The modern era of research on cilia and flagella of eukaryotic cells began in the early 1950s with the discovery of the 9+2 structure of the axoneme, quickly followed by the demonstration of flagellar ATPase activity, the demonstration of ATP-reactivated motility of membrane permeabilized flagella, and the classic study of the morphology of movement of the sea-urchin sperm flagellum by Professor Sir James Gray (1955; see Gibbons, 1981, for other references). During the past two decades, the idea that the bending of flagella and cilia is caused by active sliding between the outer doublet microtubules of the axoneme has become firmly established (Gibbons, 1981). The active sliding process derives energy from the dephosphorylation of MgATP, and this ATPase activity is associated with the dynein arms, which are attached at their basal ends to the A-tubules of the outer doublet microtubules and interact transiently with the B-tubules of the outer doublet microtubules. Since uniform activity of dynein arms distributed around the ring of outer doublet microtubules will not effectively generate bending, it has been concluded that there must be a control process that modulates the generation of active sliding by dynein arms. When planar bending is being generated, this control process should ensure that the dynein arms on one side of the axoneme are inactive while the arms on the other side of the axoneme are active. The regular propagation of bends along a flagellum then requires a regular alternation of activity of these different subsets of the dynein arms at every point along the length of the axoneme.

Cilia and flagella can be viewed as oscillators. Any pattern of oscillatory bending could be generated just by programming an appropriate oscillatory pattern of dynein arm activation. Alternatively, the characteristics of the oscillation could be established by physical parameters such as flagellar elasticity, with the activity of dynein arms serving only to overcome viscous damping of the oscillation. Flagella resemble more familiar physical oscillators in having elastic resistances that tend to restore an equilibrium configuration of the flagellum in the absence of internal activity. The magnitudes of these elastic resistances can be determined by measurements of the bending resistance of inactive flagella (Okuno, 1980). These measurements have demonstrated that the elastic resistances are large enough to play a significant role in the balance of active and resistive forces on a moving flagellum. Unlike the most familiar physical oscillators, flagella have no significant inertia because they operate at very low Reynolds numbers, where inertial forces are negligible in comparison with viscous resistances. Oscillation requires that instead of a physical inertia, there must be an effective ‘inertia’ that results from the control of dynein arm activity, such that dynein arm activity continues, and does not fall to zero, when the flagellum reaches the equilibrium configuration determined by elastic resistances. The behaviour of a flagellum as an oscillator thus involves a balancing of elastic, viscous and active forces, or bending moments. The mathematical tools for analysing the moment balance have been developed and used in several studies of possible control mechanisms (Machin, 1958; Brokaw, 1972a; Hines & Blum, 1978). The most thoroughly explored proposal is the idea that some result of active sliding, such as the curvature of the flagellum, is involved in controlling the active sliding. This establishes a ‘feedback loop’ that can lead to oscillatory bending (Machin, 1958; Brokaw, 1971, 19726). Computer simulations have been used to demonstrate that flagellar models containing a mechanism for control of active sliding by the curvature of the flagellum will oscillate and propagate bending waves (Brokaw, 1972a). However, many of the types of behaviour of real flagella have not yet been successfully duplicated by these models.

Recent kinetic studies of dynein ATPase and the interactions of dynein with microtubules reveal a complex, multi-step ATP hydrolysis cycle with many similarities to actomyosin ATPase (Johnson, 1985). Models of active shear generation by dynein cross-bridges have relied heavily on ideas proposed for muscle cross-bridges (Huxley, 1957; Eisenberg & Hill, 1985). For many purposes it is convenient to describe our ideas about dynein-microtubule interactions in terms of two states. One state, called the detached state, may actually encompass several detached states and weakly bound states in rapid equilibrium with detached states. In this state dynein neither drives nor restricts sliding between tubules. The second state is a strongly bound attached state. Dynein can enter this state in a non-equilibrium position, thus establishing a force that tends to cause sliding as the dynein changes to an equilibrium configuration that produces zero force. While the strongly bound state remains in the equilibrium configuration, it can restrict sliding between tubules, even if it is not generating force. Binding of ATP by the dynein is required to destabilize the strongly bound equilibrium state and return the dynein to the detached state. ATP hydrolysis then restores the dynein to a form in which it can reenter the strongly bound state. In muscle, there is evidence that the transition to the strongly bound state is the phosphate-release step in the ATPase cycle, and that this step may be regulated by the troponin-tropomyosin control system.

It is easy to imagine that distortion of the microtubule surface lattice accompanying flagellar bending could modify dynein binding activity, and thus provide a mechanism for the control of active sliding (Douglas, 1975). A change in dynein binding affinity of sites on the B-tubule of an outer doublet requires an energy input analogous to the energy input provided to the actomyosin control system in muscle by Ca2+ binding to troponin C. A possible source of this energy is the work done in bending the microtubule. The energy put into controlling active sliding by this mechanism will then contribute to the bending resistance of the microtubule. A preliminary estimate (Brokaw, 1982) suggested that the flagellar bending resistance required by such a mechanism would be too high. This question deserves more thorough treatment.

A recent estimate suggests that the bending resistance of an individual doublet microtubule is about E = 0·02× 109pNnm2 (Omoto & Brokaw, 1982). Assuming that the additional bending resistance that might result from a mechanism for controlling dynein-microtubule interaction is comparable in magnitude, the energy per unit length stored in a bent microtubule will then be approximately E X (curvature)2. If there is one dynein arm for every 12nm of length, the energy stored per dynein arm when the microtubule is bent to a curvature of 0·0002 radnm-1 is 10 pN nm. How does this compare with the amount of energy needed to control active sliding?

In the worst case, where every tubulin monomer (69 per 12 nm of doublet length) is a potential dynein binding site, and where the energy of bending is uniformly distributed among all of the tubulin monomers, the energy per site in the bent doublet is less than 0·2pNnm. This is negligible compared to the free energy difference of 8·7 pN nm that is required for a 10-fold reduction in the affinity of dynein (or any ligand) for a binding site. If the site is larger, and located in a region of the doublet that undergoes relatively large strain when the flagellum bends, the energy per site can be larger, but it is still going to be only a fraction of the total bending energy of 10 pN nm.

For comparison, it is instructive to look at the system for regulation of the actomyosin contractile system in skeletal muscle, as described by Hill (1983). With three myosin molecules (6 heads) per 14·3 nm of myosin filament, two troponin complexes per 38·5 nm of actin filament, and two actin filaments per myosin filament, the troponin : myosin ratio is approximately 0·5. If a change in Ca2+ concentration from 10−7M to 10−6M causes the binding of two Ca2+ per troponin, this corresponds to a free-energy change of 17 pN nm per myosin. In the actomyosin system, cooperativity between adjacent tropomyosin molecules plays a significant role in enhancing the degree of regulation of myosin cross-bridge activity that can be achieved by this free-energy change (Hill, 1983). An important point is that the energy requirement could not have been deduced just from thermodynamic principles. It is necessary to know that an efficient system has evolved in which potential myosin binding sites on seven actin monomers are covered by one tropomyosin molecule and therefore regulated by one troponin complex. It is also necessary to know about the cooperativity of tropomyosin in order to know the extent of the regulation that can be obtained from Ca2+ binding.

In flagella, this information is lacking. Therefore, it is not possible to rule out a priori the possibility that the energy to control the active sliding is obtained directly from bending, and represented in the flagellar bending resistance.

However, this idea has an important experimental consequence. The regulation of actomyosin by Ca2+ binding to troponin C can be expressed as an increase in the stability of strongly bound attached cross-bridges, generated by an increase in the rate constant for entering the strongly bound state (Hill, 1983). If microtubule bending regulates in a similar manner the stability of dynein attachment to sites on the microtubule, then dynein binding to these sites should change the configuration of the microtubule. The more efficient the system, in terms of regulating many dynein binding sites by a small amount of energy introduced by bending the microtubule, the larger should be the effect of dynein binding on the configuration of the microtubule. Experiments designed to examine the effects of dynein concentration on the configurations of outer doublet microtubules in solution might be the most effective means of testing the hypothesis that the active sliding process is regulated directly by flagellar bending in this manner.

If this version of the hypothesis is not correct, what are the alternatives? One possibility is that the information in flagellar bending is amplified with energy from another source to provide the energy needed to regulate the dynein cross-bridge cycle. This energy might be provided by ATP dephosphorylation, perhaps by one of the several dynein ATPase molecules present in each dynein arm. The direct approach to this question probably requires perfection of methods for reassembly of dynein molecules into functional arms. If this can be achieved, it may be possible to incorporate non-functional Aα or Aβ dyneins into an arm, and test which ones are required to be functional in order to cause sliding disintegration of dynein-depleted axonemes, as in the recombination experiments of Yano & Miki-Noumura (1981), or to cause an increase in the beat frequency of demembranated flagella as in the experiments of Gibbons & Gibbons (1976). This experimental method should in principle be able to identify a dynein ATPase that is not required for active sliding, but is required for flagellar oscillation and bend propagation. If these experimental techniques for in vitro reconstitution cannot be perfected, the alternative may be to construct dynein arms with non-functional dynein ATPases by genetic engineering in Chlamydomonas. However, the transformation techniques that are required for such constructions are not yet perfected.

Comparison with the troponin—tropomyosin control system in muscle may be misleading. In muscle, which must remain in a turned-off state for long periods of time, the method of choice may be a mechanism that alters the stability of actinmyosin interactions. However, to generate flagellar bending waves, it is only necessary to suppress dynein activity for short periods — approximately half a beat cycle. This could be done with a catalytic mechanism, as is more common in controlling cellular biochemistry, in which both the forward and reverse rates of a reaction step are altered by the same amount and no changes in the free energy of states in the reaction sequence are required. For example, both forward and reverse rates of a phosphate-release step might be reduced to turn off cross-bridge activity.

How could this alternative be detected experimentally? In principle, methods involving changes in free-energy levels of cross-bridge states should work independently of the beat frequency. On the other hand, a mechanism involving rate control that was optimized for normal beat frequencies might not function well at very low beat frequencies. Such a mechanism could conceivably be responsible for the threshold effect that is seen when demembranated sea-urchin, sperm flagella are reactivated at low ATP concentrations (Brokaw, 19756), but this hypothesis is difficult to evaluate because low ATP concentration will directly affect other steps in the cross-bridge cycle.

The final alternatives would be mechanisms that do not depend at all upon information about the curvature of the flagellum. I have suggested (Brokaw, 1975a, 1976) that a decrease in sliding velocity might provide a signal for switching between alternative cross-bridge systems, and presented cross-bridge models that automatically respond in this way when a resistance to sliding is encountered. Support for this idea has been provided by observations of oscillatory sliding during measurements of sliding forces in disintegrating axonemes (Kamimura & Takahashi, 1981), and by observations that very short regions of a flagellum can oscillate, showing oscillatory sliding with little obvious bending in the active region (Brokaw & Gibbons, 1973). However, extensive computations of the behaviour of flagellar models incorporating velocity-regulated active sliding systems have revealed that these models can produce stable oscillation and bend propagation in the absence of external viscous resistances, but that the behaviour becomes unstable when realistic values for the external viscous resistance are used.

In a propagating bending wave on a sea-urchin sperm flagellum, the pattern of new bend initiation involves minimal sliding in more distal portions of the flagellum (Goldstein, 1975, 1976). Consequently, in propagating bends, regions of high sliding velocity travel along the flagellum in phase with regions of high curvature. One might expect, therefore, that models in which either curvature or sliding velocity control the active sliding process would be rather similar. Results of computations with such models show that this is not the case, and that there are significant differences between results obtained with these two types of models. On the other hand, these two parameters are similar enough that models that use various linear combinations of these two parameters to control the active sliding process show no new properties.

These two parameters, curvature and sliding velocity, by no means exhaust the possibilities for parameters that might control the active sliding process. Another obvious possibility is the shear displacement, which will be out of phase with curvature and sliding velocity. A priori, one might expect this not to be a very useful control parameter. In a short flagellum, regions near the distal end of the flagellum will be the first to attain the values of shear displacement that might be required for switching between active sliding systems. As a consequence, the switching event, and the bending waves, would be expected to propagate towards the basal end of the flagellum rather than towards the tip.

Other control parameters might be the rate of bending, and the rate of change of curvature with length. These parameters will also be out of phase with curvature and sliding velocity. A control paradigm that used a combination of curvature and rate of change of curvature might be ideal for generating active shear moment with components in phase with both viscous and elastic resistances.

These possibilities are enumerated largely to show that many theoretical possibilities remain to be explored by detailed computer modelling, just to examine the formal relationships to control parameters without even considering the detailed mechanisms. An important, if not the most important, reason for this exploration is to determine whether there are differences between the behaviour of these various models that could be distinguished experimentally.

A serious impediment to progress with these modelling investigations of alternative control mechanisms has been realized as a result of recent studies of a simple curvature-controlled model (Brokaw, 1985). This work demonstrates that specifying the control of the active sliding process by curvature is not sufficient to determine the behaviour of a flagellar model. At low viscosities, the parameters of the bending waves generated by this curvature-controlled model were found to be very sensitive to minor variations in the details of bend initiation at the base of the flagellum. At high viscosities, these models tend to choose bending wave parameters that give an integral number of waves on the flagellum, because this facilitates the satisfaction of boundary conditions for moments at the ends of the flagellum. However, these properties can be over-ridden by forcing the flagellar model to operate at a designated frequency. These insights demonstrate that other control properties of a flagellum, in addition to control of active sliding by a parameter such as curvature, strongly influence the bending behaviour. It is therefore difficult to evaluate the various possibilities for controlling the active sliding process unless the other control properties are known, or at least considered.

Over the past two decades, a lot of detailed quantitative information about the behaviour of cilia and flagella has accumulated. Some of this has exploited the usefulness of ATP-reactivated, demembranated flagella, particularly those of sea-urchin spermatozoa; other work has benefited from the availability of Chlamy tomonas mutants with altered flagellar behaviour. These data should contain important clues to help us deduce the internal control mechanisms of flagella. (1) The basic bending wave parameters -frequency, mean bend angle and wavelength -of sea-urchin sperm flagella are invariant when the asymmetry and, therefore, the curvatures of the bends of the bending waves are manipulated by changing the Ca2+ concentration (Brokaw, 1979). Similarly, the beat frequency of reactivated ctenophore comb-plate cilia is independent of Ca2+ concentration changes, which can induce ciliary reversal (Nakamura & Tamm, 1985). (2) The frequency of flagellar oscillation is sensitive to mechanical loading, such as changes in viscosity (Brokaw, 1966; Mulready & Rikmenspoel, 1984). The wavelength of flagellar bending waves is especially sensitive to viscosity, and can be changed even under conditions where viscosity has little effect on beat frequency (Brokaw, 19756). (3) Several situations are known in which changes in beat frequency are accompanied by inverse changes in bend angle, indicating that sliding velocity is an invariant parameter under these conditions (Brokaw, 1980; Okuno & Brokaw, 1979; Brokaw & Luck, 1985). However, in other situations, bend angle and sliding velocity can be reduced, with frequency remaining relatively constant (Brokaw & Simonick, 1976; Asai & Brokaw, 1980). (4) Flagella containing ‘rigor bends’ can be produced in various ways (Gibbons & Gibbons, 1974; Brokaw & Simonick, 1976). When conditions for bend propagation are restored, these bends resume propagation in the normal direction. This suggests that information about the control status of the active sliding process is stored during rigor, either in the bent configuration of the flagellum or by some other mechanism.

Missing from this list are statements about the shapes of flagellar bending waves, which are commonly thought to be a direct indicator of the underlying forces responsible for generating the bending waves. This remains an area of controversy. New methods for enhancement and analysis of microscopic images of beating flagella, combined with computerized management of the large amount of data provided by these methods, will probably revolutionize this area in the next decade.

Most of the ideas mentioned in the above discussion have been around in various forms for the past 20 years. Perhaps our failure to understand flagellar bending-wave generation by now indicates a need to think about flagella in quite new and different ways. Two recent observations may serve as useful stimulants for new ways of thinking about flagella.

Allen et al. (1985) have used high-resolution video-enhanced light microscopy to demonstrate the gliding movements of native microtubules in extruded squid nerve axoplasm. When the forward gliding of a microtubule is impeded by an obstruction at its forward end, the microtubule remains in place and can generate regular, propagated, bending waves with a striking similarity to the bending waves generated by highly organized flagellar axonemes (Fig. 1). I have performed computer simulations of this phenomenon, using appropriately modified versions of the computer programs developed for analysis of flagellar bending wave propagation (Brokaw, 1972a, 1985). These simulations demonstrate that the generation of propagated bending waves by a microtubule does not require any control or modulation of the mechanism for longitudinal translation and force generation along the axis of the microtubule. The bending waves arise because the curvature of the microtubule causes the force generated along the length of the microtubule to produce bending moments. The curvature therefore enters into the generation of bending moments in a manner analogous to its role in curvature-controlled models for flagellar bending waves.

Fig. 1.

The sequence of straight and serpentine shapes assumed by a gliding microtubule (length 4-8μm) before and after it encounters an obstacle. Times are in seconds. (Reproduced from The Journal of Cell Biology (1985) 100, 1736-1752, with copyright permission of the Rockefeller University Press.)

Fig. 1.

The sequence of straight and serpentine shapes assumed by a gliding microtubule (length 4-8μm) before and after it encounters an obstacle. Times are in seconds. (Reproduced from The Journal of Cell Biology (1985) 100, 1736-1752, with copyright permission of the Rockefeller University Press.)

Could this elementary property of microtubules have been exploited in the early stages of the evolution of the eukaryotic flagellum, with the major evolutionary trend being the increasing organization of the force-producing molecules (dyneins) to produce higher forces and velocities? Does appreciation of such a possible scenario help us in thinking about the mechanism for bending wave generation by flagella? These are new questions, and the answers are not yet clear.

Another recent observation, from my own laboratory, suggests that the organization of the eukaryotic flagellar axoneme may be more subtle than previously suspected. Direct visualization of active sliding by flagellar outer doublet microtubules has most often made use of digestion by elastase or trypsin to eliminate structural constraints that prevent sliding beyond the range associated with normal oscillatory bending (e.g. see Summers & Gibbons, 1970). However, axonemes of Tetrahymena cilia frequently disintegrate by sliding without protease digestion, especially at low ATP concentrations (Warner & Zanetti, 1980). At low ATP concentrations, sea-urchin sperm flagella will also show partial sliding disintegration without protease digestion, when bending is prevented by attachment of part of the axoneme to the microscope slide surface. This disintegration typically begins by a bulging out of a bundle of doublets from the side of the axoneme, thus allowing sliding in the more distal region of the axoneme. This sliding would normally produce a principal bend near the base of the flagellum, but this bending is prevented by attachment of the basal region of the flagellum to the slide surface. Ina few cases, the restriction on bending has been transitory, and the separated bundle of doublets has been able to reassociate with the remainder of the axoneme and restore its original appearance. In the most dramatic case observed, the ‘bulge’ of separated doublets appeared to be driven off the end of the flagellum by bending and reassociation of doublets in the basal portion of the flagellum. For a brief instant, the distal third of the flagellum appeared as a ‘brush’ of three or more separated microtubule bundles. These bundles then reassociated, and the flagellum resumed normal beating, looking just like other flagella in the preparation. Unfortunately, this event was not photographed, but Fig. 2 shows another case, of a flagellum that underwent repeated dissociation and reassociation of microtubule bundles over a period of several minutes.

Fig. 2.

Selected images from a series of records showing weak beating, partial disintegration, and reassembly of a demembranated sea-urchin spermatozoon (Arbacia punctulatd) reactivated at low ATP concentration in the presence of 2mM-ADP. For methods, see Brokaw (1986). Photographed on moving film with flashes at 100 Hz. Relative image numbers within each record are indicated. The beat frequency was approximately 8 Hz.

Fig. 2.

Selected images from a series of records showing weak beating, partial disintegration, and reassembly of a demembranated sea-urchin spermatozoon (Arbacia punctulatd) reactivated at low ATP concentration in the presence of 2mM-ADP. For methods, see Brokaw (1986). Photographed on moving film with flashes at 100 Hz. Relative image numbers within each record are indicated. The beat frequency was approximately 8 Hz.

These observations, although difficult to reproduce and record, suggest that the structural constraints that maintain the axoneme are not permanent, unbreakable linkages. The observations are more consistent with the view of interdoublet linkages suggested by the observations of Warner (1983) than with earlier views of the ‘nexin’ links as permanent, highly elastic, linkages.

Taken together, these new observations suggest that the flagellum could have arisen as, and may still be, a loosely associated bundle of microtubular elements retaining and enhancing the capability for bending wave generation that is seen with individual axoplasmic microtubules. Whether this new view of the flagellum will turn out to be correct, or even helpful, remains to be tested.

Preparation of this paper has been supported by an NIH research grant, no. GM 18711.

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