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 fourstate 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 10^{9}/mm^{2}. 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 ratelimiting step of the overall attached time. Thus diaphragm and soleus myosins belong to subtype 1 rowers.
 duty ratio
 attached and detached crossbridge probabilities
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 Xray crystallographic studies on threedimensional molecular structures (4, 12, 33), by in vitro motility assays (9, 10, 20, 3840) and mutagenesis (36), by microneedles (19) and optical tweezers (6, 7, 30, 42), and by timeresolved structural studies on muscle fibers (18). Alternatively, theoretical models have contributed to a better understanding of chemomechanical transduction in molecular motors (5, 1517,33).
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 crossbridge (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 ratelimiting 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 P_{i} 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
 A
 Attached state
 CB
 Cross bridge
 D
 Detached state
 Dia
 Diaphragm
 e
 Free energy required to split one ATP molecule (5.1 × 10^{−20} J)
 f_{1}
 Peak value of the rate constant for CB attachment (s^{−1})
 g_{1} and g_{2}
 Peak values of the rate constants for CB detachment (s^{−1})
 h
 Molecular step size (11 nm)
 k_{D1} =t_{D1}^{−1}
 Rate constant of transition between states D and 1 (s^{−1})
 k_{12} =t_{12}^{−1}
 Rate constant of transition between states 1 and 2 (s^{−1})
 k_{23} =t_{23}^{−1}
 Rate constant of transition between states 2 and 3 (s^{−1})
 k_{3D} =t_{3D}^{−1}
 Rate constant of transition between states 3 and D (s^{−1})
 K_{m}
 Michaelis constant at ATP concentration ([ATP]) for which the ATPase turnover rate is halfmaximal
 l
 Distance between two actin sites (36 nm)
 L_{m}
 Michaelislike constant at [ATP] for which the fiber velocity is halfmaximal
 N*
 “Saturating” number of motors
 ω
 L_{m}/K_{m}
 Π
 Elementary force per single CB (pN)
 Ψ
 CB number per mm^{2} (× 10^{9}) at peak isometric tension
 P_{1}
 Probability of state A_{1}
 P_{2}
 Probability of state A_{2} = duty ratio =t_{12}/t_{c}
 P_{3}
 Probability of state A_{3}
 P_{A}
 Probability of CB being attached
 P_{D}
 Probability of CB being detached
 R_{max}
 Maximum turnover rate of myosin ATPase (s^{−1})
 Sol
 Soleus
 ϑ
 Average CB velocity (μm/s)
 t_{12}
 Time stroke (s)
 t_{c} = 1/R_{max}
 Overall duration of the CB time cycle (s)
 V_{max}
 Maximum unloaded muscle shortening velocity (resting muscle length/s)
 w
 Maximum mechanical work of a single CB (3.8 × 10^{−20}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 KrebsHenseleit solution, bubbled with 95% O_{2}5% CO_{2}, 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, 1ms 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 servomechanismcontrolled 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 (L _{o}) that corresponds to the peak of the isometric active tensioninitial length relationship. The initial preload (resting tension), which determined L _{o}, 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 (V _{max}, in L _{o}/s) was measured as the peak value of the contraction abruptly clamped to zero load just after the electrical stimulus. The hyperbolic tensionvelocity relationship was derived from the peak velocity (V) of 7–10 isotonic afterloaded contractions, plotted against the isotonic load level normalized per crosssectional area (P), by successive load increments, from zero load up to the isometric tension. Experimental data from the PV relationship were fitted according to Hill's equation (P + a) (V + b) = [(cP_{max}) +a] b, where a and b are the asymptotes of the hyperbola (11) and cP_{max}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 ttest after ANOVA. Pvalues < 0.05 were required to rule out the null hypothesis. Linear regression was based on the least squares method. The asymptotesa 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 A_{1}(P _{1}); and 3) the probability of state A_{2} (P _{2}). The equations of Leibler and Huse were used to calculate 1) the probability of state A_{3} (P _{3}); 2) the probability of CB being detached (P _{D});3) the probability of CB being attached (P _{A}); 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 ratelimiting 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 (E˙) and the isotonic tension (P
_{Hux}) as a function of muscle V were calculated from Huxley's equations (15). E˙ is given as
The minimum value of E˙ occurring in isometric conditions is
P
_{Hux} is given by
CB characteristics in Leibler and Huse equations.
These equations describe a stochastic minimal fourstate model, composed of one detached state (D) and three attached states (A_{1}, A_{2}, and A_{3}), that acts by means of a tightcoupling mechanism (5, 1517). In state D, the CB is detached from the fiber and binds to the nucleotide (Fig. 1). In state A_{1}, the myosin head is bound to the actin fiber. During the transition A_{1}→ A_{2}, P_{i} release from the actomyosin complex triggers the power stroke of the molecular motor. During the transition A_{2}→ A_{3}, the hydrolysis product ADP is released. In state A_{3}, 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 R_{max}, the ϑ_{o}, and the P _{D}. The t_{ij} is the transition time between states i and j (wherei and j = states 1, 2, 3, and D), andk_{ij} = t_{ij} ^{−1}is the rate constant of transition between states i andj. The probability P_{j} of the statej to occur is P_{j} =t_{ij} /t _{c} (25).
According to Leibler and Huse (25), the equations of R_{max}, ϑ_{o}, and P _{D} are as follows.
R_{max} is
The ϑ_{o} is
The P
_{D} is
By rearranging Eq. 9
, we obtained
The P
_{A} was
Ratio of the Michaelis and Michaelislike constants ω.
The turnover rate of myosin ATPase complies with a simple Michaelis law. The R_{max} for large [ATP] is given by Eq.9. K
_{m} is the Michaelis constant at [ATP] for which the ATPase rate is halfmaximal. L
_{m} is the Michaelislike constant at [ATP] for which the fiber velocity in in vitro motility assay is halfmaximal. The ratio of Michaelis and Michaelislike constants ω 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, P _{D} is high (P _{D} ≈ 1) and P _{2} is << 1. From a theoretical point of view, there are two subtypes of rowers: the D → A_{1} ratelimiting rowers and the A_{1}→ A_{2} ratelimiting rowers (25).
In the D → A_{1} ratelimiting rowers, the ratelimiting step of the time cycle is the binding step to the fiber rather than the release step of the ATP hydrolysis products. The time constantt _{D1} is much larger thant _{12} and t _{23}; (1 −P _{D}) << 1 and P _{2} << 1. The ratelimiting step of the time cycle is the transition D → A_{1}, where t _{D1} = 1/f _{1}. In the A_{1}→ A_{2}ratelimiting rowers, the time constant t _{12} is much larger than t _{23} andt _{D1}; (1 − P _{1}− P _{D}) << 1 and P _{2} << 1. The ratelimiting step of the time cycle is the transition A_{1}→ A_{2}. 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 (Table1). V _{max} was about twofold higher in Dia than in Sol. The asymptote a of the tensionvelocity relationship did not differ between the two muscles. The asymptote b was significantly higher in Dia than in Sol. The G of the tensionvelocity relationship did not differ between the two muscles (Table 1).
Calculated Data
The total number of working CB/mm^{2} was 14.0 ± 0.9 × 10^{9}/mm^{2} in Dia and 13.5 ± 2.3 × 10^{9}/mm^{2} in Sol and did not differ between the two muscles (Fig. 2). The CB unitary force (Π) did not differ between Dia and Sol (Fig. 2). The R_{max} 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 (t _{c} =1/R_{max}) was significantly shorter in Dia than in Sol (Fig.3). The time parameterst _{D1} and t _{12} on the one hand and t _{1D} and t _{23} on the other hand did not differ between Dia and Sol (Fig. 3). In both Dia and Sol, t _{D1} was much longer thant _{12} and t _{23} (Fig. 3). Consequently, in Dia and Sol muscles, most of the overall attached time was occupied by the attachment step t _{D1} = 1/f _{1}, i.e., the ratelimiting step of the overall cycle was the transition D → A_{1}. At the onset of the transition A_{3}→ A_{D}, the time for CB detachment (1/g _{2}) was significantly shorter in Dia than in Sol (Fig. 3).
In the two muscles, P _{D} was markedly high (∼98%), and P _{A} was markedly low (∼2%) (Fig. 4). Both P _{D}and P _{A} did not differ between Dia and Sol (Fig.4). In Dia and Sol, the probabilities P _{1},P _{2}, and P _{3} of the three attached states A_{1}, A_{2}, and A_{3} were ∼2 × 10^{−2}, 2 × 10^{−3}, and 2–4 × 10^{−4}, respectively (Fig. 4). Probability A_{1} did not differ between Dia and Sol. Probabilities A_{2} and A_{3} 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 fourstate 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 ratelimiting 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 lengthtension 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 × 10^{−20} J or 22 kJ/mol of CB. This is very similar to the w equal to 3.8 × 10^{−20} J = 0.75 e (where e = 5.1 × 10^{−20} 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 onehalf of the assumed value to twice the assumed value (6, 7,30, 42). Xray diffraction studies (4, 33) allow a stepsize 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 (f _{1}) and detachment (g _{1} and g _{2}) depended on h, uncertainties on h implied uncertainties on f _{1}, g _{1}, and g _{2}.
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 (1517) 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 twostate and is analytically solvable. However, Huxley's equations (15) make it possible to calculate several substeps, i.e., the attachment step (t _{D1} = 1/f _{1}), the stroke or step size (t _{12}), the detachment step (1/g _{2}), the remainder of the CB cycle (i.e.,t _{23} + t _{3D}), and the overall duration of the CB cycle (t _{c}). 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 twostate 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 crosssectional area (>10^{9}/mm^{2}) 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 halfmyosin 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).
P_{A}, P_{D}, 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 − P _{D} << 1) and the low duty ratio (P _{2} << 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 − P _{A}<< 1). The large duty ratio predicted for porters is corroborated by experimental data on kinesin (1, 13). The twoheaded 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 oft _{D1}, compared with those of the time stroke (t _{12}) and the time for ADP release (t _{23}) (Fig. 4), made it possible to classify the Dia and Sol myosins in subtype 1 of rower motors. Thus the duration of the transition t _{D1}, i.e., the attachmentstep duration, represents the ratelimiting 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 ω
In our study, the value of N* ranged from 600 to 950 (Fig. 4). The “high” number of working CB (>10^{9}/mm^{2}) 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* ≈ ω is >>1 in rowers. A high ω value is expected to be found in muscles, corresponding to L _{m} >>K _{m}. In fact, several studies corroborate this theoretical prediction. In muscular myosin motors,L _{m}, the [ATP] at halfmaximal filament velocity, ranges from 50 to 150 μmol (9, 20), whereasK _{m}, the [ATP] at R_{max}/2, ranges from 2 to 6 μmol (9, 38). On the basis of these results, ω can be estimated between 10 and 50. Other experimental data also corroborated the concept of rower molecular motors, characterized by a high ω value. Indeed, in flagellar dyneins, L _{m}has been shown to be 100 μmol (28), whereasK _{m} is <1 μmol (31). These results give a ω value of >100 in flagellar dyneins. If many motors are present (ω >> 1 and, consequently, L _{m}>> K _{m}), the fiber velocity saturates at much larger [ATP] than does the hydrolysis rate. This strongly contrasts with results observed in porter molecular motors, whereL _{m} has been estimated as ∼20 μmol for bovine brain kinesin (14), whereas K _{m} has been reported to be ∼10 μmol (8, 22). As N* ≈ ω, the estimated value would then be ω = 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. Lowprotein 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 ratelimiting step is the binding to the fiber rather than the release of the ATP hydrolysis products.
Acknowledgments
The authors thank Monique Cogan for excellent technical assistance.
Footnotes

Address for reprint requests and other correspondence: Y. Lecarpentier, LOAENSTA, Batterie de l'Yvette, 91761 Palaiseau, France (Email: lecarpen{at}enstay.ensta.fr).

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 Copyright © 2001 the American Physiological Society