Spirillum Swimming: Theory and Observations of Propulsion by the Flagellar Bundle

All flagellated micro-organisms must divide the biochemical energy they generate into the performance of biosynthetic, molecular transport and mechanical work. The relative rates at which this energy is dissipated (the power) in the various work functions is one measure of the relative survival value of the function in question. Absolute values for power cost can provide the limits within which models for the processes by which each of the three kinds of work is carried out must remain. Moreover, since energy relationships are common denominators for all cell activities, they can be used to trace the causal connexions between the three work processes.

The procedures described in this report can be used to calculate the power cost to any flagellated micro-organism of overcoming fluid resistance to propulsion. The present case is particularly illustrative because the geometry of the flagellar beat is not readily apparent, so a variety of possibilities are explored. The fact that the organism is a prokaryote imposes no limits on the calculation of dynamic quantities - force, moment, etc. - but the values generated by this analysis are of prime importance to the development of a model for prokaryote flagellar oscillation.

Calculation of power requirements for mechanical work usually proceed from a determination of the forces and moments imposed on the system at work by the resisting environment. Where force is F and swimming velocity is U the power P is readily calculated as a form of F. U in dyne cm s^-1. The energy dissipated over a period of time t a form of P. t which is related to work by the first law of thermodynamics.

The organism of interest in the present analysis is the bacterium Spirillum. The fluid-mechanical model by which forces and moments are calculated is the rotational resistive theory developed by Chwang & Wu (1971). This theory for flagellum-propelled micro-organisms was developed for three-dimensional flagellar beats from the two-dimensional theory of Gray & Hancock (1955). Chwang, Wu & Winet (1972, hereafter to be designated CWW) have in fact, used measurements of Spirillum locomotion obtained by Metzner (1921) to calculate dynamic quantities for this organism. We shall show, however, that the flagellar beat geometry interpreted from Metzner’s drawings - and consequently the resulting CWW model - are oversimplified. In the CWW model, as in all analyses based upon rotational resistive theory, a tail, flagellum, or, as is the case with Spirillum, a flagellar bundle tries to ‘whip about’ at some angular velocity ω_i, as indicated in Fig. 1. The fluid reacts to ω_i, in balancing the angular momentum, with a viscous resistance which has the effect of counter-rotating the non-flexible body and attached bundle at an angular velocity Ω. As a result of this balancing an observer sitting on the body sees the tail whipping about at ω_i, but an observer in the fixed lab system sees an apparent tail rotation velocity of

In order to produce forward motion, the body and/or bundle must propagate a travelling wave which can push the fluid continuously, generating a viscous fluid reaction in the direction of the rotation axis which we shall term the x axis. In the case of Spirillum the rigid body is already helical in shape so it can generate an x fluid force component in reaction to the rigid wave whose balance propagation velocity is Ω/k_b= c_b where k_b = 2π/ λ_b the wave constant, with λ_b the rigid helix wavelength. When the fluid forces generated by c_b and the moments of force (or torques) generated by ω and Ω are balanced, the organism moves forward (in the positive x direction) at a constant velocity U. The resulting swimming is called quasi-steady (because oscillations are present but relatively small as determined by the oscillatory Reynolds number Re _w=fl²/v which is ≪ 1) and characterizes all motion which conforms to rotational resistive theory.

In the CWW model the function of the flagellar bundle is limited to the production of ω as it is taken to transcribe a cone of half-angle α with the x axis. Implicit in this assumption is a limitation of flexibility in the bundle to a small region near the base. The alternative to bundle flexibility would be a single rotating shaft which is not conceivable for the bundle as a unit because each of its individual flagella arise from separate ‘holes’ in the cell wall (Williams & Chapman, 1961) and it is unlikely that all but one of the entire array of flagella can detach themselves from their moorings and revolve around the flagellum remaining attached. This view of the bundle beat is similar to that of Metzner (1921) with the added simplification of an unbent distal bundle end.

It is the purpose of this report to show that (1) the bundles are bent distally and (2) at least one of the bundles contributes directly to the balanced force component of the swimming bacterium by propagating a helical (or at least three-dimensional) wave of velocity C_f= ω/k_f (where k_f = 2π/ λ_f) and that the aft (posterior) bundle is the more likely candidate for this beat form.

Spirillum volutans was obtained from the American Type Culture collection (catalogue No. 19553) and a bacterium assumed to be the same species was collected from Los Angeles hyperion (sewage treatment) plant sludge. The organisms were maintained without aeration at room temperature (21−25°C) until use, whereupon they were allowed to cool to 20°C.

Spirilla to be analysed were suspended in a petroleum jelly-ringed slide preparation and photographed with a Milliken DBM-55 cine camera at speeds of 100−300 frames/s. Darkfield illumination was provided by a camera-pulsed xenon-arc lamp. Magnification was provided by a Zeiss WL microscope system fitted with a darkfield Apo 40×objective. The resulting films were analysed through tracings made on a parallax-corrected Triad film reader. All observations and recordings were made at 20°C. To ensure that organisms photographed were not unevenly influenced by wall drag, the plane of focus was maintained midway between the coverslip and the slide. One cannot assume, of course, that all wall effects were eliminated.

A total of 18 Spirilla were analysed. All motions were referenced to the lab system. Values for c_b and c_f were determined by direct measurement of wave progression or by following the progress of illuminated spots on the wave which were formed when the curved structure moved into and out of the darkfield light belt. Time measurements were facilitated by a timing light in the camera which marked the film at 120 Hz. Values for Ω and ω were determined by three complementary methods. First from rotation frequency measurements obtained from the films where, Ω = 2 π f_b and ω = 2 π f_f (with f_b and f_f the body and flagellar bundle frequencies respectively). Second, from c_b and Cf, utilizing the definitions given above. Third, the general range for Ω was verified by direct observation of swimming Spirilla wherein the regular xenon-arc was replaced by a Strobotac (General Radio) and the flash rate was decreased slowly from 600 Hz to the values at which the body appeared as two standing wave images 180 ° out of phase, and then one standing wave image. The value of Ω obtained in this manner was then com-compared with measurements from the first two methods. If the range of values agreed, the original measurements were considered verified.

Diagrams showing the parameters measured for each model are presented in Fig. 2. All symbols with ‘b’ subscripts refer to the rigid body and all symbols with f’ subscripts to the flagellar bundles regardless of their geometry. The measurements indicated in Fig. 2(a) are straightforward and easy to obtain. The quantity β_b which is the helical pitch angle is readily calculated from tan β_b = k_bh_b.

The most difficult measurement to evaluate is α, the average angle formed by the bundle with the progression or x axis. The reasons for this difficulty are clear from Fig. 2 (b). As the body helix rotates it carries the axis of the bundle beat around and it is unlikely that this axis is a linear extension of the body material axis (the long dash line in figure) because during the forward motion of the organism the bundles are bent aftward by viscous drag ; however, calculations are greatly simplified if we ignore this difference. Couple this condition with the fact that ω is always greater than Ω and it becomes clear that any representative value for α must be an average. In addition, the a observed through the microscope is a projection of the true three-dimensional angle onto the plane of view, so the only hope for obtaining a true value for any instantaneous a is limited to the instant when the tip of the body helix and the tip of the bundle emerging from it move into the plane of view simultaneously. Accordingly, α is represented by a range of values which is approximated from the maximum and minimum measured values and for calculations we make the further simplification that the flagellar bundle axis coincides with the body material axis as in Fig. 1.

Fig. 2(c) summarizes the bundle beat models to be matched against the conical beat model of 2 (b). At the left the damped helix DH, at the centre the rigid helix RH and at the right the parabola PB. In all applications of the models the values of h_f and λ_f chosen as constants were single values obtained from one organism swimming at an average velocity (note that tan β_f= h_fk_f). For the parabolic beat an estimate of the focus γ was obtained from where h_f and l = 7 μm, the approximate bundle length (Metzner, 1921), form two sides of a right triangle as illustrated in the third diagram of Fig. 2(c).

Three bundle motion geometries were examined vis-à-vis the conical beat form. In the simplest the bundle was allowed to possess sufficient flexibility to be bent into a parabola by the viscous drag on the swimming organism. The other two geometries were both helical; one replacing the stiff portion of the unbent bundle with a rigid helical bundle and the other replacing the stiff portion with a flexible helix which is assumed to have passive flexibility. Accordingly, the flexible helix would have to express the damping effects of viscous drag. Note that since resistive theory assumes quasisteady motion there is no allowance for time-dependent variations in either wave propagation or rotation about the material axis of a flagellar bundle. Accordingly, the fluiddoes not ‘recognize’ a bundle’s ‘elastic’ events beyond their expression in the form of the bundle wave. The form of the wave, in turn, becomes the only basis by which bundle elasticity may be measured (it is a parameter of the damping factor g which is discussed below). The significance of the quasi-steady assumption is evident in the small differences between the damped and flexible helix bundle models to be presented below (such differences become large only when g ≪ 1 or g ≫ 1, conditions which are not explored because they are not supported by observation).

In measuring the degree to which each model fits observations one can often take advantage of the linearity of (11−14) and (19−22) to separate the velocity from the geometric terms. The ideal resulting statements are of the form U/c (velocity ratio) and Ω/ ω (spin ratio). Where c_b the body wave propagation velocity and c_f the bundle wave propagation velocity are both present, separation is less than ideal but by solving for U, substituting the answer in the balanced torque equation and utilizing the definitions of Ω the body angular velocity and w the bundle angular velocity, the spin terms can still be retained in the form Ω/ ω (the U and c term is, in any case, called a ‘velocity ratio’). Accordingly, the velocity and spin ratios for each of the models are presented in Tables 2 and 3.

The traditional notion of the Spirillum swimming form is illustrated in the cine film sequence of Fig. 3. The swept-back appearance of both bundles conforms more to the parabolic arcs in Metzner’s (1921) drawings than to the straight rod of the CWW model (which, as we shall see, is nevertheless a reasonable approximation). Such observed geometry is consistent with the expected curvature for a real attached fibre being bent back under a uniform stress.

One can appreciate the acceptability of Metzner’s interpretation when the available data consistently confirm his observations. If, however, one makes a long exposure print of a film sequence showing the aft end of the swimming bacterium he obtains a bundle image as shown in Fig. 4 which is not unlike the three-dimensional bundlewave reported for compressed Spirillum by Jarosch (1972, his plate 14). Unfortunately, a similar print cannot be obtained for the fore end because the body reflects enough light to obscure the fore bundle. Notwithstanding this lack of data about the fore bundle, it is reasonable to assume that a beating flexible fore bundle has a geometry in the helix-to- parabola range. Tracings of the bundles carried by the specimen of Fig. 4 are presented in Fig. 5 and appear to bear this assumption out although the fore bundle is always incompletely represented. In any case, limitation of fore bundle geometry to a helix or a parabola and the aft bundle to a helix is reasonable given the data presented.

The models to be tested quantitatively against the CWW model are, then, the body with (1) one or two flexible (i.e. damped) helical bundles, or one flexible and/ or one parabolic bundle (DHPB with appropriate choices for m), (2) one or two rigid helical bundles (RH) and - for completeness - (3) one or two parabolic bundles (PB).

The adaptability of rotational resistive theory is well illustrated in the DHPB model. Since the dynamic relationships for all models are composed of linear operators, one can superpose any two of them to get a combination model. Then one can test three models (e.g. a DH, a PB and a DHPB bundle) from the combination model by adjusting m so that only terms for the model in question, and the number of bundles being considered, are present.

The curves which represent the various models and are compared with observations are all presented as functions of κ_b, a quantity which must be common to all possible models of Spirillum. Variation of x₆, in turn, is due to changes in h_b as λ_b is limited to two values, one for generating higher and one for lower model curve values. Model curves generated by the right-hand side of the spin and velocity ratio equations for RH, DHPB, PB and C (from CWW) are compared with data points generated by the lefthand side of the same equations in Figs. 6 and 7. Each curve represents a maximum or minimum (not absolute) locus of values between which the measured ratios are supposed to lie. These extrema are not absolute because although they are maxima or minima for some of the parameters which generated them, they incorporate arbitrary estimates of such parameters as κ_f and g. Other quantities generating these curves were taken from the upper and lower ends of the ranges listed in Table 1. The specific values used to generate the extrema curves are listed in Table 4. All parameters were substituted into the right side of equations (23−30) to generate the curves presented in the figures. Measurements from 18 specimens were substituted into the left side of the same equations to obtain model test values. No value for C_f (and ω) could be obtained from six specimens. As nearly all of these had κ_b values near 1·o, an average spin ratio of 0·288 was obtained from an arbitrary selected group of specimens with κ_b values near 1·0 and this Ω/ ω was used to generate the missing ω (and c_f) values from measured Ω body angular velocity values. The bundle parameters used in generating the data points of Fig. 6(a) are listed in Table 5. Averages and ranges for measured velocity and spin ratios are listed in Table 6. The large variances in the dynamic quantities are not surprising if one assumes that fine strands of debris connecting the organism with the slide are sufficient to result in lower-than-normal ratios. The highest ratios are not so easily explained, particularly when repeated examination of the original frames produced no evidence of convection. We may note in passing the rather large variance in κ_b.

Close examination of Figs. 6 (a) and (b) reveals that all bundle beat forms save the parabolic are consistent with observations of swimming Spirilla. As indicated above, there are no time-dependent (i.e. elastic) hydrodynamic differences between the DH and RH models, but there are geometric dissimilarities which result from the elasticity disparity. The terms which express the geometric dissimilarities on the right side of equations (25) and (26) tend to be small and the resulting curves in Fig. 6(a) differ by less than 1 %. The left side of the same equation, however, contains C_f coefficients which magnify more than c_b coefficients. Accordingly, more discrepancy appears between the data points of Fig. 6(a). A number of data values appears outside the model ranges in both graphs of Fig. 6. The values below the minimum extremum can be readily explained as due to abnormally low U values which were probably caused by adherence of the organism to strands of debris (a likely complication in sewage samples) which may in turn be attached to the slide or coverslip. Data values above the maximum extremum cannot be so easily explained. As indicated above, fluid convection has not been detected and appears unlikely, given xenon strobe illumination and a petroleum jelly-sealed slide. One is intuitively suspicious of a high velocity ratio such as the U/c_b value of 0·96 for one κ_b = 0·738. But repeated measurements of the recorded images produced no other value. Moreover, the present value does fit the helical models, so its existence cannot be dismissed out of hand as anomalous. Other velocity ratio data points which are significantly underestimated by model ranges may be found to lie above the PB maximum extremum curve, and, as expected, above the DHPB curves whose extrema are both too small vis-à-vis the data.

All spin ratio plots are referenced to the same Ω/ ω form of spin ratio so a direct comparison can be made of the four basic models. It is difficult to imagine data which would exceed the extrema of all but the PB and DHPB plots. Thus, it is not surprising that no data points fall outside the C, DH or RH extrema. Moreover, the points which lie outside the two sets of PB plots are below their estimated value, a condition which can be explained by the same reasoning as that presented for the similar condition in the velocity ratio plots.

Observed spin and velocity ratios remain within the presented limits for all models of Spirillum swimming save those which include the PB geometry (see Figs. 6, 7). While PB-containing models performed acceptably in predicting spin ratios, they consistently underestimated the observed velocity ratios. The PB models, then, are invalidated by their predicted velocity ratios, notwithstanding the observations of Metzner (1921). The CWW model, in turn, is invalidated by photographic evidence, so, the accuracy of its theoretical dynamic ratio predictions must be considered fortuitous.

There remain, then, the two helical models which differ so slightly in their ability to account for the data that one cannot choose between them. The true bundle wave may, in fact, be between ‘rigid’ and ‘flexible’ because there is evidence supporting both extremes. On the one hand there must be enough flexibility to allow for the rather high curvature bends near the base of the fore bundle which are detectable in Fig. 3,(a, d, g). The curling of individual flagella of a rod bacterium bearing bipolar bundles as described by Strength & Kreig (1971) also suggests a less-than-rigid flagellar structure. Given a flexible bundle one would expect, of course, that upon making the ‘decision’ to swim, a Spirillum commences a whipping motion with both bundles which would propagate a base-to-tip helical wave down each. The firings of the initial beat impulses at the two bundles would have to be asynchonous and/or of unequal magnitude - a likely condition given an anisotropic stimulus such as a chemical diffusion gradient - such that one of the two bundles would assume the lead in accelerating to a maximum ω, angular velocity. Since both bundles naturally point away from the transverse plane of the body (see Fig. 2), the lead bundle would become the aft bundle and as the body attains steady translation the viscous stresses would bend both bundles aftward in a manner similar to that detectable in Figs. 3 and 5. Implicit in this interpretation is the prediction that a monotrichous Spirillum has an aft bundle only.

On the other hand, there must be limits to flagellar flexibility which would account for the observations of Jarosch (1972) cited above that the flagella of bundles splayed out by flattening Spirillum do not exhibit the pronounced damping one would expect for a flexible body, particularly one near a wall. Nor would flexibility be consistent with the report of Krieg, Tomelty & Wells (1967) that the bundles (which they term ‘fascicles’) of non-translating Spirillum volutans treated with coordination inhibitors rotate as mirror images of one another.

The effect of viscous bending on the motion of the individual flagella is not clear. The rotating shaft concept of flagellar beat generation (Doetsch, 1966; Mussill & Jarosch, 1972; Berg & Anderson, 1973) is certainly supported by the observations that individual flagella arise from separate ‘holes’ in the cell membrane (Williams & Chapman, 1961) and the observation that the cell body of a bacterium stuck to a glass slide by its bundle ‘would undergo violent back-and-forth movements around the point of attachment’ (Strength & Krieg, 1971). Moreover, there is no basis for assuming any incompatability in accepting both an RH bundle and an individual flagellum rotation model so long as the flagella are wound in a sense opposite that of their rotation (Smith & Koffler, 1971). There is, however, some evidence that the flagella of Spirillum-like bacteria - at least Spirillum serpens - arise from a single common basal structure (Abram, 1969 as cited by Smith & Koffler, 1971). If the basal ends of all the flagella are

Each living organism has at any instant a limited supply of biochemical energy which is dissipated in the performance of chemical, mechanical, and concentration work. The mechanical work performed by a bacterium is limited to flagellar motion (with a possible exception being a flexibacterium) which is divided into work performed against elastic (internal) and work performed against hydrodynamic (external) resistance. Elastic resistance includes ‘bearing friction’ on the rotating flagellum and the stiffness of the flagellum as it relates to wave propagation. The latter is, of course, negligible in an RH-type flagellum. (As regards the stiffness of the bundle, we can say that the fact that Spirillum can reverse its swimming direction precludes a rigid-helix-parabola combination which would be inconsistent with the exchange of bundle geometries between the poles which should accompany swimming reversal.)

Hydrodynamic resistance includes not only the viscous stresses upon which rotation resistive theory is based, but also the viscous and pressure stresses which may develop because of the non-slip condition at each solid boundary of two flagella which rotate very close to one another. Berg & Anderson (1973) estimate that the value of this inter- flagellar drag is small because the spacing of the flagellar array is relatively large, but the actual spacing of the flagella has not been measured and dynamic pressure can be considerable where moving bodies are close enough to require a lubrication theory motion analysis. Hence, the question of the significance of inter-flagellar energy dissipation cannot be considered settled.

There remains now the task of showing that the power available for Spirillum propulsion is sufficient for the hydrodynamic demands of a beating helical bundle. To be sure, the energy generating process for motility is oxidative phosphorylation, but unlike the case for eukaryote motility, ATP is not the energy source (Larsen et al. 1974). Unfortunately, the nature of the true energy source is still unknown beyond its appellation ‘the intermediate form of energy for oxidative phosphorylation’ (Larsen et al. 1974). It would be instructive, accordingly, to evaluate the swimming energetics and construct a flagellar motor model based upon an ATP-energized motility. Then, when the necessary information about the true energy sources becomes available, the same procedure may be used to update the model.

The energy metabolism of Spirillum is not as well known as that of E. coli, another bacterium associated with sewage, but it shall be assumed here that their common environment has selected for a similar physiology. The ultimate generator of energy in a model cell is ATP and, accordingly, the availability of ATP for propulsion determines the energy available to flagellar bundles.

Utilizing the energy data from E. coli supplied by Lehninger (1971) one can calculate the total power capacity of a Spirillum cell as 1·38 × 10⁻³ dyne cm s^-1. Thermodynamic efficiency e can then be used to estimate the portion of total power capacity needed for the work in question. Wilkie (1974) defines ⁠, where is the rate of working and which is the rate of change of Gibbs’ free energy that generates for vertebrate voluntary muscle is about 0·66. The corresponding e for insect fibrillar muscle which exhibits high frequency contractions is about 0·32. As indicated in Table 8 the rate at which biochemical energy must be supplied for Spirillum to perform its work against external viscous resistance is about 5 × 10⁻⁸ dyne cm s^-1 which means the AC must lie between 6·65·10⁻⁸ and 1·56×10⁻⁷ dyne cm s^-1. If we can assume that is equivalent to ‘power capacity’ then the work performed against external viscous resistance requires only a small fraction of the total power capacity of the organism.

With these power data in hand we can return to the question of the internal dynamics of the bacterial flagellum. The purpose here is not to explore the internal resistance problem, which as noted previously, is beyond the scope of the present analysis - an investigation similar to that conducted by Schneider & Doetsch (1974), in which the effects of viscosity on flagellar bundle beat per se are used to indicate the magnitude of internal resistance, will be necessary - but to show how power expenditure data may be used to test the feasibility of flagellar contraction models. For the sake of this exercise we shall assume (1) that the power dissipated by internal resistance is negligible and (2) that the work performed at a contraction unit - an attachment-detachment (A-D) cycle - is the same in both vertebrate striated muscle and a prokaryote flagellum. This assumption is invalidated by the work of Larsen et al. (1974) and there is no longer a compelling reason for retaining any component of the striated muscle model in constructing a prokaryote contraction model. But since there are no available values for biochemical energy consumed by prokaryote motility, the striated muscle values will have to suffice.

The Berg & Anderson (1973) model considers a rotating shaft flagellum surrounded by an M ring of perimeter 7·07 ×10⁻⁶ cm. The flagellum rotates at 50 Hz and is driven by 3 A-D sites on the M ring each 8×10⁻⁷ cm in length. Each A-D cycle’s frequency is, then, (7·07 ×10⁻⁶/8 × 10⁻⁷) × 50 = Hz. If each A-D site generates a force of 3-7 × 10⁻⁷ dynes then thetotalpower generated by all three A-D sites, which is to say the flagellum, is 3·7 × 10⁻⁷ dyne × 8 × 10⁻⁷ cm × 3 × 442 Hz = 3·92 ×10⁻¹⁰ dyne cm s^-1. This value agrees with the estimate for flagellum power dissipation of 4×10⁻¹⁰ dyne cm s^-1 by Coakley & Holwill (1972). Unfortunately, the latter value is not a useful measure for determining the biochemical energy spent to overcome external resistance because it does not represent the total fraction of externally dissipated power supplied to each flagellum. This fraction must include power dissipated by both the flagellum and the body because the motion of the flagellar bundle used to calculate its hydrodynamic power dissipation is a consequence of both the supplied and the forces and torques exerted by the fluid on the entire organism. In eukaryote spermatozoa which exhibit a planar flagellar beat, the power fraction spent by the body is 5−20 % of the total for the organism (Brokaw, 1975). Although this fraction is significant, it is not as great as the body fraction for Spirillum as indicated in Table 8. Indeed, the average for all specimens in the present study is 61·8 ± 27·2 % for the RH and 72·5 + 27·2 % for the DH model.

To better illustrate the utility of using the total organism power expenditure for calculations of biochemical energy need as well as for construction of a rotating shaft bacterial flagellum model, we consider a specimen whose bundle frequency is similar to the example from Coakley & Holwill (1972). For this specimen we have f= 49 Hz, m = 2, κ_b = 0·738, U/c_b= 0·640, Ω/ ω = 0·266 and total power dissipated P= 5(P_RH+P_DH) = 1·4× 10⁻⁷ dyne cms^-1. If we reduce m to 1 then we have P= 1·24×1o^-7 dyne cms^-1 and if we assume each flagellum receives an equal amount of ΔG and, as they assumed, that there are 200 flagella per bundle, then the externally dissipated power fraction received by each flagellum is 6·22 × 10⁻¹⁰dyne cm s^-1. If we follow the logic used for the Berg & Anderson (1973) model, then we would have to postulate 5 A-D cycles or crossbridges which would give a power dissipation value of 6·54 × 10⁻¹⁰ dyne cm s^-1 per flagellum (including of body).

Finally, the dynamics of Spirillum swimming can serve as an example of how natural selection may lead to optimization of a biological function but does not require high efficiency. Chwang & Wu (1971) noted that the hydromechanical efficiency (Power dissipated by flagella/Power dissipated by entire organism) is maximized if a uniflagellated organism with a spherical body thickness-to-flagellum thickness ratio greater than 10 swam with a rotational beat and if one with ratio less than 6 swam with a planar beat. If we ignored the non-sperical shape of Spirillum and calculated the body- to-bundle ratio from the b_b and b_f values in Table 3, we would obtain a ration of 5−6 which suggests that this bacterium is grossly inefficient. If, however, we drew a prolate spheroid around the average Spirilum (n_bλ_b = major axis of spheroid, h_b = minor axis) and used its equivalent sphere radius in place of b_b we would obtain a body-to-bundle ratio of 3·8 :0·14 = 28. The true ratio undoubtedly lies between the values given, but is surely > 10 so it agrees quite well with the rotational resistive theory prediction. In contrast the spermatozoa of the tunicate Ciona has a body-to-tail ratio of less than 6 and it swims with a planar beat (although it has been known to rotate (Brokaw, 1975)). The predictable consequences of this diversity are summarized in Table 9. The amount of power dissipated by the Spirillum body is so large as compared to the flagellar bundle that hydromechanical efficiency - in this case 1·0−P_b/P- is quite small as Shown in Table 9. This state of affairs is reflected in the relative motion of the two cells (the last column). The more efficient spermatozoan is over 15 times faster than the Spirillum. Yet, no one could claim that the prokaryote is any the less successful for this inefficiency.

b_b material radius of rigid body

b_f material radius of flagellar bundle

C conical bundle beat model of CWW

c_b wave propagation velocity of helical body

C_f wave propagation velocity of helical flagellar bundle

C_m viscous coefficient of moment of force tangent to body

viscous coefficient of moment of force tangent to bundle

C_n viscous coefficient of force normal to body

viscous coefficient of force normal to bundle

C_s viscous coefficient of force tangent to body

viscous coefficient of force tangent to bundle

DHPB damped helix and/or parabola bundle beat model

f_b rotation frequency of body

F_b force on body

F_d force on damped helical bundle

f_f beat frequency of bundle

F_p force on parabolic bundle

F_r force on rigid helical bundle

g damping coefficient

h_b amplitude of body helix

h_f amplitude of bundle helix or distance of bundle tip from cone or parabola axis

k_b body wave parameter ;

k_f bundle wave parameter;

l length of unbent bundle

L material body length

m number of flagellar bundles

n_b number of body helical waves

n_f number of bundle helical waves

P total power dissipated; also parabolic bundle model

P_b power dissipated by swimming body

P_d power dissipated by swimming damped helical bundle

P_DH power dissipated by organism with damped helical bundle

P_DHPB power dissipated by organism with damped helical and/or parabolic bundle

P_f power dissipated by bundle

P_r power dissipated by rigid helical bundle

P_RH power dissipated by organism with rigid helical bundle

r body radius

r_b position vector for body

r_d position vector for damped helical bundle beat

RH rigid helical bundle beat model

r_p position vector for parabolic bundle beat

r_r position vector for rigid helical bundle beat

U translational velocity of organism

U volume of body

α average angle formed by bundle geometric axis with x axis

β_b pitch angle of body helix; tan^-1k_b

β_f pitch angle of bundle helix; tan¹k_f

γ parabolic coefficient

η hydromechanical efficiency

θ k_fx+ω t

Θ κ_bx − Ω t

K_btan β_b;k_bn_b

K_f tan β_f; κ_fh_f

λ_b wavelength of body helix

λ_f wavelength of bundle helix

Λ arc length

ϕ ω t

ω apparent angular velocity of rotation of flagellar bundle

ω_i whip-torque spin or induced angular velocity of rotation of flagellar bundle

Ω counter-torque spin or apparent body spin

This work was supported by National Science Foundation Grants ENG 74-23008 AOi and AEN 72-03587 AO1. We should like to thank Dr Charles J. Brokaw for his many useful conversations about this work.