Omoto for enthusiastically sharing with us many of her ideas about flagellar structure and function.

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