## ABSTRACT

Equations are developed to calculate the relative displacements of the doublet microtubules at the tip of a cilium when the microtubules twist about the axis of the organelle.

Displacements measured from electron micrographs show asymmetry (or skew) which can be matched quantitatively by the theoretical model with the appropriate selection of twist angle and orientation of the axoneme with respect to the plane of beat.

For

*Elliptio*cilia the experimental results are consistent with a planar effective stroke and a recovery stroke involving a three-dimensional bend. The plane of the effective stroke is not normal to a surface containing the central pair of microtubules but contains microtubule 2 to produce the observed skew.This model for the beat also explains the range of orientations of axoneme observed in sections through the metachronal wave.

## INTRODUCTION

The sliding microtubule model of ciliary motion is now well documented by a series of careful experiments involving geometrical tests (Satir, 1965, 1968; Warner & Satir, 1974; Sale & Satir, 1976), preparation of triton- and trypsin-treated axonemes (Summers & Gibbons, 1971, 1973), mutant analysis (Allen & Borisy, 1974; Witman, Fay & Plummer, 1975) and other criteria. General mathematical models involving sliding as the basis of motion have been formulated by Brokaw (1976), Rikmenspoel (1971), and others. The models are reasonably successful in predicting the bend parameters of certain cilia and sperm tails under various conditions such as changes in the viscosity of the surrounding medium, but none explain how cilia execute threedimensional movements. This may be attributable to the assumption made in all the models that the microtubules exhibit no twisting along the ciliary length, whereas the real axoneme is undoubtedly twisted (Satir, 1963), possibly in critical regions (Gibbons, 1975). For example, previous work utilizing a simple planar model of the ciliary beat and an untwisted axoneme has shown that there is good quantitative agree-ment between the experimentally determined positions at which the doublets terminate in the ciliary tip and the predictions of the sliding microtubule model (Satir, 1968), but the experimental curves exhibit a skew that the simple model cannot produce and that may be due to twisting of the microtubules. In this model the plane of bending is normal to the surface containing the central microtubules.

As we shall show, to reproduce the experimental curves more precisely, it is necessary to add to the model one or both of two assumptions: either that the doublets twist about the axis of the cilium, similar to the way in which rope fibres twist about the rope’s axis, or that the bend plane is inclined to that assumed in the simple model. Schreiner (1977) has presented a mathematical treatment of this problem, but his results have not yet found practical application. In the present paper we shall derive general expressions for the relative microtubule displacements as a function of the angle of twist, compare our results with earlier theoretical and experimental work and obtain information relating to the local plane of bending of an organelle and the beat pattern of an individual cilium.

## GENERAL EXPRESSIONS FOR MICROTUBULE DISPLACEMENTS

*R(s)*, where

*s*is the -position of the element measured along the cylinder from an arbitrary origin (Fig. 1). If

*Z*

_{i}is the radial distance of the ith inextensible microtubule from the neutral surface (Fig. 2), the longitudinal displacement between the two (relative to their positions in a straight cilium) is

*δ α*is the angle subtended by the arc at the centre of the circle of which it is part. Satir’s (1968) measurements are of displacements relative to microtubule 1, which for the 1th microtubule is given by

*Z*

_{i}can be expressed in terms of the radius (

*r*) of the axoneme and the angle (

*v*

_{i}) between the radius to the

*i*th filament and the radius contained by the neutral surface (Fig. 2): Equation (2) now becomes assuming the nine peripheral doublets to be equally spaced on the circumference of a circle so that the adjacent doublets subtend an angle of 2π/9 at the centre. In general, when the axoneme is twisted,

*v*

_{1}will be a function of a, the bend angle; the tip displacement can only be calculated if the form of this function is known. The general relationship between

*v*

_{1}and a may be found from a consideration of Fig. 3, which shows a short section

*δs*of the cilium with doublet 1 inclined at an angle

*β*to a generatrix of the cylinder (curvature of the element has been omitted for the sake of clarity). The arc length

*αa*can be obtained in terms of

*β*and of

*δv*

_{1}, so that

*/r*is the rate of change of

*v*

_{1}with length along the cilium. Since ∂

*v*

_{1}/∂

*s*represents the variation with distance of the angular twist of the whole axoneme relative to a reference section, it provides a convenient measure of the twist and will be designated τ. The expression for the axonemal twist then becomes where

*V*

_{10}is a reference orientation of doublet 1. The integral of equation (8) cannot be evaluated without a knowledge of the behaviour of

*R*(

*s*) and τ. If, for example,

*R(s)*and τ are constant, we may write where

*R*is the constant value of

*R(s)*. It is perhaps worth noting that equations (4) and (8) are quite general, and are applicable to two- and three-dimensional shapes of a cilium.

### Application to planar bends

*Axoneme with no twist*. If τ = o, the axonemal fibres remain parallel to the axis of the cilium along its length, and

*v*

_{1}is constant (=

*V*

_{10}, equation 9). Equation (4) can then be integrated immediately to yield the displacement

*s*

_{1i}between micro-tubules 1 and

*i:*where α is the total bend angle. This equation can be re-written in the form If

*V*

_{10}is set equal to , SO that the bend plane is perpendicular to that containing the central microtubules, equation (10) for the tip displacement

*s*

_{1i}is formally identical to that deduced by Satir (1968), and predicts that a curve of displacement against filament number should be symmetric about a point midway between filaments 5 and 6 as shown in Fig. 4(

*a*). If the bend plane is arbitrary so that , the resulting displacement curve is no longer symmetric about this point (Fig. 4

*b*).

*Axoneme with twist*. For a twisted axoneme, i.e. one where τ ≠ 0, equation (8) must be used to obtain as a function of α prior to the integration of equation (4). If

*R(s)*and

*r*are constant, equation (9) is the expression for

*v*

_{1}which is substituted into equation (4), leading, upon integration, to By setting ≠ = 0, corresponding to an untwisted axoneme, equation (10) is regained.

*i*, the filament number, and the significance of the equation may be appreciated better in the form Where In this form it is clear that, for given

*B*and

*v*

_{1}, the dependence of s

_{1i}on

*i*is identical to that displayed by equation (10) and is represented in Fig. 4(

*a*) (for ) and Fig. 4(

*b*) (for other values of

*v*

_{1}).

*s*

_{1i}with

*v*

_{10}for given values of 7 and a exhibits maximum positive and negative values as shown in Fig. 5. Differentiation of equation (13) shows that the turning points occur for

Two values (e.g. o and *π*) are sufficient to provide the two maxima referred to above and yield values of *V*_{10} appropriate to these points on the curve of Fig. 5. As can be seen from equation (12) the dependence of *s*_{1i} on τ is, in general, more complex than its dependence on *V*_{10}. The term outside the square brackets of equation (12) can be written in the form rα sin *ϕ/ ϕ* where *ϕ* = *R*τα/2. For small angles, sin *ϕ/ ϕ* remains close to unity (e.g. if *ϕ =* 30°, sin *ϕ* / *ϕ* is about 0·955), so that equation (14) can be used to find the approximate values of τ (for a given *V*_{10}) to give maximum displacements provided *R* τ α */2* remains less than about π/6. If the geometry of the system is such that *R* τ α */2* is outside the range ± π/6, and/or if high accuracy is needed, the value of τ required to give the maximum displacement, at a given value of *v*_{10}, for a specific microtubule is most conveniently found by numerical methods.

From equation (14) and the ensuing discussion it is clear that as τ is altered so also must be the value of *V*_{10} to give the maximum displacement condition for a particular microtubule. This reflects the fact that the displacement of a microtubule will depend on its path within the curved part of the flagellum. Thus a tubule which is predominantly on the outside of the bend will suffer a greater displacement than one on the inside. The extent to which a tubule remains on the outside of the bend will depend on its starting position (*v*_{10}) as well as on the twist per unit length (τ). The condition for the maximum relative displacement of a particular microtubule thus allows an infinite number of combinations of *v*_{10} and τ, all other quantities associated with the system being assumed constant. Of these combinations, that with τ = 0 gives a greater relative displacement than the others; the maximum possible displacements, calculated as projections parallel to the axis of the axoneme, are obtained in a system where no twisting of the microtubules occurs. However, because of the slow variation with *ϕ* of the sin *ϕ* / *ϕ* term referred to earlier, the curve of *s*_{1i} against τ is relatively flat in the region of τ = o (Fig. 6). An alternative treatment of the condition for maximum displacement involves maintaining *v*_{10} at a constant value, as discussed earlier, in which case *s*_{1i} is maximal for the value of τ given approximately by equation (14). The variation of *s*_{1i} with τ for this condition is also shown in Fig. 6.

### Application to three-dimensional bends

In his discussion of twisting filaments in an organelle which adopts a threedimensional configuration, Schreiner (1977) makes use of the mathematical concept of torsion used to define the linear rate at which a curve twists in space (see e.g. Coxeter, 1963). This concept is useful in discussing the response of a solid rod to three-dimensional bending, and will be considered further in the Discussion, but for the assemblage of microtubules involved in the present argument it is more convenient to consider the variation in the angle, ε, between the principal radius of curvature at a point on an organelle and the line which joins the centre pair of microtubules at that point (Fig. 7). In an axoneme with no twist each microtubule runs parallel to the central line of the organelle, so that e remains constant along its length. Where a twisted axoneme occurs, e varies with distance along the filament and the rate of change of *e* with length corresponds to τ. The principal radius of curvature at a section on a curved rod is normal to the neutral surface. Further, the line joining the centre of the flagellar cross-section to doublet 1 is, in an ideal structure, normal to the line joining the central pair of microtubules. Hence the angle *e* is equal to msed in the earlier analysis of planar bends.

The equations which describe the filament displacement in an axoneme sustaining a three-dimensional bend with constant radius of curvature (i.e. a helix) and twist are identical to those obtained for planar bends (equation (10) for τ = o, equation (12) for any value of τ). Fig. 4 thus represents the tip displacements for the nine doublet microtubules in a helical axoneme as well as in an organelle bent in a circular arc. As for the general two-dimensional case, microtubule displacements in an axoneme bent into an arbitrary three-dimensional shape require the evaluation of *v*_{1} from equation (8) before equation (4) can be integrated.

## COMPARISON OF THEORY WITH EXPERIMENT

The experimental results to be compared with the theoretical predictions are those of Satir (1968), who used electron microscopy to examine serial sections of the tips of gill cilia from the freshwater mussel *Elliptio complanatus*. Essentially similar results have been obtained by Sale & Satir (1976) for *Tetrahymena* cilia, although certain additional assumptions as to numbering the microtubules were made in this case. The beat form of *Elliptio* cilia has been determined (Satir, 1967) and can conveniently be separated into effective and recovery strokes (Fig. 8). The serial sectioning technique allowed the relative displacements of peripheral microtubules of cilia fixed in known stroke positions to be measured. For comparison with the theoretical predictions, the accumulated angle subtended by the bends in the cilium is required. This is most reliably available for those cilia in positions such that they carry a single bend (i.e. 2, 3 and 7, Fig. 8). A control preparation in which all the ciliary shafts are parallel (rather than having phase differences as in a metachronal wave) also contains a single bend which subtends a well defined angle. In the convention adopted by Satir (1967) the net angle subtended by bends on an effective-pointing cilium (i.e. one executing the recovery stroke) is positive while that on a recovery-pointing cilium is negative. The angles subtended by the bends are +97° in stroke positions 2 or 3, – 81° for position 7 and – 72° in the control preparation. Observed tip displacements (to the nearest 0·1 μ m) are shown in Fig. 9, which clearly illustrates the skew (i.e. asymmetry about the median between filaments 5 and 6) referred to in the Introduction.

Examination of equation (13) and Fig. 4 indicates that the microtubule tip displacements will exhibit skew if *v*_{1}is not equal to . *v*_{1} is the angle between the neutral surface of the organelle and the radius to microtubule 1, and if it is equal to the plane of bending contains microtubule 1 while the neutral surface contains the central microtubules. Present data indicate that, even in a twisted axoneme, the position of the central pair relative to doublet number 1 is fixed. The skew observed for cilia thus indicates that these organelles bend in such a way that the bend plane, at least in certain regions along the flagellum, is not normal to the surface containing the central pair of microtubules. This conclusion is valid whether or not there is any twisting of the axonemal microtubules and is also valid for two- and three-dimensional bending.

### (i) Skew of tip pattern with no twist

From the results presented by Satir (Fig. 9) it is possible to estimate the value of *v*_{1} needed to produce the observed skew, assuming that the microtubules do not twist about the axis of the cilium. (In this case τ = 0 and so *v*_{1} = *v*_{10}.) Maximum tip displacements consistently occur for microtubules 4 and 5 in cilia which are effectivepointing and for microtubules 6 and 7 in cilia which are recovery-pointing. Given that there is no twist of the axonemal microtubules the observations could be matched qualitatively if the bend plane contained microtubule 9 for cilia which are effective pointing and microtubule 2 for those which are recovery-pointing. The neutral surface for the two cases would thus be oriented at angles of ± 2 π/9 relative to the surface defined by the central microtubules; the positive sign would apply to recoverypointing cilia while organelles which are effective-pointing would require the negative sign. The two angles needed to explain the experimental results suggest that the effective and recovery strokes of an individual cilium occur in substantially different planes. This suggestion was made by Satir (1963) to explain the variation in orientation, sometimes by more than 90°, of the central microtubules in cilia fixed while actively beating in metachrony. However, the qualitative assessment of bend plane made on the basis of the theoretical model presented here assumes implicitly a consistent orientation of the axoneme relative to the cell surface, rather than the variable patterns observed. The skew cannot therefore be attributed solely to a change in the bend plane during the two phases of the beat cycle. Nevertheless it is interesting to note that a three-dimensional beat pattern is reported by Aiello & Sleigh (1972) for gill cilia of the marine mussel *Mytilus edulis*, with an angle of about 45° between the planes of the effective and recovery strokes.

In the case of *Elliptio* cilia, a quantitative evaluation of the microtubule displacement assuming no twisting of the microtubules, but a change in the beat planes of the effective and recovery strokes, can be made using equation (10) and known parameters associated with the cilium and its movement. Taking the radius (*r*) of the axoneme as 0·1 μ *m*(e.g. Satir, 1963) and the bend angles a specified earlier, the microtubule displacements shown in Fig. 9 are obtained. These values are of the order of magnitude observed in some experiments, although higher values have been recorded.

### (ii) Skew of tip pattern because of twist around neutral surface

Microtubule displacements similar to those observed experimentally could also be the result of a continuous twisting of the tubules along the cilium, and equation (12) can be used to obtain a quantitative assessment of the amount of twisting needed to produce the observed displacements. It is evident from equation (12) that if the microtubules twist in a straight region of the flagellum (i.e. one where *R* is infinite), *s*_{1i} = o and so the relative displacements of the microtubules will remain unaltered. It is only if twisting occurs through a bent region of the cilium that changes in the displacements will occur. The conditions for maximum microtubule displacement are discussed earlier (see equation (4)*et seq*.*)* and these can be used to obtain, for various angles of twist per unit length (*r*), the appropriate values of *v*_{10} required to yield maximum positive displacements for microtubules 4 and 5 (when α = 97°) and maximum negative displacements for microtubules 6 and 7 (when α = – 72° and – 81°). As noted earlier, maximal values of the tip displacement are obtained for τ = o if *v*_{10} is adjusted according to equation (14), but displacements of the order of those observed are obtained for values of τ as high as 20° μ m^{−1}(Fig. 10). The reference values P_{10} are those obtained at the beginning of the curved region and may thus be influenced by twisting in the straight region between the ciliary base and the curve. If *v*_{10} is kept constant at 90°, maximum tip displacements for the relevant microtubules are obtained for a value of τ near – 10° μ m^{-1} for each of the bend angles studied (Figs. 6 and 11).

### (iii) Skew of tip pattern involving three-dimensional deformation

In the discussion so far, it has been assumed that the effective and recovery strokes are planar, although the observations suggest that the planes of the two strokes differ. To achieve continuity between the effective and recovery strokes, the cilium must move in three dimensions, and it is therefore appropriate to consider the behaviour of the microtubules within an organelle deformed in this way. A convenient simplified model of the beat pattern is shown in Fig. 12 in which the effective stroke occurs in a single plane while the bend which travels along the cilium during the recovery stroke forms part of a helix. The relative displacements of the microtubules depends on then-disposition within the bend. If the local bend plane contains microtubule number 1 throughout the bend then, as noted earlier, microtubule displacements will be symmetric about the median of microtubules 5 and 6. This would give rise to the variable observed orientations of the axoneme (Satir, 1963) but not to the skew.

*θ*and radius

*R*

_{H}, the twist per unit length (or torsion) τ is given by The precise three-dimensional form of the bending in

*Elliptio*cilia has not been completely determined, so that the values of

*θ*and

*R*

_{H}in equation (15) are not known with certainty. It is, however, possible to obtain an estimate of their values by considering the electron micrography of Satir (1968) to represent helical, rather than planar, arcs and assuming a beat pattern for

*Elliptio*similar to that described for

*Mytilus*by Aiello & Sleigh (1972).

If the helix angle is taken to be 45° (see Fig. 12), since *R*_{H}*= R* cos^{2}*θ*, the radius of the helix is just one-half of the radius of curvature. It follows from equation (15), that, in this special case where *θ =* 45°, the twist per unit length is equal to the reciprocal of the radius of curvature of the arc. The value of τ for the bend angle measured for the recovery stroke (i.e. = 97°) is 16·6° μ m^{−1} and is of a size which, according to the earlier analysis, would lead to skewed tip displacements of the order of magnitude observed. Further, since the arc length is known to be 5·7 μ m (Satir, 1967) the actual twist of the axoneme through the bend can be calculated and is found to be 94°. A consideration of the idealized beat pattern leads to the conclusion that this angle would be the change in the orientation of the axoneme at the ciliary tip as the organelle moves from the effective stroke to the generation of the complete recovery stroke bend. Satir (1967) notes that the angle of 97° is the maximum bend angle subtended by an arc during the recovery stroke, so that the angle of 94° represents the maximum change to be expected in the relative orientation of the axoneme; angles between zero and 94° should also be observed. This prediction is clearly in agreement with the observations of Satir (1963), who reported relative orientations which could differ slightly in excess of 90° for cilia fixed in the metachronal wave.

## DISCUSSION

From the foregoing it is evident that the skew observed could be consistent with three types of axonemal distortion of increasing complexity and generality:

no twisting of the microtubules, but orientation such that the neutral surface does not contain the central pair;

twisting of the microtubules about the neutral surface;

three-dimensional deformation.

Since the cilia we have particularly analysed have a three-dimensional beat pattern the last, most encompassing mechanism, i.e. (c), provides the closest fit to all major parameters.

It must be remembered that the three-dimensional beat and resultant skew are themselves the products of the specific sliding interactions of the axonemal microtubules, both magnitude and direction being determined by the precise format of such interactions. There is now evidence that the polarity of active sliding is identical for all doublets around the axoneme (Sale & Satir, 1977). One consequence of this conclusion is that the doublets do not all slide synchronously; in fact a recent model reasonably consistent with observed tip patterns suggests that opposite halves of the cilium (i.e. doublets 1 –4 *v*. 6 –9) are active during different stroke stages (Satir & Sale, 1977). Asynchrony of sliding activity of this sort will readily produce a torque which could lead to the desired twist. The amount of twisting actually evolved will depend upon the effect of the sliding on other axonemal components, such as the radial spokes or inter-doublet links -in short on the entire mechanochemical feedback loop which converts sliding into bending and which is still not completely understood. The twist predicted by the theoretical model may thus have functional implications which necessitate further experimental study.

It is relevant to note that Gibbons (1975) has shown by high-voltage electron microscopy that twisting of the axoneme of sea urchin spermatozoa does occur, but only in the transitional regions between straight and curved portions of the flagellum. Measurement of his published electron micrographs indicate an average twist per unit length of about 19° μ m^{−1}, of the order of magnitude required by the theoretical arguments presented in this paper for *Elliptio* cilia. Gibbons notes that the central pair of microtubules remains normal to the bend plane throughout a curved region, but may be inclined at 45° to it in a straight region.

Further recent observations which suggest that twisting of the axoneme may occur are reported by Woolley (1977) who found the beat pattern of golden hamster sperm tails to be three-dimensional. The form of the beating indicates that the plane of a bend twists as it progresses along the flagellum, and it is possible that this is accompanied by a twisting of the axoneme. The organization of the microtubules in this organism requires further study so that the relationship between the microtubular arrangement and the local bend plane can be determined. For both this sperm and the sea urchin sperm studied by Gibbons, it would be important to obtain complete information on tip displacements to test the points developed here.

In summary, the experimental results pertaining to skewed microtubule displacement at the ciliary tip and axonemal orientation in *Elliptio* cilia can be reconciled with the known form of beat in which the effective stroke is planar but the bend propagated to produce the recovery stroke is three-dimensional (Fig. 12). The plane of the effective stroke contains microtubule 2, rather than 1, in order to produce the observed skew. In its recovery stroke, the microtubules have imparted to them a twist in their bent regions which is sufficient to account in reasonable quantitative detail for both the observed skew and variation in orientation of the axonemes.

## ACKNOWLEDGEMENTS

We are grateful to Allen Rochkind for discussions of mathematical aspects of this study. This work was supported in part by a grant from the USPHS (HL 22560). This study was begun at the Department of Physiology and Anatomy, University of California, Berkeley.

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