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
 |
ABSTRACT |
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
 |
INTRODUCTION |
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 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 |
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.
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
cross-sectional 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.

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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.
|
|
CB characteristics and energetics.
Huxley's equations (17) were applied to smooth muscle
(see APPENDIX). The rate of total energy release (
;
W/mm2) per cross-sectional area, the Huxley isotonic
tension (PHux) per cross-sectional area, and the rate of
mechanical energy (
M; W/mm2) per
cross-sectional area are expressed as a function of V
(17).
is given as
|
(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)
|
(2)
|
|
(3)
|
|
(4)
|
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
|
(5)
|
where
o is the minimum rate of total
energy release (W/mm2).
o occurs
under isometric conditions (V = 0 in Eq. 1), is equal to the product of ab (15,
17), and is given by
|
(6)
|
Thus the
is
|
(7)
|
The PHux is given by (17)
|
(8)
|
The maximum PHux (PHux,max) is reached
for V = 0 and is assumed to be equal to total
Po, which was experimentally determined. The mean
unitary force per CB in isometric conditions (
; pN) equals
PHux,max/
|
(9)
|
The mean CB velocity during the stroke size (
o;
µm/s) is
|
(10)
|
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).
M equals
PHux · V. At any given load level, the mechanical
efficiency of the muscle is defined as the ratio of
M to
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.
 |
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,
, 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).

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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.
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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
( ).
|
|
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).

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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.
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Fig. 5.
A: linear relationship between
Vmax and kcat.
B: relationship between Vmax and CB
mean velocity ( o).
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|
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 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
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 =
0.278; P = 0.381). A log
transformation before statistical analysis led to similar results.

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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.
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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.
|
|
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).
 |
DISCUSSION |
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 µm. 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 µm) 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 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 =
o (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 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
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 
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
from the theory of allometry and from Bárány's law
(1) have been previously described (28).
Importantly, these deviations have been reported in skeletal muscle
from humans, rabbits, and rats. Thus the unexplained dissociation
between Vmax and myosin ATPase activity observed
in ASM of humans, rabbits, and rats in the present study has already
been described in skeletal muscle from the same three species.
Smooth muscles have been classified into two classes, tonic and phasic
(35). Kinetics of the contractile machinery and force development have been found to differ in these two types of smooth muscle. However, the same mathematical formalism can be applied to both
of them. Mechanical differences observed in our study may be related to
the site along the airways. Isometric tension and
Vmax decreased from 0 to 6 generations of
airways (22). Thus the lower mechanical performance
observed in human smooth muscle may be partly explained by the fact
that we used bronchial samples (generations
1-3), whereas tracheae were studied in rats and
rabbits. Moreover, in a given species, shortening velocity decreases
somewhat with age (43). Thus the lower velocity observed in humans compared with the other species may be partly due to age. Age
also influences contractile protein content, calcium handling,
Na-K-ATPase activity, and receptor responsiveness (43). Finally, shortening velocity in smooth muscle is linearly related to
phosphorylation level of 20-kDa myosin light chain (14), but we have no indication of the phosphorylation level of 20-kDa myosin
light chain. However, muscle strips were studied under similar tetanic
electrical conditions of stimulation and the same experimental
conditions (temperature).
In conclusion, we proposed a new formalism of Huxley's equations
(17) specifically adapted to smooth muscle. ASM mechanical behavior of different species was characterized by CB kinetics and CB
unitary force. The present study may have potential interest for
studying CB kinetics in the ASM pathophysiology (9, 12, 23,
37), particularly airway hyperresponsiveness.
 |
APPENDIX |
In Huxley's original manuscript (17), the velocity
(v; in µm/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 µm). In Huxley's study (17),
per volume unit is expressed as a function of m, the number
of sites per volume unit. In the present study,
per
cross-sectional area is expressed as a function of
, and
s was equal to 2 µm.
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
o= ab) and
8 under isometric conditions (with V = 0 and
PHux,max = Ga), we deduced that
|
(A1)
|
Thus g1 = 2wb/ehG (Eq. 3).
Calculation of f1.
was linearized as a function of PHux. Let
|
(A2)
|
and
|
(A3)
|
By inserting A and B in Eqs. 1 and 8, Eq. 1 becomes
|
(A4)
|
and Eq. 8 becomes
|
(A5)
|
where d = 1/G.
From Eq. A4, we deduced
|
(A6)
|
and from Eq. A5, we deduced
|
(A7)
|
From Eqs. A4 and A5, we deduced
|
(A8)
|
From Eq. A8 and near isometric conditions, where
V can be neglected (V
0),
can be
linearized as a function of PHux as follows
|
(A9)
|
The slope of this relationship between
and
PHux has been shown to be equal to b; then,
b = ehf1/2w, so that
|
(A10)
|
From Eqs. 3 and A10, we deduced
|
(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
|
(A12)
|
 |
ACKNOWLEDGEMENTS |
We thank Dr. Mahmoud Zureik for statistical analysis and helpful discussion.
 |
FOOTNOTES |
Address for reprint requests and other correspondence: Y. Lecarpentier, LOA-ENSTA-Batterie de l'Yvette, 91761 Palaiseau, France (E-mail: lecarpen{at}enstay.ensta.fr).
The costs of publication of this
article were defrayed in part by the
payment of page charges. The article
must therefore be hereby marked
"advertisement"
in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
Received 21 March 2001; accepted in final form 23 August 2001.
 |
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