Myosin cross bridges in skeletal muscles: "rower" molecular motors
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《应用生理学杂志》
Service de Physiologie, Université Paris-Sud XI, Hopital Bicêtre, Assistance Publique-H?pitaux de Paris, 94275 Le Kremlin-Bicêtre; and Laboratoire d'Optique Appliquée Unité Mixte de Recherche-7639, Centre National de La Recherche Scientifique, Ecole Nationale Supérieure de Techniques Avancées-Ecole Polytechnique, Institut National de la Sante et de la Recherche Médicale, 91761 Palaiseau, France
ABSTRACT
Different classes of molecular motors, "rowers" and "porters," have been proposed to describe the chemomechanical transduction of energy. Rowers work in large assemblies and spend a large percentage of time detached from their lattice substrate. Porters behave in the opposite way. We calculated the number of myosin II cross bridges (CB) and the probabilities of attached and detached states in a minimal four-state model in slow (soleus) and fast (diaphragm) mouse skeletal muscles. In both muscles, we found that the probability of CB being detached was ~98% and the number of working CB was higher than 109/mm2. We concluded that muscular myosin II motors were classified in the category of rowers. Moreover, attachment time was higher than time stroke and time for ADP release. The duration of the transition from detached to attached states represented the rate-limiting step of the overall attached time. Thus diaphragm and soleus myosins belong to subtype 1 rowers.
keywords:duty ratio; attached and detached cross-bridge probabilities
INTRODUCTION
NEW INSIGHTS INTO THE LINK between the cascade of elementary biochemical events of the actomyosin ATPase cycle and myosin molecular motor mechanics (3, 35) have been provided by X-ray crystallographic studies on three-dimensional molecular structures (4, 12, 33), by in vitro motility assays (9, 10, 20, 38-40) and mutagenesis (36), by microneedles (19) and optical tweezers (6, 7, 30, 42), and by time-resolved structural studies on muscle fibers (18). Alternatively, theoretical models have contributed to a better understanding of chemomechanical transduction in molecular motors (5, 15-17, 33).
Two distinct classes of molecular motors, namely "rowers" and "porters," have been proposed by Leibler and Huse (25) to describe a general phenomenological theory for transduction from chemical to mechanical energy. The distinction is mainly based on the number of motors working together and the probability of the motors being in detached or attached states. Rowers such as muscular class II myosins and axonemal dyneins generally work in large assemblies. Muscular myosins spend a large fraction of time detached from the fiber (13, 35). Porters such as cytoplasmic kinesins or dyneins generally work processively (26), alone or in small groups, and spend a relatively large fraction of time attached to the fiber (1).
The aim of our study was to assess in living, isolated skeletal muscles the rower characteristics of muscular myosin (class II) molecular motors and whether or not they belong to the same rower subtype. Experimental characterization was performed in skeletal muscles from mouse, whose diaphragm (Dia) is almost exclusively composed of fast fibers and whose soleus (Sol) is almost exclusively composed of slow fibers (43). Our approach was based on simultaneous utilization of Huxley's equations (15) and the rowers vs. porters theory (25). From the experimental data, Huxley's equations were used to calculate the cross-bridge (CB) number, the turnover rate of myosin ATPase, and the rate constants for CB attachment and detachment. At the same time, the stochastic model of Leibler and Huse (25) (Fig. 1) was used to calculate 1) the probability of each state occurring and 2) the rate-limiting step of the actomyosin cycle, thus enabling determination of the rower subtype to which the myosin motor belongs. The theoretical model of Leibler and Huse is a "minimal" model, which is compatible with what is commonly established for actomyosin in mechanics and biochemistry (27), namely, 1) ATP hydrolysis induces strain, which is then transformed into mechanical energy; 2) the release of Pi triggers the release of the strain; and 3) the presence of bound ADP makes detachment of the myosin head impossible. We proposed a theoretical framework combining the equations of both Huxley (15) and Leibler and Huse (25), which were simultaneously applied to living skeletal muscles.
Glossary
AAttached state
CBCross bridge
DDetached state
DiaDiaphragm
eFree energy required to split one ATP molecule (5.1 × 1020 J)
f1Peak value of the rate constant for CB attachment (s1)
g1 and g2Peak values of the rate constants for CB detachment (s1)
hMolecular step size (11 nm)
kD1 = tD11Rate constant of transition between states D and 1 (s1)
k12 = t121Rate constant of transition between states 1 and 2 (s1)
k23 = t231Rate constant of transition between states 2 and 3 (s1)
k3D = t3D1Rate constant of transition between states 3 and D (s1)
KmMichaelis constant at ATP concentration ([ATP]) for which the ATPase turnover rate is half-maximal
lDistance between two actin sites (36 nm)
LmMichaelis-like constant at [ATP] for which the fiber velocity is half-maximal
N*"Saturating" number of motors
Lm/Km
Elementary force per single CB (pN)
CB number per mm2 (× 109) at peak isometric tension
P1Probability of state A1
P2Probability of state A2 = duty ratio = t12/tc
P3Probability of state A3
PAProbability of CB being attached
PDProbability of CB being detached
RmaxMaximum turnover rate of myosin ATPase (s1)
SolSoleus
Average CB velocity (μm/s)
t12Time stroke (s)
tc = 1/RmaxOverall duration of the CB time cycle (s)
VmaxMaximum unloaded muscle shortening velocity (resting muscle length/s)
wMaximum mechanical work of a single CB (3.8 × 1020 J); w = 0.75 e
MATERIALS AND METHODS
In one time, mechanical experiments were carried out on isolated Dia and Sol muscles of mouse. Tension and velocity were measured in living muscles throughout the overall load continuum to determine the Hill hyperbolic relationship, which is characterized by the two asymptotes (a and b) and the curvature G (11, 44). In a second time, these experimental parameters were introduced in the Huxley (15) and Leibler and Huse (25) equations. This made it possible to calculate the CB number and the probability of attached and detached states, thus enabling classification of the myosin II of skeletal muscles in either rower or porter molecular motor.
Experimental Protocol
Mounting procedure. Experiments were conducted in adult mice. After anesthesia with pentobarbital (30 mg/kg ip), muscle strips from the ventral part of the costal Dia (n = 10) and from the Sol (n = 8) were carefully dissected out from the muscles in situ. Each muscle strip was attached to an electromagnetic force transducer in a tissue chamber containing a Krebs-Henseleit solution, bubbled with 95% O2-5% CO2, and maintained at 22°C and pH 7.40. Dia and Sol muscle strips were electrically stimulated by means of two platinum electrodes delivering tetanic stimulation as follows: electrical stimulus, 1-ms duration; stimulation frequency, 50 Hz; train duration, 250 ms; train frequency, 0.17 Hz. While the lower end of the strip was held by a stationary clip at the bottom of the bath, the upper extremity of the strip was held in a spring clip, linked to an electromagnetic lever system, as previously described (23). Briefly, the load applied to the muscle was determined by means of a servomechanism-controlled current through the coil of an electromagnet. Muscular shortening induced a displacement of the lever, which modulated the light intensity of a photoelectric transducer. The equivalent moving mass of the whole system was 150 mg, and its compliance was 0.2 μm/mN. The system was linear up to 5 mm of muscle shortening. Experiments were carried out at the resting muscle length (Lo) that corresponds to the peak of the isometric active tension-initial length relationship. The initial preload (resting tension), which determined Lo, was automatically maintained constant throughout the experiment. All analyses were made from digital records of force and length obtained with a computer.
Mechanical analysis. Maximum unloaded shortening velocity of the muscle (Vmax, in Lo/s) was measured as the peak value of the contraction abruptly clamped to zero load just after the electrical stimulus. The hyperbolic tension-velocity relationship was derived from the peak velocity (V) of 7-10 isotonic afterloaded contractions, plotted against the isotonic load level normalized per cross-sectional area (P), by successive load increments, from zero load up to the isometric tension. Experimental data from the P-V relationship were fitted according to Hill's equation (P + a) (V + b) = [(cPmax) + a] b, where a and b are the asymptotes of the hyperbola (11) and cPmax is the calculated peak isometric tension for V = 0.
Statistical analysis. Data are expressed as means ± SE. Dia were compared with Sol using Student's unpaired t-test after ANOVA. P values < 0.05 were required to rule out the null hypothesis. Linear regression was based on the least squares method. The asymptotes a and b of the Hill hyperbola were calculated by multilinear regression and the least squares method.
Theoretical Background
Two theoretical approaches, i.e., that of Huxley (15) and that of Leibler and Huse (25), were combined to study the kinetic behavior of myosin CB molecular motors and determine myosin CB characteristics in living skeletal muscles (Fig. 1). Although these two models operate under two general states, either the attached state or detached state, both models allow the calculation of supplementary substates. Huxley's equations were used to calculate 1) the total CB number at peak isometric tension; 2) the probability of state A1 (P1); and 3) the probability of state A2 (P2). The equations of Leibler and Huse were used to calculate 1) the probability of state A3 (P3); 2) the probability of CB being detached (PD); 3) the probability of CB being attached (PA); and 4) the saturating number of motors (N*). This made it possible to classify myosin II into either rower or porter molecular motor types. Finally, in the case of rowers, the rate-limiting step of the actomyosin cycle was calculated to determine the rower subtype to which the myosin II motor belongs.
CB characteristics in Huxley's equations. The rate of total energy release () and the isotonic tension (PHux) as a function of muscle V were calculated from Huxley's equations (15). is given as
(1)
The is the CB number per mm2 at peak isometric tension (15). The h is the molecular step size or CB stroke size and is defined by the translocation distance of the actin filament per ATP hydrolysis, produced by the swing of the myosin head (17). The estimated value of h (11 nm) taken in our study is supported by the three-dimensional head structure of muscle myosin II (4, 33). The l is the distance between two actin sites and is equal to 36 nm (34); f1 is the peak value of the rate constant for CB attachment; and g1 and g2 (which appear in Eq. 4) are the peak values of the rate constants for CB detachment. The tilt x of the myosin head relative to actin varies from h to 0; f1 and g1 correspond to a tilt x = h, and g2 corresponds to a tilt x 0 (15). The free energy for the splitting of one ATP molecule (e) is equal to 5.1 ×1020 J (15, 44); = (f1 + g1)h/2 = b (24), where b is an asymptote of the hyperbolic tension-velocity relationship (11).
The minimum value of occurring in isometric conditions is
(2)
o is also equal to the product of the two asymptotes (ab) of the hyperbolic tension-velocity relationship (11). Determination of the asymptotes a and b was derived from mechanical data. The maximum turnover rate of myosin ATPase under isometric conditions (Rmax, in s1) is o/e
(3)
The overall duration of the CB time cycle (tc) is equal to 1/Rmax.
PHux is given by
(4)
where w = 0.75 e is the maximum mechanical work of a single CB (15). The CB number per square millimeter at peak isometric tension is then given by the equation: = PHux max/, where PHux max is the maximum value of PHux when V = 0. Then the elementary force per single CB in isometric conditions (, in pN) is
(5)
Calculations of f1, g1, and g2 have been described previously (2, 23, 24) and are given by the following equations
(6)
(7)
(8)
where Vmax is the maximum unloaded shortening velocity of the muscle and G is the curvature of the hyperbolic tension-velocity relationship (11). The average CB velocity (o, in μm/s) is given by o = o/( × ), where o is in mN · mm2 · mm1 · s1. The duration of the time stroke is t12 = h/o, and the duration of the attachment step is tD1 = 1/f1.
CB characteristics in Leibler and Huse equations. These equations describe a stochastic minimal four-state model, composed of one detached state (D) and three attached states (A1, A2, and A3), that acts by means of a tight-coupling mechanism (5, 15-17). In state D, the CB is detached from the fiber and binds to the nucleotide (Fig. 1). In state A1, the myosin head is bound to the actin fiber. During the transition A1 A2, Pi release from the actomyosin complex triggers the power stroke of the molecular motor. During the transition A2 A3, the hydrolysis product ADP is released. In state A3, the motor is still attached to the fiber, and CB detachment occurs when ATP binds to the actomyosin complex. The probability distributions of the four states are governed by equations that take into account the motor motion and the transitions between the states. Equations provide the Rmax, the o, and the PD. The tij is the transition time between states i and j (where i and j = states 1, 2, 3, and D), and kij = tij1 is the rate constant of transition between states i and j. The probability Pj of the state j to occur is Pj = tij/tc (25).
According to Leibler and Huse (25), the equations of Rmax, o, and PD are as follows.
Rmax is
(9)
where KD1 is the equilibrium constant and is equal to kD1/k1D.
The o is
(10)
with o = h/t12.
The PD is
(11)
By rearranging Eq. 10, we obtained
(12)
This implies that t12 > t23.
By rearranging Eq. 9, we obtained
(13)
From Eqs. 12 and 13, we deduced that
(14)
This equation had three positive roots. One root was t23 > t12 and must be excluded. The second root t23 t12 was also excluded, because ADP release is fast (5). The third positive root t23 << t12 was retained. Moreover, time cycle tc was equal to
(15)
where tD1 + t12 + t23 was the overall attached time and t3D was the detached time.
The PA was
(16)
The PD was
(17)
P1, P2, and P3 were the probabilities of states A1, A2, and A3, respectively; P2 = t12/tc is called the duty ratio under isometric conditions.
Ratio of the Michaelis and Michaelis-like constants . The turnover rate of myosin ATPase complies with a simple Michaelis law. The Rmax for large [ATP] is given by Eq. 9. Km is the Michaelis constant at [ATP] for which the ATPase rate is half-maximal. Lm is the Michaelis-like constant at [ATP] for which the fiber velocity in in vitro motility assay is half-maximal. The ratio of Michaelis and Michaelis-like constants for the fiber velocity and turnover rate of ATPase is defined as follows (25)
N*. For large [ATP] and in in vitro motility assay, fiber velocity increases with the number of motors N and then saturates for a number of motors equal to N*. In the equations of Leibler and Huse (25), it has been shown that
Criteria for subtypes of rowers. In all rower subtypes, PD is high (PD 1) and P2 is << 1. From a theoretical point of view, there are two subtypes of rowers: the D A1 rate-limiting rowers and the A1 A2 rate-limiting rowers (25).
In the D A1 rate-limiting rowers, the rate-limiting step of the time cycle is the binding step to the fiber rather than the release step of the ATP hydrolysis products. The time constant tD1 is much larger than t12 and t23; (1 PD) << 1 and P2 << 1. The rate-limiting step of the time cycle is the transition D A1, where tD1 = 1/f1. In the A1 A2 rate-limiting rowers, the time constant t12 is much larger than t23 and tD1; (1 P1 PD) << 1 and P2 << 1. The rate-limiting step of the time cycle is the transition A1 A2. Calculations of the probabilities of each state of the CB cycle and determination of the rower motor subtype to which skeletal myosin II belongs did not depend on the values of w, e, h, and l.
RESULTS
Experimental Data
Total isometric tension did not differ between Dia and Sol (Table 1). Vmax was about twofold higher in Dia than in Sol. The asymptote a of the tension-velocity relationship did not differ between the two muscles. The asymptote b was significantly higher in Dia than in Sol. The G of the tension-velocity relationship did not differ between the two muscles (Table 1).
Calculated Data
The total number of working CB/mm2 was 14.0 ± 0.9 × 109/mm2 in Dia and 13.5 ± 2.3 × 109/mm2 in Sol and did not differ between the two muscles (Fig. 2). The CB unitary force () did not differ between Dia and Sol (Fig. 2). The Rmax was higher in Dia than in Sol (Fig. 2). Both the overall attached and detached times were longer in Sol than in Dia (Fig. 2).
The time cycle (tc =1/Rmax) was significantly shorter in Dia than in Sol (Fig. 3). The time parameters tD1 and t12 on the one hand and t1D and t23 on the other hand did not differ between Dia and Sol (Fig. 3). In both Dia and Sol, tD1 was much longer than t12 and t23 (Fig. 3). Consequently, in Dia and Sol muscles, most of the overall attached time was occupied by the attachment step tD1 = 1/f1, i.e., the rate-limiting step of the overall cycle was the transition D A1. At the onset of the transition A3 AD, the time for CB detachment (1/g2) was significantly shorter in Dia than in Sol (Fig. 3).
In the two muscles, PD was markedly high (~98%), and PA was markedly low (~2%) (Fig. 4). Both PD and PA did not differ between Dia and Sol (Fig. 4). In Dia and Sol, the probabilities P1, P2, and P3 of the three attached states A1, A2, and A3 were ~2 × 102, 2 × 103, and 2-4 × 104, respectively (Fig. 4). Probability A1 did not differ between Dia and Sol. Probabilities A2 and A3 were significantly higher in Dia than in Sol (Fig. 4). Moreover, N* was significantly lower in Dia than in Sol (Fig. 4).
DISCUSSION
The main aim of this study was to characterize, in living skeletal muscles, the molecular motor category and subtype to which the muscular II myosin CB belong. To this end, two powerful theoretical approaches (15, 25) were combined to calculate the number and kinetics of CB in a minimal four-state model. Experimental characterization was performed in fast and slow skeletal muscles from mouse. The high probability of CB being detached, the high number of working CB, and the N* together made it possible to classify these muscular myosin motors into the category of rowers. As the rate-limiting step was the binding to the fiber rather than the release of the ATP hydrolysis products, muscular myosin CB were classified as subtype 1 of rowers in both muscles.
Values of the Constants e, h, and l in Huxley's Equations
The length-tension behavior of a CB can be determined in quick release experiments (44). The work that can be done by a CB is the area under its elastic deformation curve and is at least 3.7 × 1020 J or 22 kJ/mol of CB. This is very similar to the w equal to 3.8 × 1020 J = 0.75 e (where e = 5.1 × 1020 J) used in our study. This is of the same order of magnitude as the e in vivo (21).
The h is subject to uncertainty and could range from one-half of the assumed value to twice the assumed value (6, 7, 30, 42). X-ray diffraction studies (4, 33) allow a step-size estimate of ~10 nm, a value consistent with that predicted by Huxley and Simmons (16) and measured by Finer et al. (6). As the rate constants for attachment (f1) and detachment (g1 and g2) depended on h, uncertainties on h implied uncertainties on f1, g1, and g2.
The pitch of the polymerized actin helix, i.e., l, is 36 nm in all actin isoforms from eukaryotic cells, i.e., in both muscle and nonmuscle actins. In eukaryotic cells, sequences of actin are more highly conserved than almost any other proteins (34). It is largely admitted that the value of l is invariant and equal to 36 nm.
Combined Theoretical Models of Huxley and Leibler and Huse
These two models were combined because together they allow calculation of several biological events that cannot be calculated if the models are used separately. In particular, the Leibler and Huse model makes it possible to calculate the CB step for ADP release, whereas that of Huxley is used to calculate the CB detachment step. Both models belong to the class of tight coupling of motor functioning (15-17) and assume that transition rates are strain dependent. The two models operate under two general states, either attached state or detached state, both having the possibility of generating supplementary substates. Huxley's model is classically considered as two-state and is analytically solvable. However, Huxley's equations (15) make it possible to calculate several substeps, i.e., the attachment step (tD1 = 1/f1), the stroke or step size (t12), the detachment step (1/g2), the remainder of the CB cycle (i.e., t23 + t3D), and the overall duration of the CB cycle (tc). The Leibler and Huse model is constructed to be minimal, i.e., to include the minimum number of states that cannot be reduced if agreement is seeked with established biochemical and mechanical data for actomyosin and can be examined as a four-, three-, or even two-state model (25).
For the sake of simplicity, the Leibler and Huse model does not take into account the fact that the binding sites of the motor proteins to the fiber are discrete. However, this minimal model can be described as a periodic model (25), introducing the l, which is a basic parameter of Huxley's model.
Number of Working Molecular Motors
The first major characteristic of rowers is that they work in large assemblies of uncorrelated motors. Our results show a high number of working myosin CB per cross-sectional area (>109/mm2) in both Dia and Sol (Fig. 2). In rowers, such high numbers of working muscle myosin heads have been observed in species other than the mouse, in particular in pathophysiological conditions (23) and during development (2). This was partly due to the tight lattice of myosin thick filaments in skeletal muscle and to the fact that each half-myosin thick filament is composed of ~300 myosin heads. In our study, the CB number calculated at peak isometric tension is the ratio of total isometric tension to mean CB single force. This contrasts with the characteristics of porters such as cytoplasmic kinesins or dyneins, which work alone or in small groups (1, 41).
PA, PD, Duty Ratio
The second major characteristic of rowers predicted by the theoretical model of Leibler and Huse is the high probability of CB being detached (1 PD << 1) and the low duty ratio (P2 << 1). Our results were in agreement with these predictions (Fig. 4). A small duty ratio has been previously suggested for muscular myosin (35). The duty ratio of muscle myosin motors can be considered as the reciprocal of the minimum number of heads needed for continuous movement (13) and has been found to be small, i.e., <0.01-0.1 (39). Indeed, in gliding assays, a minimum of tens to hundreds of myosin heads are needed for continuous motility of actin filament (10). These results contrast with those observed in porter molecular motors, which are characterized by a high probability of CB being attached (1 PA << 1). The large duty ratio predicted for porters is corroborated by experimental data on kinesin (1, 13). The two-headed conventional kinesin has to remain continuously bound to the microtubule. Thus its duty ratio must be at least 0.5 to prevent the motor from diffusing away from the filament. However, a single kinesin molecule is sufficient for motility (14).
The high probability of CB being detached and the low duty ratio (Fig. 4) gave the Sol and Dia a status of rowers. Moreover, according to the criteria of Leibler and Huse (25), the high values of tD1, compared with those of the time stroke (t12) and the time for ADP release (t23) (Fig. 4), made it possible to classify the Dia and Sol myosins in subtype 1 of rower motors. Thus the duration of the transition tD1, i.e., the attachment-step duration, represents the rate-limiting step of the overall attached time. This latter finding is in agreement with other experimental and theoretical studies (5, 11, 35). Subtype 2 rower motors have not yet been reported.
N* and
In our study, the value of N* ranged from 600 to 950 (Fig. 4). The "high" number of working CB (>109/mm2) refers to rower molecular motors working in large assemblies, contrasting with the behavior of porters, which are molecular motors working alone or in small groups. The high value of N* may be due to the fact that numerous and highly organized actin and myosin filaments are involved in living muscles, whereas only one actin filament interacts with some myosin heads in in vitro motility assays. The N* for muscular myosin has been estimated to be >10 (39, 40). In both Dia and Sol, N* was >>1, as expected in rower molecular motors (25). This contrasts with the low value of N* (equal to 1 or 2) observed in kinesin (1), with the latter finding being in agreement with predicted porter behavior.
In the equations of Leibler and Huse (25), N*is >>1 in rowers. A high value is expected to be found in muscles, corresponding to Lm >> Km. In fact, several studies corroborate this theoretical prediction. In muscular myosin motors, Lm, the [ATP] at half-maximal filament velocity, ranges from 50 to 150 μmol (9, 20), whereas Km, the [ATP] at Rmax/2, ranges from 2 to 6 μmol (9, 38). On the basis of these results, can be estimated between 10 and 50. Other experimental data also corroborated the concept of rower molecular motors, characterized by a high value. Indeed, in flagellar dyneins, Lm has been shown to be 100 μmol (28), whereas Km is <1 μmol (31). These results give a value of >100 in flagellar dyneins. If many motors are present ( >> 1 and, consequently, Lm >> Km), the fiber velocity saturates at much larger [ATP] than does the hydrolysis rate. This strongly contrasts with results observed in porter molecular motors, where Lm has been estimated as ~20 μmol for bovine brain kinesin (14), whereas Km has been reported to be ~10 μmol (8, 22). As N* , the estimated value would then be = 2, in agreement with the equations of Leibler and Huse (25) and with experiments where N* is equal to 1 or 2 in conventional kinesin (1).
Rower Behavior
In molecular motors, solutions to generate movement and force are highly diversified (1, 13, 26, 29, 32, 33, 41). The strategy for cell motion adopted by muscular myosin motors offers advantages. By detaching frequently and/or for long periods of time from the fiber, large assemblies of uncorrelated myosin motors can work together without disturbing one another. Rower myosin motors must avoid working against one another. It is thus possible to minimize protein friction because the probability of CB being attached is very low (Fig. 4). Indeed, protein friction (37) is due to motors attached to the fiber, particularly in systems in which numerous motors interact with the fiber. Low-protein friction may be expected when attached time is short and when the motor detaches from the fiber as soon as the power stroke is made. Such behavior was observed in mouse Dia and Sol, in which the overall attached time was much shorter than the detached time (Fig. 2).
Conclusion
By combining two theoretical approaches, it was possible to determine the kinetics and probabilities of the different states of myosin CB in isolated skeletal muscles. Muscular class II myosin heads belong to the subtype I rower category of molecular motors, according to the criteria of Leibler and Huse (25). Thus they presented a high probability of being detached and of having a high number of working CB and a high N*, together characterizing rower behavior. In mouse, both fast and slow skeletal muscles studied belong to subtype I rowers, because the rate-limiting step is the binding to the fiber rather than the release of the ATP hydrolysis products.
ACKNOWLEDGEMENTS
The authors thank Monique Cogan for excellent technical assistance.
FOOTNOTES
Address for reprint requests and other correspondence: Y. Lecarpentier, LOA-ENSTA, Batterie de l'Yvette, 91761 Palaiseau, France (E-mail: lecarpen@enstay.ensta.fr).
The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
Received 12 January 2001; accepted in final form 26 July 2001.
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ABSTRACT
Different classes of molecular motors, "rowers" and "porters," have been proposed to describe the chemomechanical transduction of energy. Rowers work in large assemblies and spend a large percentage of time detached from their lattice substrate. Porters behave in the opposite way. We calculated the number of myosin II cross bridges (CB) and the probabilities of attached and detached states in a minimal four-state model in slow (soleus) and fast (diaphragm) mouse skeletal muscles. In both muscles, we found that the probability of CB being detached was ~98% and the number of working CB was higher than 109/mm2. We concluded that muscular myosin II motors were classified in the category of rowers. Moreover, attachment time was higher than time stroke and time for ADP release. The duration of the transition from detached to attached states represented the rate-limiting step of the overall attached time. Thus diaphragm and soleus myosins belong to subtype 1 rowers.
keywords:duty ratio; attached and detached cross-bridge probabilities
INTRODUCTION
NEW INSIGHTS INTO THE LINK between the cascade of elementary biochemical events of the actomyosin ATPase cycle and myosin molecular motor mechanics (3, 35) have been provided by X-ray crystallographic studies on three-dimensional molecular structures (4, 12, 33), by in vitro motility assays (9, 10, 20, 38-40) and mutagenesis (36), by microneedles (19) and optical tweezers (6, 7, 30, 42), and by time-resolved structural studies on muscle fibers (18). Alternatively, theoretical models have contributed to a better understanding of chemomechanical transduction in molecular motors (5, 15-17, 33).
Two distinct classes of molecular motors, namely "rowers" and "porters," have been proposed by Leibler and Huse (25) to describe a general phenomenological theory for transduction from chemical to mechanical energy. The distinction is mainly based on the number of motors working together and the probability of the motors being in detached or attached states. Rowers such as muscular class II myosins and axonemal dyneins generally work in large assemblies. Muscular myosins spend a large fraction of time detached from the fiber (13, 35). Porters such as cytoplasmic kinesins or dyneins generally work processively (26), alone or in small groups, and spend a relatively large fraction of time attached to the fiber (1).
The aim of our study was to assess in living, isolated skeletal muscles the rower characteristics of muscular myosin (class II) molecular motors and whether or not they belong to the same rower subtype. Experimental characterization was performed in skeletal muscles from mouse, whose diaphragm (Dia) is almost exclusively composed of fast fibers and whose soleus (Sol) is almost exclusively composed of slow fibers (43). Our approach was based on simultaneous utilization of Huxley's equations (15) and the rowers vs. porters theory (25). From the experimental data, Huxley's equations were used to calculate the cross-bridge (CB) number, the turnover rate of myosin ATPase, and the rate constants for CB attachment and detachment. At the same time, the stochastic model of Leibler and Huse (25) (Fig. 1) was used to calculate 1) the probability of each state occurring and 2) the rate-limiting step of the actomyosin cycle, thus enabling determination of the rower subtype to which the myosin motor belongs. The theoretical model of Leibler and Huse is a "minimal" model, which is compatible with what is commonly established for actomyosin in mechanics and biochemistry (27), namely, 1) ATP hydrolysis induces strain, which is then transformed into mechanical energy; 2) the release of Pi triggers the release of the strain; and 3) the presence of bound ADP makes detachment of the myosin head impossible. We proposed a theoretical framework combining the equations of both Huxley (15) and Leibler and Huse (25), which were simultaneously applied to living skeletal muscles.
Glossary
AAttached state
CBCross bridge
DDetached state
DiaDiaphragm
eFree energy required to split one ATP molecule (5.1 × 1020 J)
f1Peak value of the rate constant for CB attachment (s1)
g1 and g2Peak values of the rate constants for CB detachment (s1)
hMolecular step size (11 nm)
kD1 = tD11Rate constant of transition between states D and 1 (s1)
k12 = t121Rate constant of transition between states 1 and 2 (s1)
k23 = t231Rate constant of transition between states 2 and 3 (s1)
k3D = t3D1Rate constant of transition between states 3 and D (s1)
KmMichaelis constant at ATP concentration ([ATP]) for which the ATPase turnover rate is half-maximal
lDistance between two actin sites (36 nm)
LmMichaelis-like constant at [ATP] for which the fiber velocity is half-maximal
N*"Saturating" number of motors
Lm/Km
Elementary force per single CB (pN)
CB number per mm2 (× 109) at peak isometric tension
P1Probability of state A1
P2Probability of state A2 = duty ratio = t12/tc
P3Probability of state A3
PAProbability of CB being attached
PDProbability of CB being detached
RmaxMaximum turnover rate of myosin ATPase (s1)
SolSoleus
Average CB velocity (μm/s)
t12Time stroke (s)
tc = 1/RmaxOverall duration of the CB time cycle (s)
VmaxMaximum unloaded muscle shortening velocity (resting muscle length/s)
wMaximum mechanical work of a single CB (3.8 × 1020 J); w = 0.75 e
MATERIALS AND METHODS
In one time, mechanical experiments were carried out on isolated Dia and Sol muscles of mouse. Tension and velocity were measured in living muscles throughout the overall load continuum to determine the Hill hyperbolic relationship, which is characterized by the two asymptotes (a and b) and the curvature G (11, 44). In a second time, these experimental parameters were introduced in the Huxley (15) and Leibler and Huse (25) equations. This made it possible to calculate the CB number and the probability of attached and detached states, thus enabling classification of the myosin II of skeletal muscles in either rower or porter molecular motor.
Experimental Protocol
Mounting procedure. Experiments were conducted in adult mice. After anesthesia with pentobarbital (30 mg/kg ip), muscle strips from the ventral part of the costal Dia (n = 10) and from the Sol (n = 8) were carefully dissected out from the muscles in situ. Each muscle strip was attached to an electromagnetic force transducer in a tissue chamber containing a Krebs-Henseleit solution, bubbled with 95% O2-5% CO2, and maintained at 22°C and pH 7.40. Dia and Sol muscle strips were electrically stimulated by means of two platinum electrodes delivering tetanic stimulation as follows: electrical stimulus, 1-ms duration; stimulation frequency, 50 Hz; train duration, 250 ms; train frequency, 0.17 Hz. While the lower end of the strip was held by a stationary clip at the bottom of the bath, the upper extremity of the strip was held in a spring clip, linked to an electromagnetic lever system, as previously described (23). Briefly, the load applied to the muscle was determined by means of a servomechanism-controlled current through the coil of an electromagnet. Muscular shortening induced a displacement of the lever, which modulated the light intensity of a photoelectric transducer. The equivalent moving mass of the whole system was 150 mg, and its compliance was 0.2 μm/mN. The system was linear up to 5 mm of muscle shortening. Experiments were carried out at the resting muscle length (Lo) that corresponds to the peak of the isometric active tension-initial length relationship. The initial preload (resting tension), which determined Lo, was automatically maintained constant throughout the experiment. All analyses were made from digital records of force and length obtained with a computer.
Mechanical analysis. Maximum unloaded shortening velocity of the muscle (Vmax, in Lo/s) was measured as the peak value of the contraction abruptly clamped to zero load just after the electrical stimulus. The hyperbolic tension-velocity relationship was derived from the peak velocity (V) of 7-10 isotonic afterloaded contractions, plotted against the isotonic load level normalized per cross-sectional area (P), by successive load increments, from zero load up to the isometric tension. Experimental data from the P-V relationship were fitted according to Hill's equation (P + a) (V + b) = [(cPmax) + a] b, where a and b are the asymptotes of the hyperbola (11) and cPmax is the calculated peak isometric tension for V = 0.
Statistical analysis. Data are expressed as means ± SE. Dia were compared with Sol using Student's unpaired t-test after ANOVA. P values < 0.05 were required to rule out the null hypothesis. Linear regression was based on the least squares method. The asymptotes a and b of the Hill hyperbola were calculated by multilinear regression and the least squares method.
Theoretical Background
Two theoretical approaches, i.e., that of Huxley (15) and that of Leibler and Huse (25), were combined to study the kinetic behavior of myosin CB molecular motors and determine myosin CB characteristics in living skeletal muscles (Fig. 1). Although these two models operate under two general states, either the attached state or detached state, both models allow the calculation of supplementary substates. Huxley's equations were used to calculate 1) the total CB number at peak isometric tension; 2) the probability of state A1 (P1); and 3) the probability of state A2 (P2). The equations of Leibler and Huse were used to calculate 1) the probability of state A3 (P3); 2) the probability of CB being detached (PD); 3) the probability of CB being attached (PA); and 4) the saturating number of motors (N*). This made it possible to classify myosin II into either rower or porter molecular motor types. Finally, in the case of rowers, the rate-limiting step of the actomyosin cycle was calculated to determine the rower subtype to which the myosin II motor belongs.
CB characteristics in Huxley's equations. The rate of total energy release () and the isotonic tension (PHux) as a function of muscle V were calculated from Huxley's equations (15). is given as
(1)
The is the CB number per mm2 at peak isometric tension (15). The h is the molecular step size or CB stroke size and is defined by the translocation distance of the actin filament per ATP hydrolysis, produced by the swing of the myosin head (17). The estimated value of h (11 nm) taken in our study is supported by the three-dimensional head structure of muscle myosin II (4, 33). The l is the distance between two actin sites and is equal to 36 nm (34); f1 is the peak value of the rate constant for CB attachment; and g1 and g2 (which appear in Eq. 4) are the peak values of the rate constants for CB detachment. The tilt x of the myosin head relative to actin varies from h to 0; f1 and g1 correspond to a tilt x = h, and g2 corresponds to a tilt x 0 (15). The free energy for the splitting of one ATP molecule (e) is equal to 5.1 ×1020 J (15, 44); = (f1 + g1)h/2 = b (24), where b is an asymptote of the hyperbolic tension-velocity relationship (11).
The minimum value of occurring in isometric conditions is
(2)
o is also equal to the product of the two asymptotes (ab) of the hyperbolic tension-velocity relationship (11). Determination of the asymptotes a and b was derived from mechanical data. The maximum turnover rate of myosin ATPase under isometric conditions (Rmax, in s1) is o/e
(3)
The overall duration of the CB time cycle (tc) is equal to 1/Rmax.
PHux is given by
(4)
where w = 0.75 e is the maximum mechanical work of a single CB (15). The CB number per square millimeter at peak isometric tension is then given by the equation: = PHux max/, where PHux max is the maximum value of PHux when V = 0. Then the elementary force per single CB in isometric conditions (, in pN) is
(5)
Calculations of f1, g1, and g2 have been described previously (2, 23, 24) and are given by the following equations
(6)
(7)
(8)
where Vmax is the maximum unloaded shortening velocity of the muscle and G is the curvature of the hyperbolic tension-velocity relationship (11). The average CB velocity (o, in μm/s) is given by o = o/( × ), where o is in mN · mm2 · mm1 · s1. The duration of the time stroke is t12 = h/o, and the duration of the attachment step is tD1 = 1/f1.
CB characteristics in Leibler and Huse equations. These equations describe a stochastic minimal four-state model, composed of one detached state (D) and three attached states (A1, A2, and A3), that acts by means of a tight-coupling mechanism (5, 15-17). In state D, the CB is detached from the fiber and binds to the nucleotide (Fig. 1). In state A1, the myosin head is bound to the actin fiber. During the transition A1 A2, Pi release from the actomyosin complex triggers the power stroke of the molecular motor. During the transition A2 A3, the hydrolysis product ADP is released. In state A3, the motor is still attached to the fiber, and CB detachment occurs when ATP binds to the actomyosin complex. The probability distributions of the four states are governed by equations that take into account the motor motion and the transitions between the states. Equations provide the Rmax, the o, and the PD. The tij is the transition time between states i and j (where i and j = states 1, 2, 3, and D), and kij = tij1 is the rate constant of transition between states i and j. The probability Pj of the state j to occur is Pj = tij/tc (25).
According to Leibler and Huse (25), the equations of Rmax, o, and PD are as follows.
Rmax is
(9)
where KD1 is the equilibrium constant and is equal to kD1/k1D.
The o is
(10)
with o = h/t12.
The PD is
(11)
By rearranging Eq. 10, we obtained
(12)
This implies that t12 > t23.
By rearranging Eq. 9, we obtained
(13)
From Eqs. 12 and 13, we deduced that
(14)
This equation had three positive roots. One root was t23 > t12 and must be excluded. The second root t23 t12 was also excluded, because ADP release is fast (5). The third positive root t23 << t12 was retained. Moreover, time cycle tc was equal to
(15)
where tD1 + t12 + t23 was the overall attached time and t3D was the detached time.
The PA was
(16)
The PD was
(17)
P1, P2, and P3 were the probabilities of states A1, A2, and A3, respectively; P2 = t12/tc is called the duty ratio under isometric conditions.
Ratio of the Michaelis and Michaelis-like constants . The turnover rate of myosin ATPase complies with a simple Michaelis law. The Rmax for large [ATP] is given by Eq. 9. Km is the Michaelis constant at [ATP] for which the ATPase rate is half-maximal. Lm is the Michaelis-like constant at [ATP] for which the fiber velocity in in vitro motility assay is half-maximal. The ratio of Michaelis and Michaelis-like constants for the fiber velocity and turnover rate of ATPase is defined as follows (25)
N*. For large [ATP] and in in vitro motility assay, fiber velocity increases with the number of motors N and then saturates for a number of motors equal to N*. In the equations of Leibler and Huse (25), it has been shown that
Criteria for subtypes of rowers. In all rower subtypes, PD is high (PD 1) and P2 is << 1. From a theoretical point of view, there are two subtypes of rowers: the D A1 rate-limiting rowers and the A1 A2 rate-limiting rowers (25).
In the D A1 rate-limiting rowers, the rate-limiting step of the time cycle is the binding step to the fiber rather than the release step of the ATP hydrolysis products. The time constant tD1 is much larger than t12 and t23; (1 PD) << 1 and P2 << 1. The rate-limiting step of the time cycle is the transition D A1, where tD1 = 1/f1. In the A1 A2 rate-limiting rowers, the time constant t12 is much larger than t23 and tD1; (1 P1 PD) << 1 and P2 << 1. The rate-limiting step of the time cycle is the transition A1 A2. Calculations of the probabilities of each state of the CB cycle and determination of the rower motor subtype to which skeletal myosin II belongs did not depend on the values of w, e, h, and l.
RESULTS
Experimental Data
Total isometric tension did not differ between Dia and Sol (Table 1). Vmax was about twofold higher in Dia than in Sol. The asymptote a of the tension-velocity relationship did not differ between the two muscles. The asymptote b was significantly higher in Dia than in Sol. The G of the tension-velocity relationship did not differ between the two muscles (Table 1).
Calculated Data
The total number of working CB/mm2 was 14.0 ± 0.9 × 109/mm2 in Dia and 13.5 ± 2.3 × 109/mm2 in Sol and did not differ between the two muscles (Fig. 2). The CB unitary force () did not differ between Dia and Sol (Fig. 2). The Rmax was higher in Dia than in Sol (Fig. 2). Both the overall attached and detached times were longer in Sol than in Dia (Fig. 2).
The time cycle (tc =1/Rmax) was significantly shorter in Dia than in Sol (Fig. 3). The time parameters tD1 and t12 on the one hand and t1D and t23 on the other hand did not differ between Dia and Sol (Fig. 3). In both Dia and Sol, tD1 was much longer than t12 and t23 (Fig. 3). Consequently, in Dia and Sol muscles, most of the overall attached time was occupied by the attachment step tD1 = 1/f1, i.e., the rate-limiting step of the overall cycle was the transition D A1. At the onset of the transition A3 AD, the time for CB detachment (1/g2) was significantly shorter in Dia than in Sol (Fig. 3).
In the two muscles, PD was markedly high (~98%), and PA was markedly low (~2%) (Fig. 4). Both PD and PA did not differ between Dia and Sol (Fig. 4). In Dia and Sol, the probabilities P1, P2, and P3 of the three attached states A1, A2, and A3 were ~2 × 102, 2 × 103, and 2-4 × 104, respectively (Fig. 4). Probability A1 did not differ between Dia and Sol. Probabilities A2 and A3 were significantly higher in Dia than in Sol (Fig. 4). Moreover, N* was significantly lower in Dia than in Sol (Fig. 4).
DISCUSSION
The main aim of this study was to characterize, in living skeletal muscles, the molecular motor category and subtype to which the muscular II myosin CB belong. To this end, two powerful theoretical approaches (15, 25) were combined to calculate the number and kinetics of CB in a minimal four-state model. Experimental characterization was performed in fast and slow skeletal muscles from mouse. The high probability of CB being detached, the high number of working CB, and the N* together made it possible to classify these muscular myosin motors into the category of rowers. As the rate-limiting step was the binding to the fiber rather than the release of the ATP hydrolysis products, muscular myosin CB were classified as subtype 1 of rowers in both muscles.
Values of the Constants e, h, and l in Huxley's Equations
The length-tension behavior of a CB can be determined in quick release experiments (44). The work that can be done by a CB is the area under its elastic deformation curve and is at least 3.7 × 1020 J or 22 kJ/mol of CB. This is very similar to the w equal to 3.8 × 1020 J = 0.75 e (where e = 5.1 × 1020 J) used in our study. This is of the same order of magnitude as the e in vivo (21).
The h is subject to uncertainty and could range from one-half of the assumed value to twice the assumed value (6, 7, 30, 42). X-ray diffraction studies (4, 33) allow a step-size estimate of ~10 nm, a value consistent with that predicted by Huxley and Simmons (16) and measured by Finer et al. (6). As the rate constants for attachment (f1) and detachment (g1 and g2) depended on h, uncertainties on h implied uncertainties on f1, g1, and g2.
The pitch of the polymerized actin helix, i.e., l, is 36 nm in all actin isoforms from eukaryotic cells, i.e., in both muscle and nonmuscle actins. In eukaryotic cells, sequences of actin are more highly conserved than almost any other proteins (34). It is largely admitted that the value of l is invariant and equal to 36 nm.
Combined Theoretical Models of Huxley and Leibler and Huse
These two models were combined because together they allow calculation of several biological events that cannot be calculated if the models are used separately. In particular, the Leibler and Huse model makes it possible to calculate the CB step for ADP release, whereas that of Huxley is used to calculate the CB detachment step. Both models belong to the class of tight coupling of motor functioning (15-17) and assume that transition rates are strain dependent. The two models operate under two general states, either attached state or detached state, both having the possibility of generating supplementary substates. Huxley's model is classically considered as two-state and is analytically solvable. However, Huxley's equations (15) make it possible to calculate several substeps, i.e., the attachment step (tD1 = 1/f1), the stroke or step size (t12), the detachment step (1/g2), the remainder of the CB cycle (i.e., t23 + t3D), and the overall duration of the CB cycle (tc). The Leibler and Huse model is constructed to be minimal, i.e., to include the minimum number of states that cannot be reduced if agreement is seeked with established biochemical and mechanical data for actomyosin and can be examined as a four-, three-, or even two-state model (25).
For the sake of simplicity, the Leibler and Huse model does not take into account the fact that the binding sites of the motor proteins to the fiber are discrete. However, this minimal model can be described as a periodic model (25), introducing the l, which is a basic parameter of Huxley's model.
Number of Working Molecular Motors
The first major characteristic of rowers is that they work in large assemblies of uncorrelated motors. Our results show a high number of working myosin CB per cross-sectional area (>109/mm2) in both Dia and Sol (Fig. 2). In rowers, such high numbers of working muscle myosin heads have been observed in species other than the mouse, in particular in pathophysiological conditions (23) and during development (2). This was partly due to the tight lattice of myosin thick filaments in skeletal muscle and to the fact that each half-myosin thick filament is composed of ~300 myosin heads. In our study, the CB number calculated at peak isometric tension is the ratio of total isometric tension to mean CB single force. This contrasts with the characteristics of porters such as cytoplasmic kinesins or dyneins, which work alone or in small groups (1, 41).
PA, PD, Duty Ratio
The second major characteristic of rowers predicted by the theoretical model of Leibler and Huse is the high probability of CB being detached (1 PD << 1) and the low duty ratio (P2 << 1). Our results were in agreement with these predictions (Fig. 4). A small duty ratio has been previously suggested for muscular myosin (35). The duty ratio of muscle myosin motors can be considered as the reciprocal of the minimum number of heads needed for continuous movement (13) and has been found to be small, i.e., <0.01-0.1 (39). Indeed, in gliding assays, a minimum of tens to hundreds of myosin heads are needed for continuous motility of actin filament (10). These results contrast with those observed in porter molecular motors, which are characterized by a high probability of CB being attached (1 PA << 1). The large duty ratio predicted for porters is corroborated by experimental data on kinesin (1, 13). The two-headed conventional kinesin has to remain continuously bound to the microtubule. Thus its duty ratio must be at least 0.5 to prevent the motor from diffusing away from the filament. However, a single kinesin molecule is sufficient for motility (14).
The high probability of CB being detached and the low duty ratio (Fig. 4) gave the Sol and Dia a status of rowers. Moreover, according to the criteria of Leibler and Huse (25), the high values of tD1, compared with those of the time stroke (t12) and the time for ADP release (t23) (Fig. 4), made it possible to classify the Dia and Sol myosins in subtype 1 of rower motors. Thus the duration of the transition tD1, i.e., the attachment-step duration, represents the rate-limiting step of the overall attached time. This latter finding is in agreement with other experimental and theoretical studies (5, 11, 35). Subtype 2 rower motors have not yet been reported.
N* and
In our study, the value of N* ranged from 600 to 950 (Fig. 4). The "high" number of working CB (>109/mm2) refers to rower molecular motors working in large assemblies, contrasting with the behavior of porters, which are molecular motors working alone or in small groups. The high value of N* may be due to the fact that numerous and highly organized actin and myosin filaments are involved in living muscles, whereas only one actin filament interacts with some myosin heads in in vitro motility assays. The N* for muscular myosin has been estimated to be >10 (39, 40). In both Dia and Sol, N* was >>1, as expected in rower molecular motors (25). This contrasts with the low value of N* (equal to 1 or 2) observed in kinesin (1), with the latter finding being in agreement with predicted porter behavior.
In the equations of Leibler and Huse (25), N*is >>1 in rowers. A high value is expected to be found in muscles, corresponding to Lm >> Km. In fact, several studies corroborate this theoretical prediction. In muscular myosin motors, Lm, the [ATP] at half-maximal filament velocity, ranges from 50 to 150 μmol (9, 20), whereas Km, the [ATP] at Rmax/2, ranges from 2 to 6 μmol (9, 38). On the basis of these results, can be estimated between 10 and 50. Other experimental data also corroborated the concept of rower molecular motors, characterized by a high value. Indeed, in flagellar dyneins, Lm has been shown to be 100 μmol (28), whereas Km is <1 μmol (31). These results give a value of >100 in flagellar dyneins. If many motors are present ( >> 1 and, consequently, Lm >> Km), the fiber velocity saturates at much larger [ATP] than does the hydrolysis rate. This strongly contrasts with results observed in porter molecular motors, where Lm has been estimated as ~20 μmol for bovine brain kinesin (14), whereas Km has been reported to be ~10 μmol (8, 22). As N* , the estimated value would then be = 2, in agreement with the equations of Leibler and Huse (25) and with experiments where N* is equal to 1 or 2 in conventional kinesin (1).
Rower Behavior
In molecular motors, solutions to generate movement and force are highly diversified (1, 13, 26, 29, 32, 33, 41). The strategy for cell motion adopted by muscular myosin motors offers advantages. By detaching frequently and/or for long periods of time from the fiber, large assemblies of uncorrelated myosin motors can work together without disturbing one another. Rower myosin motors must avoid working against one another. It is thus possible to minimize protein friction because the probability of CB being attached is very low (Fig. 4). Indeed, protein friction (37) is due to motors attached to the fiber, particularly in systems in which numerous motors interact with the fiber. Low-protein friction may be expected when attached time is short and when the motor detaches from the fiber as soon as the power stroke is made. Such behavior was observed in mouse Dia and Sol, in which the overall attached time was much shorter than the detached time (Fig. 2).
Conclusion
By combining two theoretical approaches, it was possible to determine the kinetics and probabilities of the different states of myosin CB in isolated skeletal muscles. Muscular class II myosin heads belong to the subtype I rower category of molecular motors, according to the criteria of Leibler and Huse (25). Thus they presented a high probability of being detached and of having a high number of working CB and a high N*, together characterizing rower behavior. In mouse, both fast and slow skeletal muscles studied belong to subtype I rowers, because the rate-limiting step is the binding to the fiber rather than the release of the ATP hydrolysis products.
ACKNOWLEDGEMENTS
The authors thank Monique Cogan for excellent technical assistance.
FOOTNOTES
Address for reprint requests and other correspondence: Y. Lecarpentier, LOA-ENSTA, Batterie de l'Yvette, 91761 Palaiseau, France (E-mail: lecarpen@enstay.ensta.fr).
The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
Received 12 January 2001; accepted in final form 26 July 2001.
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