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Biophys J, August 2002, p. 858-879, Vol. 83, No. 2
The Bone and Mineral Centre, The Rayne Institute, Department of Medicine, University College London, London WC1E 6JJ, United Kingdom
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ABSTRACT |
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 TOP
 ABSTRACT
 INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
CONCLUSIONS
REFERENCES
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Many organs adapt to their mechanical environment as a result of physiological change or disease. Cells are both the detectors and effectors of this process. Though many studies have been performed in vitro to investigate the mechanisms of detection and adaptation to mechanical strains, the cellular strains remain unknown and results from different stimulation techniques cannot be compared. By combining experimental determination of cell profiles and elasticities by atomic force microscopy with finite element modeling and computational fluid dynamics, we report the cellular strain distributions exerted by common whole-cell straining techniques and from micromanipulation techniques, hence enabling their comparison. Using data from our own analyses and experiments performed by others, we examine the threshold of activation for different signal transduction processes and the strain components that they may detect. We show that modulating cell elasticity, by increasing the F-actin content of the cytoskeleton, or cellular Poisson ratio are good strategies to resist fluid shear or hydrostatic pressure. We report that stray fluid flow in some substrate-stretch systems elicits significant cellular strains. In conclusion, this technique shows promise in furthering our understanding of the interplay among mechanical forces, strain detection, gene expression, and cellular adaptation in physiology and disease.
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INTRODUCTION |
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TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
CONCLUSIONS
REFERENCES
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Many organs adapt to their mechanical
environment: new bone is synthesized in response to high exercise
regimen (Rubin and Lanyon, 1984
), cardiac and vascular smooth muscle
adapt to pump pressure (Xu, 2000), and skeletal muscle adapts to
exercise levels (Russell et al., 2000). The detection and adaptation to
mechanical strain are performed by the cells constituting these organs.
Many experiments in vitro have highlighted cellular detection and
adaptation to mechanical stimuli using a variety of devices to apply
mechanical stimulation: endothelial cells submitted to fluid flow for
24 h align with the direction of flow (Girard and Nerem, 1993),
and steady and oscillating fluid shear stress can elicit calcium
transients in a variety of cell types (Hung et al., 1996); cells
submitted to substrate stretch realign perpendicular to the direction
of strain (Hayakawa et al., 2001); chondrocytes submitted to
intermittent hydrostatic pressure increase proteoglycan synthesis
(Jortikka et al., 2000); osteoblasts increase their intracellular
calcium concentration when subjected to micropipette poking or pulling
via magnetic microbeads (Xia and Ferrier, 1992; Glogauer et al., 1995);
and endothelial cells increase gene expression of endothelin-1 when
subjected to microbead twisting (Chen et al., 2001). Methods of
applying mechanical stimulation can be broadly divided into two
categories: those that apply stimulation over the whole cell (substrate
stretch, fluid shear, intermittent hydrostatic pressure), and those
that stimulate only a small part of the cell body (microbead pulling,
microbead twisting, micropipette poking). Results obtained with one
straining system are difficult to compare to those obtained with
another. Indeed, cells are most likely to detect deformations applied
onto their structure or, in engineering terms, strain (deformation per
unit length). Knowing the strain distributions on cell surfaces would
enable results from different straining techniques to be compared to
one another, and their physiological consequences to be analyzed.
Common engineering techniques such as computational fluid dynamics
(CFD) or finite element modeling (FEM) can be used to compute the shear
stresses resulting from fluid flow or the strain distributions due to
mechanical stimulation. CFD enables velocity and pressure distributions
generated by a fluid flowing over a surface to be determined and,
therefore, shear stress distribution can be determined. CFD has been
utilized with success to investigate the flow of blood through arteries
and their bifurcations (Long et al., 2001). Barbee et al. (1995)
calculated the shear stresses due to fluid flow over an endothelial
cell monolayer whose topography had been acquired using atomic force
microscopy (AFM). Used in conjunction with FEM, this can yield the
cellular strains elicited by fluid shear stress. Indeed, FEM allows the
strain distribution due to a given set of loading and boundary
conditions applied onto a structure whose material properties are known
to be determined. FEM has been applied with success to modeling and
determining the strain distributions within whole organs such as bone
(van Rietbergen et al., 1999), cartilage (Gu et al., 1997), or the arterial wall (for a review see Simon et al. (1993)), but has seldom
been applied to individual cells due to lack of precise data on
cellular material properties or shape. Riemer-McReady and Hollister
(1997) modeled an osteocyte embedded within its lacuna to find the
strains applied to the cell by a uniform compression of the matrix in
which it was embedded. Guilak and Mow (2000) and Wu and Herzog
(2000) modeled a chondrocyte embedded within a cartilaginous matrix. In
all three cases the cells were modeled as spheres with homogenous
properties, hence ignoring potential inhomogeneities in material
properties or topology. Other finite element models have concentrated
on predicting cellular material properties from the cytoskeletal
structure (Hansen et al., 1996), predicting the rearrangement of the
cytoskeleton (Picart et al., 2000), or the evolution of the cell shape
in response to micropipette aspiration (Drury and Dembo, 1999).
Although many methods exist to measure the bulk cellular material
properties, only AFM enables the three-dimensional profile of cell
surfaces to be acquired at high resolution together with their material
property distribution (for a review see Radmacher (1997)).
In this study we combined AFM with FEM and CFD to calculate the strain distributions resulting from common whole-cell mechanical stimulation techniques. Experimentally acquired cell profiles and material property maps acquired by AFM were converted into three-dimensional finite element models. Different sets of boundary and loading conditions were applied to the cell models to simulate straining experiments (substrate stretch, fluid shear, and intermittent hydrostatic pressure). Common micromanipulation experiments (microbead pulling and twisting, micropipette poking) were modeled on a small subcellular volume and strain distributions were calculated to provide a comparison to the whole-cell straining experiments. Cellular adaptation to mechanical stresses was simulated by increasing the elastic modulus of the cells and examining its effect on the strain distributions. The different parameters pertaining to the stimulation method were varied and their effect on the strain distributions was examined. In addition, we used these models to calculate the strain magnitudes resulting from experiments by other groups and compared the strain levels needed to trigger the reported detection mechanisms and downstream cellular responses.
In conclusion, we report for the first time the application of AFM in conjunction with FEM and CFD to calculate the strain distributions in cells resulting from common methods of mechanical stimulation. The knowledge of these strain distributions will enable different straining experiments to be compared to each other. Moreover, these data should aid our understanding of whether strains induced by commonly used straining techniques are detected via different intracellular signaling pathways.
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MATERIALS AND METHODS |
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TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
CONCLUSIONS
REFERENCES
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Experimental data
Cell culture
Osteoblasts were isolated from the long bones of neonatal rats by mechanical disaggregation and cultured for 72 h at 37°C in an atmosphere of 5% CO2 in air in DMEM (Gibco Life Technologies, Paisley, UK) supplemented with 10% FCS, 2% glutamine, 2% penicillin streptomycin, 2% 1 M HEPES, pH 7.0.Immunostaining and confocal microscopy
Immunostaining was performed as described in Nesbitt and Horton (1997). Briefly, the cells were fixed in a PBS solution containing 2%
formaldehyde and 0.1% glutaraldehyde, and permeabilized in ice-cold
Triton X-100 buffer for 5 min at 4°C. They were then incubated with
monoclonal anti-paxillin (Transduction Laboratories, Lexington, KY), a
focal contact protein, for 30 min, FITC-labeled goat anti-mouse Ig
antibody (Dako, Denmark) for 30 min, and rhodamine-phalloidin (Molecular Probes Europe, Leiden, The Netherlands) for 30 min. All
coverslips were imaged with a 100× oil-immersion objective on a Leica
confocal microscope running TCS NT (Leica, Bensheim, Germany).
Fluorescent images were sequentially collected in 0.4-µm steps with
emission wavelengths of 488 and 568 nm for the FITC and TRITC
fluorophores, respectively. The images were then post-processed using
Imaris software (Bitplane Ag, Zürich, Switzerland) on an SGI
O2 workstation (SGI, Mountain View, CA).
Atomic force microscopy
A Thermomicroscopes Explorer (Thermomicroscopes, Sunnyvale, CA) interfaced onto an inverted microscope (Nikon Diaphot 300, Nikon UK, Kingston, UK) was used to acquire the material properties of the cells (Lehenkari et al., 2000). The measurements were
carried out using soft V-shaped cantilevers with pyramidal tips
(k = 0.032 N·m
1, model 1520, Thermomicroscopes) and these were calibrated in air before experimentation.
Osteoblastic cells cultured on glass coverslips were transferred to the
AFM sample holder and examined in physiological buffer (127 mM NaCl, 5 mM KCl, 2 mM MgCl2, 0.5 mM
Na2H PO4, 2 mM
CaCl2, 5 mM NaHCO3, 10 mM
glucose, 10 mM HEPES, 0.1% BSA adjusted to pH 7.4). For each cell,
force-distances curves were collected at points on a 50 × 50 or
100 × 100 grid. The approach speed used for the force-distance
curves was 5 µm·s1 to minimize
contributions of cellular viscoelasticity to the estimated cellular
elasticity (A-Hassan et al., 1998).
Material property measurement
Cellular material properties were evaluated as described in Radmacher (1997). Briefly, the cell was assumed to be a homogenous half
space and the tip conical. The force F needed to produce an
indention of depth 
in a half-plane with an elastic modulus E is (Johnson, 1985):
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(1) |
the opening angle of the conical tip and 
the local
Poisson ratio. Knowing the cantilever stiffness and by fitting the
theoretical curve to the experimental data, the elastic modulus can be
deduced (Radmacher, 1997). The cellular Poisson ratio was assumed to be
0.3, in line with experimental measurements in live cells (0.25 ± 0.05, Maniotis et al., 1997). It is necessary to choose a value of the
Poisson ratio, as experimentally acquired force-distance curves fitted
with Eq. 1 cannot yield both the cellular elasticity and Poisson ratio.
A custom-written program running under Pv-Wave (Visual Numerics,
Boulder, CO) on an SGI O2 workstation was used to
fit the force-distance curves automatically.
The spatial resolution of the material property maps could be estimated
by calculating the diameter of the tip-cell contact area. Using Eq. 1,
one can find and, assuming that the indentation is cone-shaped, the
diameter of contact is d = 2tan(). For = 30°, E = 1 kPa, F = 1 nN, = 0.3, we find d = 1.04 µm. Hence, 100 × 100 grids, which sample material properties every micrometer, are the
highest resolution that can be attained without excessive spatial
overlap in measurements.
Numerical modeling for whole-cell mechanical models
The AFM scans of 10 osteoblasts were converted into three-dimensional finite element models incorporating the experimentally measured elasticities and topographies using a custom-written program running under Pv-Wave.
Generation of whole-cell models
The cells were "virtually" extracted from their experimental substrate and plated onto a flat substrate with a Young's modulus of 4 GPa. The models (Fig. 1 F) had 50 × 50 elements in the x- and y-directions, a resolution of 2 µm, and ~7000 elements. The resolution in the z-direction was chosen to be the same as in the x- and y-directions. An additional zone 20 µm wide was added around the model to reduce boundary effects. The substrate was two elements thick. The number of elements at a given location in the cell was equal to its height divided by the z-resolution rounded up to the next integer. Most cells had between one and two elements in their height. Cell and substrate were presumed to be uniformly bound along their contact area rather than in a discrete number of points representing the cellular focal adhesion complexes (Fig. 1 A). The cells and the substrate were modeled with eight-noded parametric volumic elements. Because of the large number of different elastic moduli within a cell, the cellular distributions were grouped into 10 material property collectors with the following elasticities:
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(2) |
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Physical model
As we were interested only in the static solutions for whole-cell strains, all materials were assumed to be linear elastic and isotropic (Zhu et al., 2000). Cell and substrate had a Poisson ratio of
0.3 (Maniotis et al., 1997). Appropriate boundary conditions and forces
were applied to the models to simulate substrate stretch or
intermittent hydrostatic pressure.
The linear elastic continuum mechanics equations (Timoshenko and
Goodier, 1970) were then solved to find the strain distributions exerted on the cells. In its simplest expression, engineering strain
can be defined as the length variation dl per unit
length l ( = dl/l). Engineering strain is
usually expressed in percent variation of length or microstrain
(µ) with 1% strain = 0.01 = 10,000 µ. All of
the finite element calculations were carried out with CAST3M, a
general-purpose finite element solver with an integrated pre- and
post-processor (CAST3M, Commissariat à l'Energie Atomique,
Saclay, France, kk2000{at}semt1.smts.cea.fr., available free for
universities) and were run either on an SGI O2 or
a standard PC.
Boundary conditions
Substrate stretch. For the substrate stretch simulations, a displacement equivalent to 0.1% stretch in the x-direction was applied to one end of the substrate and the other side was constrained in the x-direction. The sides running parallel to the x-direction were constrained in the y-direction. The underside of the substrate was constrained in the z-direction (All boundary conditions are represented in Fig. 2 A).
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was varied, for the whole mesh, between 0.2 and 0.5 while applying a stretch of 0.1% along the x axis. To assess the effect of stretch direction, the simulated direction of
stretch was varied and had angles of 0°, 30°, 45°, 60°, and 90° with the x axis, while keeping stretch magnitude and
Poisson ratio constant. These analyses were carried out on one
osteoblast model only, as the strain distributions for the other cells
would vary similarly due to the linear elastic nature of the mechanical model.
Intermittent hydrostatic pressure. For the
intermittent hydrostatic pressure experiments, the underside of the
substrate was fully constrained and a hydrostatic pressure of 5 Pa was
applied to the top surface (All boundary conditions are represented in Fig. 3 A). To assess the
effect of the Poisson ratio on cellular strain distributions, it was
varied, for the whole mesh, between 0.2 and 0.5 while keeping the
pressure constant. This analysis was carried out on one cell model
only.
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Generation of osteocyte models
The material properties of osteocytes were assumed to be the same as those of osteoblasts. As osteocytes and their cavities are ellipsoidal (Marotti et al., 1992), we modeled only half of the
cell-cavity complex, thereby assuming that the other half of the cell
was perfectly identical. To simulate that osteocytes were embedded in
the bone matrix, the cell models were covered by a layer of matrix
elements forming a brick with a "mold" of the cell on the
underside. The cell and the matrix were assumed to be uniformly bound
along their surface of contact. The underside of this model was
constrained in the z-direction. A displacement equivalent to
0.1% compression in the x-direction was applied to one end
of the block of matrix and the other side was constrained in the
x-direction. The sides running parallel to the
x-direction were constrained in the y-direction.
The bone matrix was assumed to have an elasticity of 4 GPa, in
agreement with experimentally measured values (Mente and Lewis, 1989).
Numerical modeling for fluid shear simulations
To examine the strain distributions resulting from fluid flow on cells, the calculations had to be performed in two distinct steps. First, a CFD model had to be generated to calculate the flow lines and shear stresses resulting from flow over the cellular profile. Second, an FE model of the cell was generated and the shear stresses from the CFD simulation were applied to the mechanical model. The strain distribution resulting from these could be calculated. As the cellular deformations were small (<0.1%), we assumed that the cellular deformations did not significantly affect the flow lines around the cell profile, and hence we did not need to iterate the process.
Generation of the models
For the fluid flow simulations, the material property distributions and topographies were reduced to a 25 × 25 grid to reduce calculation time and were converted into a three-dimensional finite element model. An entrance and an exit, 10 µm wide, were added to reduce transitory effects. First, a CFD model of the cell and substrate surface was created with eight-noded linear volumic fluid flow elements (Fig. 4 A). The CFD model had a height of 16 µm, which was over fourfold greater than the average height perturbation introduced by the cell profile.
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Computational fluid dynamics: physical model and boundary conditions
The cell surface was subjected to a laminar flow of an incompressible viscous fluid with a parabolic profile (Fig. 4 A). We assumed that the flow on the top surface of the model was not significantly perturbed by the cell profile and therefore imposed a constant velocity umax. We assumed that the cell did not significantly perturb the flow in the transverse horizontal direction and imposed a condition of no transverse flow on the side surfaces. The velocity on the cell-substrate surface was imposed to be 0 (all boundary conditions are represented in Fig. 4 A). The CFD code (CAST3M) solved the Navier-Stokes equations (Currie, 1993) using a finite-element approach
and output the velocities and pressures for each element of the CFD
model:
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(3) |
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(4) |
):
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(5) |
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(6) |
ij the Kronecker delta, 
the density of the fluid, vi the
velocity of the fluid flow in the i direction,
ij the element of the shear tensor in
position ij, i the resulting shear stress vector in the i direction,
ei the directing vector i of the
orthonormal base of vectors, and µ the cinematic viscosity of the
fluid; and µ were assumed to be the same as for water
(respectively 1000 kg·m3 and
103 N·s·m2);
umax was adjusted to give rise to a
shear stress of 5 Pa (50 dyn·cm2) on a flat
substrate and was 0.046 m·s1.
Finite element modeling: physical model and boundary conditions
All materials were assumed to be linear elastic and isotropic. The Poisson ratio was assumed to be 0.3. The model was fully constrained on the bottom surface and the shear stresses derived from the CFD calculation were multiplied by the surface normals and imposed onto the surface nodes (Fig. 4 B).Variation of the physical parameters
To assess the effect of the cinematic viscosity µ on the cellular strains, its value was varied between 5 × 104 and 4 × 103
N·s·m2 while keeping all other parameters
constant. The effect of cell height on cellular strain was assessed by
varying it between 50 and 200% of the original height, all other
parameters being constant. Finally, the influence of the cellular
Poisson ratio was evaluated by varying it, for the whole mesh, between
0.2 and 0.5 while keeping the other variables constant. These analyses
were only performed for one cell model, as the other cell models would
show identical trends.
Numerical modeling for micromanipulation models
Micromanipulation techniques apply mechanical stimuli only to small areas of the cell inducing large local strains. Therefore, finer meshes are needed to calculate the strain distributions with reasonable accuracy. Modeling the whole cell with a suitable mesh refinement would be unpractical, as the memory space and calculation time needed become inordinate.
Based on our experience in modeling cellular indentation by a spherical
AFM tip (Charras et al., 2001), we decided to model only a small volume
of the cell and assume that the material was isotropic and linear
elastic. The Young's modulus was chosen to be 1 kPa, the lower value
for cellular material properties, hence giving an upper bound of the
cellular strains. The Poisson ratio was chosen to be 0.3 (Maniotis et
al., 1997).
Magnetic microbead pulling
In magnetic microbead pulling experiments, ferromagnetic beads coated with collagen are sprinkled over a cell layer and left to settle for 30 min. The collagen-coated beads bind to the cells via integrin cell adhesion receptors and the cells are washed several times to remove unbound beads. During the experiment a magnetic field is applied to the cells and the microbeads are displaced vertically, pulling the cell with a force of 4 pN over the area of contact (Glogauer and Ferrier, 1998).
To determine the area of contact of the magnetic microbeads with the
cell surface, we assumed that when the microbeads contact the cells
they indent the cellular material with a force
Fw equal to their weight minus the
buoyancy force, thus creating an indentation of radius a
(Johnson, 1985).
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(7) |
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(8) |
the Poisson ratio,
E the elastic modulus, g the universal
gravitational constant (9.81 kg·m·s2),
Fe3O4 the volumic mass of magnetite (4897 kg·m3), and H2O the
volumic mass of water (103
kg·m3). The radius of the beads was chosen to
be 2 µm (Glogauer and Ferrier, 1998) yielding a radius of contact of
0.12 µm.
Taking the symmetries of the problem into account, only one-quarter of
the space was modeled (Fig. 5
A). A box 2 µm long in the x- and
y-directions and 3 µm thick was meshed with 20-noded parametric volumic elements with a higher density in the area of
contact (shown in green in Fig. 5 A), and particularly at
the boundary between the tethered and untethered region. The model was
constrained in displacement in the z-direction at its base and in the x- and y-directions, respectively, on
the y- and x-sides in which the pulling was
applied (Fig. 5 A). The other sides were left free. To
simulate magnetic pulling, a uniform force was applied to the top
surface nodes in the area of contact. The area of contact was
constrained in the x- and y-directions to
simulate integrin linkages between the cell surface and the bead. The
experimentally controllable parameters are the force applied (through
the magnetic field), and the radius of the microbeads.
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= 0.3.
Microbead twisting
During microbead twisting experiments the ferromagnetic beads are tethered to the cells following a similar protocol as for microbead pulling, but are coated with RGD, an integrin receptor agonist (Wang and Ingber, 1994). The beads are then magnetized using a large magnetic
pulse and a much smaller magnetic field is used to make them rotate and
apply a torque onto the cell cytoskeleton via integrin receptors (Wang
and Ingber, 1994).
The area of contact of the beads with the cell surface was calculated
as for magnetic bead pulling. The problem only has one plane of
symmetry and accordingly, we modeled one half-space (Fig. 6 A). A 4 µm × 2 µm
box in the x- and y-directions and 3 µm thick was meshed with 20-noded parametric volumic elements with a higher density in the area of contact (shown in green in Fig. 6 A)
and particularly at the boundary between the region in contact and the
free region. The model was constrained in displacement in the
z-direction at its base, in the y-direction on
the plane of symmetry, and in the x- and
y-directions, respectively, on the y- and
x-sides on the other vertical surfaces (Fig. 6
A). To simulate the magnetic twisting, the nodes on the
surface of contact were rotated by an angle around the y
axis of the contact disk. The force and pressure applied could be
calculated by extracting the resultant of this imposed displacement.
The area of contact was constrained in the x- and
y-directions to simulate integrin linkages between the cell
surface and the bead. The read-out variable for these experiments is
the angle of rotation and the control is effected via the pressure
applied. Wang and Ingber (1994) report a maximal pressure of 4 Pa
applied on the area of contact.
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= 0.3. The
effect of the radius of contact was assessed by varying it between 0.1 and 0.6 µm with p = 4 Pa, E = 1 kPa, and = 0.3.
Micropipette poking
Micropipette poking consists of lightly indenting the cell surface with a blunt micropipette. We assumed that this was similar to indenting the cell surface with a rigid spherical indentor with a radius of 1.5-3 µm. This experiment was modeled similarly to cellular indentation by a glass sphere using AFM (Charras et al., 2001). Taking the symmetries of
the problem into account, only one-quarter of the space was modeled. A
box 15 µm in length in the x- and y-directions and 4 µm thick was meshed with 20-noded parametric volumic elements, with a higher density of elements in the area of indentation and particularly at the boundary between the indented (shown in green in
Fig. 7 A) and the free
regions. The model was constrained in displacement in the
z-direction at its base and in the x- and y-directions, respectively, on the y- and
x-sides in which the indentation was performed (Fig. 7
A). The other sides were left free. The cellular material
was assumed to be linear elastic and isotropic. As the main control in
this system is the depth of indentation , we used this as an input
parameter in our calculations rather than applied force.
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= 1 µm, and R = 3 µm. The depth of indentation was
varied between 0.5 and 1.5 µm and the cellular strains resulting
calculated with E = 1 kPa, = 0.3, and
R = 3 µm. The radius of the pipette was varied
between 1 and 3 µm with E = 1 kPa, = 0.3, and = 0.5 µm.
Sensitivity of mechano-detection mechanisms
To investigate the sensitivity of the different
mechano-detection mechanisms, we predicted the cellular strains used in
mechanical stimulation studies by our and other groups, in which the
detection mechanism was examined (Table 5). The strains were
calculated as previously described assuming that all cell types have
similar elasticities (reviewed in Lehenkari et al., 1999) and using
E = 1 kPa for the micromanipulation studies.
Adaptation to mechanical strain
To examine the efficiency of increasing cellular elasticity to
adapt to sustained mechanical strain, we changed the cellular material
properties and examined the effect on cellular strain distributions.
For whole cells, the material properties were increased by a given
percentage. For the micromanipulation models of microbead pulling and
twisting the elasticity was varied between 0.5 and 10 kPa. The effects
of cellular elasticity on strain distributions due to micropipette
poking were not assessed, as this technique offers no control over the
force applied. However, a comparable study has been performed for
AFM-indentation with spherical tips in Charras et al. (2001).
Cellular strain resulting from stray fluid flow
Based on results from studies by Schaffer et al. (1994) giving
the substrate strains in a variety of stretching systems and Brown et
al. (1998) giving the stresses resulting from stray fluid flow in those
systems, we compared the cellular strains due to the intended
stimulation (stretch) and those due to the unintended stimulation
(fluid flow) using our FEM and CFD results. The fluid shear stresses
induced by the three systems studied were two orders of magnitude lower
than those commonly used to mechanically stimulate cells (2-5 Pa), and
hence we ignored them. However, the fluid normal stresses reached
significant magnitudes (up to 75 Pa). As cells are very close to being
a flat surface, we approximated the normal stresses due to fluid flow
by a hydrostatic pressure of similar magnitude.
Statistics and curve-fitting
Average, maximal, and median strains were compared with a Student's t-test and the results were deemed significant for p < 0.05. All curve-fitting was performed using Kaleidagraph (Synergy Software, Reading, PA) on a PC.
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RESULTS |
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TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
CONCLUSIONS
REFERENCES
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Experimentally measured material properties of osteoblasts
Topography and material properties (Fig. 1, B and D) were acquired using AFM. The cell was clearly identifiable as a softer area on a hard substrate (Fig. 1 D). The cellular elasticities varied between 1 kPa in the nuclear area and 100 kPa in the cytoplasmic skirt, and did not correlate with cell height (Fig. 1, B and D). Stress fibers could clearly be seen as "stiffer lines" spanning the cell from side to side (Fig. 1 D); these correlate with features in the AFM phase image (Fig. 1 C) and show a distribution similar to actin stress bundles identified with rhodamine-phalloidin staining (Fig. 1 A). The cellular elasticity frequency distribution for the 10 cells used fitted a Gaussian curve centered on 14 kPa (Fig. 1 E). Finite element models of the cells were created (for example, in Fig. 1 F from data in Fig. 1, B and D).
Strain distributions and magnitudes
Whole-cell models
Substrate stretch produced maximal strains along the axis of stretch. The strain distributions were very homogenous along the x axis with average and median absolute strains of 1030 µ, and maximal absolute strains of 1270 µ (Table
1). Strains in the y- and
z-directions were significantly lower than in the
x-direction. Most of the cellular strains were close to the
imposed stretch and higher strains were situated in the vicinity of
stress fibers (Fig. 2 A). The evolution of maximum, median,
and average absolute strains (averaged over 10 cells) could be
predicted for commonly used values of substrate stretch (Fig. 2
B).
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zz) of up to 14% and average vertical
strains of 3.7%.
Fluid shear produced maximal strains in the vertical direction (Table
1). Maximal strains were 10-fold higher than average or median strains,
pointing to a heterogeneous strain distribution (Table 1 and Fig. 4
B). The vertical fluid shear stresses
(z, Fig. 4 A) were lower upstream,
where pressure build-up counters the traction force due to shear
stresses, and larger downstream, where shear stresses and pressure have
additive effects. The vertical cellular strains (Fig. 4 B)
were tensile and maximal downstream from the cell apex, in areas with a
low elastic modulus that coincided with the location of the nucleus.
Maximal, average, and median strains (averaged over 10 cells) could be
calculated for commonly used values of shear stress (Fig. 4
C).
The maximal strains exerted by the three modes of stimulation were
close to one another with a 0.1% stretch producing maximal absolute
strains of 1270 µ, a 5 Pa hydrostatic pressure producing maximal
absolute strains of 700 µ, and a 5 Pa shear stress producing maximal absolute strains of 1080 µ (Table 1). Fluid flow and hydrostatic pressure exerted significantly different maximum and median
vertical strains (p = 0.04 and p = 0.01), but not average vertical strains (p = 0.70).
Maximal absolute vertical strains due to fluid flow were significantly
lower than the maximal absolute horizontal strains produced by
substrate stretch (p < 0.001).
Micromanipulation models
For commonly used values of the stimuli and an elasticity of 1 kPa, the micromanipulation techniques applied strains in excess of 5%, which was on average one order of magnitude higher than those applied by whole-cell techniques (Tables 1 and 2). The radial strain distribution (rr) for microbead pulling (Fig. 5 B) showed the presence of high tensile and short-range
surface strains at the border between the region tethered to the bead and the free region (Fig. 5 A). The cellular strains
increased linearly with applied force (Fig. 5 E). The
maximal vertical strains were the largest strain component, followed by
the maximal radial strains (Fig. 5 E).
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xx) for
microbead twisting (Fig. 6 B) showed the presence of high
tensile strains on the surface on either side of the border between the
region where the bead is bound to the cell and the free region (Fig. 6
A). The cellular strains increased quasi-linearly with
increasing angular motion (Fig. 6 C). Maximal and minimal
values for each strain component were of similar magnitude (Fig. 6
C). The bead radius, and hence the radius of indentation,
only had a marginal effect on cellular strains (data not shown).
The radial strain distribution (rr) (Fig. 7
B) for micropipette poking showed a large tensile component
on the surface at the border between the indented and free region (Fig.
7 A). The vertical strain distribution
(zz) (Fig. 7 C) showed a large
compressive component under the indented region. The cellular strains
varied linearly with indentation depth (Fig. 7 D).
Effect of material property changes
Whole-cell models
When substrate stretch was applied, a twofold increase in the cellular elasticities had no influence on the strains exerted on the cell (change <0.1%, Fig. 2 D). However, when the cellular elasticities were increased twofold and hydrostatic pressure or fluid shear were applied, the cellular strains were reduced by 50 and 33%, respectively (Figs. 3 C and 4 D).Micromanipulation models
For microbead pulling and twisting, all components of cellular strain decreased dramatically with increasing elasticity (Figs. 5 C and 6 F, respectively) for a fixed stimulus. These decreases fitted well with a power-law (r2 > 0.95 except "zz min" for microbead pulling).
In microbead pulling experiments the vertical displacement of the bead
for a force of 4 pN decreased from 21 nm for E = 0.5 kPa to 2.8 nm for E = 10 kPa. The vertical displacement
as a function of elasticity fitted well with a power-law
(r2 = 0.99, Fig. 6 F).
During microbead twisting, the angular rotation of the bead for a
pressure of 4 Pa decreased from 12° for E = 0.5 kPa
to 1.5° for E = 10 kPa. The angular motion as a
function of elasticity fitted well with a power-law
(r2 = 0.99, Fig. 5 E).
Effect of Poisson ratio changes
Whole-cell models
When substrate stretch was applied, the Poisson ratio had little effect on average or median strains.xx was
reduced by only 15% when was changed from 0.2 to 0.5 (Fig. 2
E). However, in models of hydrostatic pressure (Fig. 3
D) or fluid shear (similar evolution to Fig. 3 D,
data not shown), the Poisson ratio was an important factor and it
reduced the maximal, median, and average maximal strains by 93% when
it was varied from 0.2 to 0.5. This is due to the predominantly
compressive nature of these mechanical stimulations.
Micromanipulation models
The Poisson ratio had little influence on the order of magnitude in microbead pulling experiments (Fig. 5 D). The maximal radial and tangential strain decreased by 85% when was varied between 0.2 and 0.5. The two largest strain components in absolute value "zz max" and
"rr min" varied by 
56% and +74%,
respectively, when was changed from 0.2 to 0.5.
The magnitude of strains elicited by micropipette poking was not very
sensitive to changes in Poisson ratio (Fig. 7 E). The maximal radial strains (rr) were reduced by a
maximum of 40% and the maximal tangential strains
(tt) by 94% when was varied from 0.2 to
0.5. In microbead twisting experiments (Fig. 6 D), the
Poisson ratio had a dramatic influence around = 0.4 where all
components of strain were amplified to an order of magnitude above
their value for other values of .
Effect of the direction of application of stimulus
In substrate stretch models, the stretch direction had no effect
on the maximal strain (11 along the direction
of stretch) introducing only a 7% variation in its magnitude (Fig. 2
C). In fluid shear models, rotating the direction of flow by
90° increased the maximal vertical strain
(zz) by 12% and reduced the average and
median vertical strains by 5 and 8%, respectively (data not shown).
Effect of fluid flow parameters
The cellular strains varied linearly with the value of the cinematic viscosity µ (Fig. 4 E). Increasing cell height by 100% increased the maximal strains by only 7% (Fig. 4 F).
Osteocytes
Osteocytes embedded in a block of matrix compressed by 0.1% (1000 µ) were submitted to maximal vertical strains
(zz) of up to 1% (10,000 µ, Table 3).
Hence, cellular strains were amplified 10-fold compared to matrix
strains. Average and median vertical strains were 2-fold larger than
cellular strains in the direction of stretch
(xx).
Cellular strain elicited by stray fluid flow
The cellular strains elicited by stray fluid flow induced
additional average cellular strains of up to 2810 µ (0.28%) and maximal cellular strains of up to 10,500 µ (1.05%), which
represented respectively 13% and 39% of the strains induced by
substrate stretch (Flexercell system, Table
4).
The MIT system induced average additional strains of 4% and maximal
strains of 14% of the value imposed by substrate stretch. The fluid
flow-induced strains reached 1420 µ (0.14%) on average and a
maximum of 5310 µ (0.53%) in the Providence system.
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DISCUSSION |
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TOP
ABSTRACT
INTRODUCTION
MATERIALS AND METHODS
RESULTS
DISCUSSION
CONCLUSIONS
REFERENCES
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In this paper we combine AFM measurements of elasticities and profiles of live osteoblasts with CFD and FEM to predict, for the first time, the cellular strain distributions resulting from common whole-cell stimulation methods, such as substrate stretch, intermittent hydrostatic pressure application, or fluid shear. In addition, we give the strain distributions resulting from mechanical stimulation by micromanipulation techniques, such as microbead pulling, microbead twisting, or micropipette poking. In all cases we examined the effect of the relevant mechanical parameters on the strain distributions, and provide the magnitudes of cellular strains for commonly used values of the control parameters for each of the experimental conditions. We have examined, for each experimental condition, whether increasing cellular material properties may be a good strategy for adapting to sustained mechanical strain, by assessing the impact of this modification upon the cellular strain distributions. By combining our modeling data with experimental data from other groups (Table 5), we examine the magnitudes of strains needed to activate different strain detection mechanisms. We also underline the necessity to design new substrate stretching devices that reduce stray fluid flow. We believe that this work will help further our understanding of cellular detection of, and adaptation to, mechanical strain.
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Cellular strain detection mechanisms
Cells can detect strains through a variety of mechanisms that, as
a first step, involve stretch-activated cation channels, integrin
transmembrane receptors, G-proteins, or tyrosine kinases (Sachs and
Morris, 1998; Gudi et al., 1998; Banes et al., 1995; Malek and Izumo,
1996). The threshold strain of activation and the strain
component to which they are sensitive may be different. We examined
experiments performed by other groups investigating the mechanisms
involved in the detection of mechanical stimulation and calculated the
strain distributions that they applied (Table 5).
All of the micromanipulation techniques had both a high radial tensile
component (>1%) on the cell surface (Figs. 5 B, 6
B, 7 B), compatible with a membrane stretch
detection mechanism (Table 5). The large vertical component
(>5%) (tensile for microbead pulling, mixed for microbead twisting,
and compressive for micropipette prodding; Fig. 7 C) could
be detected by cytoskeleton-based mechanisms, such as via tyrosine
kinases (Glogauer et al., 1997; Chen et al., 2001). Furthermore, most
groups report inhibition of cellular reactions when exposed to
stretch-activated channel blocking agents, Gd3+
(Sachs and Morris, 1998), or Grammostola spatulata venom
(Suchyna et al., 2000). However, the possibility that the high levels
of strain applied by these techniques over very small areas induce cell
damage or membrane rupture should not be excluded, especially in the
case of micropipette prodding.
Detection of mechanical stimuli in cells subjected to intermittent
hydrostatic pressure have been reported to involve stretch-activated channels and integrins (Lee et al., 2000). The magnitude of the in-plane strains resulting from hydrostatic pressure was similar to
that resulting from micromanipulation techniques, and hence the
detection may be mediated by stretch-activated channels.
Reported detection mechanisms for substrate stretch are varied
and are thought to involve stretch-activated channels, tyrosine kinases, integrins, and the cytoskeleton (Table 5). FE models revealed that the cells were subjected to strains close to those imposed on the substrate (in agreement with experimental data by Caille
et al., 1998) and, for the experiments reported in Table 5
(except Peake et al., 2000), the strain magnitude was similar to that
detected by stretch-activated channels or tyrosine kinases in
micromanipulation experiments. Furthermore, recent studies point toward
the detection of substrate stretch through mechano-sensitive channels
activated by intercellular tension applied through adherens-junctions (Ko et al., 2001). In contrast, some experiments report effects of
substrate stretch on osteoblasts for low strains, around 0.1% (Peake
et al., 2000; Zaman et al., 1997; Fermor et al., 1998).
However, one must be circumspect when interpreting substrate stretch
data, as many systems also induce stray fluid flow (Table 4 and Brown
et al., 1998; You et al., 2000). Indeed, the maximal cellular strains
induced by stray fluid flow reached 40% of the value applied by
substrate stretch (Table 5); hence, cellular reactions obtained
in such systems may be mediated not only by substrate stretch, but also
by fluid flow. Owan et al. (1997) showed that the stray fluid flow
resulting from a four-point bending system (such as that used by Peake
et al., 2000) was sufficient to elicit cellular reactions. This
underlines the necessity of designing new substrate stretch systems
(e.g., You et al., 2000) and may explain differences in results between
systems that elicit large stray fluid flow and those that do not.
In fluid shear stress experiments an equal number of groups have
reported mechanisms involving, or not involving, stretch-activated cation channels for similar values of fluid shear. The strain levels
for fluid shear experiments, and in particular the in-plane strains,
which may be detected by stretch-activated channels, are two to three
orders of magnitude lower than those elicited by other stimulation
techniques. Hence, either the stretch-activated cation channels are
particularly sensitive to fluid shear or another mechanism is utilized.
Recently, it has been shown that G-proteins reconstituted within
phospholipid vesicles increased their GTPase activity in response to
fluid shear (Gudi et al., 1998). GTPase activity also increased with
increasing vesicle membrane fluidity. During fluid shear, cell membrane
fluidity in living cells increased with the onset of fluid flow in the
upstream cellular region (Butler et al., 2001). Taken together with the
very low magnitudes of strain induced and the nonspecificity of
Gd3+ (Sachs and Morris, 1998), these data may
point toward a detection mechanism relying on an increase in membrane
fluidity detected by G-proteins with Gd3+ having
an effect on mechanisms downstream from these.
In summary, whereas, for commonly used values of the parameters, most
stimulation methods elicited strains of comparable magnitude, fluid
shear experiments generated far lower cellular strains and may trigger
an entirely different detection mechanism. Moreover, several mechanisms
may co-exist on each cell type and be sensitive to different components
of the strain distribution. Chen et al. (2001) and Glogauer et al.
(1997) report that stretch-activated channels and tyrosine kinases act
cooperatively to mediate responses to microbead twisting and pulling.
Hayakawa et al. (2001) report that whereas stretch-activated channels
govern whole-cell reorientation in response to substrate stretch, they
do not seem to be involved in the reorientation of actin stress fibers
within the cell. Such co-existence of detection mechanisms may be
beneficial to the cell in fine tuning cellular responses, such as the
transcription of different genes to various ranges of cellular strain.
Importance of the cellular Poisson ratio
The cellular Poisson ratio remains unknown, though experiments by
Maniotis et al. (1997) report a Poisson ratio of 0.25 ± 0.05. The
Poisson ratio had little influence on the magnitude of strains exerted
in microbead pulling, micropipette poking, or substrate stretch
experiments (Figs. 2 E, 5 D, 7 E).
However, it was of crucial importance in hydrostatic compression or
fluid shear experiments (Fig. 3 D and data not shown), which
apply predominantly compressive stresses. When the cell tended to
incompressibility, the magnitude of cellular strains was greatly
reduced. Interestingly, for microbead twisting experiments, the Poisson
ratio appeared to have a dramatic effect around = 0.4, where
the strains elicited were increased by one order of magnitude. These
results are of particular interest as cells may be able to modulate
their Poisson ratio by altering their intracellular architecture. In
support of this, Maniotis et al. (1997) showed that specific disruption of cytoskeletal fibers increased the cellular Poisson ratio to ~0.5,
bringing the cell closer to an incompressible gel.
Cellular adaptation to mechanical perturbation
Cellular adaptation to mechanical stimuli has been reported in
many instances, and our study may explain how those adaptations help
the cell withstand sustained mechanical stimulation. When exposed to
fluid shear stress, cells and their F-actin stress fibers realign with
the direction of flow (Girard and Nerem, 1993). Furthermore, cells
submitted to sustained flow have higher elasticities than unstrained
cells (Sato et al., 2000). Realignment of the cell body with the
direction of flow serves to reduce the shear stresses (5%) acting on
the cell (Barbee et al., 1995) and increasing the cellular material
properties reduces the strains to which the cell is subjected (50%,
Fig. 4 D). In cells subjected to a long period of microbead
pulling, Glogauer et al. (1997) reported the formation of actin
plaques, with higher elasticities under the microbeads. Our study shows
that increasing cellular elasticity in response to microbead
stimulation is a very efficient method for reducing cellular strains
(Fig. 5 C). Furthermore, addition of actin need only be
restricted to a small area because the radial strains decay very
rapidly (Fig. 5 B). Similarly, our models predict that cells
exposed to long periods of increased intermittent hydrostatic pressure
could counter the increased strains by increasing their elasticity. In
agreement with this, cardiomyocytes from animals with cardiac pressure
overload had elasticities twofold higher than those from control
animals (Tagawa et al., 1997). However, several groups report a
complete disassembly of the cytoskeleton in response to intermittent
hydrostatic pressure (Haskin et al., 1993; Parkkinen et al., 1995),
though this may be due to the excessive magnitude of the pressure
applied (>4 MPa). Indeed, to reduce the in-plane strains to values
below 1% for a 4 MPa pressure, the cell would need to increase its
elasticity by several orders of magnitude. Hence, a different mechanism
of adaptation may be necessary and disassembly of the cytoskeleton
could be the first step. There is evidence that cytoskeletal tensegrity
enables cells to accommodate changes in volume (Guilak, 1994) giving
them a continuum scale Poisson ratio that is not 0.5 (Maniotis et al., 1997 report a value of 0.25 ± 0.05). Disassembly of the
cytoskeleton would change the cell into a fluid surrounded by a lipid
bilayer; this would make it quasi-incompressible, thereby drastically
reducing the magnitude of strains to which it would be subjected
(Maniotis et al., 1997; Fig. 3 D). In substrate stretch
experiments, cellular elasticity had little influence on cellular
strains (Fig. 2 D). This is due to the fact that the cells
are tethered to the substrate (Fig. 1 A) and hence any
displacement imposed on the substrate is imposed on the cells. In this
case, it is important to consider the discrete, rather than continuum,
nature of the cell cytoskeleton. Indeed, the cell is only anchored to
its substrate at a number of discrete points (focal adhesion complexes,
shown in green, Fig. 1 A) that often coincide with the
extremities of the cellular actin stress fibers (shown in red, Fig. 1
A). If the fibers run along the direction of stretch, the
distance between the two extremities will be increased and the fibers
will exert a vertical compression on the underlying structures. In
contrast, if the fibers reorganize such that they run perpendicular to
the direction of stretch, the distance between the two extremities
stays unchanged and no compression is applied (Hayakawa et al., 2001).
Thus, cellular adaptation can be explained in terms of cytoprotective
responses geared at shielding the cell from unusual strain magnitudes
(Ko and McCulloch, 2000).
Strain detection in osteocytes
The strains to which osteocytes were subjected reached 1% and
were an order of magnitude larger than the strain imposed on the matrix
(Table 3), confirming earlier results by Riemer-McReady and Hollister
(1997). Based on micro-manipulation studies and our estimates of the
cellular strains resulting from these, this magnitude of strain would
be sufficient to activate stretch-activated cation channels (Table
5). However, there is still much debate as to the nature of the
stimulus to which osteocytes respond. Predicted loading-induced shear
stresses in the osteocyte lacunae (0.8-3 Pa, Weinbaum et al. (1994))
have been found to have excitatory effects on osteocytes in vitro (You
et al., 2000; Ajubi et al., 1999). However, in cyclically loaded bone
explants, Gd3+ abolished loading-related
increases in nitric oxide and prostaglandin PGI2
(Rawlinson et al., 1996), which would point to a mechanism mediated by
stretch-activated channels. However, the same group (Rawlinson et al.,
2000) also showed that pertussis toxin, a blocker of G-proteins,
inhibits loading-related increases in prostaglandins PGE2 and PGI2, thus
pointing to a detection of the fluid flow within the bone. Hence, both
signaling mechanisms could realistically be involved and may co-exist
in osteocytes to mediate the detection of, and responses to, bone deformation.
Limitations of the integrated measurement and modeling process
Combining AFM measurements with modeling techniques has enabled us to compare the strains elicited by different
