| HOME | HELP | FEEDBACK | SUBSCRIPTIONS | ARCHIVE | SEARCH | TABLE OF CONTENTS |
* Department of Bioengineering, University of California, San Diego, La Jolla, California 92093 and Department of Biomedical Engineering, Boston University, Boston, Massachusetts 02215 USA
Correspondence: Address reprint requests to Mark F. Coughlin, Physiology Dept., Harvard School of Public Health, 665 Huntington Avenue, Boston, Massachusetts 02115. Tel.: 617-432-2610; E-mail: mcoughli{at}hsph.harvard.edu.
 |
ABSTRACT |
|---|
 TOP ABSTRACT INTRODUCTIONMODEL FORMULATIONSMODEL IMPLEMENTATIONRESULTSDISCUSSIONSUMMARYACKNOWLEDGEMENTSREFERENCES |
|---|
158 pN, and the nonlinear mechanical response during CP originates from filament kinematics. The MTC and MBM simulations revealed that the model is incomplete, however, these simulations show cable tension as a key determinant of the model response. |
INTRODUCTION |
|---|
|
TOPABSTRACTINTRODUCTIONMODEL FORMULATIONSMODEL IMPLEMENTATIONRESULTSDISCUSSIONSUMMARYACKNOWLEDGEMENTSREFERENCES |
|---|
; Elson, 1988; Singhvi et al., 1994; Chen et al., 1997; Chicurel et al., 1998; Janmey, 1998). A growing body of evidence indicates that the primary control of adherent cell shape and deformability is exerted at the level of the cytoskeletal (CSK) filaments. For example, cell shape is determined by both internal structures dense in CSK filaments (e.g., pseudopodia) and transmembrane linkages connecting CSK filaments and extracellular matrix proteins (e.g., at focal adhesion complexes). Moreover, cell deformability assessed by mechanical measurement is largely determined by the composition, architecture, and force in the underlying CSK filaments. In particular, an interconnected network of actin filaments provides the main force-bearing CSK structure within anchorage dependent cells. A complete description of CSK filament function in the cell remains elusive. To understand how physical forces regulate biological function, the microstructural mechanisms by which the CSK filaments give rise to macroscopic cellular properties must be addressed.
An engineering approach to CSK mechanics has provided new tools to address the mechanisms by which cells resist deformation (Stamenovi and Wang, 2000). We recently used a conceptual cable network model to connect microstructrual CSK parameters to macroscopic properties of adherent cells (Stamenovi and Coughlin, 1999). The actin CSK was depicted as a network of randomly oriented tension-supporting cables. A key feature of the model was that actin filaments supported a prestress, i.e., a preexisting tensile stress provided either by the cell contractile apparatus or cell distension on the substrate. Transparent mathematical expressions related CSK prestress and elastic modulus to microstructural parameters characterizing CSK forces and architecture. A merit of the prestressed cable network as a model of CSK mechanics is that some details of the CSK microstructure need not be explicitly specified. However, this generality also limits the model's predictive capacity. For example, an expression for the cell elastic modulus was obtained without designating the nature of the cable interconnections, but the strongest prediction was then a lower bound. To obtain more definite predictions of the CSK forces and elastic properties requires additional intricacies of the CSK microstructure to be postulated. Previously, interconnected elastic cable networks with cables represented by linear elastic springs provided quantitative predictions of erythrocyte elastic properties (Hansen et al., 1996) and remarkable correspondence to the erythrocyte CSK response in micropipette aspiration experiments (Discher et al., 1998). However, it is not clear that cable network models that describe the mechanics of suspended cells are as appropriate for anchorage dependent cells. The purpose of this investigation was to examine the possibility that cable networks can qualitatively and quantitatively predict the mechanical response of anchorage dependent cells subjected to various mechanical perturbations.
In the current study, two prestressed cable networks were examined as models of the adherent cell actin CSK. In one model, the cables were organized into a planar lattice of regular hexagons, and in the other, the cables were organized into a planar lattice of equilateral triangles. The geometric parameters and cable elastic properties were assigned based on data from the literature. The models were deformed to mimic cell poking (CP), magnetic twisting cytometry (MTC), and magnetic bead microrheometry (MBM) measurements on living adherent cells. The predicted responses were compared to published data. The models were used to identify mechanisms that underlie CSK deformability and to predict CSK forces and elastic properties.
|
MODEL FORMULATIONS |
|---|
|
TOPABSTRACTINTRODUCTIONMODEL FORMULATIONSMODEL IMPLEMENTATIONRESULTSDISCUSSIONSUMMARYACKNOWLEDGEMENTSREFERENCES |
|---|
 | (1) |
 | (2) |
|
). Second, the actin filament bending stiffness is small (Gittes et al., 1993), and their lateral dimension is typically much smaller than their length. This suggests that individual actin filaments cannot support substantial compressive loads without collapsing (buckling). Thus, the cables were assumed to be linearly elastic (Hookean) and support only tensile forces. The cable force-extension relationship is given by
 | (3) |
The five model parameters: l0, S, Ec, Ac and lr were assigned values based on data in the literature. Because there is no direct observation of l0 in living cells, l0 was estimated from an effective CSK pore size (Dp) in adherent cells. Experimental observations of the actin CSK in adherent endothelial and fibroblast cells reveal a network with Dp 
50–200 nm (Luby-Phelps et al., 1986; Amos and Amos, 1991; Janmey, 1996; Satcher et al., 1997; Caille et al., 1998). In this study, calculations were performed with the intermediate value Dp = 100 nm. It was assumed that each hexagon and triangle enclosed a pore with diameter 
and 
, respectively (Fig. 1, insets) in the initial configuration. It follows that l0 = 57.7 and 173.2 nm for the hexagonal and triangular networks, respectively. The model area S was chosen to match to two representative measurements of endothelial cell projected area of 350 and 1640 µm2, corresponding to round and spread cells (Wang and Ingber, 1994). It follows from Eqs. 1 and 2 and Dp = 100 nm that N = 116 and 251 for round and spread cells in the hexagonal network model, respectively, and N = 66 and 144 round and spread cells in the triangular network, respectively. The cable properties were determined from published values of the Young's modulus Ec = 2.8 GPa and cross-sectional area Ac = 18.8 nm2 of individual actin filaments (Gittes et al., 1993). The final free model parameter lr is determined by specifying the cable force in the initial configuration F0 using Eq. 3, lr = l0EcAc/(F0 + EcAc). Only F0 between 0 and the highest measured yield force for individual actin filaments of FMAX 600 pN (Tsuda et al., 1996) were examined.
The position of the pin-joints inside the attachments was determined by equilibrium equations. A Cartesian coordinate system OXYZ was placed at the center of the model with the Z-axis normal to the plane of the network. A pin-joint was in equilibrium when the cable forces exerted on the joint were balanced. Force balance in the X-direction for a pin-joint with coordinates (x0, y0, z0) and nc cables terminating at pin-joints with coordinates (xi, yi, zi), i = 1, ..., nc, is given by
 | (4) |
is the length of the cable spanning the joints at (x0, y0, z0) and (xi, yi, zi), and nc = 3 and 6 for the hexagonal and triangular networks, respectively. Equations for force balance in the Y- and Z-directions are obtained from Eq. 4 by interchanging the X-coordinates with Y- and Z-coordinates, respectively.
Model equilibrium was established by iteratively solving the equilibrium equations (Eq. 4) for each pin-joint (except the attachments) until all equilibrium equations were simultaneously satisfied. The criteria dictating pin-joint equilibrium was a net force less than 10-6 pN in each coordinate direction. The coordinates of pin-joints not satisfying the equilibrium criteria were adjusted using a Newton-Raphson method (Gerald and Wheatley, 1989). All computations were performed on the Power Challenge Array facility of the Boston University Scientific Computation and Visualization Center.
|
MODEL IMPLEMENTATION |
|---|
|
TOPABSTRACTINTRODUCTIONMODEL FORMULATIONSMODEL IMPLEMENTATIONRESULTSDISCUSSIONSUMMARYACKNOWLEDGEMENTSREFERENCES |
|---|
Cell poking
The CP measurement technique has been used to assess the mechanical properties of various adherent cells (McConnaughey and Petersen, 1980; Petersen et al., 1981; Petersen et al., 1982; Elson et al., 1983; Zahalak et al., 1990). In studies of fibroblast cell deformability (Petersen et al., 1982; Elson et al., 1983; Cai et al., 1998), a cylindrical 2 µm-diameter glass stylus indented the cell surface. The relationship between the stylus indentation depth and force revealed primarily the subcortical CSK network mechanical properties (Petersen et al., 1981; Petersen et al., 1982). An important feature of the CP technique is that the stylus does not directly bind to CSK components or associated proteins.
To simulate the CP technique in the cable network model, the stylus was idealized as a frictionless, rigid circular cylinder (punch) with diameter d = 2 µm. The stylus was applied at the center of the model. The indentation of the stylus (u) on adherent cells was mimicked in the model by imposing the following constraints on the positions of the pin-joints: (1) the vertical Z-coordinates of the pin-joints under the punch were such that z 
-u and (2) the X- and Y-coordinates of the pin-joints located outside the punch diameter were not allowed to penetrate the side of the punch, x2 + y2 > (D/2)2 when z > -u. Similarly, cables were not allowed to penetrate the surface of the punch. Cables connecting pin-joints under and outside the punch were stretched over the edge of the punch through an intermediate point with coordinates (xp, yp, u) (Fig. 2 A). The coordinates of the intermediate point were found such that the net circumferential force due to the cable segments under and outside the punch were balanced. The position of all the pin-joints was determined by Eq. 4. It was sufficient to compute the equilibrium positions for only one-twelfth of all pin-joints because of model symmetry.
|
 | (5) |
is the distance between the pin-joint with coordinates (xi, yi, zi) located outside the punch and the point with coordinates (xp, yp, u) on the edge of the punch, and C is the number of cables crossing the edge of the punch. The X- and Y-components of the cable forces exerted on the punch balance by model symmetry. The relationship between u and W was compared to data obtained from CP measurements during the indentation of adherent fibroblast cells.
Magnetic twisting cytometry
The MTC measurement technique has been used to assess the mechanical properties of various adherent cells (Wang et al., 1993; Wang and Ingber, 1994; Hubmayr et al., 1996; Cai et al., 1998). In studies of endothelial cell deformability, ligand coated magnetic beads with 1.4 or 5.5 µm diameter were bound to the cells via integrin surface receptors (Wang et al., 1993; Wang and Ingber, 1994). The attached beads were polarized, and an applied magnetic field induced a magnetic moment causing bead rotation. The relationship between the bead rotation and magnetic moment was used to assess the cellular mechanical properties. An important feature of the MTC technique is the interaction between the bead and CSK. The intracellular domain of the bead bound integrin receptors accumulate focal adhesion proteins and interact with the CSK (Wang et al., 1993). As a consequence, the main intracellular structures probed by MTC are both the cortical CSK network and possibly filaments deeper within the cytoplasm.
To implement the MTC technique in the cable network model, the cell bound magnetic beads were idealized as rigid spheres. It was assumed that one bead was bound to the model; it was half embedded in the network and centered within the attachments in the undeformed configuration. Although the bead was considered rigid for bead rotation, the pin-joints and cables in the undeformed configuration were allowed to freely penetrate the bead surface (Fig. 3). This simplified scheme for the interaction of the bead and network allowed the initial network configuration to be planar and regular. If the network interacted with the bead surface, then an initial force would be required to embed the bead in the network and the regular network geometry would be distorted in the initial configuration. The pin-joints falling within the bead boundary (bound pin-joints) were distinguished from those outside the bead boundary (free pin-joints). The rotation of a surface bound magnetic bead was mimicked in the model by collectively rotating the bound pin-joints as a rigid body through an angle 
about the Z-axis through the bead center. The equilibrium positions of the free pin-joints were determined from Eq. 4. Thus, the positions of the bound pin-joints were prescribed whereas the positions of the free pin-joints were computed.
|
 | (6) |
); shear stress 
was defined as the moment per bead volume = MZ/V, and was regarded as a shear strain. It was sufficient to compute the equilibrium positions for only one-twelfth of the free pin-joints because of model symmetry.
Magnetic bead microrheometry
Magnetic bead microrheometry (MBM) has been used to study the mechanics of adherent endothelial and fibroblast cells (Bausch et al., 1998; Bausch et al., 2001). The MBM technique has similarities to MTC. Magnetic beads with 4.5 µm-diameter were bound to integrin receptors on the cell surface and polarized in a magnetic field. A strong gradient in magnetic field induced a pulling force in each bead causing bead displacement. The relationship between the bead displacement and pulling force revealed the CSK mechanical properties. As noted for MTC, the surface bound beads connect to the CSK through a collection of accessory proteins.
To implement MBM in the network model, the bead was modeled as a rigid sphere with frictionless, impenetrable surface. It was assumed that a single bead was bound to the network, the bead was centered within the attachments, and half the bead was embedded in the network as described for MTC. To mimic the pulling of an attached magnetic bead, the bound pin-joints were displaced as a rigid body in the XY-plane by an amount 
. Thus, the positions of the bound pin-joints in the deformed configuration were prescribed, and the positions of the free pin-joints in the deformed configuration were determined by Eq. 4.
In the equilibrium configuration, the network exerts a force on the bead in two ways. The first force is due to the cables connecting bound and free pin-joints that stretch as a result of and exert a pulling force on the displaced bead. A second force arises from free pin-joints that accumulate on the bead surface as a result of . Because the bead is impenetrable, those free pin-joints will exert a pushing force on the bead. The net force on the bead 
was the sum of the pushing and pulling forces in the direction of . For a bead in the deformed configuration, the component of in the X-direction (X) was given by
 | (7) |
and 
, respectively, Li is the cable length, Ti is the pushing force exerted by pin-joint i on the bead, C is the number of cables connecting free and bound pin-joints, and P is the number of pin-joints exerting a pushing force on the bead. It was sufficient to consider only half the model to determine the equilibrium positions of the free pin-joints. |
RESULTS |
|---|
|
TOPABSTRACTINTRODUCTIONMODEL FORMULATIONSMODEL IMPLEMENTATIONRESULTSDISCUSSIONSUMMARYACKNOWLEDGEMENTSREFERENCES |
|---|
; Elson et al., 1983). For a given u, W increased with increasing F0 in both models. That is, resistance to indentation increased with increasing F0. This trend is consistent with the observed increase in stiffness in fibroblast cells containing constitutively active myosin light chain kinase (Cai et al., 1998), which is thought to increase actin filament tension. The quantitative predictions of W from both models in the spread configuration are in excellent agreement with experimental data from the literature (Fig. 4 A and Fig. 5 A). The measured W falls between the predicted upper and lower bounds for all u and u < 2.0 µm in the hexagonal and triangular models, respectively. There is also good quantitative correspondence from the round configuration but over a narrower range of u than was found in the spread configuration (Fig. 4 B and Fig. 5 B). The measured data fall within the predicted upper and lower bounds for u 1.2 and 0.7 µm in the hexagonal and triangular models, respectively.
|
|
|
Magnetic twisting cytometry
In both the hexagonal and triangular networks, in the round and spread configurations, shear stress increases nonlinearly with increasing (Fig. 7 A). This strain hardening is consistent with MTC measurements on adherent endothelial cells (Wang et al., 1993; Wang and Ingber, 1994). For a given , increases with increasing F0 in both models. This is consistent with the observed increase in endothelial cell rigidity after mechanical (Pourati et al., 1998) and pharmacological (Cai et al., 1998) increases in CSK tensile forces. The quantitative predictions of from all models exceed the measured values by several orders of magnitude (Fig. 7 B). Moreover, the model is more deformable in the spread than round configuration and predicts an increase in deformability with increasing bead size (Coughlin, 2000). Both these trends are opposite to the observed behavior (Wang and Ingber, 1994).
|
Magnetic bead microrheometry
In both the hexagonal (Fig. 8) and triangular (not shown) networks in the round and spread (not shown) configurations, increased with increasing which is consistent with MBM measurements on adherent fibroblast (Bausch et al., 1998) and endothelial cells (Bausch et al., 2001). The dependence of on is nearly linear, which is consistent with the measured data. For a given , increased with increasing with F0. This observation agrees with the decrease in endothelial cell deformability after pharmacological activation (Bausch et al., 2001). The quantitative predictions of from all models exceed the measured values by several orders of magnitude (Fig. 8).
|
|
DISCUSSION |
|---|
|
TOPABSTRACTINTRODUCTIONMODEL FORMULATIONSMODEL IMPLEMENTATIONRESULTSDISCUSSIONSUMMARYACKNOWLEDGEMENTSREFERENCES |
|---|
; Stamenovi et al., 1996; Coughlin and Stamenovi, 1997; Hansen et al., 1997; Coughlin and Stamenovi, 1998; Discher et al., 1998; Stamenovi and Coughlin, 1999; Stamenovi and Coughlin, 2000). In particular, cable network models of the spectrin CSK of suspended red blood cells predict the response to micropipette aspiration. In this study, network models were applied to the mechanics of the adherent cell CSK. The features of the models were based on published ultrastructural observations and mechanical measurements from both living cells and individual CSK filaments. The CSK elasticity was attributed solely to a dense network of interconnected actin filaments. This view is consistent with measurements on several adherent cell types showing that actin filaments determine the cell mechanical response and ultrastructural observations of the adherent cell CSK showing a preponderance of actin filaments where these cells are probed. The representation of individual actin filaments in the model as stiff, linear elastic cables follows from their observed tensile response (Kojima et al., 1994) and a detailed analysis of their mechanical properties (Gittes et al., 1993). The many interactions observed between interconnected filaments in the CSK (whether chemical, stearic, or other) were represented in the model by frictionless pin-joints, and the resulting network porosity and filament segment length was 101–102 nm. Many actin cross-linking proteins have been identified in a variety of adherent cells, and ultrastructural observations confirm the appearance of a dense, interconnected network permeating the cell cytoplasm. Fixed pin-joints in the model represented local regions of firm adhesion between an adherent cell and rigid substrate. The model was deformed to mimic CSK deformation imposed during CP, MTC, and MBM measurements on living adherent cells.
The model best captured the qualitative and quantitative features of the adherent cell mechanical response to CP deformation. The model exhibits strain hardening and decreased deformability with increasing F0. Both of these findings are consistent with observations from CP measurements on adherent fibroblast cells (Petersen et al., 1982; Elson et al., 1983; Cai et al., 1998). Moreover, the measured CP data fall between the predicted upper and lower bounds of W. Taken together, these findings support the assertion that the mechanisms determining model deformability are present within living adherent cells. That is, the prestressed peripheral actin CSK is a key determinant of adherent fibroblast cell deformability and filament kinematics is responsible for the nonlinear mechanical response. The model revealed that externally applied forces are transmitted throughout the actin CSK, with peripheral filaments contributing to resistance to local deformation. This differs from continuum based predictions where the deformation rapidly decays in the radial direction from the point of load application (Sokolnikoff, 1946). The model was unable to quantitatively predict the response of adherent cells to MTC or MBM. However, some qualitative features of the cell response were captured by the model. Thus, the cable network has limited applicability in describing the mechanics of adherent cells. Despite those limitations, filament tension was identified as a key determinant in the cell response to MTC and MBM.
A surprising aspect of this analysis is the dichotomy in the ability of the model to qualitatively and quantitatively capture the CP response while failing to predict many facets of the MTC or MBM response. Many of the important differences in the measurement techniques that may explain these discrepancies are accounted for in the model, e.g., probe size, mode of deformation, extent of deformation. However, some others are not. A feature common to MTC and MBM that is not found in CP is the probable recruitment of CSK and associated proteins to the probe. In the case of CP, forces to resist punch indentation emanate from the cortical CSK which is accurately modeled by a planar network. On the other hand, in the case of MTC and MBM, accumulation of focal adhesion proteins may bring additional CSK elements (e.g., more filaments) or structures (e.g., stress fibers) to the probe. The ensuing deformation is resisted by both the cortical CSK and the recruited CSK components. This additional structure may introduce new mechanisms or components that contribute to cell deformability but are not included in the models. For example, forces may be transmitted deeper into the cytoplasm where the CSK network is more disorganized and heterogeneous than the subcortical CSK. Also, beads thought to only rotate in MTC measurements may also translate (Fabry et al., 2001), and the translating beads in MBM measurements may also rotate. Including these additional motions in the models would likely lower the predicted moments and pulling forces in MTC and MBM measurements, respectively, but not enough to account for the several orders of magnitude. In addition to in-plane twisting as a model for MTC reported here, twisting of the beads out of the plane of the network was also examined. Twisting out of the plane of the network corresponds more closely to the actual motion of the beads in MTC measurements. The bound pin-joints were rotated as a rigid body through an angle about an axis in the plane of the network and perpendicular to an axis of mirror symmetry. The equilibrium positions of the free pin-joints were determined from Eq. 4 as described. The shear stress was computed as before. The hexagonal model in the round configuration predicts the nonlinear increase in with increasing observed in MTC measurements, and, for a given , the model predicts increasing with increasing F0 (Fig. 9). The quantitative predictions of are one-half to one-third of the predicted values of the corresponding model for in-plane twisting, however, they still far exceed the measured values.
|
) and compared to the predictions at those u for the hexagonal network in the spread configuration. The minimum least squares difference between the measured data and the model prediction was found for F0 158 pN. This suggests the actin filament tension in the peripheral CSK in adherent fibroblast cells is 158 pN.
To formulate computationally tractable models that still retain the major features of the CSK mechanical response, a number of factors known to contribute to cell deformability were not included in the model formulations. The models do not include dissipative phenomena, such as viscous losses. Measurements of various cell types and measurement techniques exhibit hysteresis or loss mechanisms (Petersen et al., 1982; Elson et al., 1983; Shroff et al., 1995; Bausch et al., 1998; Maksym et al., 2000). Also, the models do not include mechanical contributions from other structures such as membranes, microtubules, intermediate filaments, or the nucleus. Several studies indicate the contribution of membrane forces to resisting deformation imposed during mechanical measurements accounts for 30% (Wang et al., 1993) or substantially less (Petersen et al., 1981) of the cell resistance to deformation. However, the model does not include bending resistance to deformation of the membrane cortex (i.e., the membrane and peripheral CSK) which is a key mechanism in micropipette aspiration experiments of suspended neutrophils (Zhelev et al., 1994) and appears implicitly in network models of the spectrin network of suspended erythrocytes (Discher et al., 1998). Because the membrane cortex of adherent cells is a curved and bendable structure, local curvature in the region of the probes may explain the quantitative disparity in the model predictions. Smaller probes may induce more bending deformation than larger probes. Interestingly, the model successfully predicted the fibroblast cell response to CP, which features a smaller probe than either MTC or MBM. In the case of MTC measurements, the apparent endothelial cell resistance to deformation decreased by 30% when probed with 25% smaller diameter beads (1.4 µm vs. 5.5 µm) (Wang and Ingber, 1994) suggesting other mechanisms may be more prevalent than bending of the cell cortex. The hexagonal model deformed to mimic MTC predicts an increased resistance to deformation by 50% with 1.4 µm-diameter beads (Coughlin, 2000). No data is available from measurements with different probe sizes using CP or MBM. Loss of microtubule integrity is less damaging to cell deformability than loss of actin filament integrity in adherent fibroblast cells (Petersen et al., 1981). In fact, measurements from fibroblast cells using AFM show regions rich in microtubules correspond to soft regions of the cell surface. The nucleus influences measures of cellular deformability (Petersen et al., 1982; Elson et al., 1983). For small deformations, the influence of the nucleus may be minimal because the initial resistance to CP deformation is nearly constant both over and away from the nucleus (Elson et al., 1983). Additionally, dynamics of polymer filaments were ignored in the models. It is known that the CSK undergoes perpetual turnover and rearrangement mediated by several mechanisms (e.g., filament polymerization, depolymerization, severing). Also, based on the results of this analysis, actin filaments in the cortical CSK must support a prestress of the order 102 pN without breaking for the duration of the cell poking measurements. In studies measuring the actin filament yield force, the duration the filament was held under tension was not given particular attention. However, actin filaments were reported to support a tensile force near the yield force for more than 10 min (Kishino and Yanagida, 1988). A latter study showed measurements made over shorter time scales of 20 s (Tsuda et al., 1996). Both times are longer than the indentation time of 2.5 s in cell poking measurements. Thus, it might not be unreasonable to assume that actin filaments behave like elastic cables on those time scales.
|
SUMMARY |
|---|
|
TOPABSTRACTINTRODUCTIONMODEL FORMULATIONSMODEL IMPLEMENTATIONRESULTSDISCUSSIONSUMMARYACKNOWLEDGEMENTSREFERENCES |
|---|
|
ACKNOWLEDGEMENTS |
|---|
|
TOPABSTRACTINTRODUCTIONMODEL FORMULATIONSMODEL IMPLEMENTATIONRESULTSDISCUSSIONSUMMARYACKNOWLEDGEMENTSREFERENCES |
|---|
Submitted on May 23, 2002; accepted for publication October 7, 2002.
|
REFERENCES |
|---|
|
TOPABSTRACTINTRODUCTIONMODEL FORMULATIONSMODEL IMPLEMENTATIONRESULTSDISCUSSIONSUMMARYACKNOWLEDGEMENTSREFERENCES |
|---|
Bausch, A. R., U. Hellerer, M. Essler, M. Aepfelbacher, and E. Sackmann. 2001. Rapid stiffening of integrin receptor-actin linkages in endothelial cells stimulated with thrombin: a magnetic bead microrheology study. Biophys. J. 80:2649–2657.
Bausch, A. R., F. Ziemann, A. A. Boulbitch, K. Jacobson, and E. Sackmann. 1998. Local measurements of viscoelastic parameters of adherent cell surfaces by magnetic bead microrheometry. Biophys. J. 75:2038–2049.
Cai, S., L. Pestic-Dragovich, M. E. O'Donnell, N. Wang, D. E. Ingber, E. Elson, and P. De Lanerolle. 1998. Regulation of cytoskeletal mechanics and cell growth by myosin light chain phosphorylation. Am. J. Physiol. 275:C1349–C1356.[Medline]
Caille, N., Y. Tardy, and J. J. Meister. 1998. Assessment of strain field in endothelial cells subjected to uniaxial deformation of their substrate. Ann. Biomed. Eng. 26:409–416.[Medline]
Chen, C. S., M. Mrksich, S. Huang, G. M. Whitesides, and D. E. Ingber. 1997. Geometric control of cell life and death. Science. 276:1425–1428.
Chicurel, M. E., C. S. Chen, and D. E. Ingber. 1998. Cellular control lies in the balance of forces. Curr. Opin. Cell Biol. 10:232–239.[Medline]
Coughlin, M. F. 2000. The Microstructural Basis of Cellular Deformability. Dissertation. Department of Biomedical Engineering, Boston University, Boston.
Coughlin, M. F., and D. Stamenovi. 1997. A tensegrity structure with buckling compression elements: applications to cell mechanics. J. Appl. Mech. 64:480–486.
Coughlin, M. F., and D. Stamenovi. 1998. A tensegrity model of the cytoskeleton in spread and round cells. J. Biomech. Eng. 120:770–777.[Medline]
Discher, D. E., D. H. Boal, and S. K. Boey. 1998. Simulations of the erythrocyte cytoskeleton at large deformation. II. Micropipette aspiration. Biophys. J. 75:1584–1597.
Elson, E. L. 1988. Cellular mechanics as an indicator of cytoskeletal structure and function. In Annual Review of Biophysics and Biophysical Chemistry. D. M. Engelman, C. R. Cantor, and T. D. Pollard, editors. Annual Reviews Inc., Palo Alto. 397–430.
Elson, E. L., B. B. Daily, W. B. McConnaughey, C. Pasternak, and N. O. Petersen. 1983. Measurement of forces which determine the shapes of adherent cells in culture. In Frontiers in Biochemical and Biophysical Studies of Proteins and Membranes. T.-Y. Liu, S. Sakakibara, A. N. Schechter, K. Yagi, K. Yajima, and K. T. Yasunobu, editors. Elsevier Science Publishing Co., Inc., New York. 399–411.
Fabry, B., G. N. Maksym, J. P. Butler, M. Glogauer, D. Navajas, and J. J. Fredberg. 2001. Scaling the microrheology of living cells. Phys. Rev. Lett. 87:148102.[Medline]
Folkman, J., and A. Mascona. 1978. Role of cell shape in growth control. Nature. 273:345–349.[Medline]
Gerald, C. F., and P. O. Wheatley. 1989. Applied Numerical Analysis. Addison-Wesley Publishing Company, Inc., Reading.
Gittes, F., B. Mickey, J. Nettleton, and J. Howard. 1993. Flexural rigidity of microtubules and actin filaments measured from thermal fluctuations in shape. J. Cell Biol. 120:923–934.
Hansen, J. C., R. Skalak, S. Chien, and A. Hoger. 1996. An elastic network model based on the structure of the red blood cell membrane skeleton. Biophys. J. 70:146–166.
Hansen, J. C., R. Skalak, S. Chien, and A. Hoger. 1997. Influence of network topology on the elasticity of the red blood cell membrane skeleton. Biophys. J. 72:2369–2381.
Hubmayr, R. D., S. A. Shore, J. J. Fredberg, E. Planus, R. A. Panettieri, W. Moller, J. Heyder, and N. Wang. 1996. Pharmacological activation changes stiffness of cultured human airway smooth muscle cells. Am. J. Physiol. 271:C1660–C1668.[Medline]
Janmey, P. A. 1996. Coping with cellular stress: the mechanical resistance of porous protein networks. Biophys. J. 71:3–5.
Janmey, P. A. 1998. The cytoskeleton and cell signaling: component localization and mechanical coupling. Physiol. Rev. 78:763–781.
Kishino, A., and T. Yanagida. 1988. Force measurements by micromanipulation of a single actin filament by glass needles. Nature. 334:74–76.[Medline]
Kojima, H., A. Ishijima, and T. Yanagida. 1994. Direct measurement of stiffness of single actin filaments with and without tropomyosin by in vitro nanomanipulation. Proc. Natl. Acad. Sci. USA. 91:12962–12966.
Luby-Phelps, K., D. L. Taylor, and F. Lanni. 1986. Probing the structure of cytoplasm. J. Cell Biol. 102:2015–2022.
Maksym, G. N., B. Fabry, J. P. Butler, D. Navajas, D. J. Tschumperlin, J. D. Laporte, and J. J. Fredberg. 2000. Mechanical properties of cultured human airway smooth muscle cells from 0.05 to 0.4 Hz. J. Appl. Physiol. 89:1619–1632.
McConnaughey, W. B., and N. O. Petersen. 1980. Cell poker: an apparatus for stress-strain measurements on living cells. Rev. Sci. Instrum. 51:575–580.[Medline]
Petersen, N. O., W. B. McConnaughey, and E. L. Elson. 1981. Investigations of structural determinants of cell shape. Comm. Molec. Cell. Biophys. 1:135–147.
Petersen, N. O., W. B. McConnaughey, and E. L. Elson. 1982. Dependence of locally measured cellular deformability on position on the cell, temperature, and cytochalasin B. Proc. Natl. Acad. Sci. USA. 79:5327–5331.
Pourati, J., A. Maniotis, D. Spiegel, J. L. Schaffer, J. P. Butler, J. J. Fredberg, D. E. Ingber, D. Stamenovi, and N. Wang. 1998. Is cytoskeletal tension a major determinant of cell deformability in adherent endothelial cells? Am. J. Physiol. 274:C1283–C1289.[Medline]
Satcher, R., C. F. Dewey, Jr., and J. H. Hartwig. 1997. Mechanical remodeling of the endothelial surface and actin cytoskeleton induced by fluid flow. Microcirculation. 4:439–453.[Medline]
Shroff, S. G., D. R. Saner, and R. Lal. 1995. Dynamic micromechanical properties of cultured rat atrial myocytes measured by atomic force microscopy. Am. J. Physiol. 269:C286–C292.[Medline]
Singhvi, R., A. Kumar, G. P. Lopez, G. N. Stephanopoulos, D. I. C. Wang, G. M. Whitesides, and D. E. Ingber. 1994. Engineering cell shape and function. Science. 264:696–698.
Sokolnikoff, I. S. 1946. Mathematical Theory of Elasticity. Krieger Publishing Company, Malabar.
Stamenovi, D., and M. F. Coughlin. 2000. A quantitative model of cellular elasticity based on tensegrity. J. Biomech. Eng. 122:39–43.[Medline]
Stamenovi, D., and M. F. Coughlin. 1999. The role of prestress and architecture of the cytoskeleton and deformability of cytoskeletal filaments in mechanics of adherent cells: a quantitative study. J. Theor. Biol. 201:63–74.[Medline]
Stamenovi, D., J. J. Fredberg, N. Wang, J. P. Butler, and D. E. Ingber. 1996. A microstructural approach to cytoskeletal mechanics based on tensegrity. J. Theor. Biol. 181:125–136.[Medline]
Stamenovi, D., and N. Wang. 2000. Cellular responses to mechanical stress: engineering approaches to cytoskeletal mechanics. J. Appl. Physiol. 89:2085–2090.
Tsuda, Y., H. Yasutake, A. Ishijima, and T. Yanagida. 1996. Torsional rigidity of single actin filaments and actin-actin bond breaking force under torsion measured directly by in vitro micromanipulation. Proc. Natl. Acad. Sci. USA. 93:12937–12942.
Wang, N., J. P. Butler, and D. E. Ingber. 1993. Mechanotransduction across the cell surface and through the cytoskeleton. Science. 260:1124–1127.
Wang, N., and D. E. Ingber. 1994. Control of cytoskeletal mechanics by extracellular matrix, cell shape, and mechanical tension. Biophys. J. 66:2181–2189.
Zahalak, G. I., W. B. McConnaughey, and E. L. Elson. 1990. Determination of cellular mechanical properties by cell poking, with an application to leukocytes. J. Biomech. Eng. 112:283–294.[Medline]
Zhelev, D. V., D. Needham, and R. M. Hochmuth. 1994. Role of the membrane cortex in neutrophil deformation in small pipettes. Biophys. J. 67:696–705.
This article has been cited by other articles:
 |
|
 |
|
  R. Paul, P. Heil, J. P. Spatz, and U. S. Schwarz Propagation of Mechanical Stress through the Actin Cytoskeleton toward Focal Adhesions: Model and Experiment Biophys. J., February 15, 2008; 94(4): 1470 - 1482. [Abstract] [Full Text] [PDF]
|
|
|
|
 |
|
 N. E. Vlahakis and R. D. Hubmayr Cellular Stress Failure in Ventilator-injured Lungs Am. J. Respir. Crit. Care Med., June 15, 2005; 171(12): 1328 - 1342. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
B. A. Smith, B. Tolloczko, J. G. Martin, and P. Grutter Probing the Viscoelastic Behavior of Cultured Airway Smooth Muscle Cells with Atomic Force Microscopy: Stiffening Induced by Contractile Agonist Biophys. J., April 1, 2005; 88(4): 2994 - 3007. [Abstract] [Full Text] [PDF]
|
|
|
|
 |
|
 H. Huang, R. D. Kamm, and R. T. Lee Cell mechanics and mechanotransduction: pathways, probes, and physiology Am J Physiol Cell Physiol, July 1, 2004; 287(1): C1 - C11. [Abstract] [Full Text] [PDF]
|
|
|
|
 |
|
 D. E. Ingber Tensegrity I. Cell structure and hierarchical systems biology J. Cell Sci., April 1, 2003; 116(7): 1157 - 1173. [Abstract] [Full Text] [PDF]
|
|
| ||||||||||||
