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The stretching stiffness of Red Blood Cells (RBCs) was investigated using a combination of an AC dielectrophoretic apparatus and a single-beam optical tweezer. The experiments were performed at 10 MHz, a frequency high enough to avoid conductivity losses, but below the second turnover point between positive and negative dielectrophoresis. By measuring the geometrical parameters of single healthy human RBCs as a function of the applied voltage, the elastic modulus of RBCs was determined (µ = 1.80 ± 0.5 µN/m) and compared with similar values of the literature got by other techniques. The method is expected to be an easy-to-use, alternative tool to determine the mechano-elastic properties of living cells, and, on this basis, to distinguish healthy and diseased cells
© 2014 Optical Society of America
1. Introduction
Measuring the elastic properties of (human) cells has recently got to the forefront of biomedical research [1, 2]. The deformability of cells is sensitively matched to their physiological role and plays crucially important roles to maintain their function, via e.g. mechanochemical coupling mechanisms [3]. Changes in the elastic properties of the cells are often related to malfunctions; therefore they may be used as a cell marker and a diagnostic parameter for the underlying diseases.
Perhaps the most extensively investigated cells in this respect are human red blood cells (RBCs, called also erythrocytes). RBCs must normally be rather elastic, as they are subjected to large deformations during their travel through the microvasculature [4]. A contiguous network of protein molecules (called spectrins) connected to the phospholipid membrane of RBCs is considered to be the main determinant of their elastic properties [4]. The membrane elasticity of RBCs depends on physical and chemical parameters (e.g., temperature [5] and pH [6]) of the medium [7], and on physiological conditions like e.g. aging [8]. The mechanical properties of RBCs have been investigated by various methods, such as micropipette aspiration [9], laser diffractoscopy [10], filtration-based techniques [11], micro-channel methods [12] and atomic force microscopy [8]. However, most of these techniques face serious practical problems. For instance, a mechanical contact of the probe leads to adhesion and active cellular response, while the requirement of special preparation and nonphysiological handling leads to possible experimental artefacts [13].
Hence, we performed elasticity measurements on single RBCs by a combination of laser tweezers and dielectrophoretic (DEP) force techniques. A special advantage of this approach is the lack of mechanical contact with the cells. The movement of cells in a non-uniform electric field is called dielectrophoresis (DEP), which is proportional to the dipole moment and the gradient of the electric field [14]. If the cell moves towards higher field strengths, it is referred to as positive DEP (pDEP), and away from it is referred to as negative DEP (nDEP). In the usual case, when the permittivity of the cell is higher than that of the medium, the induced electric dipole moment vector of the dielectric cell is aligned parallel with the electric field, and the cell is attracted towards higher field strengths (pDEP). If the external field varies linearly over the characteristic spherical cell dimensions, the polarization of a dielectric sphere in an electric field can be represented by replacing the cell with an equivalent effective dipole located at the center of the cell. At lower frequencies (less than 1kHz), DEP behaviour is influenced by the cell surface charge [15] but at higher frequencies (between ca. 100 kHz and 10 MHz, depending also on the cell size), the polarizability is dominated by the dielectric constants of the cell interior, the cell membrane and the outer medium, leading to a pDEP. At even higher frequencies, usually, there is a second turnover point above which again an nDEP is observed [16]. When a prolate, oblate or triaxial ellipsoidal cell is polarized in a (nonuniform) electric field, the longest symmetry axis will be aligned parallel to the field lines, so DEP can also be used to rotate a particle, or trap it as well. Similarly to DEP, optical tweezers can also be used to trap particles via balancing of gradient and scattering forces. By optical tweezers, it is possible to position a cell close to the electrode (or anywhere between the electrodes).
Our experiments were performed on red blood cells trapped in a laser tweezer, and subjected to a pDEP by applying an alternating electric field (10 MHz) via a pair of triangular electrodes. From the deformation of RBCs, we give a quantitative measure of their elasticity, and discuss the general implications of the results.
2. Materials and methods
2.1Theoretical background of the DEP force stretching the red blood cells
When a RBC is situated in a low-conductivity PBS solution in an inhomogeneous electric field of amplitude E, the cell will be deformed by a DEP force:
Where υ = (1/6) πab2, a and b are the major and minor axes of red blood cells. The bracketed expression in Eq. (1) is called the Clausius-Mossotti factor (CM), containing the complex permittivities of the cell body and the phosphate buffer solution (ε*RBCs and ε*PBS, respectively). In the 100 kHz-10MHz regime, most cells and molecules show positive DEP in low-conductivity media [16–18]. The exact value of the dielectrophoretic force depends on the position of the particle, and unless the electrode shape is particularly simple, it does not have an analytical representation. The geometry of the electrodes also has a significant effect on the DEP force and, hence, the behavior of the particles.By optical tweezers, it is possible to trap RBCs at the desired position. The gradient force in the case of optical tweezers resembles the dielectrophoretic force, and the derivative of the light intensity with respect to the distance from the optical axis in the focal plane represents the gradient the trapping force is proportional with:
where ω is the radius of the beam waist, at which the light intensity drops to 1/e of its axial value, r is the radial position measured from the centre of the trap, and Kr is a proportionality constant, depending amongst others, on the relative refractive index to that of the medium (nr). In the case of red blood cells, where nr > 1, the gradient force keeps the particle in the centre of the laser focus.When cells are trapped by the combination of dielectrophoretic and laser-tweezer forces, the two gradient forces keep balance with each other, and deform the trapped RBCs. Hence, the elastic shear modulus (µ) of the cell membrane can be calculated according to Eq. (3)
where b is the minor axis of the prolate, if the membrane area expansion modulus is considerably higher, as usual [19], therefore its contribution is assumed to be neglected.2.2 Computer modelling of the electric field
The simulations were done in the COMSOL Multiphysics 4.3a programming environment. The liquid volume over the electrodes was considered to be 200µm x 50µm x 20µm for the calculations. Two equilateral triangular electrodes of 100 µm side and 1µm height (thickness) were defined, and the tips were separated by 100 µm. They were considered as pure chromium and the data of built-in material database of COMSOL was used for both the chromium and the surrounding water solution. A predefined tetrahedral mesh was used to set extra fine size, where the maximum element size was 10.5µm, and the minimum element size was 0.45µm, with a maximum growth rate of 1.35. For the calculations, the electrostatics module was used with the stationary Maxwell`s equations. The electric field strength was calculated in 1 µm steps in the X and Y directions and in 0.5µm steps in the Z direction, then the values were exported in ASCII format. The gradient of the field strength squared was calculated in the centre of the RBC (x = 15 µm, y = 0, z = 5 µm) by the MATLAB scientific programming language, v7.7.
2.3 Experimental analysis of red blood cells
The DEP chamber consisted of a pair of electrodes (1 µm thickness and 100 µm gap in between) laid down on glass by a sputtering technique. The electrodes used in this experiment were of triangular shape and the space between them was filled with electrolyte. The gap between two electrodes was 100 µm and the length of each side of the triangular electrodes was also 100 µm. The voltage on the electrodes was applied by a function generator (33250A, Agilent, USA), and monitored by an oscilloscope (9344, LeCroy, France). We used a sinusoidal alternating electric field with peak voltages from 2 V to 8 V, and a frequency of 10 MHz. The sample was put in an inverted microscope (AxioVert S100, Zeiss, Jena, Germany) equipped with a laser tweezer (ThorLabs, laser wavelength 945nm) and a camera (AxioCam HRc, Zeiss). Laser light with an optical axis perpendicular to the substrate was focussed in between the electrodes forming an optical tweezers to trap RBCs.
The erythrocytes were collected from healthy donors and washed three times by isotonic and nontoxic solution, phosphate buffered saline (PBS). The whole blood with PBS was diluted 1: 1000 vol. /vol. in a hypotonic medium of 10% mannitol solution. Under these conditions, the red blood cells do not aggregate; swell to spheroids, and haemolyse within 2.45 h. The average diameter of the RBCs in this medium was ca. 6.0 µm. The average volume of RBCs was 111.35 fL with a surface area of about 112 µm2. The conductivity of this solution was 17,000 µmhos. One drop of the diluted RBCs was poured on the surface between the electrodes and covered by a cover glass. First, we trapped individual RBCs by optical tweezers in the vicinity of one of the electrode tips (the centre of the cell was at 15 µm from the tip and 5µm from the bottom of the cuvettes) and then gradually applied voltage across the electrodes by a function generator. The shape of the individual RBCs was monitored visually by the microscope.
3. Results and discussion
After trapping RBCs by optical tweezers between the electrodes, we applied voltages from 2 to 8 V. Due to the triangular shape of the electrodes, the electric field was strongly inhomogeneous and exerted a DEP force on the cells that was compensated by the trapping force of the laser tweezers. At the same time, the shape of RBCs changed from spherical to prolate-like. Under such conditions, the induced electric dipole moment of the prolate-shaped RBCs tends to align them parallel with their long axis to the electric field. At each applied voltage, we measured both the major and minor axes of RBCs in micrometers, through acquired images using the AXIOVISION software (Rel. 4.6.3.0, Zeiss) (Table 1).
Table 1. Experimental data for RBC stretching. The original diameter of the cell (without electric field) was 5.97( ± 0.1) µm. The data are the result of 5 independent measurements.
Note that, within the error of the measurements, the volume of the cells remained constant, and for small deformations (i.e., at 2 and 3 V), the Poisson number (-Δb/Δa) was determined to be close to 0.5 showing that RBCs can be considered as quasi-incompressible elastic bodies. Besides the geometry of RBCs, the dielectrophoretic force acting on them depends also on the real part of the Clausius-Mossotti (CM) factor and the gradient of the square of the electric field strength, Eq. (1).
The membrane of RBCs has a complex permittivity due to its conductivity. For frequencies between 1 MHz and 10 MHz, however, the electric field can also couple capacitatively across the membrane so that membrane conductivity plays a minor role as compared to its capacity. Above 10 MHz, the impedance becomes very low because all the current is shunted through the membrane of RBCs (that has a high electrical resistance at DC). With the parameters of our experimental conditions, (buffer conductivity = 0.124 S/m at pH = 7.3, cytoplasm conductivity = 0.312 S/m, membrane thickness = 8 nm, medium dielectric constant = 80, cytosol dielectric constant = 60, frequency = 10MHz) the sign of the real part of CM factor is positive (pDEP), and its value was calculated to be 0.8.
In order to determine the spatial distribution of the electric field between the electrodes, computer simulations were carried out using the parameters of the experiments as input values. Figure 1 shows some characteristic field lines to illustrate the geometry of the electric field strength. The values of the simulation space were then exported, and the gradient of the effective electric field strengths squared was calculated for the position of the cell, at a 10 V effective input voltage:
Using all these values, the dielectrophoretic force by Eq. (1) could be calculated at each voltage applied, and a relationship between FDEP and the cell geometry was determined in the Fig. 2.
Fig. 2 Dependence of the minor axis of the prolate on FDEP. The line shows the result of a linear fit to the first 5 points.
The slope of the line fitted to the first 5 data points in Fig. 2 was –11.29 ± 3 pN/μm, from which the shear modulus could be determined according to Eq. (3): µ = 1.80 ± 0.5 μN/m. (Assuming a constant volume (111.35 fL) for the RBC, the same quantity can be derived within the experimental error and also from the long-axis values of the prolate: µ = 1.83 ± 0.5 μN/m.) A deviation from the initial slope above 10 pN is observed in concert with the results of [19].
There are a number of data published in the literature for the elastic properties of healthy red blood cells. The values for the shear modulus determined by the micropipette aspiration method are in the range of 6-10 µN/m [5, 20–22]. Techniques based on (dual-beam) optical tweezers [4, 23], magnetic twisting cytometry [24], have reported µ values between 10 and 20 µN/m. (Note that the shear modulus measured in our experiments should be distinguished from the membrane area expansion modulus. For the latter quantity, the data values and their deviation are considerably bigger reported between ca. 20 µN/m and 400 mN/m [20].)
Recent dynamic membrane fluctuation measurements [7, 25], on the other hand, have reported on µ values between 1 and 10 µN/m. The lower values observed here were explained by the nonlinear dependence of the shear modulus on the mechanical stress, yielding lower µ values for lower, and higher ones for higher stresses applied (such as in the micropipette measurements and in those laser tweezers experiments that use microbeads attached to the membrane). However, lower values for the shear modulus of the membrane and the underlying spectrin skeleton (µ = 1.4 - 2.5 μN/m) have also been reported by independent laser tweezers experiments using silica microbeads, as well [4, 19].
Our method represents an independent technique from the previous ones and the results are consistent with the smaller values for the elastic modulus of RBCs. Given its sensitivity and the lack of mechanical contact with the cells, we consider our method an easy-to-use alternative tool to determine the mechano-elastic properties of living cells, and on this basis, to distinguish healthy and diseased cells, or monitoring the effects of aging [8], crucial both for the preservation of blood for medical issues and for forensic applications.
Acknowledgments
This paper was partially supported by: Sectoral Operational Programme Human Resources Development, financed by the European Social Fund and by the Romanian Government under the contract number POSDRU/89/1.5/S/64109, by CNCSIS-UEFISCSU, project number 1193 PNII-Idei code 1195/2008, and by a Hungarian research grant TAMOP-4.2.2.A-11/1/KONV-2012- 0060. M.M. Haque was a recipient of an ITC fellowship sponsored by BRC, Szeged. We thank A. Sobetkii from MGM STAR CONSTRUCT SRL (Bucharest, Romania) for producing the dielectrophoresis chambers.
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