A Quantitative Model of Cellular Elasticity Based on Tensegrity

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1 A Quantitative Model of Cellular Elasticity Based on Tensegrity Dimitrije Stamenović Mark F. Coughlin Department of Biomedical Engineering, Boston University, Boston, MA A tensegrity structure composed of six struts interconnected with 24 elastic cables is used as a quantitative model of the steady-state elastic response of cells, with the struts and cables representing microtubules and actin filaments, respectively. The model is stretched uniaxially and the Young s modulus (E 0 ) is obtained from the initial slope of the stress versus strain curve of an equivalent continuum. It is found that E 0 is directly proportional to the pre-existing tension in the cables (or compression in the struts) and inversely proportional to the cable (or strut) length square. This relationship is used to predict the upper and lower bounds of E 0 of cells, assuming that the cable tension equals the yield force of actin ( 400 pn) for the upper bound, and that the strut compression equals the critical buckling force of microtubules for the lower bound. The cable (or strut) length is determined from the assumption that model dimensions match the diameter of probes used in standard mechanical tests on cells. Predicted values are compared to reported data for the Young s modulus of various cells. If the probe diameter is greater than or equal to 3 m, these data are closer to the lower bound than to the upper bound. This, in turn, suggests that microtubules of the CSK carry initial compression that exceeds their critical buckling force (order of pn), but is much smaller than the yield force of actin. If the probe diameter is less than or equal to 2 m, experimental data fall outside the region defined by the upper and lower bounds. S Introduction Tensegrity architecture has been proposed as a model of deformability of adherent cells cf. 1. According to this hypothesis, the cytoskeleton CSK is organized as a network of interconnected tension-bearing elements e.g., actin filaments and isolated compression-bearing elements e.g., microtubules in the aim of providing shape stability to the entire cell. It has been shown that under mechanical stresses, adherent cells display features that are distinguished properties of tensegrity structures, most notably that the apparent elastic modulus of the cell increases or decreases in response to increasing or decreasing tension in the actin CSK 2,3. We recently carried out a formal structural analysis of a simple tensegrity model to identify unifying principles that might underlie cellular deformability 4,5. The model consists of six compression elements struts, interconnected with 24 tension elements cables by frictionless pin joints Fig. 1. The cables were assumed to be linearly elastic Hookean whereas the struts were assumed to either be rigid 4 or to buckle under compression 5. The cables and struts represented actin filaments and microtubules, respectively. These studies identified tension in actin filaments, CSK architecture, and buckling of microtubules as key contributors to cellular elasticity. However, a quantitative comparison between model predictions and data from mechanical measurements on cells has not yet been made. In the present study, the six-strut tensegrity model was used as a basis for quantitative predictions of steady-state elastic properties of cells. This mechanistic approach did not consider the effect of CSK filament dynamics on the cell elastic response. An expression for Young s modulus was derived from the model using an equivalent continuum approximation. This expression revealed how forces in CSK filaments and geometry of the CSK might come into play in determining cellular elasticity. By combining this expression with data for mechanical properties of actin filaments and microtubules reported in the literature, we predicted the lower and upper bounds of the Young s modulus. Predictions Contributed by the Bioengineering Division for publication in the JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received by the Bioengineering Division July 14, 1998, revised manuscript received October 5, Associate Technical Editor: V. T. Turitto. were compared to representative data for the Young s modulus obtained previously from mechanical measurements on living adherent cells. Model The simple six-strut tensegrity structure, shown in Fig. 1, was chosen as a model of cell elastic steady-state behavior based on its previous success as a qualitative model of the CSK. The model is subjected to uniaxial extension such that a pair of parallel struts are pulled apart by a force T/2 applied at the end points Fig. 1. The governing equations of the model were derived previously 5 and are given below. Equilibrium equations are the following: s X L BB s X T 2F AB 2F l AC AB l AC F AB s Y l AB F BC L CC s Y L AA s Z s Z F AC F AC P AA F AB L AA l AB F AC L AA s Z l AC L BB s X L BB P BB F AB F AB (1a) (1b) (1c) (1d) (1e) L CC L CC s Y P CC F AC F (1f) AC where F AB, F AC, and F BC are forces in cables AB, AC, and BC, and l AB, l AC, and are the corresponding cable lengths, s X, s Y, and s Z are distances between the pairs of parallel struts along the X, Y, and Z axes, respectively, P AA, P BB, and P CC are forces in struts AA, BB, and CC, and L AA, L BB, and L CC are the corresponding strut lengths Fig. 1. Equations 1a and 1d describe the balance of forces at joint A in the X and Y directions, Journal of Biomechanical Engineering Copyright 2000 by ASME FEBRUARY 2000, Vol. 122 Õ 39

2 Fig. 1 Six-strut tensegrity structure. Struts: AA, BB, CC; cables: AB, AC, BC. Stretching force TÕ2 is applied at the endpoints A. Fig. 2 Young s modulus E 0 versus strut length. Lines: predicted upper bound UB and lower bound LB of E 0 ; dots: data from mechanical measurements on endothelial Endo, epithelial Epi, fibroblast Fibro, and smooth muscle SM cells that are either spread S, round R, or exposed to shear flow F. Eqs. 1b and 1e the balance of forces at B in the Y and Z directions, and Eqs. 1c and 1f the balance of forces at C in the Z and X directions, respectively. From model geometry it follows that: l AB 1 2 L BB s X 2 s 2 2 Y L AA (2) l AC 1 2 s X 2 L 2 CC L AA s Z 2 (3) 1 2 L BB 2 L CC s Y 2 2 s Z At the reference state before the application of T, l AB l AC l 0, L AA L BB L CC, and s X s Y s Z s 0 /2. Taking these into account, it follows from Eqs. 2 4 that l 0 3/8. It is assumed that the cables are linearly elastic Hookean of stiffness k and resting length l r (l r l 0 ) and they support only tensile forces. Thus, the force in a cable F is given as follows: F k l l r if l l r (5) 0 if l l r At the reference state, the cables may carry an initial tension, F 0 k(l 0 l r ). The corresponding compression force in the strut is P 0. From Eqs. 1d 1f, it follows that F 0 1/6P 0. The struts are assumed to be slender and either rigid 4 or elastic 5. In the latter case, the struts were viewed as pin-ended Euler columns that exhibit a post-buckling equilibrium behavior described by a continuous compression force versus strut chord length relationship. The stretching force versus extension T versus s X relationship was obtained from Eqs We used this relationship to obtain the Young s modulus using an equivalent continuum approximation. We assumed that the work of the stretching force T on an incremental extension s X per unit reference volume V 0 of the structure equals the work of uniaxial stress X on an incremental (4) change in uniaxial strain e X of an elastic continuum beam, T s X /V 0 X e X. Here, V 0 5L 3 0 /16 is the volume enclosed by the model at the reference state i.e., the volume defined by the triangular surfaces ABC, ABA, BCB, CAC; Fig. 1 and e X s X /s 0, where s 0 /2. Taking these into account, and defining the Young s modulus as E 0 d X /de X 0, we obtained that E 0 (0.8/ )dt/ds X 0. By substituting the T versus s X relationship into this expression, we obtained the following: E F P (6) where 0 is the initial cable strain, 0 (l 0 l r )/l r. In the case where the struts buckle, dt/ds X 0 is obtained numerically 5. Results The upper and lower bounds of E 0 were obtained using the following criteria. For the upper bound, it was assumed that the initial cable force F 0 corresponds to the yield force of actin filaments whereas for the lower bound, it was assumed that the initial strut force P 0 corresponds to the critical buckling force of microtubules. Estimates of the Upper Bound of the Young s Modulus. Tsuda et al. 6 showed that on the average, the maximum tensile force an isolated actin filament can support before it breaks is F 0,max pn. They estimated the Young s modulus of an actin filament with effective radius r a 2.8 nm to be Y a 1.8 GPa. Using these values we calculated the yield strain 0,max F 0,max /Y a r a 2 and obtained that 0,max 0.9 percent. Substituting the values for F 0,max and 0,max into Eq. 6, we obtained the upper bound of E 0, for ranging from 1 to 6 m Fig. 2. Estimates of the Lower Bound of the Young s Modulus. The lowest critical Euler buckling force of the pin-ended strut is: P 0,cr 2 B m 2 (7) 40 Õ Vol. 122, FEBRUARY 2000 Transactions of the ASME

3 Fig. 3 Critical buckling force of a microtubule P 0,max versus length relationship obtained from Eq. 7 for a bending stiffness B m Ä21.5 pn" m 2 where B m is the bending stiffness of isolated microtubules, B m 21.5 pn m 2 7. By substituting this value into Eq. 7, we obtain estimates of P 0,cr for ranging from 1 to 6 m Fig. 3. The corresponding values of F 0 were obtained from the relationship F 0 1/6P 0. Since values of F 0 were much smaller than the yield value F 0,max 400 pn, the corresponding strains 0 were also much smaller than the yield strain 0,max 0.9 percent and therefore, according to Eq. 6, their contribution to E 0 is negligible. Taking this into account and substituting P 0,cr from Eq. 7 into Eq. 6, we obtained the lower bound of E 0 Fig. 2 as follows: Comparison With Experimental Data E B m 4 (8) The expression for E 0, Eq. 6, which was used as a basis for quantitative estimates of cellular steady-state behavior, contains three independent parameters, F 0 or P 0, 0, and. Two of those F 0, 0 or P 0, 0 were determined from previous measurements of mechanical properties of isolated actin filaments 6 and microtubules 7, and one ( ) was left free. To compare our predictions for the upper and lower bounds of E 0 with reported data for cell elastic moduli obtained by standard mechanical tests micropipette aspiration, cell poking, and magnetic twisting cytometry, it was necessary to choose a characteristic length that could represent. Our choice was based on an ad hoc assumption that dimensions of the model match dimensions of the probes 8, i.e., the model was assumed to describe local deformation of the CSK. Thus, we set the reference strut length to be equal to the probe diameter. Sato et al. 9 performed a series of micropipette aspiration tests on suspended porcine endothelial cells. The cells were grown to confluence and either detached and probed i.e, control, round cells or exposed to prolonged shear flow before being tested. The average inner diameter of the micropipettes was 3 m. It was found that the Young s modulus ranges from 750 dyn/cm 2 in control cells to up to 1575 dyn/cm 2 in cells exposed to shear stress of 20 dyn/cm 2 over 24 h. In comparison with our predictions for the lower and upper bounds of E 0 at 3.0 m, the lowest measured value was by a factor of 4.5 greater than the lower bound, whereas the highest measured value was by a factor of 4 smaller than the upper bound Fig. 2. Petersen et al. 10 used cell poking poker diameter 2 m to probe elastic properties of cultured mouse fibroblasts. From their measurements, we calculated a Young s modulus of 16,000 dyn/cm 2 appendix. This was 9 percent greater than the upper bound predicted for 2.0 m Fig. 2. Magnetic twisting cytometry has been used to probe CSK mechanical properties by twisting small magnetic beads bonded to integrin receptors on the cell apical surface. The apparent shear modulus was measured as a function of applied stress for the following cells in culture: endothelial cells 11,12 smooth muscle cells 2,12, and epithelial cells 13. To compare these data with our model predictions for E 0, we first extrapolated the shear modulus versus applied stress curves to zero stress, to obtain the initial shear modulus. We next assumed that the CSK was isotropic and incompressible Poisson s ratio of 0.5 and obtained the initial Young s modulus 3 shear modulus. From data for round and spread endothelial cells measured with 5.5- m-dia beads, we estimated the Young s modulus of 22 and 45 dyn/cm 2, respectively. These values were at least 50 percent greater than the lower bound and by at least an order of magnitude smaller than the upper bound of E 0 for 5.5 m Fig. 2. From data for spread endothelial cells measured with 1.4- m-dia beads, we estimated the Young s modulus of 40 dyn/cm 2. This was nearly two orders of magnitude smaller than the lower bound for 1.4 m Fig. 2. Following the same steps, we estimated the Young s modulus for smooth muscle cells and epithelial cells. For smooth muscle cells 5.5- m-dia beads, we obtained 115 dyn/cm 2. This was a factor of eight greater than the lower bound and nearly twice that smaller than the upper bound of E 0 for 5.5 m Fig. 2. For subconfluent epithelial cells 4- m-dia beads, the Young s modulus was 75 dyn/cm 2. This was 40 percent greater than the lower bound and by an order of magnitude less than the upper bound for 4.0 m Fig. 2. Discussion In this study we showed that the simple six-strut tensegrity structure that was used previously only as a qualitative model of the CSK 1,4, could also provide quantitative predictions of the steady-state response of cells. A simple, mathematically transparent expression Eq. 6 describing the relationship between macroscopic and microscopic determinants of cell elastic behavior was obtained. Despite consistency between model predictions and data from the literature, model assumptions need to be critically evaluated. Critique. Two key assumptions of this study are the choice of the six-strut model as representative of cell steady-state elastic behavior and the choice of the probe diameter as a characteristic length of the model. These are ad hoc assumptions and have no rationally based justification. Nevertheless, in the absence of quantitative data for CSK microstructural geometry and considering that Young s moduli determined by different techniques in different types of cells fall, in general, within the predicted lower and upper bounds, our assumptions may not be unreasonable. These issues are further discussed below. To investigate the effect of model geometry on predicted E 0, we considered the possibility that a representative microstructural unit is composed of several six-strut units. For example, if these units are connected in series a better description of the aspired portion of the cell in micropipette measurements, E 0 would be the same as for a single unit, providing each unit has the same. However, a more complex topological arrangement of the units could yield a different value of E 0. The constitutive equation of cables, Eq. 5, is consistent with the observed behavior of isolated actin filaments in the sense that the actin tensile stiffness remains constant i.e., the force versus extension relationship is linear over a wide range of applied ten- Journal of Biomechanical Engineering FEBRUARY 2000, Vol. 122 Õ 41

4 sile forces 14. Actin filaments can also support small compression, a feature not included in Eq. 5. This, however, has no effect on the model predictions since during uniaxial stretching of the six-strut model all cables remain under tension 4. At m, the upper and lower bounds of E 0 would coincide. This was obtained from Eq. 7, for P 0,cr 1/6F 0,max, where F 0,max 400 pn. Thus, predictions of our model cannot be compared to data obtained from techniques where probe dimensions approach m. This may explain the huge discrepancy between the data from magnetic twisting cytometry and the model predictions obtained at 1.4 m Fig. 2. If 1.4 m, P 0,cr 108 pn Fig. 3, which may be well above the force that the actin CSK exerts on microtubules. Another concern regarding is the equivalent continuum assumption, i.e., at what length scale does this assumption fail? The pore size of the actin CSK of endothelial cells is 0.1 m 15. Thus, for the continuum assumption to hold, 0.1 m. Since the probes examined here are at least 1.4 m in diameter, this condition seems to be satisfied. The equivalent continuum approximation implicitly assumed material isotropy and incompressibility. On the other hand, the six-strut model is neither isotropic nor incompressible. We estimated the degree of model anisotropy as follows. The model geometry has a cubic symmetry. Assuming that the equivalent continuum has the same material symmetry, we calculated the average over all directions of the upper and lower bounds of E 0, where all directions are equally probable. The average values differ by less than 5 percent from the values predicted from Eq. 6. The volume of the six-strut model changes during uniaxial stretching, although this change is very little for small strains, which justifies the assumption of incompressibility. There are indications, however, that the CSK may be compressible 16. This would imply that Young s moduli calculated from the data from magnetic twisting cytometry and cell poking experiments are overestimates. These overestimates, however, would not exceed 50 percent, assuming Poisson s ratio of 0 for high compressibility instead of 0.5 for incompressibility. There are factors that are not included in our model and that are known to affect cellular deformability. These include the contributions of cytoplasmic viscosity and pressure turgor, intermediate filaments, and stress fibers. Cytoplasmic viscosity, however, does not contribute to steady-state elastic moduli of cells, whereas turgor may balance part of the tension in the actin CSK and thus reduce the compression in microtubules, making the cell less deformable. The role of intermediate filaments as stress-bearing components of the CSK is not well understood. It appears that they carry significant stresses only for strains that exceed 20 percent 17. Thus, their deformability should have little effect on the mechanical response of cells during small deformation. However, intermediate filaments may stabilize microtubules by preventing buckling 18, and thus reduce cell deformability. Based on geometry of actin stress fibers, it is feasible that they support compression and thus assume the same role as microtubules in the scheme of cellular tensegrity. Unlike microtubules, which have been observed to buckle under compression 19, to our knowledge, no such evidence exists for actin stress fibers. Moreover, the data of Satcher et al. 15 show no evidence of stress fibers in the apical region of endothelial cells. Biophysical Implications. Data for the Young s modulus obtained from micropipette aspiration and magnetic twisting cytometry are, in general, closer to the predicted lower bound than to the predicted upper bound of E 0 Fig. 2. This is reasonable to expect since it is unlikely that actin filaments in a living cell carry tension that approaches the yield force of 400 pn and that we used to determine the upper bound. On the other hand, the fact that experimentally determined values of Young s moduli are greater than the lower bound of E 0 suggests that CSK microtubules carry forces that are greater than their critical buckling force Fig. 3. A number of morphological measurements support this assertion. For example, Kaech et al. 19 showed in living transformed epithelial cells that when microtubules push against the actin CSK, they buckle even before any external force is applied to the cell. We previously modeled this situation by considering a six-strut tensegrity model in which the struts buckle 5. The mechanical properties of cables and struts were assigned to match the mechanical properties measured in isolated actin filaments and microtubules, respectively 7. By comparing predictions for E 0 from the buckling strut model with the predicted lower bound of E 0, we found, for a given, the former to be greater than the latter. For example, for a model with initially buckled struts whose resting length is 5 m, E 0 42 dyn/cm 2, corresponding to P 0 18 pn. These values are greater than the lower bound of E 0 of 22 dyn/cm 2 Fig. 2 and corresponding P 0,cr 3.5 pn Fig. 3 obtained for 5 m. Cell poking on fibroblast cells 2- m-diam poker 10 yielded a Young s modulus greater than the upper bound of E 0 16,000 versus 14,660 dyn/cm 2, Fig. 2. This discrepancy is in part due to the contribution of cytoplasmic viscosity to cell short time mechanical response 2.5 s measured during cell poking. Young s moduli determined from micropipette aspiration and magnetic twisting cytometry are measured under steady-state conditions when the contribution of viscous forces is negligible. Another reason could be that the stress applied in cell poking is much larger than either in magnetometry or micropipette aspiration measurements. The assumption that the characteristic length of the model equals the probe diameter has a potentially interesting implication. The range of characteristic lengths of the model ( 1 6 m) is smaller than the persistence length of isolated actin filaments 7, which implies that the actin filaments behave as rods rather than cables. Unlike cables, rodlike filaments are capable of supporting compression, twisting, and bending. This is not unlike the description of biopolymer networks in which long filaments are segmented by frequent entanglements. When the distance between entanglement points is shorter then the persistence length of the filament, the segments appear straight and the network response is determined by stretching and bending of the filament segments 20. If, however, these were the principal modes of deformation of CSK actin filaments, predicted values of E 0 would be by at least an order of magnitude greater than experimentally obtained values 21. This issue is discussed in the following section. Thus, even if CSK actin filaments are rodlike, it appears that during small deformation of the cell their mechanical role is to carry initial tension of the CSK, conferring in that way cell shape stability, i.e., they behave as tensed cables. Comparison With Other Microstructural Models. Satcher and Dewey 21 used a microstructural approach to estimate static elastic moduli of the CSK of endothelial cells quantitatively. These authors assumed that these properties are primarily determined by bending of actin filaments organized as a network of interconnected struts. They predicted that the Young s modulus of the actin network is on the order of 10 5 dyn/cm 2. Taking into account this value and the value for the actin bending stiffness, the model predicts the length of the actin filament to be on the order of 10 1 nm. Both results seem to be unrealistic. First, the predicted value for the Young s modulus of 10 5 dyn/cm 2 is at least an order of magnitude greater than the values obtained from the mechanical measurements in cells considered in this study. Second, the length of actin filaments on the order of 10 1 nm appears too short, equal to the length of several actin monomers, compared to the 10 2 nm pore size of the actin CSK in endothelial cells 15. These discrepancies suggest that during small deformation of the cell, CSK actin filaments do not bend. Recently, Wendling et al. 18 used the six-strut model to study large deformation of cells. They argued that the model can be used to describe the cell elastic response providing that the scale 42 Õ Vol. 122, FEBRUARY 2000 Transactions of the ASME

5 is appropriate to the experimental conditions. Our study showed that this is might not be true for characteristic lengths below 2 m. Summary. The six-strut tensegrity model was used to quantitatively predict the static elastic modulus of the cells. Assuming that the model dimensions were equal to the diameter of various cell mechanical probes, the upper and lower bounds of the cell Young s modulus were predicted. For the probe diameter ranging from 3.0 and 5.5 m, experimentally determined values of Young s modulus fell within the limits predicted by the model and were closer to the lower bound than to the upper bound. If the probe diameter was less or equal to 2 m, experimental data fell outside the predicted range. Since the upper bound was determined by the yield tension of actin 400 pn and the lower bound by the critical buckling force of microtubules ( pn), it was concluded that buckling of microtubles was an important determinant of cellular elasticity, whereas the role of the actin filaments was to carry initial tension conferring load supporting capability to the CSK. Acknowledgments This study was supported by National Heart, Lung, and Blood Institute Grant No. HL Appendix For the cell poking measurements, the Young s modulus E is estimated according to the following equation: W 2ur E 1 2 (A1) where W is the force exerted by the poker on the cell surface, u is the corresponding indentation, r is the poker radius, is Poisson s ratio, and is a function of the ratio of r and cell thickness underneath the poker h, r/h. This analysis is based on the assumption that the cell is an infinite, isotropic, incompressible 0.5 elastic layer bonded to a flat rigid surface and that surface tension effects are negligible. For r h, can be approximated by the following series 22 : O 5 (A2) Taking the values for W/u 0.6 dyn/cm, which is found to be uniform across the cell surface, r 1 m, and h from the nuclear and perinuclear regions between 3.4 and 4.4 m from the cell poking measurements on fibroblast cells 10, 0.5, and using the first five terms from Eq. A2, we obtained from Eq. A1 the mean E 16,000 dyn/cm 2. References 1 Ingber, D. E., 1993, Cellular Tensegrity: Defining New Rules of Biological Design That Govern the Cytoskeleton, J. Cell. Sci., 104, pp Hubmayr, R. D., Shore, S. A., Fredberg, J. J., Planus, E., Panettieri, R. A., Jr., Moller, W., Heyder, J., and Wang, N., 1996, Pharmacological Activation Changes Stiffness of Cultured Human Airway Smooth Muscle Cells, Am. J. 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W., and Gordon, R., 1990, Intermediate Filaments May Prevent Buckling of Compressively Loaded Microtubules, J. Biomech. Eng., 112, pp Kaech, S., Ludin, B., and Matus, A., 1996, Cytoskeletal Plasticity in Cells Expressing Neuronal Microtubule-Associated Proteins, Neuron, 17, pp MacKintosh, F. C., Käs, J., and Janmey, P. A., 1995, Elasticity of Semiflexible Biopolymer Networks, Phys. Rev. Lett., 75, pp Satcher, R. L., Jr., and Dewey, C. F., Jr., 1996, Theoretical Estimates of Mechanical Properties of the Endothelial Cell Cytoskeleton, Biophys. J., 71, pp Duszyk, M., Schwab, B., III, Zahalak, G. I., Qian, H., and Elson, E. L., 1989, Cell Poking: Quantitative Analysis of Indentation of Thick Viscoelastic Layers, Biophys. J., 55, pp Journal of Biomechanical Engineering FEBRUARY 2000, Vol. 122 Õ 43