Sliding velocity between outer doublet microtubules of sea-urchin sperm axonemes

A sliding movement between doublet microtubules was first reported by Summers & Gibbons (1971), who observed that doublet tubules are extruded from trypsin-treated axonemes of sea-urchin sperm flagella on addition of ATP. Sale & Satir (1977) showed that each doublet in Tetrahymena cilia actively slides relative to its neighbours in a single direction on addition of ATP, the polarity of force generation of the dynein arms being from base to tip. A rabbit serum prepared against the Fragment A portion of dynein 1, inhibited not only the undulatory movement of Triton-model spermatozoa (Gibbons, Ogawa & Gibbons, 1976) but also ATP-driven sliding movement (Masuda, Ogawa & Miki-Noumura, 1978). We found also that axonemes without outer dynein arms which were extracted with NaCl, were able to extrude outer doublets in the presence of ATP and trypsin, using sea-urchin sperm flagella. The sliding speed was slower than that of control axonemes having both arms (Hata et al. 1980).

These observations indicated that the bending movement of cilia and flagella results basically from these active sliding movements between adjacent doublet tubules in the axonemes and that this sliding movement may be generated by mechanochemical interaction between outer and inner dynein arms with the adjacent doublet tubules. Recently, the attachment of the dynein arms to the B-tubules and their ATP-induced detachment from the B-tubules in Tetrahymena cilia, was observed by Takahashi & Tonomura (1978).

Considering that the sliding movement may be induced by an ATP-driven mechano-chemical cycle involving interaction of dynein arms with the adjacent doublet tubules, the velocity of the sliding movement would also be expected to be dependent upon the ATP concentration present. Gibbons (1974) reported preliminarily that the sliding speed between adjacent tubules is about 10 μm/s in 1 mM ATP and 0·8 μm/s in 0·01 mM ATP at 25 °C.

Although we have previously reported a dependency of sliding speed on ATP concentration (Hata, Yano & Miki-Noumura, 1979), in this paper we have attempted to measure it more carefully and precisely at a constant temperature, 25 ± 1 °C, while examining several factors affecting the velocity, in order to compare the dependency of sliding velocity on ATP concentration with that of the ATPase activity of the axonemes.

The sperm, of sea urchins, Pseudocentrotus depressus and Hemicentrotus pulcherrimus, was obtained by injecting 0·5 M KC1 into the body cavity. After centrifugation at 4000 g for 5 min, the sperm was kept at 0 °C, and used within 2 days.

Sperm were suspended in 20 times the volume of axoneme isolation medium (0·15 M KC1, 2 mM MgSO₄, 2 mM CaCl₂, 0 5 mM EDTA, 1 mM dithiothreitol, 0·25 % Triton X-100 and 10 mM Tris-HCl buffer, pH 8·3), and stirred gently for 20 min at room temperature to solubilize cell membranes and to separate tails from sperm heads. Precipitated sperm heads were discarded after centrifugation at 1000 g for 10 min. Axonemes in the supernatant were collected by centrifugation at 10000 g for 10 min, washed once with ATP-free reactivation medium (015 M KC1, 2 2 mM MgSO₄, 0·5 mM CaCl₂, 2 mM EGTA, 1 mM dithiothreitol, 2% polyethylene glycol and 10 mM Tris-HCl buffer, pH 8·4) and suspended in the same medium. The total length of each axoneme was 30-40 μm. In order to obtain shorter fragments of the axonemes, 2 ml of axoneme suspension were homogenized with a motor-driven Teflon-glass homogenizer at 0 °C. Axoneme fragments of various lengths were obtained by changing the number of strokes in the homogenizing process.

To prepare ‘axonemes previously digested with trypsin’, an axoneme suspension of approximately 1 mg/ml was treated with trypsin in a final concentration of 2 μg/mi at 25 °C. The weight ratio of protein to enzyme was 500:1. The digestion was terminated by adding a 10-fold concentration of soybean trypsin inhibitor at different times.

The non-digested axonemes in reactivation medium, were placed on a slide and covered with a coverslip coated on opposite edges with Vaseline. A small volume of ATP and 2 μg/ml trypsin in the reactivation medium was drawn continuously under the coverslip with a piece of filter paper. Alternatively, the trypsin-digested axonemes were added to a small volume of ATP in reactivation medium. The concentration of MgATP was calculated, taking into account the concentrations of the other constituents in the reactivation medium. Observations were made in a room maintained at 25 ± 1 °C.

We used axoneme samples having a sliding percentage (ratio of number of axonemes having sliding movement to all axonemes) of more than 95 % within 1 min after adding ATP and trypsin.

The sliding speed of the axonemes was observed to remain constant in the 2 species of sea urchin.

A dark-field microscope was used for observation. The sliding process was recorded with a CTC-9000 Night Vision Camera system including a video tape-recorder. The images on a picture monitor from the recorded video tape were photographed on Tri-X film with a Nikon FL camera equipped with a motor-driven apparatus. Time and the length of the tubules extruded from the axonemes were measured directly from the image on the picture monitor. Careful observations were necessary in measuring the sliding length, to exclude cases in which a doublet tubule was sliding on adjacent tubules, which were themselves sliding relative to other doublet tubules at the same time, because the apparent values of the velocity in these cases were incorrect. Using the optical and recording system described above, we could easily distinguish this situation from that which gave a true value for the velocity. Sliding speeds of more than 20 axonemes under each set of conditions were measured and averaged. The sliding velocity is given as the mean value and standard deviation.

Demembranated axonemes of approximately o-i mg/ml in polyethylene glycol-free reactivation medium and various concentration of ATP were incubated at 25 °C. Inorganic phosphate liberated within 1 min after the incubation was measured according to the method of Murphy & Riley (1962) with some modifications (Hayashi, 1976).

Protein concentration was determined according to the method of Lowry, Rosebrough, Farr & Randall (1951), using bovine serum albumin as the standard.

The sliding velocity of outer doublet microtubules from demembranated axonemes was affected by several factors, namely, method of preparation of the axonemes, trypsin and polyethylene glycol concentrations, length of the axonemes, and temperature. In order to measure the velocity more exactly and to determine the dependency of sliding velocity on ATP concentration, we first checked the above factors at 25 ± 1 °C.

The demembranated axonemes were digested with trypsin to destroy connecting structures between doublet microtubules, such as nexin or radial spokes. Two procedures for trypsin digestion were tried: in one the axonemes were digested before-hand in a test tube and the digestion was terminated with soybean trypsin inhibitor before observation under the dark-field microscope; in the other the axonemes were digested with simultaneous addition of ATP and trypsin on a slide during observation under the microscope.

When digested axonemes were mixed with 1 mM ATP in the reactivation medium on a slide under the microscope, the axonemes digested for 30 s did not show any sliding movement. Sliding percentage (ratio of number of axonemes demonstrating sliding movement to the total number of axonemes) was 20% for axonemes digested for 60 s and increased to about 90% for axonemes digested for 90 s. The axonemes digested for 120 s divided into groups of doublets without demonstrating any sliding movement, as if the axonemes were, so to speak, peeled apart in the presence of ATP. Therefore, we concluded that digestion for 90 s with 2 μg/ml trypsin was optimal for further experimentation.

Fig. 1A shows the typical sliding movement of axonemes, previously digested for 90 s with trypsin, in the presence of ATP. When ATP diffused into the axonemes they showed a twitching movement for a moment and then appeared to lose their rigidity. Then one or more outer doublet microtubules in a bundle were extruded from the both ends of the axonemes at the same time, as shown in Fig. 1 A.

Various amounts of trypsin and 1 mM ATP were added simultaneously to the demembranated axonemes on a slide and drawn continuously under the coverslip with a piece of filter paper. When the trypsin concentration was 1 μg/ml, the sliding speed was slower than at 2 μg/ml trypsin in the range of ATP concentration from 20μM to 1·5 mM, and the sliding percentage was also lower. At more than 3 μg/ml trypsin, most axonemes disintegrated into bundles without showing active sliding movement, which might be caused by the excess digestion. Although some axonemes extruded outer doublets, values of the velocity were widely scattered, as shown in Fig. 2. These observations led to the conclusion that a trypsin concentration of 2 μg/ml was appropriate in this experiment.

When 1 mM ATP and 2 μg/ml trypsin were added simultaneously to the axonemes and drawn under the coverslip with filter paper, the axonemes seemed to become less rigid after the twitching movement. Then, after 10 s or so, the axonemes began to extrude outer doublet microtubules. The sliding process shown by these axonemes was different in some respects from that of the trypsin-digested axonemes. The axonemes extruded some outer doublets in a bundle of tubules, from which one or more doublets separated successively. The sliding speed was determined by measuring the distance of the sliding edge of doublet microtubules from the adjacent doublet in the axoneme, and the time. In Fig. 1B arrows indicate the sliding edge of outer doublet microtubules. No difference in sliding velocity was found between one or two microtubules and their bundles.

When only trypsin was added to the axonemes on the slide, after 1 min or more the axonemes disintegrated longitudinally into bundles of outer doublets, showing no sliding movement (Fig. 3). The process was clearly different from that of axonemes in the presence of ATP and trypsin. These observations indicated that the simultaneous addition of ATP and trypsin induced active sliding movement between adjacent doublet microtubules in the axonemes. The sliding process shown by the trypsin-digested axonemes was compared with that seen with non-digested axonemes, to which ATP and trypsin were added simultaneously. In the latter process it was easier to measure the length of the extruded part from the end of the axonemes, because the extruded part was nearly straight and the outer doublets telescoped out successively.

Sliding velocity of the axonemes was measured in reactivation medium containing various concentrations of polyethylene glycol (PEG).

Axonemes in reactivation medium containing from o to 5 % PEG were placed on a slide. A small volume of ATP and 2 μg/ml trypsin in reactivation medium containing each concentration of PEG were added to the axonemes on a slide, and the sliding movement was observed. As shown in Fig. 4, the velocity was almost constant from o to 2 % PEG and then decreased a little in proportion to the increase of PEG beyond 2%. It took more time to induce sliding movement in the axonemes in the presence of PEG, probably because ATP and trypsin diffused more slowly into the axonemes in the more viscous reactivation medium.

Comparing ATP concentration dependency of the sliding velocity in the presence or absence of 2% PEG in the reactivation medium, sliding speeds were found to be similar in the ATP concentration range of 20 μM to 1 mM ATP. However, the presence of 2 % PEG in the reactivation medium had the advantage of protecting the structure of the axonemes and reducing the Brownian movement of particles on the slide under the microscope.

The total length of axonemes which we prepared from sea-urchin sperm flagella was around 30-40 pm and the mean length was 33·1± 4·0μm. By homogenizing whole axonemes shorter fragments of axonemes were obtained. When the number of homogenizing strokes was changed, various lengths of axoneme fragments could be prepared. Table 1 shows the relationship between number of strokes and length of axoneme fragments. The axonemes homogenized with 8–16 strokes were divided longitudinally, which suggested that the axonemes were injured mechanically.

Sliding speed was examined using axoneme fragments from 10 to 28 μm in length.

The shorter fragments had similar values to those of longer ones at less than 200 μM ATP. At i mM ATP the speed shown by the shorter ones that of the longer ones, as shown in Fig. 5.

ATP concentration dependency of the sliding speed has been reported briefly elsewhere (Hata et al. 1980a). However, we attempted here to measure it again more exactly at a constant temperature, 25 ± 1 °C, while examining several other factors affecting speed, as described above, in order to examine the relationship between the sliding velocity of the axonemes and ATPase activity. From these observations adequate conditions for measuring the speed were determined: that is, sliding movement in demembranated whole axonemes in reactivation medium containing 2% PEG being induced by simultaneous addition of ATP and 2 μg/ml trypsin on a slide under the microscope.

Before measuring the sliding speed at various concentrations of ATP, the speed during the first 3 s after addition of various concentrations of ATP and 2 μg/ml trypsin was confirmed to be constant as shown in Fig. 6.

Sliding speed was measured at various concentrations of ATP and 2 μg/ml trypsin in the reactivation medium. As shown in Fig. 7, the velocity increased gradually from 2-3 to 14μm/s with increasing ATP concentration from 20 to 1 mM (18-4-874μM MgATP). The velocities were 8·0±0·8μm/s at 200μM ATP (183μM MgATP), n-0± 1·5 μm/s at 500 μM ATP (451 μM MgATP) and 14·0± 1·1 μm/s at 1 mM ATP (874 μM MgATP), respectively. V_max was about 14 μm/s at 1 mM ATP. Fig. 8 shows the relationship between ATP concentration and the sliding velocity in a double reciprocal plot, in which the curve was non-linear.

When ATP and trypsin were added to the axonemes, the axonemes showed a twitching movement at first. Then, after about 10 s, the axonemes extruded two or three bundles of outer doublets. The extruded bundles sometimes showed a bending motion and then separated from each other successively. In the range of 100 μM to 1 mM ATP, the extruded bundles of outer doublets separated one after another, telescoping out along the length of the doublets. At 50 μM ATP the axonemes divided at first into a few bundles of outer doublets. These bundles showed a beating motion of about 5 Hz, and doublets were extruded gradually from their ends. At concentrations of from 20 to 30 μM ATP the doublets could not slide along their whole length and only walked a little on the adjacent doublet, showing the bending motion. The sliding movement quickly stopped at 10 μM ATP, after partial extrusion of the doublets from the axonemes. As these responses of axonemes to ATP depended so exactly upon the concentration of the added ATP, we were able to estimate roughly the concentration of ATP from observing the axoneme response.

The minimum concentration of ATP needed to induce sliding movement in trypsin-digested axonemes was 4 μM ATP. It was difficult to determine the minimum concentration of ATP needed in the case of the non-digested axonemes with the simultaneous addition of ATP and trypsin, because it took more time to induce the movement at a lower concentration of ATP, i.e. less than 10 μM ATP, during which the trypsin destroyed the axoneme structure.

To determine ATP concentration exactly, the axonemes were previously suspended in reactivation medium containing a definite concentration of ATP. After 1 min, the axonemes were mixed with the same concentration of ATP and 2μg/ml trypsin, and then sliding speed was examined. In the range of ATP concentration from 20 to 200 μM, the axonemes previously suspended in the same concentration of ATP showed similar velocities to those of axonemes which were not suspended in ATP. The sliding percentage decreased to about 50% at 200 μM ATP. The sliding movement of the axonemes could not be induced at more than 300 μM ATP. After 1 h or so, such axonemes extruded the outer doublets in a manner similar to the axonemes not suspended in ATP, which might be due to the decrease of ATP concentration. Some axonemes divided longitudinally during suspension in ATP without trypsin. Therefore, we could not measure the sliding velocity at more than 200μM ATP concentration, using axonemes previously suspended in ATP.

Using shorter fragments of around 15 am (15·0 ± 3·7 μm) in length, the dependency of sliding velocity upon ATP concentration was measured. The velocities had values similar to those of the whole axonemes, in the range of 20–200 μM ATP. The shorter fragments showed a slower speed than the longer ones at 1 mM ATP, that is, the F_max was 12·4+1·1μm/s for the shorter fragments, while the F_max was 14μm/s for the whole axonemes, as shown in Fig. 7.

In order to study the relationship between ATP hydrolysis and sliding velocity, we tried measuring ATPase activity using the same axonemes in the same reactivation medium, in which sliding movement of the axonemes could be induced. Only polyethylene glycol (PEG) was omitted from the reactivation medium, because PEG had an inhibitory effect on colour development due to liberated inorganic phosphate. We performed this measurement in the absence of trypsin. It was difficult to measure ATPase activity exactly in the presence of trypsin, which brought about other effects such as solubilization of dynein arms from the extruded doublet tubules, digestion of dynein molecules and doublet microtubules, etc. To compare ATPase activity with sliding movement, ATP hydrolysis by ATPase was measured as soon as possible after addition of ATP. ATP hydrolysis was measured during the first 1 min after addition of various concentrations of ATP. As shown in Fig. 9 the activity gradually reached maximal value at 1 mM ATP and then dropped slightly. This relationship was expressed in a double reciprocal plot, as shown in Fig. 10, which indicated that the relationship between ATP concentration and ATP hydrolysis did not obey Michaelis-Menten kinetics. ATP hydrolysis was also surveyed during the initial 5 s of the reaction after addition of ATP: this resembled the activity of ATPase during the first 1 min. One possibility examined was that the liberated ADP had an inhibitory effect on ATPase activity, which caused the non-linear relationship between ATP concentration and hydrolysis activity. Measurement of ATP hydrolysis at various concentrations of ATP was done preliminarily, keeping a constant concentration of ATP by a pyruvic acid and pyruvate kinase-coupled system, in which ADP was transformed into ATP as soon as ADP was liberated. However, ATPase activity in this system also was expressed in a non-linear curve substantially similar to those described above.

The sliding velocity of outer doublets was measured using axonemes prepared under 2 sets of conditions: adding ATP to trypsin-digested axonemes, and adding ATP and trypsin simultaneously to non-digested axonemes.

When 1 mM ATP in the reactivation medium was added to the digested axonemes, the sliding movement occurred at the same time among several outer doublets, which were in coiled form according to an inherent tendency. Adding 1 mM ATP and 2 μg/ml trypsin simultaneously, the non-digested axonemes extruded some outer doublets as bundles, from which one or more outer doublets telescoped out successively. Sliding velocity could be determined by measuring the sliding length of the doublets and the time required. It was easier to measure the sliding length along the non-digested axonemes in the presence of ATP and trypsin, than that of the trypsin-digested axonemes. Furthermore, the axonemes digested with 2μg/ml trypsin for 90 s developed a new band in the dynein region on 4% polyacrylamide gel electrophoresis. The appearance of the new band indicated that trypsin digestion for a longer period may do some damage to the structure of the axonemes, especially to the dynein molecules.

Adequate trypsin concentration was determined to be 2 μg/ml, based on the observations of decrease of sliding percentage at less than 2 μg/ml and of increase of the disintegration phenomenon without sliding movement at more than 3 μg/ml trypsin concentration.

The effect of viscosity of the reactivation medium was examined using PEG. Sliding velocity was almost constant in the range of 0·2% PEG concentration, and became slower in proportion to the increase of PEG beyond 2%. In accord with these observations, we decided to use a reactivation medium containing 2% PEG. Gibbons & Gibbons (1972) reported that ATPase activity coupled with bending movement of sperm tails decreased slightly at more than 2% PEG concentration.

Shorter-length fragments of axonemes were obtained by homogenizing whole axonemes and the sliding velocity was measured. The shorter fragments had velocity values similar to those of longer ones at less than 200 μM ATP. Although we cannot definitely specify the reason, the shorter ones showed a slightly slower speed at 1 mM ATP than the longer ones. Some doubt remained whether or not the shorter fragments may receive more injury than the longer ones in the homogenizing process.

On the basis of these observations, the sliding speed of the outer doublets from demembranated whole axonemes was measured in reactivation medium containing 2% PEG with the simultaneous addition of ATP and 2μg/ml trypsin to the slide at 25 ± 1 °C. Before measuring the sliding velocity at each concentration of ATP, the sliding speed during the 5 s after addition of ATP and 2 μg/ml trypsin was surveyed and confirmed to be almost constant during these 5 s.

We measured the sliding velocity at various concentrations from 20 μM to 2 mM ATP. The velocity increased in proportion to the increase in concentration of added ATP. The velocities were 8 μm/s at 200 μM ATP and 11 μm/s at 500 μM ATP. At 1 mM ATP, the velocity reached a maximal value of 14μm/s. To determine ATP concentration exactly, some axonemes were suspended previously in reactivation medium containing ATP. After adding the same concentration of ATP and 2 μg/ml trypsin the sliding velocity of the axonemes was measured. However, several points described below, led to the conclusion that it was adequate in this experiment to measure sliding velocity of the non-suspended axonemes by simultaneously adding ATP and trypsin on the slide. The axonemes showed similar values of sliding velocity in the range of ATP concentration from 20 to 200 /μM to those of axonemes previously suspended in ATP. The axonemes suspended in greater than 300 μM ATP could not be induced to extrude tubules by adding ATP and trypsin. As we added ATP and trypsin continuously to the non-suspended axonemes on the slide until trypsin digestion began to take effect on the axonemes after about 30 s, the added ATP diffused sufficiently to the axonemes on the slide. So the doubt about the exact concentration of ATP around the axonemes can be removed.

In our previous work (Hata et al. 1979), we reported on the ATP concentration dependency of tubule extrusion and gave a maximal value for the velocity of 7·8 μm/s. The reason why the values were lower than those reported here may be attributed to the different methods of preparation of the axonemes, the effect of lower temperature, shorter length of axonemes, etc. Since the trypsin-digested axonemes responded by extruding tubules as soon as ATP diffused into the axonemes, it could be difficult to determine ‘true’ ATP concentration on the slide, especially at the beginning of the sliding movement. Therefore we measured the sliding velocity here more exactly using a constant temperature of 25 ± 1 °C and examining several factors affecting the sliding velocity.

The minimum concentration of ATP required to induce sliding movement was determined to be 4 μ M ATP, using axonemes previously digested by trypsin. We could not determine the minimum concentration of ATP in the case of the non-digested axonemes because with simultaneous addition of ATP and trypsin more time was required to induce the sliding movement at less than 10 μ M ATP. Trypsin destroyed the structure of the axonemes before ATP could induce the extrusion of the tubules.

Gibbons (1974) made a preliminary report of a sliding speed of 10 μ m/s in 1 mM ATP and o-8 μ m/s in 10 μ M ATP at 25 °C, and compared these values with those predicted using a sliding filament model. For example, demembranated flagella in1 mM ATP at 25 °C propagate waves at a bend angle of 2 radians and a beat frequency of 30/s, and this corresponded to a velocity of 8 · 4 μ m/s. However, assuming that the distance between adjacent doublet microtubules is 48-55 run, instead of the smaller value of about 35 nm reported by Gibbons, we get a velocity of 12 – 14 μ m/s or so, which is close to the maximal value reported here.

Gibbons & Gibbons (1972) reported that the beat frequency of Triton-extracted sperm varied in concentrations from 4 μ M to 4 mM ATP in the reactivation medium. They attempted to relate the changes in motility to changes in the rate of ATP hydrolysis. Within the range of acceptable experimental error, the movement-coupled ATPase activity appears to be proportional to beat frequency under each set of conditions, except the value of movement-coupled ATPase in 1 mM ATP. They also prepared double reciprocal plots demonstrating the dependency of beat frequency on ATP concentration. In the range of ATP concentrations from 35 μ M to 1 mM the points fitted a straight line.

In the present paper, we have attempted to study the relationship of sliding velocity to dynein ATPase activity in axonemes. ATPase activity and sliding velocity of the axonemes were both measured using the same axoneme samples and the same reactivation medium.

ATPase activity reported here was not ‘movement-coupled ATPase activity’ according to the definition of Brokaw & Benedict (1968) and Gibbons & Gibbons (1972). Trypsin digestion necessary for inducing sliding movement brought about some secondary effects on ATPase activity, that is, solubilization of dynein arms from the extruded doublet tubules, digestion of dynein ATPase, doublet tubules, etc. These effects consequently brought about values of ATPase activity too widely scattered to provide any firm values. Therefore we used non-digested axonemes as the axoneme sample for ATPase activity measurements in these experiments.

Figs. 7 and 9 show that both the sliding velocity and ATPase activity depended upon ATP concentration. The relationship between ATPase activity and sliding velocity is expressed in Fig. 11, which indicates that velocity and ATPase activity are closely correlated. The enzyme kinetic properties of the axoneme ATPase are similar to those of bound dynein ATPase under some conditions reported by Gibbons & Fronk (1972), in which a double reciprocal plot was also non-linear. This complex enzyme kinetic behaviour might be characteristic of axoneme ATPase assayed under conditions that would induce sliding movement in axonemes. As Fig. 11 shows a close relationship between ATPase activity and sliding movement, sliding-movement-coupled ATPase activity might well be reflected in the ATPase activity reported here. Although the reason why both curves representing ATPase activity and sliding velocity are non-linear in a double reciprocal plot cannot be stated at present, this complexity may result from co-operative interaction between the enzymic units of the dynein, or from the presence of 2 ATPase’s having different K_m values as has been pointed out already by Gibbons & Fronk (1972).

The ATPase of trypsin-treated axonemes of Tetrahymena cilia was reported to have 2 apparent K_m values, 12 · 7 and 1 · 0 μ M. The dependency of the turbidity change of the axonemes on ATP concentration had an apparent of 1 μ M, which agreed with the K_m of ATPase in the lower ATP concentration range (Takahashi & Tonomura, 1978).

Assuming briefly that the ATPase had 2 different K_m values, addition of which would fit the observed non-linear curve of ATPase activity, we tried to obtain them and found them to be 1 and 300 μ M respectively, using the weighted least-square procedure in computer simulation. Although we do not know what actual meaning these calculated K_m values have for flagellar movement, it is very interesting that the K_m for the beat frequency of flagella was 210 μ M (Gibbons & Gibbons, 1972) and that rigor waves were released in the presence of 1 μ M ATP (Gibbons & Gibbons, 1974). Further study of this point is in progress now.

The authors wish to thank Dr S. T. Ohnishi of Hahnemann Medical College in Philadelphia, Dr R. Kamiya of Nagoya University and Dr Y. Hiramoto of the Tokyo Institute of Technology for valuable suggestions, Dr M. Yoneda of Kyoto University for computer simulation, and Dr S. Yoshino of Nagoya University for calculation of MgATP concentration. Thanks are also due to the Director and Staff of Tateyama and Sugashima Marine Biological Laboratory for kind help during the course of this work.

This work was supported in part by grants from the Ministry of Education of Japan.