2.1.

Sample Preparation

Cells were prepared with the same procedures as previously described in Ref. 24. Fresh blood was collected from a mouse, and 10 μl of it was diluted by 50-fold with a low K⁺ solution (Mm: KCl 2, NaCl 145, HEPE-Na 10, MgCl₂ 0.15, and 1 mg/ml BSA). This low K⁺ solution works as a surrounding medium for the observation of an RBC in our QPM system. After being washed twice, normal discoid shape RBCs were adjusted to an appropriate experimental concentration, and then abnormal spherical shape RBCs were prepared by adding 1.3 mM Ca²⁺ and 1 mM of A23187 to the low-K⁺ solution. This dehydrates RBCs and acts as a Ca²⁺ ionophore to transport ions across the lipid bilayer of RBC membrane.³¹ All experiments were carried out at 25°C.

2.2.

Quantitative Phase Imaging

We have used the QPM system proposed by Professor Michael Feld's group at MIT (Ref. 14) to obtain a time series of quantitative phase images of RBC membrane fluctuations. The QPM system has provided a quantitative phase image with subnanometer optical path length sensitivity. Quantitative 2D phase information ϕ(x,y) can be easily converted into a 2D thickness profile t(x,y) by using the equation: ϕ(x, y) = 2π/λ · Δn · t(x, y), where λ is the center wavelength of the light source, Δn is the difference in refractive index between the RBC and the surrounding medium. The basic configuration of the QPM is a combination between a Mach–Zehnder interferometer with a laser diode as an optical source and an optical microscope with a charge coupled device (CCD) as an image capturing device. The center wavelength of the laser diode is 633 nm, and the optical magnification of the imaging system is calibrated to be ∼110 for the imaging system. After using appropriate high frequency filtering and the Hilbert transformation of a recorded interferogram captured by the CCD, we can obtain a quantitative phase image ϕ(x,y). The 2D thickness profile t(x,y) of a cell is calculated from ϕ(x,y), where the refractive index difference Δn between the RBCs and the surrounding medium of our low K⁺ solution is estimated to be 0.06.¹⁹

2.3.

Detrended Fluctuation Analysis

The following DFA algorithm has been previously described by Peng ²⁵ For a given data series of length N, we calculate a new integrated time series y(k) defined with

2.4.

Statistical Analysis

Data were analyzed by ANOVA using STATVIEW 5.0.1 software (SAS Institute. Inc.) and summarized as the mean ± SEM. Statistical significance was accepted for cases with p < 0.05.

3.1.

DFA Computation of Time Series RBC Thickness Data Obtained by QPM

In order to analyze RBC membrane fluctuations, we have measured 2000 sequential thickness images of the normal discoid shape and the abnormal spherical shape RBCs. These cells were observed for 20 s with an imaging speed of 100 frames/s.

Figure 1a shows a typical 2D thickness image for a normal discocyte RBC. The color bar on the right side of the figure indicates the thickness of a cell in micrometers. Figure 1b shows temporal thickness variations at two different points indicated by two arrows in the measured thickness image of Fig. 1a. The upper graph shows temporal fluctuations in thickness for a pixel on the membrane of the normal discocyte RBC, while the lower graph shows those for a pixel on the surrounding medium outside of the discocyte. The standard deviation (SD) of the amplitude of the membrane vibrations is 55.4 nm in the upper graph of Fig. 1b, while the SD of the background medium is shown to be 10.3 nm in the lower graph. These results indicate that the RBC membrane undulation amplitude is significantly larger than the background noise. Since we have a very small refractive index difference of 0.06 between the cell and the surrounding medium, 10.3 nm error in thickness t(x,y) corresponds to 0.62 nm (≈ λ/1000) measurement sensitivity in optical path length difference, which is defined as the product of refractive index difference Δn and thickness t(x,y).

Fig. 1

(a) Thickness image of normal RBCs with QPM with a colorbar in micrometers. (b) Temporal thickness variations for a pixel inside a cell (upper plot) and a pixel outside a cell (lower plot); two corresponding pixels are indicated by two arrows in (a).

Figure 2a shows the same flicking data at one point on the RBC membrane that is the same as the one shown in the upper plot of Fig. 1b. Since 100 images were taken in 1 s, we have 2000 data points for the whole observation period of 20 s. We have used this data for time series data t(i) defined in Eq. 1. Integrated time series y(k) were calculated using Eq. 1 and are plotted with the line (1) in Fig. 2b. The line (2) in Fig. 2b shows the representative example of local trend y _n(k). The local trendy _n(k) was fit in each box with a length of 200 data points using a least-square line fit. Finally, we obtained a double log graph between log F(n) and log n by repeating the calculation shown in Eq. 2. Figure 2c shows the plot of log F(n) with respect to log n. The fractal scaling exponent α for this vibrational motion is calculated to be 0.91 from this log-log plot. This suggests that the flicking motion of the membrane of a normal discocyte RBC has long-range correlation properties. These results were in good correspondence to the previous study of normal RBCs using PCM.²⁷

Fig. 2

(a) Typical thickness fluctuation data B(i) for a pixel on the membrane of a normal RBC. (b) The integrated time series y(k) depicted as line (1) and the local trend y _n(k) depicted as line (2). (c) Log-log plot for F(n) and n.

3.2.

Spatial Correlation Properties of Membrane Fluctuations in Discocyte and Spherocyte RBCs

From a series of measured thickness images for an RBC, we obtained temporal thickness fluctuation data that were similar to the data shown in Fig. 2a for each pixel in the series of measured images. From the thickness fluctuation data, we have calculated the fractal scaling exponent α for each pixel with the procedures explained in Sec. 3.1. As a result, a 2D map of α was constructed for an RBC from a series of thickness images obtained by QPM.

Figures 3a and 3d are thickness images of discocyte and spherocyte RBCs, respectively. Figure 3b shows a 2D map of α for the normal discocyte RBC, while Fig. 3e is the 2D map of α for the abnormal spherocyte RBC. Statistical properties of the calculated α for discocyte and spherocyte RBCs can be clearly seen by two histograms of α displayed in Figs. 3c and 3f. Calculated α values from the thickness data of the discocyte are distributed between 0.7 and 1.0, and their mean value is 0.87. These results reveal that there exists a long-range correlation in the membrane fluctuations of a normal discocyte RBC. As illustrated in Fig. 3e, calculated α values from the thickness data of the spherocyte are within a range from 0.85 to 1.2. Its mean value is 0.99. These results indicate that detrended fluctuation analysis for the dynamical properties of membrane flicking in the abnormal spherical shape RBC is more or like random walk.

Fig. 3

Averaged thickness images [(a) and (d)] and their corresponding fractal scaling exponent maps [(b) and (e)] for a discocyte and a spherocyte RBCs. Histogram of fractal scaling exponent α for the discocyte RBC, (c) and the spherocyte RBC (f).

We have also measured fractal scaling exponent α for the background medium in our system without a sample cell. Figure 4a shows a 2D map of fractal scaling exponent α for an area without a cell indicated by the white dashed square in Fig. 1a. Figure 4b is the histogram of α for the pixels within the 2D map of Fig. 4a. It shows that 96.5% of the measured α for the background fluid without a sample ranges from 0.46 to 0.54. The average value of the measured exponents for the background fluid is approximately 0.5, which demonstrates that the time series of background fluid fluctuation is an uncorrelated white noise.

Fig. 4

(a) Fractal scaling exponent map of background fluid outside a cell indicated with a white dashed box in Fig. 1a. (b) Histogram of fractal scaling exponent α for the background fluid.

We have repeated the above measurement of the fractal scaling exponent α for 10 different normal discocyte RBCs and 10 abnormal spherocyte RBCs, as well as for the background fluids of these RBC samples. Figure 5 shows the box plot representation of measured fractal scaling exponent distributions for discocytes, spherocytes, and background fluids. The mean and the SD of measured exponent values (mean ± SD) for the discoctyes and the spherocytes are 0.86 ± 0.03 and 0.99 ± 0.02, respectively. And, those for the background fluids are 0.49 ± 0.03. This finding suggests a significant difference in the long-range scaling behavior between healthy (discocyte) and diseased states (spherocyte). It is well known that a vibrational motion is unbounded or like random walk when the fractal scaling exponent α is larger than 1.²⁵ We have also calculated the percentage of pixels whose scaling exponent is larger than 1 (α > 1) for these two different types of RBCs. 2.3% of pixels in discocytes have their scaling exponent larger than 1, while 46.2% of pixels in spherocytes have α > 1. These results show that the fluctuation motion of the membrane of a spherocyte resembles an unbounded random walk more than that of a discocyte. Studies on membrane fluctuations in discocytes and spherocytes have been previously reported by using a QPM system. Popescu showed an increase in membrane tension as a discocyte was changed to a spherocyte.¹⁸ Also, Park recently reported significant increases in the shear and the bending moduli of the membrane associated with morphology changes in an RBC.³² In this paper, we have demonstrated a relationship between the temporal correlation properties of membrane fluctuation in an RBC and its morphological shape (discoid and spherical shapes). Variances in p-value between discocytes and spherocytes displayed in Fig. 5 clearly verify differences between these two morphological shapes of an RBC (p < 0.0001).

Fig. 5

Box plots for the distributions of the measured fractal scaling exponents of 10 normal discocyte RBCs, 10 abnormal spherocyte RBCs, and 10 background fluids (*** p < 0.0001, discocytes versus spherocytes).

Our findings are of great importance for the relationship between correlation properties of membrane fluctuations and metabolic driving force depending on ATP. We suggest a scenario that the increase of membrane rigidity in the absence of ATP forces to have a larger fractal scaling exponent α in spherocytes, which means correlation property with a random walk or a Brownian motion. We believe that these results of higher stiffness and a larger fractal scaling exponent are results of low deformability and ATP deletion in spherocytes due to A23187 induced calcium accumulation. Based on these findings, we could assume that the transition from discocytes to spherocytes causes the change of the mechanical characteristics and the correlation properties of RBC membrane fluctuations. In this regard, our study has verified that QPM can be effectively used with the detrended fluctuation analysis for a time series motion of absolute thickness undulations in an RBC by showing the significant difference in fractal scaling exponent between discocytes and spherocytes.