‘Steady’ Swimming Kinematics of Tiger Musky, an Esociform Accelerator, and Rainbow Trout, a Generalist Cruiser

Juveniles and adults of most fish species are more or less fusiform (see Nelson, 1976; Bone & Marshal, 1982), and their ‘steady’ swimming mode appears to fall towards the subcarangiform and carangiform centre of the thunniform-anguilliform continuum (see Breder, 1926; Bainbridge, 1963; Hunter & Zweifel, 1971; Lighthill, 1975; Webb, 1975; Aleyev, 1977; Kayan et al. 1978; Blake, 1983; Videler, 1985). [Steady swimming is periodic propulsion using cyclically repeating propulsor movements where, although forward motion is not at constant velocity, variations in velocity are small and due to variation in thrust within propulsion cycles (Weihs & Webb, 1983).] The body and fin morphology of these fish is quite variable, ranging from the esociform shape of typical accelerators to the chaetodontiform shape of fish typically found in structurally complex habitats (Webb, 1984a,b). These morphologies are associated with differences in body and fin area and area distribution along the body length. Many analyses of the mechanics and energetics of swimming show that swimming movements increase drag compared with that of an equivalent rigid body (see Brett, 1964; Alexander, 1967; Lighthill, 1971; Webb, 1973,1975; Blake, 1983), especially in the tail region (Hunter & Zweifel, 1971; Webb, 1973, 1982, 1984a,b). Since drag is a surface force, variations in area are expected to affect the performance capabilities of the different morphologies of subcarangiform swimmers (Webb, 1984a,b).

Only a small range of morphologies of subcarangiform swimmers has been studied. To predict the effects of the diverse morphologies of subcarangiform swimmers on their locomotor performance, we must consider other factors that influence the force balance, especially lateral recoil of the anterior body, the yawing response of the body to the lateral component of the normal force developed by the caudal fin. These recoil movements may increase resistance and reduce net thrust and efficiency as a result of the generation of a vortex force upstream of the trailing edge, carrying momentum poorly correlated with that edge’s motion (Lighthill, 1970, 1977). Lighthill (1977) showed that the cube of maximum depth of the body anterior to the tail is a morphological characteristic associated with recoil damping. Body depth is especially small in esociform fishes, so that the anticipated drag penalty due to the large caudal area of this morphology during steady swimming should be greatly exacerbated by energy losses associated with recoil movements.

The purpose of the experiments described here was to assess the consequences of the esociform morphology in steady swimming through a comparison of the kinematics, thrust coefficients and power requirements of tiger musky, an esociform fish, with rainbow trout, a generalist cruiser. It was expected that thrust coefficients and power requirements of musky would greatly exceed those of trout and be associated not only with the relatively larger caudal area of the former but also with very much larger lateral excursions (recoil) of the anterior body. The former hypothesis is supported, but expectations on recoil are rejected.

Tiger musky (hybrid male Esox lucius × female Esox masquinongy) and rainbow trout (Salmo gairdneri) were obtained from local hatcheries and acclimated to laboratory conditions for at least 4 weeks before use. Fish were held in 110-1 tanks, supplied with a continuous flow of water (200% replacement per day). Dissolved oxygen was maintained at air saturation levels using air stones. Fish were fed maintenance rations of live goldfish (musky) and Purina trout chow (trout).

Individuals were taken at random from stock and placed in a recirculating flume modifed from Vogel & LaBarbera (1978). The working section of the flume was 122cm in length, with a 45 cm entry length and a 60cm observation section. The entry section contained straightener grids (1·2cm square egg-crate), which also delineated the upstream limit of the observation section. The downstream limit was an electrified grid (6V d.c.). The flume had a square cross-section (15 cm ×15 cm), and a detachable lid that remained in place throughout experiments. The flow velocity was calculated from a tachometer reading of a multitoothed gear attached to the shaft of the propeller regulating the flow. The tachometer was calibrated using an Ottmeter. The top and back of the observation section were lined with reflective material (Scotchlite) marked with a 5cm grid. The front and bottom were clear Plexiglas. A 45° mirror located beneath the flume allowed simultaneous bottom and side views of a fish.

Each fish was left in the flume overnight (18–20 h) at a flow velocity of 5–10 cm s⁻¹. Fish were then swum at a range of speeds and samples of swimming movements were recorded on ciné film (200 frames s⁻¹) and video tape (60 fields s⁻¹).

Methods were designed to obtain records of steady swimming over as wide a speed range as possible. Trout were tested in a standard increasing velocity test, with speed increments of approximately 5cms⁻¹ every 2min (Brett, 1964; Farlinger & Beamish, 1977). A different method was used for musky, since this fish has little slow oxidative muscle, and hence quickly fatigues at speeds requiring body/caudal fin propulsion. Musky were tested by increasing the flow rate from 5 or 10cms⁻¹ to a test velocity over approximately Is. The test velocity was maintained for 30–60s, and then quickly returned to the starting level. This procedure was repeated every 30 min to test musky at speeds ranging up to 90cms⁻¹, with test speeds applied in a random sequence.

At the end of each experiment, total length, mass, wetted surface area and myotomal muscle mass were measured (Table 1). Wetted surface area was measured as the sum of the circumferences, determined using thread, of the body and median fins at 1-cm intervals along the body length. Body and fin depths (spans) along the body length were measured from tracings of the lateral projection of the body and extended median fins. Myotomal muscle was dissected from the skeleton and skin and weighed.

Ciné film and video tape were analysed frame by frame to measure the following kinematic variables at each speed; tail beat frequency, maximum tail beat amplitude and caudal fin trailing edge depth. The depth of the anterior fins of musky exceeded that of the caudal fin trailing edge when the anterior fins contribute to mean thrust (Wu, 1971). Therefore, maximum amplitudes and depths were measured for the trailing edge of the anterior median fins. The length of the propulsive wave was measured as the distance between nodes of superimposed images of the body at mirror-image positions during the tail beat cycle (Webb et al. 1984). Selected sequences spanning the range of observed swimming speeds were used to measure maximum lateral amplitudes at 20–30 positions along the body length, and the angle subtended by the trailing edge and the plane of lateral movement of the trailing edge. This angle, θ, affects rates of energy loss to the wake (Lighthill, 1971).

Kinematics were only analysed for fish swimming steadily, using sequences where fish swam for five successive and continuous tail beats, during which the swimming speed did not vary by more than 5% of the average. In addition, sequences were only analysed when fish swam with the trailing edge more than one tail depth from both walls. Under these circumstances, the wall-to-fin-gap-span ratio was greater than unity, when ground (wall) effects are small (Lighthill, 1979; Alexander, 1983). These criteria are more rigorous than usually applied in analysing swimming mechanics, and resulted in the rejection of much material. However, they were considered necessary to ensure that results were not confounded by unsteady swimming and wall effects.

Data were analysed using non-linear regression methods on untransformed data to estimate regression coefficients using the MINFIT method (Kirsch & Lane, 1979).

Thrust power for subcarangiform swimmers is usually calculated from reaction theories (e.g. Lighthill, 1975; Wu, 1977) because the acceleration reaction force dominates thrust development at intermediate Reynolds number, high reduced frequency, and for low aspect ratio fins (see Daniel & Webb, 1987). I used the results from regression analyses with the large-amplitude slender-body theory of Lighthill (1971). The mean thrust power, T_p, generated by periodic movements of any sharp trailing edge during steady swimming is:

where m is the added mass per unit length, equal to π ρB²/4, W is the lateral velocity of the trailing edge, calculated from πfa/1·414, w is the velocity increment given to the water, calculated as W(1 — u/c), c is the velocity of the propulsive wave, given by fλ, a is the tail beat amplitude, B is the trailing edge span, f is the tail beat frequency, u is the swimming speed, λ is the length of the propulsive wave, ρ is the density of water and θ is the angle subtended by the tail to the lateral beat plane.

The mean thrust power generated from the caudal fin trailing edge was calculated directly from equation 1. Contributions of upstream sharp trailing edges of greater depth than the caudal fin were calculated as the average thrust for the upstream fins less that which would be cancelled by the overlapping depth of the downstream fin.

The non-dimensional thrust coefficient, Ct, is the most important measure of thrust power requirements for comparing the fish. It was obtained by equating thrust and drag, and using the standard Newtonian equation (Hoerner, 1965) for the latter:

where Cd is the drag coefficient, Ct is the thrust coefficient and S is the wetted surface area.

Thrust and drag coefficients are usually expressed as functions of the non-dimensional Reynolds number, Re:

where L is the total length of the fish and v is the kinematic viscosity of water. Froude efficiency, η, is given by (Lighthill, 1969);

Factors affecting the energy costs associated with lateral recoil have been described by Lighthill (1977). He calculated a ratio, ζ, between the maximum sideforce of the tail and the lateral drag and added mass forces of the body resisting recoil.

where F is the maximum of the periodically varying force on the tail, ω is the radian frequency of the tail, 2πf, B_max is the maximum span of the anterior body and k is a constant, 0·074 for carangiform fish.

Lighthill’s analysis leading to equation 5 strictly applies only to carangiform swimmers, which can be modelled as a tail, generating the side force, and attached by a narrow, flexible caudal peduncle to a rigid body which responds to the side force. Nevertheless, subcarangiform swimmers are similar to carangiform swimmers in that the side force is concentrated caudally where the largest increases in amplitude occur. Therefore, f and B_max are also the important factors reducing recoil, even in subcarangiform swimmers, and their effectiveness in recoil reduction will vary with the magnitude of f²B_max³.

Lighthill also showed how the energy losses associated with the lateral resistance of the body may be calculated. The lateral drag is the same as that of a plate of equal projected area as a body element and moving normal to the flow, even for the periodic motions of the body. Therefore, this component was calculated from the Newtonian drag equation and standard drag coefficients (Hoerner, 1965; Lighthill, 1977) from data for lateral amplitudes, periods and surface areas along the body length. The total inertia of the body and water added mass, periodically accelerated by lateral movements, approximates to ρ ×span², the various numerical constants summing to unity (Lighthill, 1977). The inertial forces associated with this mass were calculated for the accelerations in periodic motions from data for lateral periodic motions and the span distribution along the body length. The energy expended on the lateral drag and acceleration reaction forces were calculated and summed for Lem strips along the anterior two-thirds of the body length.

The wetted surface area, S, of musky was 59% that of trout (Table 1). However, the fish differed in absolute size. Therefore, a non-dimensional specific area was calculated, S/V²/³, where V is the volume (mass divided by fish density, which is 1 for these neutrally buoyant fish). This normalized area was 11·6 for musky, 20% larger than that for trout, 9·7.

Area and span (Fig. 1) were not distributed along the body in the same way for the two species. Compared with trout, musky had a larger proportion of total area located caudally (Fig. 1A,B), associated with the more caudal positioning and larger span of the median fins. Most growth in amplitude of lateral movements occurred over the caudal third of the body (see Fig. 6). 38% of the total wetted surface area was concentrated in this region for musky, and 31% for trout.

The myotomal muscle of both trout and musky made up 52% of body mass (Table 1). The amount of muscle relative to the surface area can be non-dimensionalized as the ratio of muscle mass²/³ to surface area. The value for musky was 0·056, 20% smaller than that for trout, 0·067.

Tail beat frequency increased linearly with swimming speed (Fig. 2A,B; Table 2) as usual for swimming organisms (Webb et al. 1984; Wassersug & Hoff, 1985). Tail beat frequency increased with speed at approximately the same rate in trout and musky, but values for musky were approximately 2 Hz greater than those for trout.

Tail beat amplitude increased linearly with swimming speed for trout, but a power function gave a slightly better description of the relationship for musky (Fig. 2C,D; Table 2). Tail beat amplitudes of trout were larger than those of musky. The tail beat amplitudes of musky were always less than 0·16L. This is small compared with values for other actinopterygians, which are, typically about 0·2L (Hunter & Zweifel, 1971).

Trout had larger tail beat amplitudes but smaller tail beat frequencies than musky, implying that these two kinematic variables were inversely related for the two species. Products of frequency and specific amplitude at any speed were similar, and could not be distinguished statistically (ANOVA, P<0·01).

The depth of the caudal fin was independent of swimming speed for musky, but showed a significant increase with speed for trout (Fig. 3; Table 2). Non-dimensional specific tail depths, normalized for total fish length, were lower for musky, 0·18, than for trout, 0·2–0·24. The depth of the anterior median fins of musky exceeded the depth of the re-entrant caudal fin by a factor of 1·4, independent of swimming speed.

The length of the propulsive wave was independent of swimming speed, averaging 17·9 ±0·3 cm and 15·0 ±0·6cm (X±2S.E.) for trout and musky, respectively. Non-dimensional specific wavelengths normalized by total fish length were 0·9 for trout and 0·8 for musky. These results are in the range of values reported elsewhere (Gray, 1933; Bainbridge, 1963; Webb, 1973; Wardle, 1975; Videler, 1981, 1985; Webb et al. 1984).

The orientation of the trailing edge to its plane of beating, θ, affects the magnitude of thrust by reducing the thrust component of the normal force acting on the tail while increasing the lateral force component and energy losses to the wake. The relevant measure of changes in θon forces and energy losses is cosθ (Lighthill, 1971, 1975).

Cosθ showed substantial variation during a single tail beat. There were no statistical differences between trout and musky in the relationship between cosθ and lateral displacement of the trailing edge (Fig. 4), although values tended to be lower for musky than for trout.

Average values of cosθ during propulsive cycles increased with swimming speed for both trout and musky, and for the trailing edge of the musky tail and of the anterior median fins (Fig. 5; Table 2). Values of cosθ were similar for the two musky trailing edges (t-test; P<0·01), and values for trout were slightly larger than those for musky.

To compare lateral amplitudes of body movements between musky and trout, amplitudes along the body were normalized for the trailing edge (maximum) value and position along the body was normalized for length (Fig. 6). Normalized amplitudes along the body were independent of swimming speed, and data were combined for all speeds to obtain the averaged curves shown in Fig. 6.

Lateral movements of the body anterior to the caudal peduncle were smaller in musky than in trout. For example, at a speed of 50 cm s⁻¹, the centre of mass of the fish body, located in musky at 0·41L (measured from the nose) and in trout at 0·35L (Webb, 1978), moved through amplitudes of 0·6cm for musky and 1·1 cm for trout.

Lateral amplitudes decreased from the nose to a minimum value at 0·04L in musky and 0·14L in trout. Amplitudes posterior to this minimum increased continuously along the body length of musky but not of trout. In trout, amplitudes increased to a body position of 0·4L, decreased slightly more posteriorly, and then increased again over the peduncular region to reach maximum values at the caudal fin trailing edge (Fig. 6). Therefore, the initial amplitude peak seen in trout occurs in the region of the anterior median fins (Figs 1,6).

Thrust power increased with speed for trout and musky (Fig. 7). Over most of the speed range, thrust requirements for musky were slightly larger than for trout, but in view of the normal experimental error associated with each parameter used in equation 1, thrust values must be considered indistinguishable between the two sets of fish tested.

Comparisons between thrust requirements for the two groups of fish must ultimately be based on non-dimensional thrust coefficients (equation 2) and Reynolds number (equation 3). Thrust coefficients decreased with Reynolds number for both species (Fig. 8), but values for musky averaged 1·55 times those of trout for speeds from 10 to 100 cm s⁻¹.

Froude efficiency (equation 4) increased with swimming speed (Fig. 9), as usual for swimmers. Efficiencies were larger for trout, reflecting their lower tail beat frequencies compared with musky.

The energy expended against the lateral resistance was calculated for the drag component from the data in Figs 1A,B and 6 and for the added mass component from the data in Figs 1C,D, 2 and 6. The rate of working was calculated and compared with the total power calculated from equation 1. The lateral recoil work was as large as 60% of the total rate of working (Table 3), calculated from Lighthill’s reaction model, but was relatively smaller at higher speeds where cosθ tended to be larger (Fig. 5). The lateral rate of working for musky is estimated to be substantially lower than for trout, as expected from the low recoil amplitudes for musky. Further quantification of the differences is not considered justified in view of the approximate nature of the theory, especially as applied to subcarangi-form swimmers.

On the basis of the Newtonian drag equation (for example, equation 2), wetted surface area is the morphological factor correlated with the magnitude of drag. Indeed, performance or kinematics at a given speed is often assumed to be proportional to area (Hill, 1950; Bainbridge, 1961; Schmidt-Nielsen, 1984). This is not supported by the present results. Power requirements were similar for the two fish, because although the musky has a wetted surface area that is 59% smaller than that of trout, it is associated with larger drag coefficients.

In practice, drag coefficients are expected to vary along the length of a body, so that the mean drag coefficients for various species should vary where area is unequally distributed along the body length. For a rigid body, the boundary layer becomes thicker with increasing distance from the leading edge. As a result, the shear stress in the boundary layer and the local drag coefficient decrease along the body length (Hoerner, 1965; Schlichting, 1968). In contrast, the boundary layer on a swimming fish is believed to be modified by locomotor movements (Lighthill, 1971), which appear to increase local drag coefficients in the caudal region by one or two orders of magnitude (Webb, 1973).

The present results support the idea that local effects due to locomotor movements increase mean thrust (= drag) coefficients since these coefficients are larger for musky, with a larger proportion of the total surface area located caudally. The caudal region represents 31% and 38% of the total wetted surface area of trout and musky, respectively. If the drag coefficient of this caudal region were one order of magnitude larger than that of the anterior body, a conservative but realistic approximation (Webb, 1973), the average drag coefficient of musky should be 1·45 times that of trout, of the same order as the observed ratio of 1·55.

Lateral recoil was smaller for musky than for trout, as shown by the lower lateral amplitudes along the length of the body (Fig. 6; Table 2). This was contrary to expectations based on the smaller value of B_max³ (equation 5) for musky. Futhermore, the somewhat smaller values of cos# of musky (Fig. 5) would exacerbate the situation. Differences in the phase relationships between cos# and lateral displacement could be used to reduce recoil in musky compared with that in trout, but differences were not found (Fig. 4). Therefore, some other mechanism(s) must be involved in reducing recoil for musky.

According to Lighthill’s (1977) analysis of recoil (see equation 5), an esociform fish could compensate for its small B_max³ and damp the expected recoil either by reducing the side force, F, or by increasing tail beat frequency, f. The former option is clearly not available (Fig. 7). However, musky have higher tail beat frequencies at a given speed than trout. Thus at 10 cm s⁻¹, f was 4·4 Hz for musky and 1·9 Hz for trout, so that f² for musky was over five times larger than for trout. At 80cm s ^{− 1}, f was 7·7Hz and 5·5 Hz for musky and trout, respectively, so f² was twice as great for musky.

Increases in tail beat frequency cannot be made without compensating changes in other kinematic variables to maintain a given thrust at any speed. Musky could reduce the length of the propulsive wave to permit a higher tail beat frequency. In practice, wavelength cannot vary much, as it affects w (equation 1), the velocity given to the water, which must remain large enough to generate thrust (Lighthill, 1970). Therefore, although the specific wavelength of musky is among the smallest reported for subcarangiform fish (Webb, 1975), it is still relatively large at 89% that of trout. Consequently, the lower propulsive wavelength of musky would have a small effect in ameliorating the much larger differences in f².

Larger tail beat frequencies could be offset by smaller trailing edge depths. Musky had a smaller trailing edge than trout, but this made possible a large contribution from the anterior median fins.

Tail beat amplitude is the most likely kinematic variable to compensate for high tail beat frequencies. As observed for musky, amplitude must decrease to compensate for higher frequencies. Furthermore, musky tail beat amplitudes were low compared with typical values for fish. Thus, the high tail beat frequency and low tail beat amplitude at any swimming speed of musky, when compared with values for other aquatic vertebrates, probably represents a mechanism for reducing recoil movements of the anterior of the body.

Irrespective of the contribution of higher tail beat frequencies to damping recoil, there are efficiency costs. Froude efficiency is reduced (Fig. 9). However, Froude efficiencies are large for both trout and musky, and do not vary by more than about 10%. This is probably a small penalty compared with possible increases in power requirements and performance that could occur without recoil control.

The increased tail beat frequency of musky is not adequate to reduce recoil to the full extent observed. The major kinematic and morphological factors damping recoil are contained in the product f²B_max³ (equation 5). B_max has values of 4·9 cm for trout and 2·1 cm for musky so that B_max³ of trout is nearly 13 times greater than that of musky. The 2-to 5-fold difference in f² is clearly inadequate to compensate. Therefore, some additional non-hydrodynamic factor must be involved in reducing recoil.

I suggest that the myotomal muscles are actively involved in recoil control. Contraction of the anterior myotomes contralateral to, and out of phase with, those moving the tail could actively suppress yawing, resulting in head movements in the opposite direction to the recoil force (Lighthill, 1970). Recordings from the muscles of fish and tadpoles show that the requisite contraction patterns occur (Blight, 1977).

The postulated involvement of the muscles in stabilizing yawing would not be without cost to performance. The total rate of working required to swim at a given speed will not be reduced, but rather a portion would be shifted internally to the muscles, presumably with reduced surplus power to overcome drag (Lighthill, 1977). This would reduce maximum sprint performance. Unfortunately, such a reduction in performance would be extremely difficult to show experimentally. Comparative experiments would be required, when performance would depend not only on functional morphology, but also on such anatomical and physiological factors as proportions of slow oxidative and fast glycolytic fibres, fibre trajectories, buffering capacity of tissues and enzyme kinetics (Greer-Walker & Pull, 1975; Alexander, 1969; Klar et al. 1979; Hochachka & Somero, 1984; Davie et al. 1986).

On the basis of existing slender-body theory (Lighthill, 1975), certain morphologies can be defined that are optimal for maximizing performance in steady or unsteady swimming. These morphologies are sufficiently different that forms optimal for one swimming pattern are expected to carry large performance penalties in another (Lighthill, 1975; Weihs & Webb, 1983). This leads to the concept of mutually exclusive optimal designs for various activities that provides a useful framework for understanding the general biology of aquatic vertebrates (Weihs & Webb, 1983; Webb, 1982, 1984a,b). The extrapolations from theory, however, have never been explicitly tested. The goal of the present experiments was to determine if the esociform morphology carried theoretically expected costs for steady swimming by comparing tiger musky and trout. It would, of course, be more desirable to test tiger musky against a thunnid, but the large differences in size and habitat preclude such direct comparison.

In general, musky was expected to have higher thrust (= drag) coefficients and power requirements, attributable to the relatively larger surface area and recoil movements of the esociform morphology compared with the generalist cruiser. Experiments were performed on fish of similar lengths, since length is the appropriate independent scaling variable for dynamically similar steady swimming with small linear and angular accelerations (Daniel & Webb, 1987).

The experiments indeed showed differences in steady swimming mechanics between the two morphologies, implying costs for the esociform morphology in steady swimming. However, differences were less obvious than expected from theory, and sometimes opposite to expectations, with the result that the origins and nature of any costs must now be qualified.

Only the expectation of higher thrust coefficients for musky was substantiated. The smaller absolute area of musky versus trout of similar lengths together with lower body recoil of musky resulted in similar power requirements.

Differences in kinematics occurred. However, larger tail beat frequencies and amplitudes, to meet the anticipated higher power requirements of musky, and-greater body recoil were not seen. The kinematic differences between musky and trout can be considered collectively to be mechanisms for hydrodynamic damping of body recoil by musky, probably the major factor leading to power requirements that were lower than anticipated. However, hydrodynamic damping alone cannot explain the observed low recoil of musky; an active mechanism is probably required. Irrespective of the mechanism, energy must be expended for damping recoil and this will ultimately appear as some reduction in performance, probably at maximum cruising and maximum sprint speeds. This would represent a cost of the esociform morphology for any fish where swimming at aerobic and anaerobic limits was an important component of behaviour.

However, such high steady swimming performance using the body and caudal fin may not be very important for esociform fish. Paired fins and propellar movements of the caudal fin are most commonly used to hold station, supplemented by slow subcarangiform swimming to new ambush sites. Foraging is by accelerating lunges at prey. These typically terminate in less than 1 – 4 s, and more often in tens to hundreds of milliseconds as prey quickly reach some sort of refuge (Weihs & Webb, 1983). In general, therefore, steady swimming is likely to be less important than acceleration. As such, esociform behaviour may shift any steady swimming penalty due to its morphology to an area of activity least relevant to its overall biology.

The differences associated with the esociform and generalist cruiser morphologies discussed above are for fish of similar length. Other measures of size, such as mass and volume, could be important. Mass roughly represents energy reserve levels, a factor affecting overwinter survival for many temperate-zone fish species, spawning capacity for any batch of eggs, etc. Assuming isometric scaling, musky of equal mass to the trout used here would have a length of 26 cm, a total wetted surface area of 196 cm², and hence about twice the thrust power requirements. Therefore, where mass (volume) is of selective importance, then the esociform morphology would carry a large cost with respect to steady swimming.

In general, therefore, morphologically correlated differences are found between esociforms and generalist cruisers in steady swimming.

This work was supported by the National Science Foundation, Regulatory Biology, grant number PCM-84-01650.