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Research Papers

# Axisymmetric Optical-Trap Measurement of Red Blood Cell Membrane Elasticity

[+] Author and Article Information
Alexandre Lewalle1

Randall Division of Cell and Molecular Biophysics, King’s College London, New Hunt’s House, Guy’s Campus, London SE1 1UL, UK

Kim H. Parker

Department of Bioengineering, Imperial College London, London SW7 2AZ, UK

1

Corresponding author. Present address: London Centre for Nanotechnology, University College London, London WC1H 0AH, UK.

J Biomech Eng 133(1), 011007 (Dec 23, 2010) (9 pages) doi:10.1115/1.4003127 History: Received March 04, 2010; Revised November 15, 2010; Posted November 29, 2010; Published December 23, 2010; Online December 23, 2010

## Abstract

The elastic properties of the cell membrane play a crucial role in determining the equilibrium shape of the cell, as well as its response to the external forces it experiences in its physiological environment. Red blood cells are a favored system for studying membrane properties because of their simple structure: a lipid bilayer coupled to a membrane cytoskeleton and no cytoplasmic cytoskeleton. An optical trap is used to stretch a red blood cell, fixed to a glass surface, along its symmetry axis by pulling on a micron-sized latex bead that is bound at the center of the exposed cell dimple. The system, at equilibrium, shows Hookean behavior with a spring constant of $1.5×10−6 N/m$ over a $1–2 μm$ range of extension. This choice of simple experimental geometry preserves the axial symmetry of the native cell throughout the stretch, probes membrane deformations in the small-extension regime, and facilitates theoretical analysis. The axisymmetry makes the experiment amenable to simulation using a simple model that makes no a priori assumption on the relative importance of shear and bending in membrane deformations. We use an iterative relaxation algorithm to solve for the geometrical configuration of the membrane at mechanical equilibrium for a range of applied forces. We obtain estimates for the out-of-plane membrane bending modulus $B≈1×10−19 Nm$ and an upper limit to the in-plane shear modulus $H<2×10−6 N/m$. The partial agreement of these results with other published values may serve to highlight the dependence of the cell’s resistance to deformation on the scale and geometry of the deformation.

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## Figures

Figure 1

Summary of published experimental results for B and H. Estimates given by Eqs. 13,14 are compared with experiments based on thermal motion (3-4,6,5,27,7), micropipette aspiration (2,20), atomic force pulling (21), optical-tweezers pulling (10-11), and deformation induced by high-frequency electric fields (28). The result of the present experiment (B=(1.1±0.9)×10−19 N m, H<∼1×10−6 N/m) is shown for comparison.

Figure 2

Experiment concept. (a) A red blood cell settles and becomes fixed to the glass coverslip. (b) The cell is positioned under a bead in the optical trap. The bead becomes bound to the membrane by biotin-avidin interactions. The radiation force pushes the lower dimple downward onto the glass, as discussed in the text. The inset shows the bright-field image of the bead, captured by camera C2 (Fig. 4). (c) Displacing the coverslip by a distance zstage results in a displacement zbead of the bead and hence in an external force F applied locally on the cell membrane. The change in appearance of the bead image (inset) provides a means to measure zbead (see Sec. 2). (d) The “linear spring” model for the forces acting on the bead, with spring constants kcell and ktrap.

Figure 3

Red blood cell bound to the cover slip, as imaged by high-resolution camera C2 under (a) broad-band bright-field illumination and (b) narrow-band 415 nm illumination (the absorption peak of hemoglobin). The inset plot is a semiquantitative estimate of the cell thickness profile (path length through hemoglobin), measured along the diameter of the image and assuming Beer’s law. ((c) and (d)) The configuration of the lower cell membrane is imaged by reflection interference contrast microscopy, with the trapping laser switched either (c) off or (d) on. The disappearance of the interference fringes in (d) signifies that the lower dimple of the cell collapses onto the glass surface.

Figure 4

Optical setup. The dotted line indicates the path of the trapping laser (Nd:YAG 1064 nm), introduced into the objective via dichroic mirror M1. The dashed line shows the path of the xenon arc lamp illumination for bright-field imaging by cameras C1 (low resolution) and C2 (high resolution). BS is a 10–90% beam splitter and F2 a YAG rejection filter. Small displacements of lens L2 of the magnifying telescope shift the position of the imaging plane at the sample without significantly altering the magnification. Mirror M2 may be removed to allow the imaging of the condenser back-focal plane onto the photodiode detector, as required for measuring the stiffness of the optical trap in the axial direction (see text).

Figure 5

(a) The response of Sz (the ratio of the intensity over a square of area (0.1 μm)2 at the center of the bead image to the background intensity) to the piezo-controlled displacement of the microscope stage, with the trapping laser either off or on. As shown in the series of snapshots along the vertical axis, the intensity profile of the bead image varies as a function of the position of the bead relative to the focal plane of the trapping objective. This dependence is exploited to provide a measure of the bead position. In the “trap off” case, the cell is not deformed and the bead follows the stage exactly. The resulting curve Szcal is used to calibrate the bead position, zbead. The optical trap exerts a restraining force on the bead, resulting in a smaller gradient, dSz/dzstage. The intersection of the two curves marks the point at which the laser exerts no net force on the cell. (b) The force-extension curve F(ϵ) corresponding to the bead displacement data in (a), showing Hookean behavior (stiffness kcell=dF/dϵ=1.7×10−6 N/m) when the cell is stretched (ϵ>0) and a diverging stiffness as the cell is compressed (ϵ<0), most likely due to the nonlinearity of the optical trap (see Sec. 3). The inset shows a histogram of kcell values measured for an ensemble of 74 cells (mean kcell=1.5×10−6 N/m, SD=1.4×10−6 N/m).

Figure 6

Force-extension characteristics of the optical trap, measured in an agarose gel (see text). (a) Bead displacement as a function of stage displacement, for a bead embedded in the soft agarose gel in the absence of a cell. At the origin (zbead=zstage=0), the gradient and scattering forces cancel out. The different shades of gray of the curves correspond to different positions of the imaging focal plane (see text). (b) The force-extension curve for the trap, deduced from the data in (a). The histogram inset and the vertical gray band indicate the offset in the zero-force point (measured via Sz) caused by the membrane force Fmem. The horizontal gray band shows the range of Fmem estimated from laser intensity measurements. (c) Comparison of an experimental cell force-extension curve (solid black line) with a simulated curves (dashed line), based on the trap result in (b), assuming a perfectly Hookean membrane force, for Fmem=6.5 pN.

Figure 7

Simulated nondimensional force-extension curves g(C) for a standard cell for C=0, 10, 50, 100, 200, 500, and 1000. The initial (unstrained) configuration is that of the standard cell, where the ratio of the thinnest to the thickest regions is 0.31. The inset shows the calculated profile of the cell for the range of deformations spanning ϵ=−0.9 μm and 1.2 μm.

Figure 8

(a) Nondimensional force-extension curves g(C) for C=0, 10, 50, 100, 200, 500, and 1000, for a range of different initial membrane configurations, indicated by the cell profiles above, with thickness ratios of 0.15, 0.31, 0.46, and 0.62 (b) The dimensionless gradient g(C)≡dF/dϵ×a2/B, measured at ϵ=0 for the curves in (a). The symbols identify the initial membrane profile shown above (a). (c) The “map” of H versus B, generated by equating the gradient values in (b) with the experimentally measured average gradient kcell=1.5×10−6 N/m. The straight line passing through the origin represents C=100.

Figure 9

Sketch of the deformed cell defining the coordinates, forces, and moments acting on the membrane. The cell is axisymmetric about the Z axis with radius R and circumferential angle θ. The distance along the meridian is S. ϕ is the angle of the local normal to the membrane relative to the axis of symmetry. P is the transmural pressure acting normal to the membrane, N is the in-plane stress resultant, Q is the out-of-plane shear stress resultant, and M is the bending moment. For clarity, only the meridional components of N and M are shown.

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