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VASCULAR BIOLOGY
1Department of Environmental Health, Harvard School of Public Health, Boston; 2Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts; 3Department of Mechanical Engineering, University of Wisconsin-Madison, Madison, Wisconsin; and 4Department of Biological Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts
Submitted 7 November 2006 ; accepted in final form 7 April 2007
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ABSTRACT |
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 TOP ABSTRACT MATERIALS AND METHODSRESULTSDISCUSSIONGRANTSREFERENCES |
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storage modulus; loss modulus; magnetic twisting cytometry; erythrocyte; viscoelasticity; rheology
Knowledge of RBC deformability has come from measurements on populations of cells as well as single-cell tests. Single-cell measurements include micropipette aspiration (MA), optical tweezers (OT), and high-frequency electrical deformation (ED) tests, which apply quasi-static loads and can achieve large deformations (6, 8, 12, 29). These tests have characterized the deformability of the RBC in terms of the shear, bending, and area expansion moduli of the cell membrane, as well as relaxation times. However, little is known about the mechanical properties of the RBC and its associated response to time-dependent loads. Few measurements of the viscoelastic storage and loss moduli of RBCs have been done on cell suspensions (11, 42), but no measurements of these moduli have yet been obtained at the single-cell level.
Here we used optical magnetic twisting cytometry (MTC) to apply direct mechanical loading on the isolated RBC. By applying a torque to a ferrimagnetic microbead tightly bound to the cell surface, the loading was manipulated in a precisely controlled manner. Using this approach, we studied responses to step and sinusoidal loadings. From these measurements, we determined elastic (storage) and frictional (loss) moduli of the single, isolated RBC over wide ranges of forcing time scale and forcing amplitude. We found unexpected dependencies of storage and loss moduli upon frequency that are inconsistent with previous models incorporating discrete relaxation times. Moreover, a finite element (FE) model of static deformations suggests that, in the experiments, membrane bending dominated membrane shear. While mechanisms accounting for RBC dynamics remain unclear, methods described here open new windows into the study of connections among the mechanical, chemical, and biological characteristics of the RBC in health and in disease.
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MATERIALS AND METHODS |
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TOPABSTRACTMATERIALS AND METHODSRESULTSDISCUSSIONGRANTSREFERENCES |
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RBC isolation.
RBCs were obtained by drawing 
150 µl of blood from healthy donors by a fingertip needle prick. The blood was diluted in 600 µl of PBS (pH = 7.4) and then washed three times by centrifugation to isolate the RBCs. As the buffer has a physiological osmolarity ( 270 mosM), the RBCs retain their natural flat, biconcave, disk-like shape.
Experimental protocols. Clean, 35-mm glass bottom wells (MatTek, Ashland, MA) were treated with 0.1 mg/ml poly-L-lysine for 1 h at room temperature (T = 22°C). At this concentration, the RBCs adhere firmly to the wells, but retain their biconcave shape (27). RBCs were allowed to adhere for 10 min to the wells, before the beads were added.
Ferrimagnetic microbeads and binding.
Ferrimagnetic beads (solid Fe3O4, 2.27 µm ± 11.3% diameter) were produced as described previously (18). Beads (1 mg) were coated with ConA by incubating overnight at 4°C in ConA solution (1 mg/ml). ConA is not blood group specific but has an affinity for terminal 
-D-mannosyl and -D-glucosyl residues. A wide variety of serum and membrane glycoproteins have a "core oligosaccharide" structure, which includes -linked mannose residues, such as band 3. Hence, ConA-coated beads are firmly attached to the spectrin network directly due to the binding through transmembrane glycoproteins and indirectly due to the coupling of the lipid bilayer to the spectrin (37).
Coated beads were rinsed several times in PBS and then added to the RBC preparation in the well. The volume of beads added was controlled such that there was approximately one bead per RBC. The preparation was incubated with the beads for 10 min, allowing adhesion of microbeads to the RBC membrane. Wells were then washed twice with PBS to remove unbound beads and cells. All measurements were made on beads located near the center of the cell (Fig. 1) and were at room temperature.
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MTC with optical detection. The experimental setup for MTC with optical detection has been described in detail elsewhere (18, 19). Briefly, after the cells were adhered on the bottom of a glass well and the beads attached, the well was placed on the stage of an inverted microscope (Leica DM IRBE, Leica Microsystems, Weitzler, Germany) and viewed under bright field with an oil immersion x63 objective and optically magnified x1.5. Beads were first magnetized horizontally and then subjected to a vertical magnetic field (Fig. 2). This magnetic field induces a specific (mechanical) torque, Ts(t), on the bead.
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 | (1) |
is the angle of the bead's magnetic moment relative to the original magnetization direction, and c is the bead constant, expressed as torque per unit bead volume per Gauss. The bead constant is determined by placing beads in a fluid of known viscosity and then measuring the angular velocity that results from an applied magnetic field (48). The bead constant in this study was c = 0.225 Pa/G. As was small, the cos term in Eq. 1 was assumed to be unity when computing the specific torque from the applied magnetic twisting field. This results in errors of <5% for small bead rotations ( < 18°) or small lateral bead displacements (d < 450 nm) in the case of a pivoting motion of the bead around a contact point at the bead perimeter. The magnetic twisting field creates a torque that acts upon the adherent bead and causes both a rotation and a lateral translation (Fig. 2). The magnitude of the rotation and translation is a function of the magnitude of the applied torque, the geometry of the cell and the bead attachment, and the mechanical properties of the cell. Both rotation and translation result from the loading, because the bead is only attached near its base, and hence the torque applied to the bead is transferred to the cell as a bending moment supported by both normal and tangential tractions. The bead motions were tracked through images captured by a charge-coupled device camera (JAI CV-M10, Gloatrup, Denmark) with an exposure time of 0.1 ms and acquisition frequency of 24 Hz. From recorded images, the bead position d(t) was computed using an intensity-weighted center-of-mass algorithm that provides an accuracy in d(t) of better than 5 nm (19).
Mechanical tests. Two types of forcing were used: sinusoidal and step functions. Sinusoidal forcing spanned 10–1 to 102 Hz and amplitudes from 0.5 to 4 G. As described previously (19), heterodyning was used to sample the displacement response in tests where the sinusoidal forcing frequency was >4 Hz. Step functions had amplitudes between 1 and 4 G. The displacement response to the step functions was sampled at intervals of 50 ms.
Measurement of the complex elastic modulus.
Two methods were used to extract the storage and loss moduli from the magnetic torque Ts(t) and corresponding lateral bead motion d(t). In the limit that the system becomes linear, these methods reduce to the same standard definitions for a linear viscoelastic system. The first method follows the approach of Fredberg and Stamenovic (21) for sinusoidal analysis of a nonlinear system and defines the apparent complex elastic moduli g* as
 | (2) |
, loss tangent 
, stiffness g', and loss modulus g''' then become:
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 | (4) |
 | (5) |
is the radian frequency. The second method used to extract the storage and loss modulus assumes the system is linear. The components of the complex elastic modulus are computed from the amplitudes of the applied torque, 
Ts/2, and displacement response, d/2, and the phase difference between the two signals . The values of Ts, d, and were determined by fitting
 | (6) |
 | (7) |
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Computation of creep from measurements of the complex elastic modulus.
We measured the step response directly as described above, and also computed the step response from the Laplace transform of the response measured in the frequency domain. Let J(t) be the creep response for unit step loading, and let 
(s) be its Laplace transform in variable s. Let 
(s) be the complex elastic modulus at complex frequency s = i. The creep response and the complex modulus are related by
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(i) are described reasonably well by a power law in frequency with a constant offset. To satisfy the Kramers-Kronig relationship (3), this experimental observation implies that, in Laplace space, we have
 | (9) |
 | (10) |
0), a Taylor expansion about s = 
and subsequent inversion suffices to yield
 | (11) |
(.) is the gamma function. This is a convergent power series for all t, but is useful only for small t, because the number of terms necessary for a good approximation rises extremely rapidly for large times. For long times, we perform a Taylor expansion about s = 0. The two leading terms and their inversion yield the asymptotic behavior of the creep response,
 | (12) |
0.024 s. FE modeling. To approximate the deformation field induced in the RBC by magnetic bead twisting, we developed a FE model. Based on the geometry of the RBC (13), the cell membrane was meshed with 24,000 four-noded, bilinear, reduced-integration shell elements; its displacement was constrained such that the interior cell volume was constant. The membrane constitutive law was based on a strain energy potential used previously to model finite deformations of the RBC (8); in its application here, where the strains are small, this is essentially equivalent to a linear elastic two-dimensional (2D) membrane with given shear and bending (B) moduli. The boundary conditions imposed were as follows. The bottom surface of the RBC (interior to the lowermost circle of symmetry) was held fixed. The displacement of nodes adherent to the bead (bead nodes) were constrained to move in rigid translation or rotation with the bead, which was modeled as a rigid solid. The tractions on all other nodes were set to zero. Although the net force applied to the bead was zero, a nonzero torque resulted in lateral (in-plane) displacement, with an apparent stiffness given by the ratio of the applied torque to the lateral displacement. A parametric study was performed to examine the effects of bending and shear moduli, as well as the degree of binding between the bead and the cell on the overall torque-displacement relationship measured in the experiments.
Technical limitations. Optical MTC is a tool that brings with it important technical advantages, such as the ability to probe the single cell with very small deformations and over wide ranges of time scale and amplitude. However, these advantages bring with them other technical limitations that can contribute to measurement variability. For example, the contact area between the cell and the bead, as well as the location at which the bead is attached to the cell, are not controlled and therefore vary from cell to cell. These shortcomings are mitigated to a large degree by the ability to measure a large number of cells and the ability to select cells in which the bead is centered. We found no significant correlation between bead location and stiffness, or cell size and stiffness. Moreover, the overall cell-to-cell variability was quite small.
Similarly, the FE analysis had several technical limitations. The nature of RBC viscoelasticity that we determined experimentally does not conform to any simple or known viscoelastic behavior and, as such, cannot be modeled from first principles. As such, we limited the FE analysis here to the purely elastic (i.e., static) case, but this is not a severe limitation, as it was observed that, over most of the experimental range, the loss modulus was much smaller than the storage modulus. We reasoned, therefore, that this model should provide 1) a reasonable approximation of the deformation field induced by magnetic twisting, and specifically the extent to which the deformation is localized to the neighborhood of the bead; and 2) a basis of comparison to previous static measurements of RBC deformability.
A second limitation is found in the specification of the membrane elastic properties. In the dimensional reduction of the real membrane to a 2D equivalent, we began with the assumption of isotropy and incompressibility. Such a material has only one elastic modulus, and, for a given thickness, the 2D equivalent shear modulus, area expansion modulus, and bending modulus are not independent but are fully determined. For material properties that approximate the 2D shear and bending moduli, this turns out to underestimate the area expansion modulus, and hence the apparent stiffness. However, since the response is 1) significantly dominated by bending, as opposed to surface strain, and 2) the area dilation is small compared with surface shear, this error is thought to be negligible.
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RESULTS |
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TOPABSTRACTMATERIALS AND METHODSRESULTSDISCUSSIONGRANTSREFERENCES |
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fa), with the exponent a = 0.64, fit g''' data reasonably well.
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2.5 times the contact radius, thus showing that elastic strain energy density is modestly localized to the neighborhood of the bead.
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DISCUSSION |
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TOPABSTRACTMATERIALS AND METHODSRESULTSDISCUSSIONGRANTSREFERENCES |
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For smaller torque amplitudes, the torque-displacement loops were nearly elliptical, and harmonics were small (Fig. 6), but, for larger applied torques, nonelliptical loops were observed, and harmonics became more pronounced (Fig. 6). Such behavior represents evidence of small strain-hardening nonlinear responses. Potential factors leading to these nonlinearities are, first, geometrical. The torque applied to the bead varies as the cosine of the angle between the magnetization and the applied field (Eq. 1); if the bead pivoted around a contact point at the bead perimeter, this effect would result in errors of no more than 10% for large bead rotations ( < 26°) or lateral bead displacements (d < 570 nm). Another geometrical factor that may contribute is the large deformations of the thin shell structure of the cell. The deformation of the shell structure is complex (Fig. 8), and one would expect that, when the deformations become large, the change in geometry will result in a nonlinear relationship between the applied load and displacement. Other possible factors for the observed nonlinear behavior are related to the intrinsic material properties of the cell membrane itself. In this context, OT tests have demonstrated that the effective shear modulus of the red cell membrane increases dramatically at large deformations (8, 36). In the studies reported here, we saw a linear range followed by slight strain hardening, but no evidence of strain softening or yielding. In the remainder of the DISCUSSION, we restrict attention to experiments in which deformations were small and the response was linear.
Direct measurement of the creep function (Fig. 7, black data points) compared favorably with that computed from direct measurements of the complex modulus (Fig. 7, gray solid line), suggesting that, when the torque amplitude is small, a linear response function in either the time or Fourier domain is sufficient to characterize the rheological properties of the RBC. Moreover, by transforming data from the Fourier domain (Fig. 5) into the time domain (Fig. 7, gray solid line), we were able to compare our observations to step responses of the RBC reported previously in the literature (Fig. 7). Compared with previous measurements, our direct measurements were close to those reported by Hochmuth et al. (31) (Fig. 7, light gray circles), but somewhat faster than those reported by the others. One possible explanation for these differences may be associated with differences in loading. By the nature of our bead twisting protocol, the applied torque and induced rotation vectors lie in the plane of the apical surface of the cell, which, in turn, implies that the loading is supported by both shear and bending, but predominantly by bending of the membrane. To the extent that simple bending is associated with less dissipation than loadings involving isotropic membrane tension and associated 2D bulk dilation, or normal stress gradients associated with bulk three-dimensional transport of intracellular material, responses might be more rapid. The idea that bending stiffness is important in this context is supported by the observation of a parallel elastic behavior as seen especially in the low-frequency regime in Fig. 5 (and as modeled by Eq. 9) and is supported by the FE simulations, which correspond to low-frequency regime.
Creep and relaxation responses, such as those shown in Fig. 7, have been conventionally modeled as a function of time with an exponential decay (14–16). That is to say, all previous models of RBC dynamics are based on the assumption that dynamic responses can be characterized by one or a few characteristic relaxation times. While some studies [e.g., Dao et al. (8), diamond points in Fig. 7] report good agreement with an exponential behavior, virtually none of these have followed the response over a time course that is sufficiently long to distinguish between exponential behavior vs. other possibilities [e.g., Hochmuth et al. (31) (Fig. 7, gray circles) and Henon et al. (28) (Fig. 7, black circle with gray center)]. Nonetheless, all of these studies used single exponentials to characterize the first 300–400 ms and, for MA and ED, and OT experiments, respectively, have suggested time constants of 0.10–0.13 s and 0.13–0.25 s (12, 36). By contrast, data reported here firmly establish that, in the case of MTC, at least, the dynamic response of the RBC exhibits no distinct time constants. Rather, the dynamic response follows a power law decay. We cannot rule out that MTC yields power law response, whereas MA, OT, and ED tests probe different facets of RBC dynamics and yield responses that conform to an exponential decay. Alternatively, we propose here that relaxation times estimated previously from data, and any models that rest upon them, are artifactual. This is not to say that the estimated value of a time constant is inaccurate, but rather to say that, in such a system, the very concept of a time constant is meaningless (49). The artifact, we suggest, arises from forcing an exponential fit to data of limited temporal duration that, in actuality, defines a power law response (49).
In connection with power law responses, the frequency dependence of the elastic and frictional moduli (Figs. 5 and 6) of the RBC differed dramatically from that of spread adherent somatic cells, such as smooth muscle cells, fibroblasts, and epithelial cells, that have been measured using the same technique (19, 32, 40) and other techniques (1, 9, 10, 43). In a variety of adherent cell types, measurements of g' and g''' vs. frequency typically exhibit weak power law behaviors, with a viscous Newtonian contribution arising only at very high frequencies. By contrast, g' of the RBC was nearly independent of frequency up to 30 Hz (Fig. 6), and g''' increased with frequency as a power law throughout the measurement range, but the exponent (0.64) was much greater than that seen in somatic cells, but systematically weaker than expected for a Newtonian fluid. The frequency dependence of g' and g''' observed in RBCs (Figs. 5 and 6) also differed from that measured in in vitro systems, such as cross-linked actin gels (23, 46, 50).
What factors might account for the frequency dependence of g' and g'''? The RBC membrane has an outer phospholipid bilayer attached to an inner 2D cytoskeleton network (2, 24, 25, 34). The links of the network consist of flexible spectrin molecules (R 80–100 nm), cross-linked at the network nodes by a complex containing a short (30 nm) actin filament, band-4.1, and other proteins (2, 45). The spectrin cytoskeleton of RBC is slightly stretched and under tension, even in the un-deformed state (20, 51). We speculate that, on the one hand, the 2D spectrin prestressed network is probably responsible for the elastic-like response and the apparent storage moduli being insensitive to frequency. On the other hand, the origin of the power law behavior of g''' is uncertain. However, the membrane is thought to be the dominant source of viscous dissipation, because the membrane viscous dissipation is two orders of magnitude greater than that in the internal hemoglobin solution (14, 30). Moreover, hemoglobin solutions behave like Newtonian fluids (7), and so the RBC intracellular milieu is unlikely to yield the non-Newtonian behavior we find in g'''.
Interlayer coupling of the lipid bilayer (17), including the potential effects of lipid "flip-flop" (41), has been proposed as a candidate for viscous loss mechanisms, especially in circumstances where membrane curvature is rapidly changing. It is unlikely that the global effect of transmission of cumulative tension and associated area expansion is responsible, however, since our bead movements are more localized compared with, for example, vesicle tethering phenomena. Nevertheless, the interlayer coupling of the bilayer immediately bound to the surface of the bead in our experiments remains a potential source of dissipation, despite uncertainty regarding its frequency-dependent behavior.
Lipid bilayer fluctuations driven by thermal forces are also a potential loss mechanism (25, 52). On the one hand, Gov's theoretical analyses are based on a model that replaces the spectrin network with a rigid shell separated from the bilayer, but with an induced effective surface tension and harmonic potential. As our beads are firmly attached to the spectrin network, it is unclear that bilayer fluctuations taken over a rigid cytoskeleton, even with equivalent bilayer mechanics, will capture the features of the dissipation we observe. On the other hand, all approaches that relate spontaneous fluctuation phenomena directly to dissipation seen in driven motion rely on some form of the fluctuation-dissipation theorem (FDT). FDT is strictly valid only in systems at thermodynamic equilibrium and is known to be strongly violated in other cell types (4).
To our knowledge, length scales and time scales over which FDT might hold for the RBC are unknown. If FDT does hold, then the anomalous frequency dependence of f–5/3 predicted by Gov et al. (25) for the mean square normal displacement in Fourier space would translate into f2/3 for the frequency dependence of g''', which is close to the behavior that we observe (Fig. 5). This coincidence of predicted vs. measured exponents points to the importance of future studies to examine systematically the prefactor, which may then elucidate whether spontaneous fluctuations of the lipid bilayer may be linked to the dissipation observe for driven deformations measured with beads firmly connected to the spectrin network.
Spectrin network fluctuations driven by metabolic and thermal sources (26, 33, 47) can also be a candidate for viscous loss mechanism, although the sources are found to be important only at low frequencies (44). At high frequencies, it has been shown that fluctuations in semiflexible polymers driven by thermal forces leads to viscoelasticity that scales with frequency with a universal exponent of 3/4 (9, 22, 38), a behavior that is not observed in experiments reported here. Nevertheless, interlayer coupling, bilayer fluctuations, and spectrin fluctuations remain candidates and potential mechanisms that can lead to the observed viscous dissipation.
Conclusions. Here we have reported elastic and frictional moduli of the isolated RBC over a wide range of forcing time scale and forcing amplitude. Except at very high frequencies, elastic stresses dominated and were independent of frequency. Frictional stresses depended on frequency but not as predicted by any simple model. This frictional behavior was non-Newtonian and suggests origins different than surface viscosity of lipid membrane components or the viscous hemoglobin solution (14, 15, 30). It is unclear if the frequency dependence of g''' is related to membrane molecular fluctuations or metabolic activity controlled by intracellular concentrations of ATP (25, 26, 47). Further exploration of these processes could shed light on the underlying viscoelastic deformation characteristics of the RBC. Most important, the data reported here indicate that a power-law fluid taken in parallel with a Hookean stiffness is an appropriate phenomenological model to describe dynamic responses of the RBC in the linear range.
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ACKNOWLEDGMENTS |
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FOOTNOTES |
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The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
* M. Puig-de-Morales-Marinkovic and K. T. Turner contributed equally to this work. 
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/2, hence Ts(t) is proportional to the magnetic field [Ts(t) = c·H]. B: stress in phase g' (real) and out of phase g''' (imaginary) with bead displacement, expressed in complex plane. 
