INTRODUCTION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

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 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.

View larger version (10K): [in this window] [in a new window]   Fig. 1.   This model groups chemical and conformational states of the cross-bridge (CB) cycle into 4 effective states, with 1 detached state (D) and 3 attached states (A1, A2, and A3). In state D, the CB is detached from the fiber and binds to the nucleotide. This state may include 2 substates: 1 with bound ATP and the other with hydrolyzed nucleotide ADP Pi. In state A1, the myosin head attaches the nucleotide and binds to the actin fiber with the rate constant for attachment f1. The power stroke is triggered by Pi release during the transition A1 A2 and is provided by relaxation of elastic strain. The hydrolysis product ADP is released during transition A2 A3. The motor must pass through state A3 before detaching. CB detachment occurs when ATP binds to the actomyosin complex, and the rate constant for detachment is g2 [time for CB detachment (1/g2)]. tD1, Time for CB attachment; t1D, time for transition A1 AD; t12, time stroke; t23, time for ADP release. See text for definitions of t32, tD3, and t3D.

Glossary

A	Attached state
CB	Cross bridge
D	Detached state
Dia	Diaphragm
e	Free energy required to split one ATP molecule (5.1 × 1020 J)
f1	Peak value of the rate constant for CB attachment (s1)
g1 and g2	Peak values of the rate constants for CB detachment (s1)
h	Molecular step size (11 nm)
kD1 = tD11	Rate constant of transition between states D and 1 (s1)
k12 = t121	Rate constant of transition between states 1 and 2 (s1)
k23 = t231	Rate constant of transition between states 2 and 3 (s1)
k3D = t3D1	Rate constant of transition between states 3 and D (s1)
Km	Michaelis constant at ATP concentration ([ATP]) for which the ATPase turnover rate is half-maximal
l	Distance between two actin sites (36 nm)
Lm	Michaelis-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
P1	Probability of state A1
P2	Probability of state A2 = duty ratio = t12/tc
P3	Probability of state A3
PA	Probability of CB being attached
PD	Probability of CB being detached
Rmax	Maximum turnover rate of myosin ATPase (s1)
Sol	Soleus
	Average CB velocity (µm/s)
t12	Time stroke (s)
tc = 1/Rmax	Overall duration of the CB time cycle (s)
Vmax	Maximum unloaded muscle shortening velocity (resting muscle length/s)
w	Maximum mechanical work of a single CB (3.8 × 1020 J); w = 0.75 e

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

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 (L_o) that corresponds to the peak of the isometric active tension-initial 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 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) = [(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 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

CB characteristics in Huxley's equations. The rate of total energy release () and the isotonic tension (P_Hux) 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).

(2)

1

(3)

(4)

(5)

(6)

(7)

(8)

2

1

1

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, P_i 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 R_max, the _o, and the P_D. The t_ij is the transition time between states i and j (where i and j = states 1, 2, 3, and D), and k_ij = t_ij¹ is the rate constant of transition between states i and j. The probability P_j of the state j to occur is P_j = t_ij/t_c (25).

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

Ratio of the Michaelis and Michaelis-like constants . 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 half-maximal. L_m 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, P_D is high (P_D  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).

	RESULTS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

Experimental Data

Calculated Data

View larger version (14K): [in this window] [in a new window]   Fig. 2.   Values (means ± SE) obtained in diaphragm (Dia) and soleus (Sol). Top left: CB number per mm2; top middle: CB single force; top right: CB attached time; bottom left: maximum turnover rate of myosin ATPase (Rmax); bottom right: CB detached time. Significant difference between Dia and Sol: * P < 0.05; ** P < 0.01.

The time cycle (t_c =1/R_max) was significantly shorter in Dia than in Sol (Fig. 3). The time parameters t_D1 and t₁₂ on the one hand and t_1D and t₂₃ on the other hand did not differ between Dia and Sol (Fig. 3). In both Dia and Sol, t_D1 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 t_D1 = 1/f₁, i.e., the rate-limiting step of the overall cycle was the transition D  A₁. At the onset of the transition A₃ A_D, the time for CB detachment (1/g₂) was significantly shorter in Dia than in Sol (Fig. 3).

View larger version (15K): [in this window] [in a new window]   Fig. 3.   Values (means ± SE) obtained in Dia and Sol. Top left: CB time cycle; top middle: t12; top right: t23; bottom left: t1D; bottom middle: tD1; bottom right: 1/g2. Significant difference between Dia and Sol: * P < 0.05; *** P < 0.001.

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 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).

View larger version (19K): [in this window] [in a new window]   Fig. 4.   Values (mean ± SE) obtained in Dia and Sol. Top left: A1 probability; top middle: A2 probability; top right: A3 probability; bottom left: probability of CB being attached; bottom middle: probability of CB being detached; bottom right: saturating number (N*). Significant difference between Dia and Sol: * P < 0.05; ** P < 0.01.

	DISCUSSION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

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

20

20

20

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

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

P_A, P_D, Duty Ratio

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 t_D1, 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 t_D1, 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 the equations of Leibler and Huse (25), N*   is >>1 in rowers. A high  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 half-maximal filament velocity, ranges from 50 to 150 µmol (9, 20), whereas K_m, the [ATP] at R_max/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, L_m has been shown to be 100 µmol (28), whereas K_m is <1 µmol (31). These results give a  value of >100 in flagellar dyneins. If many motors are present ( >> 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, where L_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*  , 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

Conclusion