## Abstract

Mechanical properties of spontaneously contracting isolated nonpregnant human myometrium (NPHM) were investigated throughout the whole continuum of load from zero load up to isometry. This made it possible to assess the three-dimensional tension-velocity-length (T-V-L) relationship characterizing the level of contractility and to determine crossbridge (CB) kinetics of myosin molecular motors. Seventy-seven muscle strips were obtained from hysterectomy in 42 nonpregnant patients. Contraction and relaxation parameters were measured during spontaneous mechanical activity. The isotonic tension-peak velocity (T-V) relationship was hyperbolic in 30 cases and nonhyperbolic in 47 cases. When the T-V relationship was hyperbolic, the Huxley formalism could be used to calculate CB kinetics and CB unitary force. At the whole muscle level and for a given isotonic load level, part of the V-L phase plane showed a common pathway, so that a given instantaneous length corresponded to only one possible instantaneous velocity, independent of time and initial length. At the molecular level, rate constants for CB attachment and detachment were dramatically low, ∼100 times lower than those of striated muscles, and ∼5 to 10 times lower than those of other smooth muscles. The CB unitary force was ∼1.4 ± 0.1 pN. NPHM shared similar basic contractile properties with striated muscles, reflected in the three-dimensional T-V-L relationship characterizing the contractile level. Low CB attachment and detachment rate constants made it possible to generate normal CB unitary force and normal muscle tension in NPHM, even though it contracted extremely slowly compared with other muscles.

- crossbridge kinetics
- uterus
- contractility
- relaxation

over the past two decades, numerous studies have described the cellular and molecular mechanisms that regulate the contractile properties of myometrium (1, 9, 16, 22, 25, 32, 33, 46, 47). The mechanics of isolated nonpregnant human myometrium (NPHM) obtained from hysterectomy have been studied mainly under isometric conditions (26, 30, 44, 45, 49). On the other hand, isotonic mechanical properties over the entire load continuum have not been intensively investigated in NPHM. This also includes the relationships between tension (T) and shortening (L), velocity (V) and shortening, and finally tension and velocity. In their pioneering work conducted in rabbit uterine muscle, Csapo and Goodall (6) established a hyperbolic isotonic tension-peak velocity (T-V) relationship, as previously observed in striated muscles (12). Importantly, characteristics of Hill's T-V relationship can be introduced in Huxley's equations (13) to calculate kinetics and unitary force of crossbridge (CB) molecular motors. Of all theoretical models, Huxley's CB model remains the most commonly accepted for calculating CB molecular motor properties in striated and smooth muscles (3, 18). We first investigated the basic mechanical contractile properties of isolated NPHM strips obtained from hysterectomy, including *1*) the preload- and time-independent three-dimensional T-V-L relationship; *2*) the inverse linear relationship between maximum amplitude of shortening length and isotonic load level; and *3*) the mechanical coupling between contraction and relaxation at low and heavy loads. We then studied the mechanics and energetics of CB molecular motors, applying Huxley's formalism to NPHM.

## MATERIALS AND METHODS

#### Human biological samples.

Myometrial tissues were obtained from 42 nonpregnant women (ages ranging from 35 to 91 yr) undergoing hysterectomy at Meaux Maternity Hospital mostly for benign gynecological conditions. For 35 women, 2 muscle strips were dissected and for 7 women only one strip was dissected (total: 77 muscle strips). Patients gave written, informed consent with approval of Local Ethical Committee, Comité Consultatif sur le Traitement de l'Information en matière de Recherche dans le domaine de la Santé (CCTIRS) 08.284bis, Direction Générale de la Santé (DGS) 2008–0161, and Ministère de l'Enseignement Supérieur et de la Recherche DC 2008–333.

#### Mechanical experiments.

Mechanical studies were performed on isolated NPHM. Myometrial samples were collected from normal zones of the uterine corpus, i.e., free of macroscopic abnormalities. The largest structures within the myometrium are fasciculi, which are visible to the unaided eye and can be dissected free. Fasciculi were dissected just under the serosa. Myometrial strips were suspended in a 100-ml bath chamber containing a Krebs-Henseleit solution (in mM): 118 NaCl, 4.7 KCl, 1.2 MgSO_{4}·7 H_{2}O, 1.1 KH_{2}PO_{4}, 24 NaHCO_{3}, 2.5 CaCl_{2}·6H_{2}O, and 4.5 glucose and maintained at 31°C, pH 7.4, and bubbled with a gas mixture (95% O_{2}-5% CO_{2}). The lower end of each strip was held in a stationary clip at the bottom of the bath, and the upper end was held in a spring clip attached to an electromagnetic lever system. The preparations were allowed to equilibrate for 30–90 min with a resting force of ∼10 mN, until the appearance of regular spontaneous contractions. At the end of the equilibration period, all muscle strips that did not develop spontaneous contractions were discarded. About 95% of muscle strips yielded spontaneous contractions. Experiments were carried out at the initial resting length (Lo) corresponding to maximum isometric active (total minus resting) tension. At the end of the study, the cross-sectional area (CSA) was calculated from the ratio of fresh muscle weight and length at Lo, assuming a muscle density of 1.06 g/ml.

#### Custom-built electromagnetic system (17).

Force generation was based on the electromagnetic principle: a coil carrying a constant current and suspended in a constant magnetic field developed a well-determined torque, coupled to a 30-mm lever. Hence, a precision current source determined the force at the tip of the lever from 0 mN up to a total of 140 mN. The error of the measured force was <0.1%. The equivalent moving mass of the whole system was 155 mg, and its compliance was 0.25 μm/mN. The displacement of the lever was measured by means of an opto-electronic transducer. A small diaphragm on the lever modulated the light from a tiny infrared-LED falling on a photodiode with a daylight filter which blocked visible light but passed the near-infrared radiation. The photodiode current was converted to a voltage and filtered with an active third order low pass filter (fc = 150 c/s) with optimal step response. The resulting length range was 5 mm at a full scale output voltage of 10 V full (error 1%) with a noise floor <2.5 μm p-p.

#### Mechanical analysis.

Mechanical recording began when spontaneous contractions occurred at a quasi-constant period (200 s). Standard mechanical parameters of the muscle strips were measured from three contractions at Lo. *Contraction 2* (Fig. 1), which shortened with preload only made it possible to measure the maximum amplitude of shortening length (max delta L in % Lo). From this contraction were measured maximum shortening velocity at preload (maxVc), maximum lengthening velocity at preload (maxVr) and time-to-peak shortening (TPS). Contraction 1 (Fig. 1) was clamped from preload to zero load just after the onset of spontaneous activity to determine maximum unloaded shortening velocity (Vmax). *Contraction 6* (Fig. 1) was fully isometric: Fo was the peak isometric force and was normalized per muscle cross-sectional area in mm^{2} of muscle (isometric tension To). From *contraction 6* were measured the time-to-peak tension (TPT), the positive peak of first tension derivative (+dT/d*t*), and the negative peak of first tension derivative (−dT/d*t*). Ratio R1 = (maxVc/maxVr) measured the coupling between contraction and relaxation at low load and ratio R2 = (+dT/d*t*)/(−dT/d*t*) measured the contraction-relaxation coupling at heavy load. The period of spontaneous contractions was also recorded. Velocities were expressed in Lo/s, tension in mN/mm^{2}, first tension derivatives in mN·s^{−1}·mm ^{−2}, length in %Lo, time indexes (TPT, TPS, and contraction duration; spontaneous period of contraction) in seconds, and CSA in mm^{2}.

#### Hyperbolic or nonhyperbolic behavior of T-V relationship.

The T-V relationship was derived from the peak velocity (V) of 6 to 10 isotonic afterloaded contractions, plotted against the isotonic load level normalized per CSA (T), by successive load increments, from zero load up to isometric tension (To; Fig. 1). In 30 muscle strips (from 18 women), the T-V relationship was fitted with a hyperbola [“hyperbolic (“h”) group], according to Hill′ s equation (T + a) (V + b) = (To + a) b, where −a and −b are the asymptotes of the hyperbola (12) (Table. 1). The G curvature of Hill′ s equation is equal to To/a = Vmax /b, where To is the peak isometric tension at Lo, and Vmax the maximum unloaded shortening velocity (6, 12). In 47 muscle strips (from 24 women), the T-V relationship was not hyperbolic (“nh” group; Table. 1). In the h group, for 12 women, 2 strips were dissected, and for 6 women, only 1 strip was dissected. In the nh group, for 23 women, 2 strips were dissected, and for 1 woman, only one strip was dissected. We have never observed a “h” and a “nh” T-V relationship in separate strips of tissue from the same woman. CSA did not differ between the h and nh groups: 1.41 ± 0.05 and 1.55 ± 0.05 mm^{2}, respectively. Resting muscle length (Lo), which induces peak isometric tension did not differ between the h and nh groups: 11.9 ± 0.4 and 12.9 ± 0.5 mm, respectively.

#### CB unitary force, CB kinetics, and energetics (appendix).

Huxley's equations can be applied to smooth muscle as proposed in his princeps study (13, 17). They were used to calculate the rate constants for CB attachment (f_{1} in s^{−1}) and CB detachment (g_{1} and g_{2} in s^{−1}), maximum turnover rate of myosin ATPase under isometric conditions (kcat in s^{−1}), and maximum CB efficiency (Effmax) (18). CB mechanical parameters were calculated only in the h group because asymptotes and G curvature of the hyperbola have to be used in Huxley's equations to calculate CB kinetics. In the nh group, CB properties could not be determined by the Huxley formalism. A schematic CB cycle is described in Fig. 2*A*.

#### Statistical analysis.

Data are expressed as means ± SE. The h group was compared with the nh group and the menopause group was compared with the nonmenopause group, 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.

## RESULTS

#### Basic mechanics of isolated NPHM.

For three-dimensional T-V-L relationship, in spontaneous contractions, instantaneous shortening length and tension vs. time were recorded at six different load levels (Fig. 1, *A* and *B*). Spontaneous shortening length and tension vs. time traces were resynchronized according to the onset of tension development (Fig. 1, *C* and *D*). This presentation was important for assessment of the mechanical properties of relaxation. In Figs. 3, *A* and *B*, shortening length and tension of three spontaneous contractions were recorded vs. time with the same isotonic total load level but with three different initial lengths. During the last part of the isotonic shortening phase, the three velocity-length (V-L) phase planes (Fig. 3*C*) show a common pathway. The onset of the common pathway was indicated by “(1)” and the end of the common pathway by “(2)” in Figs. 3*C* and 4*C*. The end of the common pathway approximately corresponded to TPS. Along this common V-L pathway, one and only one value of instantaneous shortening velocity corresponded to a given level of instantaneous shortening length (indicated by the line “a” in Fig. 3, *A* and *C*), regardless of the initial length of each contraction. Moreover, if the three contractions were resynchronized as in Fig. 1, *C* and *D*, a given value of instantaneous shortening would occur at different times along the common pathway. Thus the common V-L pathway was independent of initial length (or preload) and time. For a given isotonic load level, there was a one-to-one relationship between length and velocity, independent of time. This basic mechanical result was corroborated by means of a complementary technique using load clamping. In Fig. 3, *D–F*, in the second contraction, NPHM began to contract with a given isotonic load level and was abruptly clamped to a new load level corresponding to that of the first contraction. After an oscillation due to recoil of series elastic elements (Fig. 3*D*, *second trace*), the V-L phase planes of the two contractions became merged (indicated by the line “b”). Thus, for a given isotonic load level, shortening velocity adapted quasi-instantaneously to the shortening length, regardless of time. The protocols described in Fig. 3, *A–C*, can be provided throughout the whole load continuum from zero load up to isometry leading to a three-dimensional presentation of the V-L phase planes (Fig. 4*C*). Six V-L phase planes, which correspond to the same isotonic load levels used in Fig. 1, are presented in three dimensions as a function of tension in Fig. 4*C*. The level of contractility can be defined as the time- and initial length-independent part of the three-dimensional T-V-L relationship that is delineated by the two lines (1) and (2) in Fig. 4*C*. The dotted line (1) of Fig. 4*C* demarcated the onset of the common pathway of the V-L phase planes that can be seen in Fig. 3*C*. The dotted line (2) of Fig. 4*C* demarcated the end of the common pathway of the V-L phase planes.

#### Quantitative mechanics of the isolated NPHM.

There were no differences between the h and nh groups with regard body mass index (BMI), uterus weight, systolic arterial pressure, diastolic arterial pressure, glycemia, and creatininemia (Table 1). Figure 4*A* shows the T-V relationship. Load levels of the six contractions correspond to those used in Fig. 1. The hyperbolic or nonhyperbolic shape of the T-V relationship did not depend on age. Three women (84, 90, and 91 yr old) exhibited spontaneous mechanical activity. The T-V relationship was hyperbolic for two of these women, (84 and 91 yr old, respectively), and nonhyperbolic for the third woman who was 90 yr old.

Standard parameters are presented according to the hyperbolic or nonhyperbolic shape of the T-V relationship. Total isometric tension (To), Vmax, and max delta L were significantly lower in the nh group than in the h group (Fig. 5, *A*, *B*, and *C*, respectively). Conversely, the period of spontaneous contraction (ranging from 105 to 300 s), contraction duration, TPS, and TPT did not differ between the two groups h and nh (Fig. 6, *A*, *B*, *C*, and *D*, respectively). The putative role of menopause was also examined. Among the 42 women studied, 20 women were menopaused and 22 were nonmenopaused. The three parameters, To, Vmax, and max delta L, did not differ between muscle samples (*n* = 37) from menopaused women and muscle samples (*n* = 40) from nonmenopaused women (Fig. 5, *D*, *E*, and *F*).

#### Linear relationship between maximum amplitude of shortening length (max delta L) and isotonic load level.

Total tension (TT) as a function of shortening length at several isotonic load levels is presented in Fig. 4*B*. There was a linear relationship between max delta L and isotonic tension obtained at various load levels. The end shortening length was linearly related to the level of isotonic load level. Both the slope and the ordinate of this F-V relationship were significantly higher in the h group than in the nh group (Figs. 4*B* and 7, *A* and *B*).

All muscles studied presented the Frank-Starling phenomenon: as preload increased, total isometric tension and active isometric tension increased (Fig. 4*D*). Resting tension at Lo did not differ between the h and nh groups (Fig. 7*C*). However, the resting tension to total tension ratio at Lo was significantly lower in the h group than in the nh group (Fig. 7*D*).

#### CB kinetics, mechanics, and energetics in the h group.

These properties are presented in Table 2. Asymptote values (“−a” and “−b”) and G curvature of the T-V relationship are required for calculations of CB kinetics. CB mechanical properties, which were calculated from Huxley's equations, were thus determined only in the h group. CB unitary force was 1.37 ± 0.07 pN and maximum efficiency was 36.1 ± 4.2% (Table. 2). There was a hyperbolic relationship between G curvature and CB unitary force (Fig. 2*C*; see appendix). CB kinetics (attachment rate constant f_{1}, detachment rate constants g_{1} and g_{2}, maximum turnover rate of myosin ATPase) and CB number per CSA were also presented in Table 2. There was no relationship between total tension and CB unitary force (Fig. 2*B*) and between kcat and CB unitary force (Fig. 2*D*).

#### Relaxation parameters and contraction-relaxation coupling.

Isotonic parameters maxVc and maxVr (Fig. 8, *A* and *B*) were significantly higher in the h group than in the nh group. However, the ratio R1 = (maxVc)/(maxVr) did not differ between the two groups (Fig. 8*C*). Isometric parameters +dT/mm^{2} and −dT/mm^{2} (Fig. 8, *D* and *E*) were significantly higher in the h group than in the nh group. However, the ratio R2 = (+dT/mm^{2})/(−dT/mm^{2}) did not differ between the two groups (Fig. 8*F*). The resynchronized presentation of length and tension curves as a function of time at various load levels presented in Fig. 1, *C* and *D*, showed that the isometric relaxation phase of each afterloaded contraction (*contractions 1* to *5*) occurred after the isometric phase of the fully isometric contraction (6). This behavior characterizes a load-independent relaxation, typically observed in heart muscle when the sarcoplasmic reticulum (SR) is impaired or not functional.

## DISCUSSION

Mechanical properties were investigated in spontaneously contracting NPHM, applied to isolated muscle and to elementary CB myosin molecular motors. Mechanical properties were studied under isotonic and isometric conditions. Basic macroscopic mechanical properties found in NPHM showed similarities to sarcomeric skeletal and cardiac muscles, i.e., *1*) the Frank-Starling phenomenon; *2*) the linear relationship between end shortening length and isotonic load level; *3*) in ∼40% of cases, the hyperbolic shape of the tension-peak velocity relationship made it possible to calculate CB unitary force and CB kinetics; and *4*) the time- and initial length independent three-dimensional T-V-L relationship that characterizes the level of contractility. Thus these mechanical properties suggest some common cellular processes of actomyosin sliding.

Like striated muscles, smooth muscles exhibit an optimum initial length (Lo) for isometric tension development (31, 38). In myometrium, the Frank-Starling mechanism has been initially observed in rabbit uterine muscle (6) and human uterine muscle (34) (Fig. 4*D*). In smooth muscle, it has been observed that length dependence of the suprabasal oxygen consumption is parallel to that of the active isometric force, strengthening the argument by analogy that the sliding filament model is applicable to smooth muscle (13, 31).

Importantly in NPHM, we observed a linear relationship between the isotonic load level and the maximum amplitude of shortening (Fig. 4*B*). The slope of this relationship was higher in the h group than in the nh group (Fig. 7*A*). The ordinate was also higher in the h group than in the nh group (Fig. 7*B*). The slope of this relationship appeared to be a powerful mechanical indicator because it integrated the mechanical behavior throughout the whole load continuum and can thus be used to quantify muscle performance independently of the load level. This basic mechanical property was reminiscent of the curvilinear end-systolic pressure-volume relationship observed in heart muscle (39).

#### Three-dimensional T-V-L relationship.

The relationship between instantaneous velocity and instantaneous length for a given isotonic load level, initially described in heart muscle (4), was also found in NPHM (Fig. 4*C*). In heart muscle, simultaneous measurements of the four main variables, i.e., tension (T), shortening velocity (V), shortening length (L), and time, have been used to define the level of contractile performance. Cardiac contractility has been defined as the time- and initial length-independent part of the T-V-L diagram (4). Thus, during a given part of the contraction phase and for a given isotonic load level, instantaneous shortening velocity shows a one-to-one time-invariant function of shortening length, independently of initial length (Figs. 3*C* and 4*C*). Any change in contractility is the result of a shift in the three-dimensional T-V-L relationship (4) (Fig. 4*C*). This property has also been observed in skeletal muscles and at the sarcomere level (5), suggesting that common intracellular mechanisms underlie regulation of contractile processes in both smooth and striated muscles, independently of the sarcomeric structure.

#### Standard mechanical parameters.

The force or tension of myometrium spontaneous contractions has been measured in numerous studies in humans (15, 30, 40, 41, 44, 45, 49). In Table 3 are summarized mean values or ranges of standard mechanical parameters (tension, Vmax) previously reported in other muscle types (4, 18). Thus, among smooth muscles, myometrium was one of the slowest muscles but has a relatively high capacity to develop force and shortening. The period and time course duration of spontaneous mechanical activity were of the same order of magnitude in our study as in studies conducted in human myometrium (15, 30, 49).

#### Hill hyperbolic T-V relationship.

The hyperbolic relationship between isotonic load level and peak velocity observed in numerous smooth and striated muscles is characterized by Hill's equation (3, 12, 17). This is particularly important with regard to the applicability of Huxley's theory to smooth muscle. In his princeps study, Huxley (13) has proposed that his theory could be applied to smooth muscle. In our study, the hyperbolic fit was observed in 30 out of 77 NPHM. Thus Hill's parameters, i.e., −a and −b asymptotes and G curvature, previously reported in other muscle types are summarized in Table 3 (14, 18).

#### CB unitary force and CB kinetics (see appendix).

CB unitary force and CB kinetics have been determined in human, rabbit, and rat tracheal smooth muscles. (3, 18) The mean CB unitary force values in our study (Table 2) were of the same order of magnitude as those previously measured by means of a laser trap in both smooth and skeletal muscles (8, 10) and in intact skeletal muscles (18). Rate constants of CB attachment (f_{1}) and CB detachment (g_{1} and g_{2}) were dramatically low in NPHM (Table 2). These values can be compared with those previously published in other muscle types and summarized in Table 3 (14, 18). The CB kinetics allowed NPHM to contract extremely slowly and to generate a CB unitary force that was similar to that of striated muscles. Using Huxley's equations, kcat appeared to be much lower in NPHM (Table 2) than in other muscles (3, 17) (Table 3). In myometrium, the role of the regulatory 20-kDa myosin light chain kinase appears to be important (16, 21, 23, 45) and might modulate myometrium CB kinetics.

#### Contraction-relaxation coupling in NPHM.

Cellular mechanisms underlying myometrium relaxation have been largely investigated (24, 27–29, 36, 41, 48, 50). In our study, the parameters of relaxation at low load (maxVr) and at heavy load (−dT/d*t* max per mm^{2}) were dramatically low if they are compared with other smooth muscles (2). However, they decreased in same proportion than contraction parameters. The coupling between contraction and relaxation has not been extensively described in smooth muscles. Ratio R2 has been found to be of the same order of magnitude in a recent study (11) and in ours. R2 increases in the presence of 1 and 10 nM oxytocin due to increase of positive peak isometric +dF/d*t* and decrease of negative peak isometric −dF/d*t*. These authors have proposed the product of this index and peak active force for characterizing the contractility per se of uterine samples with different connective tissue content. Ratio R1 has been previously used in cardiac muscle as a reflect of the SR function (19). In normal mammalian cardiac muscle, R1 is <1 and tends to 1 or even may become >1 when the SR function is impaired, as observed during cardiac hypertrophy (19) or after destruction of the SR by means of detergent (Brig 58) (20). The value of R1 > 1 observed in both h and nh groups (Fig. 8*C*) suggests that the SR did not play a major role in Ca^{2+} uptake during relaxation of spontaneous contractions in the NPHM. The resynchronized presentation of length and tension curves as a function of time at various load levels presented in Fig. 1, *C* and *D*, showed that the isometric relaxation phase of each afterloaded contraction (*contractions 1* to *5*) occurred after the isometric phase of the fully isometric contraction (*contraction 6*). This behavior characterizes a load-independent relaxation, typically observed in heart muscle when the SR is impaired or not functional. (19, 20). In NPHM, the exact contribution of SR Ca^{2+} release mechanism remains uncompletely understood (27). No changes have been observed in SR Ca^{2+} movement during spontaneous activity of uterus (37), which strongly suggests that the myogenic phasic contractions mainly depend on Ca^{2+} influx through voltage-gated L-type Ca^{2+} channels (15, 35). Importantly, in the absence of external Ca^{2+}, there is no Ca^{2+} transient and no spontaneous contraction (49). No impairment of spontaneous activity occurs when Ca^{2+}-induced Ca^{2+}-release is blocked by ryanodyne (40). All these results associated with the load independence of relaxation strongly suggest a minor role for the SR during spontaneous contractions of NPHM.

In summary, in NPHM, basic mechanical properties obtained from the three-dimensional T-L-V relationship were similar to those observed in striated muscles. If hyperbolic, the T-V relationship could be used to calculate CB unitary force and CB kinetics. The low CB kinetics resulted in dramatically slow contractile processes, but with the generation of high tension.

## GRANTS

We obtained laboratory facilities from the Centre de Recherche Clinique de l'Hôpital de Meaux, France.

## DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the author(s).

## ACKNOWLEDGMENTS

We thank Dr. Christian Allard, Président de la Commission Médicale d'Etablissement de l'Hôpital de Meaux, France; Dr. Michel Grivaux, Directeur du Centre de Recherche Clinique de l'Hôpital de Meaux; Cindy Mathis; and Dr. Roland Jeny, Chief of the Department of Obstetrics and Gynecology, Esquirol Hospital, Saint Maurice, France.

## APPENDIX 1

In the CB cycle, six different conformational states were individualized (Fig. 2*D*). Three states were detached states (D1, D2, and D3), i.e., CB was detached from actin, and three states were attached states (A1, A2, and A3), i.e., CB was attached to actin. Step A1 was a weakly bound stateand A2 was a strongly bound state (7). P_{A1}, P_{A2}, and P_{A3} were the probabilities of states A1, A2, and A3, respectively. P_{D1}, P_{D2} and P_{D3} were the probabilities of states D1, D2, and D3, respectively.In A. Huxley's equations, the rate of total energy release (*EHux*) and isotonic tension (*PHux*) as a function of muscle velocity (*V*) are given by A. Huxley's equations (13):
*N* is the cycling CB number per mm^{2} at peak isometric tension; f_{1} is the peak value of the rate constant for CB attachment; and g_{1} and g_{2} are the peak values of the rate constants for CB detachment (Fig. 2*D*). The tilt of the myosin head relative to actin varies from 0 to h; f_{1} and g_{1} correspond to a tilt = 0, and g_{2} corresponds to a tilt > h; ϕ = (f_{1} + g_{1})h/2 = b.

The w is the maximum mechanical work of a single CB (w/e = 0.75) (13), and e is the free energy required to split one ATP molecule. According to Huxley's theory, one ATP molecule is split per CB cycle. The standard free energy ΔG°'_{ATP} has been found to be −32 kJ/mol at pH 7.0 and 37°C (43). However, in vivo, the value of ΔG°'_{ATP} is approximately −60 kJ/mol, thus value used for e was 10 ^{−19} J.

The molecular step size h is defined by the translocation distance of the actin filament per ATP hydrolysis, produced by the swing of the myosin head. In vitro single-head myosin produces approximately half the displacement (5 nm) of the in vivo double-head myosin (10 nm) during a unitary interaction with actin (42). The parameter ℓ is the distance between successive actin sites with which any myosin site can combine with actin. According to in vivo conditions and Huxley conditions (ℓ >> h) (13), the values of h and ℓ were h =10 nm and ℓ = 28.6 nm (close to the semi helicoidal turn of the actin filament). Calculations of f_{1,} g_{1}, and g_{2} were based on the following equations (18):

The number of active CBs (*N*) is equal to the ratio of the peak isometric tension and the elementary CB force (π). The rate of mechanical work (W_{M}) is given by the equation: W_{M} = *PHux* × V. At any given load level, the mechanical efficiency (Eff) of the muscle is defined as the ratio of W_{M} to *EHux*, i.e., Eff = W_{M}/*EHu*x and Effmax is the peak value of Eff.

- Copyright © 2011 the American Physiological Society