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1,1 Department of Biomedical Engineering, Boston University, Boston 02215; and 2 Physiology Program, Department of Environmental Health, Harvard School of Public Health, Boston, Massachusetts 02115
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ABSTRACT |
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 TOP
 ABSTRACT
 INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
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We investigated the effect of the cytoskeletal prestress (P) on the elastic and frictional properties of cultured human airway smooth muscle cells during oscillatory loading; P is preexisting tensile stress in the actin cytoskeleton generated by the cell contractile apparatus. We oscillated (0.1 Hz, 6 Pa peak to peak) small ferromagnetic beads bound to integrin receptors and computed the storage (elastic) modulus (G') and the loss (frictional) modulus (G") from the applied torque and the corresponding bead rotation. All measurements were done at baseline and after cells were treated with graded doses of either histamine (0.1, 1, 10 µM) or isoproterenol (0.01, 0.1, 1, 10 µM). Values for P for these concentrations were taken from a previous study (Wang et al., Am J Physiol Cell Physiol, in press). It was found that G' and G", as well as P, increased/decreased with increasing doses of histamine/isoproterenol. Both G' and G" exhibited linear dependences on P: G'(Pa) = 0.20P + 82 and G"(Pa) = 0.05P + 32. The dependence of G' on P is consistent with our previous findings and with the behavior of stress-supported structures. The dependence of G" on P is a novel finding. It could be attributed to a variety of mechanisms. Some of those mechanisms are discussed in detail. We concluded that, in addition to the central mechanisms by which stress-supported structures develop mechanical stresses, other mechanisms might need to be invoked to fully explain the observed dependence of the cell mechanical properties on the state of cell contractility.
storage modulus; loss modulus; oscillatory cytometry; actin network; cytoskeletal mechanics
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INTRODUCTION |
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TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
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IT IS WELL ESTABLISHED THAT mechanical behavior of adherent cells can be characterized as viscoelastic. Mechanical measurements on various types of adherent cells, using various techniques, have shown that adherent cells exhibit creep in response to applied mechanical stress (2, 7, 19, 28, 29), stress relaxation in response to applied mechanical strain (27), and hysteresis in response to cyclic loading (4, 12, 18) and that elastic and frictional stresses within the cell in response to applied cyclic loading are dependent on the loading frequency (4, 12, 20). All these manifestations are believed to be related to intrinsic viscoelastic properties of various molecules that comprise the cytoskeleton (CSK), liquid cytoplasm, cell membrane, and nucleus. On the other hand, adherent cells also exhibit features that characterize stress-supported structures (32). The prime characteristic of those structures is that their mechanical properties are strongly influenced by the preexisting tensile stress (termed "prestress") carried by their structural components. In the past, investigations have been focused on the effect of the prestress on the elastic response of the cell. Little is known how the prestress may affect frictional stress within the cell. There is a good reason to believe that the CSK prestress may affect frictional properties of the cell. It has been shown recently that, during oscillatory loading, at low frequencies (0.05-0.4 Hz) frictional losses reside within the CSK and not within the liquid cytoplasm (12). The purpose of this study was to elucidate the latter problem.
We have recently shown that the elastic stiffness of cultured airway smooth muscle cells increases linearly with increasing prestress (32). The prestress, defined as a preexisting contractile stress carried by the actin network of the CSK before application of any external load, was modulated pharmacologically by graded doses of either a relaxant agonist (isoproterenol) or a contractile agonist (histamine). We attributed this dependence of the stiffness on the prestress to principal mechanisms of stress-supported structures that are described below.
The mechanisms by which stress-supported structures develop mechanical stresses to resist distortion of their shape are changes in spacing, reorientation, and elongation of their structural components (6, 21). The greater the prestress carried by the structural elements, the smaller the distortion that the structure must undergo before attaining an equilibrium configuration. In other words, as the prestress increases, the less deformable (i.e., the more rigid) the structure is. Among these three mechanisms, changes in spacing and in reorientation are associated only with redistribution of the prestress within the network. Because the prestress is static stress, it follows that these two mechanisms contribute only to the static, i.e., elastic stress within the network. In contrast, the change in length of structural elements can directly contribute to both elastic and frictional stresses of the network, depending on the rheological properties of structural elements and the type of applied load to the network. If the structural elements were viscoelastic, then dynamic shortening and lengthening of those elements would produce both elastic and frictional stresses.
In adherent cells, the prestress is carried by the actin network and is partly balanced by microtubules that are supported by intermediate filaments (25). These three filamentous networks of the CSK are known to exhibit viscoelastic behavior in vitro (8). Thus one should expect that viscoelastic properties of the CSK network would be affected by changes in the CSK prestress. In this study we investigated this idea.
We used the oscillatory magnetic twisting technique (12) to measure the dynamic properties (mechanical impedance) of cultured airway smooth muscle cells whose contractility was modulated by graded doses of either isoproterenol or histamine. We found that both elastic and frictional components of the impedance increased with increasing cell contraction and decreased with increasing cell relaxation in a dose-dependent fashion. The central mechanisms of the stress-supported structures could fully account only for the observed dependence of the cell elastic properties on cell contractility. However, to explain dependences of both elastic and frictional properties on cell contractility, additional mechanisms may need to be invoked. Some of those mechanisms were discussed in detail.
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MATERIALS AND METHODS |
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TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
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The working hypothesis of this study is that the CSK of adherent cells is a stress-supported structure in which the prestress is carried primarily by the actin network. This does not rule out possible contributions of microtubules and intermediate filaments to the cell mechanical response because these two structures contribute to balancing the prestress and stabilizing the CSK (25). To get insight into how the CSK prestress affects elastic and frictional properties of the cell, we measured the oscillatory response of cultured airway smooth muscle cells treated with graded doses of contractile or relaxant agonists. Data from the oscillatory measurements were correlated with data for prestress measured previously in airway smooth muscle cells (32).
We used the oscillatory magnetic cytometry technique to measure the dynamic modulus (mechanical impedance) in cultured airway smooth muscle cells at different states of cell contraction. This technique can separate the contributions of elastic and frictional stresses to the impedance (12).
Cell culture. Human airway smooth muscle (HASM) cells (passage 3-6) were used for all experiments. These cells maintain smooth muscle cell morphology and physiological responsiveness to agonists until at least passage 8 (16). The reason we used HASM cells was that their contractility could be modulated pharmacologically in a dose-dependent fashion (7, 32). After cells reached confluence in plastic dishes, they were serum deprived for 48 h before being trypsinized. The cells were then plated in a serum-free medium on collagen-1-coated (0.2 mg/ml) dishes (96-well plate, Immunon II, Dynetec) (20,000 cells per well, subconfluent).
Oscillatory magnetic twisting cytometry.
This technique applies a twisting, sinusoidally varying magnetic field
to ferromagnetic beads attached to integrin receptors on the cell
apical surface (12). Integrins are linked to the actin
network of the CSK through a series of linking proteins. A vertical
magnetic field (0.1 Hz, 6 Pa peak to peak) was applied after the beads
were magnetized at 45° from the horizontal direction and resulting
bead rotation (ranging from 0.052 to 0.157 rad for 10
5 M
histamine and 105 M isoproterenol, respectively; baseline
strain was ~0.087 rad) was determined by measuring the oscillating
remnant magnetic field produced by the beads. The dynamic modulus
(G*) was defined in the frequency domain as a complex ratio
of the twisting specific torque and the corresponding angle of
rotation. The in-phase component of G* was the storage
(elastic) modulus (G'), and the out-of-phase was the loss
(frictional) modulus (G"). This method greatly reduces the
effects of heterogeneous bead rotations and abolishes contribution of
plastic deformation. Arg-Gly-Asp (RGD)-coated
ferromagnetic microbeads (4.5-µm diameter; average 2 beads per
cell) were added to the cells plated on collagen-1-coated wells for 20 min. Unbound beads were washed away with serum-free medium, and then
the magnetic twisting was performed 6-10 h after plating.
Protocol. All measurements were done at 0.1 Hz and with the applied specific torque of 6 Pa unless indicated otherwise. Measurements were done at baseline and after cells were treated with either graded doses of constricting agonist, histamine (0.1, 1, 10 µM), or graded doses of relaxant, isoproterenol (0.01, 0.1, 1, 10 µM), 1 min after administration of each dose. To investigate the relative contributions of actin filaments, microtubules, and intermediate filaments to the elastic and frictional properties of the CSK, we selectively disrupted those networks by adding cytochalasin D (1 µg/ml), colchicine (1 µM), and acrylamide (4 mM), respectively. Cytochalasin D was added at baseline for 30 min. Colchicine was added after a saturated dose of histamine (10 µM). We have shown previously that in maximally activated HASM cells (10 µM histamine) addition of colchicine does not cause further cell contraction (31). Measurements were done after 1 min of histamine and after 15 min of colchicine. Acrylamide was added at baseline for 45 min. On disruption of each of those networks, the impedance measurements were repeated as described above. To investigate the cumulative contribution of the actin, microtubule, and intermediate filament networks to the elastic and frictional CSK properties, we measured the impedance in HASM cells in which all three filamentous networks were disrupted by a combination of cytochalasin D (1 µg/ml), colchicine (10 µM), and acrylamide (10 µM) for 30 min. To test for linearity of the oscillatory response, measurements were done at baseline for a series of specific torque amplitudes of 1, 2, 4, 6, and 8 Pa.
In this study, we used data for the prestress from our laboratory's recent report (32). A brief description of those measurements and definitions are as follows. The prestress was measured in cultured HASM cells by using the traction microscopy technique (3, 17). This technique is used to measure traction at the cell-substrate interface. The substrate is an elastic polyacrylamide gel block that is used as a strain gauge to detect contraction of cells that adhere to the gel. The prestress is defined as the net contractile force transmitted by the actin network across a cross-sectional area of the cell. It was determined from measured traction by considering the mechanical balance of traction forces and prestress forces on a section of the cell. Measurements were done at baseline and after cells were stimulated or relaxed by exactly the same doses of histamine and isoproterenol as in the impedance measurements described above. For details, see Ref. 32. Statistically significant differences between groups of data were assessed by the t-tests. The statistical significance of the dependence of measured G' and G" on the prestress and on the amplitude of forcing was assessed using a two-way ANOVA. In both tests, the differences with p < 0.05 were considered as significant.| |
RESULTS |
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TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
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In general, both G' and G" increased on
average with increasing doses of histamine (Fig.
1) and decreased with increasing doses of
isoproterenol (Fig. 2). Changes in
G' were greater than changes in G": from the
baseline to 10 µM histamine G' increased by ~65%
whereas G" increased by ~33% (Fig. 1); both G'
and G" decreased between the baseline and 10 µM
isoproterenol by ~50%, although at the highest dose they exhibited a
slight increase (Fig. 2). Taken together, these results showed that
stimulating or relaxing the cell contractile apparatus had a greater
effect on the elastic stiffness than on the frictional losses of the cell.
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We also calculated the coefficient of hysteretic damping
(hysteresivity; 
) = G"/G' and found that
it changed little with drug doses, 
0.3 (Fig.
3); tended to decrease with
increasing doses of histamine (Fig. 3A) and slightly
increased with increasing doses of isoproterenol (Fig. 3B).
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To find out how G', G", and changed with the
prestress P, we used data for P as function of
the graded dozes of histamine and isoproterenol that we obtained in our
previous study (32). According to those data, P
increases with increasing doses of histamine and decreases with
increasing doses of isoproterenol (Fig.
4). We found that both G' and
G" increased with increasing P but that
G' exhibited a much greater dependence than G";
G' (Pa) 0.20P + 82 (r2 = 0.97, p = 5.5 × 106, ANOVA) and G" (Pa)
0.05P + 32 (r2 = 0.94, p = 3.95 × 105, ANOVA) (Fig.
5A). On the other hand, exhibited a slight but a significant hyperbolic decrease with
increasing P; = 2464/(P + 7502),
with P in Pa (r2 = 0.62; for
coefficients of the hyperbola p < 0.04) (Fig. 5B).
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Changes of the specific torque amplitude from 1 to 8 Pa caused both
G' and G" to increase systematically (Fig.
6) (p < 0.012, ANOVA). However, the
increase of G' and G" at each amplitude was not
significant (p > 0.09). Disruption of microtubules and
intermediate filaments caused minor (
15%) but not statistically
significant changes in both G' and G" (p > 0.14) (Fig. 7). Disruption of the actin
network caused a ~30% decrease of both G' and
G" (p < 0.05) (Fig. 7). Disruption of all three major
CSK networks (actin filaments, microtubules, and intermediate
filaments) caused a substantial decrease in both G'
and G" by ~63 and ~50%, respectively (p < 0.05)
(Fig. 7).
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DISCUSSION |
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TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
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The most significant result of this study is that both elastic and frictional components of the HASM cell impedance depend on of the status of cell contractility, i.e., both G' and G" increase linearly with increasing contractile prestress P. The linear dependence of G' on P has been observed previously (32) and could be explained by the mechanisms that are embodied in the stress-supported structure. The dependence of G" on P is a novel finding. There are a number of mechanisms that may explain this behavior, including the mechanisms of stress-supported structures, CSK remodeling nonlinear rheological properties of the CSK filament networks, friction that arises at the interface of sliding CSK filaments or at filament junctions, myosin cross-bridge kinetics, and myosin light chain phosphorylation. Before addressing those mechanisms in more detail, we first critically evaluate major assumptions and potential experimental artifacts.
A key assumption is that the dynamic behavior of HASM cells is primarily determined by the actin network of the CSK. Several findings are consistent with this assumption. First, disruption of microtubules and intermediate filaments caused minor and nonsignificant changes in G' and G" (Fig. 7). This, in turn, suggests that microtubules and intermediate filaments of the CSK are not important determinants of G' and G", at least not at the low frequency (0.1 Hz) used in this study. This is consistent with our previous finding that microtubules, supported by intermediate filaments, balance a relatively small fraction (~14%) of P (25). Disruption of actin filaments, however, caused a substantial decrease in both G' and G" relative to the basal values. This, in turn, suggests that the actin network is a key determinant of both G' and G" (Fig. 7). Furthermore, disruption of all three filamentous networks of the CSK caused an even greater decrease of both G' and G" (Fig. 7). This is consistent with previous measurements of stiffness in endothelial cells (30). The fact that the changes caused by selective disruption of each of those networks did not add up to the changes caused by disruption of all three networks suggests that there is a mechanical synergy between them. On the basis of the above, we concluded that the assumption that the actin network of the CSK plays the principal role in determining the cell dynamic response was reasonable.
In calculating values of G' and G" from the twisting measurements, we assumed that on average ~30% of the volume of twisting beads was internalized in the cell. If, however, we assumed that only 10% of the bead volume was internalized, the values of G' and G" would be higher by nearly a factor of three relative to the values obtained with the assumed internalization of 30%. We have no data to show the degree of bead internalization for individual beads in individual HASM cells. Thus the values of G' and G" should be taken with caution. However, after the beads are added for 0.5-2 h (the duration of the experiments for each well) in these HASM cells, it is estimated that the average internalization is ~30-40% (12). Regardless of the degree of bead internalization, the qualitative dependences of G' and G" on P (Fig. 5A) would not change, only the slopes may increase or decrease depending on the degree of internalization.
The contribution of cytoplasmic viscosity to G" was not taken into account. However, it may not be significant at 0.1 Hz. Fabry et al. (4) showed that in HASM cells the viscous contribution of the cytoplasm to G" becomes evident only above 10 Hz. Maksym et al. (12) showed that in HASM cells oscillated from 0.02 to 0.4 Hz, frictional and elastic stresses are primarily developed within the CSK, not the cytoplasm. The cytoplasmic viscosity alone cannot explain the observed dependence of G" on P. The reason is that, in general, viscosity does not depend on the pressure in the liquid. Thus we believe that the frictional stress, as well as the elastic stress, arises from within the CSK.
Potential mechanisms. To analyze the contribution of the mechanisms of the stress-supported structures, we consider the following microstructural model of the CSK. The CSK was depicted as a network of initially tensed, pin-joined, and randomly oriented cables. The cables played the role of CSK filaments. The initial tension in the cables corresponded to the force generated by the cell contractile apparatus. The cables were assumed to be viscoelastic and joints frictionless.
As mentioned above, tensed filaments of the network develop mechanical stress through their change in spacing, reorientation, and lengthening. To obtain a quantitative description of these mechanisms, we utilized an approach that we used previously in studies of cell and pulmonary mechanics (22, 23) (APPENDIX). The model predicted (see APPENDIX, Eq. A5) the following relationship between the storage modulus G' and the loss modulus G" of the network and the prestress (P), the storage (E') and loss (E") moduli of individual network filaments and their volumetric fraction (
)
|
(1a) |
|
(1b) |
and on their elastic properties E'. According to Eq. 1b, G"
does not explicitly depend on P. Therefore, in order for
G" to increase with P as observed (Fig.
5A), either has to increase with P, which
would suggest agonist-induced CSK remodeling, and/or E" has
to increase with P, which would suggest a nonlinear
viscoelastic behavior of the CSK filaments. In this study, we did not
consider the effect of forcing frequency on G' and
G".
According to Eqs. 1a and 1b, the hysteresivity
= G"/G' is
|
(2) |
= E"/E', i.e., it is independent of
P, which differs from the prediction of Eq. 2. The reasons for this discrepancy are twofold. First, we previously assumed ad hoc that the dynamic modulus G* is directly
proportional to P. This implies that only changes in spacing
and reorientation and not stretching of the filaments contribute to the
CSK resistance to dynamic loading. Our second assumption was that
P itself is dynamic, i.e., it varied with frequency.
However, data in the present study suggest that all three mechanisms
are likely to contribute to the mechanical properties of the CSK.
Moreover, the experimental procedure in which the cell was first
stimulated or relaxed and then subjected to oscillatory loading is
better described by the model in which the structure is first
statically prestressed and then oscillated around the state of static
prestress. Thus we believe that the present model provides a more
realistic picture of the cell behavior during oscillatory measurements
than the one in our previous study.
It is known that the actin CSK undergoes agonist-induced remodeling
(13, 26). However, it has been shown that by blocking actomyosin force generation HASM cell stiffness was ablated in response to contractile stimulation, whereas the actin polymerization was not altered (1). This, in turn, suggests that the
actomyosin motor and not actin polymerization is the dominant mechanism
that determines cell mechanical properties.
The amplitude dependences of G' and G" of HASM
cells (Fig. 6) are indicative of their nonlinear behavior. This
behavior may result from intrinsic rheological nonlinearities of CSK
filamentous biopolymers. If so, then these nonlinearities may also
account for the observed dependence of G' and G"
on P. The only major stress-bearing CSK filaments that are
known to exhibit nonlinear stress-strain behavior are intermediate
filaments. Rheological measurements in vitro on vimentin gels
(8) and on keratin gels (11) show stiffening
in response to applied load. This is consistent with the observed
positive amplitude dependence of G' and G" of HASM cells (Fig. 6). On the other hand, our data show that disruption of the intermediate filament network has minor effects on both G' and G" of the cell (Fig. 7). These data also
indicate that the actin network plays the dominant role in determining
cell's G' and G" (Fig. 7). However, data from
the literature show that isolated actin filaments exhibit constant
stiffness over a wide range of applied tensile load, indicating a
linear behavior (10). On the basis of the above, we
concluded that rheological nonlinearities of CSK filaments are not
likely to contribute significantly to the observed dependences
G' and G" on P (Fig. 5A).
However, we cannot rule out the possibility that other proteins, such
as titin, which provides passive elasticity to muscle cells, or
cross-linking proteins such as plectin, may contribute significantly to
the HASM cell mechanical behavior.
Direct mechanical interaction between CSK filaments may cause
frictional loss. Friction may arise from the filament contact and at
the filament junctions. Mijailovich et al. (14) showed that an increase in contact stress between two sliding fibers in
apposition would lead to an increase in both storage and loss moduli of
the fiber network. It is likely that an increase in cell contractile
stress may cause an increase in the contact stress between various CSK
filaments, which, in turn, may explain the observed dependences of
G' and G" on P (Fig. 5A).
Alternatively, an increase in cell prestress may cause a shift in the
frequency response of G". This, in turn, might lead to the
dependence of G" on P.
The observed increases in the prestress, elastic, and frictional
properties with increased HASM cell contractility may also be explained
by myosin cross-bridge kinetics. Fredberg et al. (5)
showed that in uniaxially oscillated tracheal smooth muscle strips
contractile force, stiffness, and frictional properties of the muscle
increase with increasing number of attached cross bridges. As the
number of attached cross bridges increases in response to increasing
doses of contractile agonists, the observed dependences in Fig.
5A could be nothing more than the reflection of the myosin
cross-bridge kinetics. However, the frequency response of uniaxially
stretched muscle shows that the stiffness increases with increasing
frequency and reaches a plateau as the frequency reaches and exceeds
bridge cycling rates (9). In contrast, the frequency
response of the smooth muscle cells is characterized by a weak power
low of stiffness on frequency over a very wide range of
frequencies (4). Thus we do not believe that the myosin cross-bridge kinetics is entirely responsible for the observed dependences of G' and G" on P
(Fig. 5A).
On the basis of theoretical analysis of myosin cross-bridge perturbed
equilibrium, Mijailovich et al. (15) predicted that, at a
given frequency and given forcing amplitude, mean tensile force,
stiffness, and frictional losses of the airway smooth muscle increase
with increasing myosin light-chain phosphorylation. To the extent that
maximal stimulation with histamine (10 µM) has been shown to cause an
increase phosphorylation in HASM cells (31), the observed
increases in G' and G" with increasing
P (Fig. 5A) could be nothing more than the result
of increased phosphorylation. However, Mijailovich's model
predicts that muscle increases with phosphorylation
(15), whereas we observed that in HASM cells decreased
with increasing doses of histamine (Fig. 3A).
It is noteworthy that the slopes of G' and G" vs.
P relationships (Fig. 5A) are much less than
those of the G' and G" vs. applied stress
relationships (Fig. 6). It is important to distinguish the prestress
from the applied local stress. The former is the internal global cell
contractile stress, and the latter is the externally applied local
stress. The G' and G" vs. P
relationships reflect the dependence of elastic and frictional
properties on the mechanical status of the cell, i.e., the shape
stability of the cell. In contrast, the G' and G"
vs. applied stress relationships reflect the dependence of
elastic and frictional properties on external mechanical perturbation,
an entirely different phenomenon (24).
In summary, results of this study showed that the mechanical response
of HASM cells at 0.1 Hz is dominated by the elastic stresses but that
viscous stresses are substantial. Both elastic and frictional stresses
are influenced by cell contractility, but it appears that the
mechanisms that determine those stresses are different. The elastic
properties of the cell appear to be determined primarily by the central
mechanisms by which the stress-supported CSK network resists shape
distortion. However, the frictional stress within the CSK and its
dependence on the cell contractility might not be fully explained by
these mechanisms. Additional mechanisms might need to be invoked to
fully explain the dependence of the dynamic response of HASM cells on
pharmacological modulations of cell contractility.
A better understanding of the HASM cellular response to mechanically
applied stimuli is essential for the treatment of conditions such as
acute respiratory distress syndrome or ventilator-induced lung injury.
Results of this study advance this understanding by explaining the
effect of priming the cell, prestressing the cell, and mechanically
stimulating the cell.
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APPENDIX |
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TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
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The network is transected by an arbitrary plane. The total
stress (T) is defined as the sum of all tensile force
transmitted by the filaments across a trans-sectional area
(A) per unit area (23). A filament caries
tensile force F and lies at angle 
from the normal to
A. The number of elements intersecting the surface is
denoted n. Thus
|
(A1) |
cos
= 3/(8
) and n/A = NL/8V where L is the filament
length, N is the total number of the filaments in the
network, and V is the volume of the network (i.e., cell
volume). In this case, the T equals the prestress
P. Thus Eq. A1 becomes
|
(A2) |
T due to small distortion is
obtained by taking small variations of Eq. A1
|
(A3) |
Suppose that the small distortion is a pure shear deformation (i.e., no
volume change) and that T is the principal stress and
e is the corresponding principal strain, both
perpendicular to A. Then it follows that
A/A = 
e,
Fsin = 2Fe/5, and (dF/dL)cosL = 4(dF/dL)Le/15
(22). By substituting these values into Eq. A3
and taking into account Eq. A2, we obtained that
|
(A4) |
T) and strain (e) are harmonic
(sinusoidal), then the term in parenthesis represents dynamic shear
modulus (mechanical impedance G*) and
dF/dL is the dynamic stiffness of the actin
filament. Inertial effects are considered negligible. For convenience,
we define the volumetric fraction () of the actin filaments in the
network as = NLS/V and the dynamic
modulus of the individual actin fiber as E* = (dF/dL)L/S where
S is the cross-sectional area of the filament. Then it
follows from Eq. A4 that the dynamic modulus of the network is
|
(A5) |
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ACKNOWLEDGEMENTS |
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We thank Dr. Srboljub Mijailovich for fruitful discussions.
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FOOTNOTES |
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This work was supported by National Heart, Lung, and Blood Institute Grants HL-65371 and HL-33009.
Address for reprint requests and other correspondence: D. Stamenovi, Dept. of Biomedical Engineering, Boston Univ., 44 Cummington St., Boston, MA 02215 (E-mail:
dimitrij{at}engc.bu.edu).
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.
10.1152/japplphysiol.00782.2001
Received 25 July 2001; accepted in final form 3 December 2001.
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REFERENCES |
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TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
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