Am J Physiol Lung Cell Mol Physiol
282: L83–L90, 2002.
.
Myosin cross-bridge kinetics in airway smooth muscle:
a comparative study of humans, rats, and rabbits
Y. LECARPENTIER, F.-X. BLANC, S. SALMERON, J.-C. POURNY,
D. CHEMLA, AND C. COIRAULT
Services de Physiologie et de Médecine Interne, Hôpital de Bicêtre, Assistance Publique-Hôpitaux de
Paris, Unité de Formation et de Recherche Paris XI, 94275 Le Kremlin-Bicêtre; Laboratoire
d’Optique Appliquée-Ecole Nationale Supérieure des Techniques Avancées-Centre National de la
Recherche Scientifique-Unité Mixte de Recherche 76 39-Ecole Polytechnique-Institut National
de la Santé et de la Recherche Médicale, 91761 Palaiseau, France
Lecarpentier, Y., F.-X. Blanc, S. Salmeron, J.-C.
Pourny, D. Chemla, and C. Coirault. Myosin cross-bridge
kinetics in airway smooth muscle: a comparative study of
humans, rats, and rabbits. Am J Physiol Lung Cell Mol
Physiol 282: L83–L90, 2002.—To analyze the kinetics and
unitary force of cross bridges (CBs) in airway smooth muscle
(ASM), we proposed a new formalism of Huxley’s equations
adapted to nonsarcomeric muscles (Huxley AF. Prog Biophys
Biophys Chem 7: 255–318, 1957). These equations were applied to ASM from rabbits, rats, and humans (n ⫽ 12/group).
We tested the hypothesis that species differences in whole
ASM mechanics were related to differences in CB mechanics.
We calculated the total CB number per square millimeter at
peak isometric tension (⌿ ⫻109), CB unitary force (⌸), and
the rate constants for CB attachment (f1) and detachment (g1
and g2). Total tension, ⌿, and ⌸ were significantly higher in
rabbits than in humans and rats. Values of ⌸ were 8.6 ⫾ 0.1
pN in rabbits, 7.6 ⫾ 0.3 pN in humans, and 7.7 ⫾ 0.2 pN in
rats. Values of ⌿ were 4.0 ⫾ 0.5 in rabbits, 1.2 ⫾ 0.1 in
humans, and 1.9 ⫾ 0.2 in rats; f1 was lower in humans than
in rabbits and rats; g2 was higher in rabbits than in rats and
in rats than in humans. In conclusion, ASM mechanical
behavior of different species was characterized by specific CB
kinetics and CB unitary force.
molecular motors; trachea; bronchi
biochemical events governing the actomyosin
ATPase cycle and myosin molecular motor mechanics. Among the theoretical models, Huxley’s CB
model (17) is the most commonly accepted one for
sarcomeric muscle and can be used to calculate CB
kinetics on the basis of the mechanics of muscle
strips. This model has been previously applied to
nonsarcomeric muscle such as swine carotid artery
(14) and bovine trachea (12).
The aim of our study was to propose a formalism for
Huxley’s equations (17) applied to smooth muscle devoid of sarcomeric structure. In doing so, we were able
to calculate the number, unitary force, and kinetics of
myosin CBs. Then, these equations were applied to
airway smooth muscle (ASM) from different species,
namely rabbits, rats, and humans. We tested the hypothesis that species differences in whole muscle force
generation and/or velocity were related to differences
in the unitary force and/or kinetics of myosin molecular
motors. Our results are discussed in the light of previous myosin head determinations of unitary force and
kinetics reported in both smooth (7, 13, 19, 29, 30, 38),
and skeletal (4, 11, 20, 21, 25, 27) muscles.
MATERIALS AND METHODS
IN SMOOTH MUSCLE AS IN STRIATED MUSCLE,
myosin cross
bridges (CBs) represent unitary force generators.
Mechanical performance depends, in turn, on both
the number and unitary force of myosin molecular
motors. New insights into actin and myosin head
behavior (5) have been provided by novel methodologies such as molecular structural biology (7, 16, 27),
in vitro motility assays (42), optical tweezers (11, 13,
19, 25), glass needles (41, 42), spectroscopy (39), and
mutagenesis (29, 38). Theoretical models have also
contributed to a better understanding of energy
transduction in molecular motors (8, 15, 17). Taken
together, these approaches provide an integrative
overview of the chemomechanical coupling between
Address for reprint requests and other correspondence: Y. Lecarpentier, LOA-ENSTA-Batterie de l’Yvette, 91761 Palaiseau, France
(E-mail: [email protected]).
http://www.ajplung.org
Rabbit and rat tracheal smooth muscle preparations. We
studied 12 samples from rabbits and 12 samples from rats (1
sample/animal). Care of the animals conformed to the recommendations of the Helsinki Declaration. After anesthesia
with intraperitoneal pentobarbital sodium (100 mg/kg), the
trachea was immediately removed. A tracheal ring consisting
of four (from rabbits) or five (from rats) tracheal segments
was carefully dissected. The rings were opened with a dorsal
midline section through the cartilage to obtain strips of the
posterior membranous portion of the trachea as previously
described (2). The body weights of the Sprague-Dawley rats
and New Zealand White rabbits were 381 ⫾ 37 g and 3.56 ⫾
0.05 kg, respectively. The ages of the rats and rabbits were 8
and 13 wk, respectively.
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.
1040-0605/02 $5.00 Copyright © 2002 the American Physiological Society
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Received 21 March 2001; accepted in final form 23 August 2001
L84
SMOOTH MUSCLE MYOSIN CROSS BRIDGES
the rate of mechanical energy (ẆM; W/mm2) per cross-sectional area are expressed as a function of V (17). Ė is given as
Ė ⫽ ⌿e
冋
(1)
where ⌿ is the CB number per square millimeter (⫻109) at
peak isometric tension; f1 is the maximum value of the rate
constant for CB attachment (s⫺1) (17); g1 and g2 (g2 appears
in Eq. 2) are the peak values of the rate constants for CB
detachment (s⫺1) (17); h is the CB step size (11 nm) (7, 11, 13,
27), defined by the translocation distance of the actin filament per ATP hydrolysis and produced by the swing of the
myosin head; e is the free energy required to split one ATP
molecule (5.1 ⫻ 10⫺20 J) (8, 17, 44); l is the distance between
two actin sites (36 nm) (32); and ⌽ ⫽ ( f1 ⫹ g1)h/2 ⫽ b (17).
Calculations of f1, g1, and g2 are given by the following
equations (see APPENDIX and Refs. 4, 20, 21)
2V max
g2 ⫽
(2)
h
g1 ⫽
f1 ⫽
⫺ g1 ⫹
2wb
ehG
(3)
冑g 12 ⫹ 4g 1 g 2
(4)
2
where w is the maximum mechanical work of a unitary CB
(3.8 ⫻ 10⫺20 J) (17, 44).
The maximum turnover rate of myosin ATPase per site
under isometric conditions (kcat; s⫺1) is
k cat ⫽ Ėo /⌿e ⫽
h f1 g1
2l f1 ⫹ g1
(5)
where Ėo is the minimum rate of total energy release
(W/mm2). Ėo occurs under isometric conditions (V ⫽ 0 in
Eq. 1), is equal to the product of ab (15, 17), and is given by
Ė o ⫽ ab ⫽ ⌿e
h f1 g1
2l f1 ⫹ g1
(6)
Thus the ⌿ is
⌿ ⫽ Ė o /共ekcat兲
(7)
The PHux is given by (17)
P Hux ⫽
⌿w
f1
䡠
l f1 ⫹ g1
⫻
再
冋 冉 冊 册冎
V
1 f1 ⫹ g1
1 ⫺ 共1 ⫺ e ⫺ ⌽/V兲 1 ⫹
⌽
2
g2
2
(8)
V
⌽
The maximum PHux (PHux,max) is reached for V ⫽ 0 and is
assumed to be equal to total Po, which was experimentally
Fig. 1. Top: shortening length (L) as a
function of time at various load levels
from 0 to isometry in rabbit, rat, and
human airway smooth muscle. Bottom:
tension as a function of time in the
same contractions. Lo, initial optimal
length; EFS, electrical field stimulation.
AJP-Lung Cell Mol Physiol • VOL
册
V
h f1
g1 ⫹ f1 共1 ⫺ e ⫺ ⌽/V兲
2l f1 ⫹ g1
⌽
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Human bronchial smooth muscle preparations. Bronchial
smooth muscle rings (generations 1–3) were obtained from
patients undergoing lobectomy or pneumonectomy to remove
lung carcinoma (n ⫽ 12; 1 sample/patient). The age of the
patients was 61.3 ⫾ 3.8 yr. The body weight was 69.8 ⫾ 5.2
kg. None of the patients had a history of atopy or asthma nor
were they chronically treated with bronchodilators. Immediately after surgical resection, macroscopically tumor-free tissue was put into a 500-ml airtight container filled with a
Krebs-Henseleit solution (in mM: 118 NaCl, 4.7 KCl, 1.2
MgSO4, 1.1 KH2PO4, 24 NaHCO3, 2.5 CaCl2 and 4.5 glucose)
bubbled with a gas mixture of 95% O2-5% CO2. After rapid
transport to the laboratory at room temperature, bronchial
rings were cut longitudinally and free from alveolar or ganglionic tissue (3).
In all ASM preparations, the epithelium was not removed. Each muscle strip was then suspended vertically in
a bath containing the same Krebs-Henseleit solution bubbled with 95% O2-5% CO2 and maintained at 37°C and pH
7.4. 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 (21). Supramaximal electrical field stimulation (30 V/cm, 50-Hz alternating current, 10-ms pulse duration, 12-s train duration)
was provided through two platinum electrodes every 5
min. Experiments were conducted after a 1-h equilibration
period. The optimal initial length (Lo; mm) was defined as
the resting muscle length corresponding to the maximum
active tension.
Mechanical analysis. The tension-velocity (P-V) relationship (37) was derived from the peak velocity (V; Lo /s) of
6–10 isotonic afterloaded contractions plotted against the
isotonic force level normalized per cross-sectional area (P)
and by successive load increments from zero load up to the
total isometric tension (Po; Fig. 1). The mean muscle crosssectional area is the ratio of muscle weight to Lo. The P-V
relationship was fitted according to Hill’s (15) equation
(P ⫹ a)(V ⫹ b) ⫽ [Po ⫹ a]b, where ⫺a (mN/mm2) and ⫺b
(Lo/s) are the asymptotes of the hyperbola as determined
by multilinear regression. For each muscle strip, the P-V
relationship was accurately fitted by a hyperbola (each r ⬎
0.98). The curvature of Hill’s equation (G; dimensionless)
is equal to Po /a ⫽ Vmax/b (44), where Vmax is the unloaded
muscle shortening velocity that is measured by means of
the zero-load clamp technique (24). Velocity is expressed
as optimal initial length per second, tension is expressed in
millinewtons per square millimeter, and time is expressed
in seconds.
CB characteristics and energetics. Huxley’s equations (17)
were applied to smooth muscle (see APPENDIX). The rate of
total energy release (Ė; W/mm2) per cross-sectional area, the
Huxley isotonic tension (PHux) per cross-sectional area, and
SMOOTH MUSCLE MYOSIN CROSS BRIDGES
L85
determined. The mean unitary force per CB in isometric
conditions (⌸; pN) equals PHux,max/⌿
⌸⫽
w f1
l f1 ⫹ g1
(9)
The mean CB velocity during the stroke size (␽o; ␮m/s) is
␽ o ⫽ 共Ėo /⌿⌸兲/Lo
(10)
RESULTS
The mathematical formulation needed to apply Huxley’s equations (17) to smooth muscle and to calculate
CB rate constants, ⌸, and ⌿ is described in APPENDIX.
This allowed estimation of the CB characteristics of
ASM in the three species under study. Total tension, ⌸,
Fig. 2. Top left: total isometric tension. Top right: cross-bridge (CB)
unitary force (⌸). Bottom left: total CB number per square millimeter
(⌿ ⫻109). Bottom right: unloaded shortening velocity (Vmax). Values
are means ⫾ SE. NS, not significant. * P ⬍ 0.05. ** P ⬍ 0.01.
AJP-Lung Cell Mol Physiol • VOL
Fig. 3. A: linear relationship between total tension and ⌿. Total
tension ⫽ (8.8⌿ ⫺ 1.3) ⫻ 109; the partial correlation coefficient
adjusted for species was r ⫽ 0.99 (P ⬍ 0.001). B: relationship
between total tension and CB unitary force (⌸).
and ⌿ were higher in rabbits than in humans and rats
but did not differ between humans and rats (Fig. 2).
There was a linear relationship between total tension
and ⌿ (Fig. 3); the partial correlation coefficient adjusted for species was r ⫽ 0.990 (P ⫽ 0.001). The
Pearson correlation coefficients indicated a significant
correlation between total tension and ⌿ in humans
(r ⫽ 0.917; P ⫽ 0.001), rats (r ⫽ 0.983; P ⫽ 0.001), and
rabbits (r ⫽ 0.996; P ⫽ 0.001). Conversely, no linear
relationship was observed between total tension and ⌸
(Fig. 3). Vmax was higher in rabbits than in rats and in
rats than in humans (Fig. 2).
The f1 was lower in humans than in rabbits and rats
but did not differ between rabbits and rats (Fig. 4). The
g2 differed significantly in the three species and was
higher in rabbits than in rats and in rats than in
humans (Fig. 4). Both the kcat and g1 were higher in
rats than in humans and did not differ between rats
and rabbits (Fig. 4). However, g1 did not differ between
humans and rabbits and kcat was lower in humans
than in rabbits (Fig. 4). There was a linear relationship
between Vmax and kcat (Fig. 5); the partial correlation
coefficient adjusted for species was r ⫽ 0.480 (P ⫽
0.004). The Pearson correlation coefficients indicated a
significant correlation between Vmax and kcat in rats
(r ⫽ 0.614; P ⫽ 0.034) and rabbits (r ⫽ 0.745; P ⫽
0.005) but not in humans (r ⫽ 0.004; P ⫽ 0.991).
Peak efficiency was higher in rabbits than in humans and rats but did not differ between humans and
rats (Fig. 6). There was a linear relationship between
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The time stroke is equal to h/␽o. The time cycle is equal to
1/kcat. The duty ratio is equal to time stroke/time cycle (36).
ẆM equals PHux 䡠 V. At any given load level, the mechanical
efficiency of the muscle is defined as the ratio of ẆM to Ė and
Effmax is the maximum value of efficiency.
Values of Huxley’s equation constants. A stroke size of 11
nm has been determined by means of optical tweezers (11,
13) and is supported by the three-dimensional structure of
the crystallized myosin head (7, 27). The distance between
two actin sites is equal to 36 nm (32). The free energy
required to split one ATP molecule per contraction site is
5.1 ⫻ 10⫺20 J and the maximum mechanical work of a single
CB is equal to 0.75e, so that it is 3.8 ⫻ 10⫺20 J (17, 44).
Statistical analysis. Data are expressed as means ⫾ SE.
The parameters of the three species were compared with
analysis of variance. Univariate associations between quantitative parameters were assessed with correlation coefficients. Partial correlation coefficients adjusted for species
(introduced as two dummy variables) are also reported. Pearson correlation coefficients for each species (rabbit, rat, and
human) were used for intraspecies analysis.
L86
SMOOTH MUSCLE MYOSIN CROSS BRIDGES
Effmax and both ⌸ and kcat (Fig. 7). For the relationship
between Effmax and ⌸ (Fig. 7), the partial correlation
coefficient adjusted for species was r ⫽ ⫺0.982 (P ⫽
0.001). The Pearson correlation coefficients indicated a
significant correlation between Effmax and ⌸ in humans (r ⫽ 0.986; P ⫽ 0.001), rats (r ⫽ 0.987; P ⫽
0.001), and rabbits (r ⫽ 0.981; P ⫽ 0.001). For the
Fig. 5. A: linear relationship between Vmax and kcat. B: relationship
between Vmax and CB mean velocity (␽o).
AJP-Lung Cell Mol Physiol • VOL
Fig. 6. Top left: peak mechanical efficiency (Effmax). Top right: time
stroke. Bottom left: ␽o. Bottom right: time cycle. Values are means ⫾
SE. * P ⬍ 0.05.
relationship between Effmax and kcat (Fig. 7), the partial correlation coefficient adjusted for species was r ⫽
⫺0.684 (P ⫽ 0.001). The Pearson correlation coefficients indicated a significant correlation between Effmax and kcat in humans (r ⫽ ⫺0.892; P ⫽ 0.001) and
rats (r ⫽ ⫺0.858; P ⫽ 0.001) but not in rabbits (r ⫽
Fig. 7. A: linear relationship between Effmax and ⌸. Effmax ⫽
5.39⌸ ⫺ 6.08; the partial correlation coefficient adjusted for species
was r ⫽ 0.98 (P ⬍ 0.001). B: relationship between Effmax and kcat.
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Fig. 4. Top left: maximum value of the rate constant for CB attachment ( f1). Top right: maximum value of the rate constant for CB
detachment (g1). Bottom left: maximum value of the rate constant for
CB detachment (g2). Bottom right: maximum turnover rate of myosin
ATPase (kcat). Values are means ⫾ SE. * P ⬍ 0.05.
SMOOTH MUSCLE MYOSIN CROSS BRIDGES
⫺0.278; P ⫽ 0.381). A log transformation before statistical analysis led to similar results.
Time stroke did not differ between the three species
(Fig. 6). Mean ␽o was higher in humans than in the two
other species but did not differ between rabbits and
rats (Fig. 6). No relationship was observed between ␽o
and Vmax (Fig. 5). Time cycle did not differ between
rabbits and rats but was higher in humans than in the
two other species (Fig. 6). The duty ratio (time stroke/
time cycle) was significantly higher in rabbits and rats
(0.010 ⫾ 0.001 each) than in humans (0.002 ⫾ 0.001;
P ⬍ 0.01). The G of the P-V relationship was higher in
rabbits (4.5 ⫾ 0.2) than in humans (3.0 ⫾ 0.3) and rats
(2.8 ⫾ 0.2; both P ⬍ 0.05).
In a new mathematical approach, we applied Huxley’s equations (17) to ASM from different species. We
then used these equations to compare CB number,
unitary force, and kinetics in rabbit, rat, and human
ASM.
Application of Huxley’s equations to smooth muscle.
In our study, we proposed a formalism of Huxley’s
equations (17), adapted to smooth muscle devoid of
sarcomeric structure. In his princeps study, Huxley
wrote: “It is natural to ask whether the mechanism
proposed here for striated muscle could account also for
the contraction of smooth muscle. On general grounds,
it is to be expected that the mechanism is fundamentally the same in both types, so that it would be unsatisfactory to postulate for one type a mechanism that
clearly cannot exist in the other. . .” The ultrastructure
of smooth muscle strongly differs from that of striated
muscle. In particular, there is no Z-line structure, even
if the attachment of actin filaments to dense bodies is
reminiscent of that found at Z lines of striated muscle
(10, 18). In Huxley’s equations, “s” represents the sarcomere length. In smooth muscle, the precise ultrastructural substratum for s remains uncertain. In the
present study, s is equal to 2 ␮m. Huxley’s equations
have been previously applied to smooth muscle in
swine carotid artery (14) and bovine trachea (12), with
a fixed value of the parameter of muscle pseudoperiodicity (s ⫽ 2.2 ␮m) in both studies.
Two other ultrastructural parameters appear in
Huxley’s equations (17), i.e., the distance between two
actin sites and the unitary displacement step or power
stroke. The monomer G-actin units are arranged on a
nonintegral helix, with subunits repeated at 5.5 nm
along two chains that twist around, with crossover
points 36 nm apart. The pitch of the polymerized actin
helix, i.e., the distance between two actin sites, 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 (32). Both smooth and skeletal muscle myosins produce similar unitary displacement (i.e., similar power stroke) of ⬃10–11 nm when
measured with optical tweezers (11, 13, 19). For these
reasons, a power stroke value of 11 nm was chosen in
AJP-Lung Cell Mol Physiol • VOL
our study. Moreover, the power stroke has recently
been estimated to be on the same order of magnitude
on the basis of crystallography analysis of the myosin
motor domain of smooth muscle (7).
The Huxley original theoretical model (17) has been
validated by using the experimental data of Hill (15)
with the values of the asymptotes ⫺a and ⫺b of the P-V
relationship, the product ab ⫽ Ėo (i.e., the maintenance
heat), G of the P-V relationship ⫽ Po /a ⫽ g2/(f1 ⫹ g1),
⌽ ⫽ b, and w/e ⫽ 0.75 (15, 17). In tracheal smooth
muscle, Mitchell and Stephens (24) have shown that
Vmax values mathematically derived from conventional
isotonic afterloaded force-velocity curves [as in Hill’s
(15) study] are valid estimates of zero-load velocity
because they are not significantly different from values
obtained by direct measurement with the zero-load
clamp technique (as in our study). The values of the
asymptotes ⫺a and ⫺b and consequently of G are not
different in the two types of force-velocity curves. The
maximum work (w) done by one CB has been determined in quick release experiments and is at least
3.7 ⫻ 10⫺20 J (44), which is a value very close to that
used in our study.
In skeletal myosin filament, there is a bipolar helical
arrangement of CBs of opposite polarity, which project
from each side of a central bare zone. In smooth myosin
filaments, CBs along a rodlike filament with no central
bare zone project in opposite directions on opposite
sides of the filament. The myosin heads along an entire
side have the same polarity along the entire length of
the filament (45). The Huxley (17) formalism applied
on the ribbonlike myosin filament structure may induce an additional factor of 2 compared with the current calculations.
Unitary force of myosin head and muscle total force.
The CB unitary force values in our study (Fig. 2) were
on the same order of magnitude as those previously
measured by means of the laser trap in both smooth
and skeletal muscles (11, 13, 25) and in intact skeletal
muscle (4, 20, 21). Total force per cross-sectional area,
i.e., the product of CB unitary force and CB number per
square millimeter, appears to be slightly lower in ASM
than in skeletal muscle. Because the CB myosin unitary force is of the same order of magnitude in smooth
and skeletal muscles (13), this difference may be partly
explained by a lower myosin concentration in smooth
muscle than in skeletal muscle (26). Accordingly, our
results show that the total number of active CBs was
lower in ASM compared with that previously reported in
skeletal muscles (4, 20, 21). In addition, we found that CB
number per square millimeter was an important determinant of total tension in ASM as attested to by the
linear relationship between total tension and CB number
per square millimeter (Fig. 3). No relationship was observed between total tension and CB unitary force.
Vmax and ␽o. Vmax and ␽o were ⬃10-fold lower in
smooth than in skeletal muscle, as previously reported
(3, 9, 37). Accordingly, in in vitro motility assays,
purified smooth muscle myosin also propels actin filament at one-tenth the velocity of skeletal muscle myosin (42). Our results showed no relationship between
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DISCUSSION
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SMOOTH MUSCLE MYOSIN CROSS BRIDGES
AJP-Lung Cell Mol Physiol • VOL
value of total tension observed in rabbit ASM compared with that seen in humans and rats was due to
higher ⌿ and ⌸ values (Fig. 2). Interestingly, Effmax
and ⌸ were of the same order of magnitude as the
values observed in skeletal muscle (Figs. 2 and 6) (4,
20, 21). This is probably due to the fact that the three
rate constants for CB attachment and detachment
were proportionally decreased in smooth compared
with skeletal muscle (4, 20, 21). As in skeletal muscle,
peak efficiency was linearly related to CB unitary force
(Fig. 7) (4, 20, 21). In smooth and skeletal muscles,
paralogous and orthologous myosin II heavy chain isoforms lead to a considerable range of variations in
shortening velocity, CB kinetics, and myosin ATPase
activity compared with the relatively small range of
variations in efficiency and CB unitary force. Even if
significant changes in CB unitary force were observed
in our study, total tension appeared to be strongly
linked to the total number of CBs (Fig. 3).
Structural differences, particularly with respect to
the ATP binding pocket, the actin binding site, and the
light chains, may represent the molecular basis for
differences in mechanical performance between species. Some authors have also discussed the role of
flexible loops (29, 36). Increases have been observed in
the ADP release rate, actin-activated ATPase activity,
and the rate of actin filament sliding in an in vitro
motility assay involving an insert of seven amino acids
in a flexible loop (the 25- to 50-kDa loop) on the surface
of the smooth muscle myosin head near the nucleotide
binding pocket (29). The role of regulatory (RLC) and
essential light chains has also been discussed (40).
Unlike striated muscle myosin, smooth muscle myosin
requires the RLC to be phosphorylated to act as a
molecular motor. RLC-deficient myosin moves slowly
in in vitro motility assays and has a low actin-activated
ATPase activity (40). However, differences in flexible
loops and/or light chains in rabbit, rat, and human
ASM have not been precisely identified. It remains to
be determined whether such differences contribute to
the differences in CB properties observed in our study.
Allometric factors. Allometry helps to explain species
differences in function of physiological variables that
have a time-related component. Small animals have
higher power-to-weight ratios than large animals. In
skeletal muscle, it has been shown that isometric tension exhibits no dependence on animal body size, but
Vmax in both fast and slow fibers and maximum power
output in fast fibers vary with the ⫺1⁄8 power of body
size (31). In the present study, we did not find any
relationship between time-related parameters (Vmax
and kcat) on the one hand and body size on the other
hand. By calculation of the partial correlation coefficient, we observed a global linear relationship between
Vmax and kcat, which is in agreement with Bárány’s law
(1). However, some discrepancies appeared in relation
to this law. As expected from allometric properties,
Vmax and kcat were lower in humans than in the two
other species (rats and rabbits). Conversely, Vmax was
higher in rabbits than in rats and there was no difference in kcat between rats and rabbits. Other deviations
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Vmax of the whole muscle and mean ␽o (Fig. 5). This
suggests that the determinants of these two parameters are different. Vmax depends on the g2 (17). It has
been suggested that the overall detachment step may
be limited by ADP release from the ATP binding pocket
of the myosin head (8, 34). Conversely, ␽o may be
inversely related to the time stroke duration (36).
Thus, in ASM, the higher ␽o was associated with the
shorter time stroke (Fig. 6).
CB kinetics and kcat. Our results showed a longer
time cycle (⫽ 1/kcat) and time stroke in ASM (Fig. 6)
than that previously reported in skeletal muscle (4, 20,
21). In smooth muscle, the duration of two major steps
of the CB cycle (i.e., 1/f1 and 1/g2) was roughly 10 times
longer than in skeletal muscle. These results corroborate the fact that the duration of the ATPase cycle is
longer for smooth than for skeletal muscle (34). Moreover, the mean attached time is longer in smooth than
in skeletal muscle myosins (13). A dephosphorylated
“latch-bridge” model has been proposed to explain the
mechanics and energetics of smooth muscle. Dephosphorylation may produce a noncycling latch bridge that
has a slow detachment rate (6). Huxley’s formalism
(17) has been adapted to the latch-bridge model (14).
This model can be used to quantitatively predict stress
maintenance with reduced phosphorylation, CB cycling rates, and ATP consumption. The apparent rate
constants f(K) and g(K) represent the average behavior
of the CB population in smooth muscle and vary linearly as a function of the phosphorylation level. Taking
into account the proportionality constants, the rate
constant values for CB attachment (f1) and detachment (g1 and g2) found by Hai and Murphy (14) are on
the same order of magnitude as those in our study (Fig.
4). It is widely accepted that the slow cycling rate of
latch CBs in smooth muscle contributes to the high
economy of tension maintenance during prolonged isometric contraction (6, 33). A similar approach has recently been used by Fredberg et al. (12), who suggested
that excessive airway narrowing in asthma may be
associated with the destabilization of dynamic processes and the resulting collapse back to static equilibrium. Because smooth and skeletal muscle myosins do
not markedly differ in their unitary force and power
stroke but rather in their kinetics (13), enzymatic and
mechanical differences may be partly due to functional
differences and/or differences in their primary amino
acid sequence.
Differences in CB mechanics and kinetics between the
species studied. In our study, Vmax was particularly low
in human ASM compared with the values in rabbit and
rat ASM (Fig. 2) (3, 23) as well as in canine (37) and
mouse (9) tracheae. CB velocity was higher in human
ASM than in the two other species, corresponding to a
shorter time stroke (Fig. 6) and suggesting that Vmax
and ␽o are modulated by different molecular mechanisms. Differences in ⌸ in rabbit, rat, and human ASM
were due to differences in CB rate constants for attachment (f1) and detachment (g1 and g2). All rate constants (f1, g1, and g2) and kcat were lower in human
ASM than in the two other species (Fig. 4). The higher
L89
SMOOTH MUSCLE MYOSIN CROSS BRIDGES
Thus g1 ⫽ 2wb/ehG (Eq. 3).
Calculation of f1. Ė was linearized as a function of PHux.
Let
A⫽
V
共1 ⫺ e ⌽/V 兲
⌽
(A2)
f1
2l共 f 1 ⫹ g 1 兲
(A3)
and
B⫽
By inserting A and B in Eqs. 1 and 8, Eq. 1 becomes
E
⫽ g1 ⫹ f1A
eh⌿B
and Eq. 8 becomes
冉
(A1)
AJP-Lung Cell Mol Physiol • VOL
(A5)
where d ⫽ 1/G.
From Eq. A4, we deduced
A⫽
g1
E
⫺
mehBf1 f1
(A6)
and from Eq. A5, we deduced
A⫽
共2w⌿B ⫺ PHux兲共2⌽兲
共2w⌿B兲共2⌽ ⫹ d2V兲
(A7)
From Eqs. A4 and A5, we deduced
Ė ⫽
2⌽共eh⌿Bf1兲
eh⌽f1
⫹ g1共eh⌿B兲 ⫺
PHux
2⌽ ⫹ Vd2
w共2⌽ ⫹ Vd2兲
(A8)
From Eq. A8 and near isometric conditions, where V can be
neglected (V ⬇ 0), Ė can be linearized as a function of PHux as
follows
Ė ⫽
⌿ehf1 ehf1
⫺
PHux
2l
2w
(A9)
The slope of this relationship between Ė and PHux has been
shown to be equal to b; then, b ⫽ ehf1/2w, so that
f1 ⫽
f 1g 1
2lab Ga2l
⫽
⫽
g1
f 1 ⫹ g 1 ⌿eh 2⌿w
冊
P Hux
1 d2V
⫽1⫺A 1⫹
2w⌿B
2 ⌽
APPENDIX
In Huxley’s original manuscript (17), the velocity (v; in
␮m/s) with which the actin filament is sliding past the myosin filament is proportional to the rate of isotonic muscle
shortening (V) according to v ⫽ (s/2) ⫻ V and ⌽ ⫽ ( f1 ⫹
g1)h/s ⫽ b (where V, ⌽, and b are in s⫺1 and s is in ␮m). In
Huxley’s study (17), Ė per volume unit is expressed as a
function of m, the number of sites per volume unit. In the
present study, Ė per cross-sectional area is expressed as a
function of ⌿, and s was equal to 2 ␮m.
Calculation of g2 (detachment rate constant at the end of
the power stroke). Because ⌽ ⫽ ( f1 ⫹ g1)h/2, ⌽ ⫽ b, G ⫽
g2 /( f1 ⫹ g1) (17), and Vmax ⫽ Gb, it is deduced that g2 ⫽
2Vmax/h (Eq. 2).
Calculation of g1 (detachment rate constant at the onset of
the power stroke). From Eqs. 6 (with Ėo⫽ ab) and 8 under
isometric conditions (with V ⫽ 0 and PHux,max ⫽ Ga), we
deduced that
(A4)
2wb
eh
(A10)
From Eqs. 3 and A10, we deduced
2w g1G f1
⫽
⫽
eh
b
b
(A11)
Thus f1 ⫽ g1G.
Because G ⫽ g2 /( f1 ⫹ g1) (17) and f1 ⫽ g1G, by solving the
quadratic equation f12 ⫹ g1f1 ⫺ g1g2 ⫽ 0, we obtained f1 as a
function of g1 and g2
f1 ⫽
⫺ g1 ⫹
冑g 12 ⫹ 4g 1g 2
2
(A12)
We thank Dr. Mahmoud Zureik for statistical analysis and helpful
discussion.
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