A Novel Technique of Quantifying Flexural Stiffness of Rod-Like Structures

Abstract

In cellular and molecular biomechanics, extensional stiffness of rod-like structures such as leukocyte microvilli can be easily measured with many techniques, but not many techniques are available for measuring their flexural stiffness. In this paper, we report a novel technique of measuring the flexural stiffness of rod-like structures. This technique is based on image deconvolution and, as an example, it was used for determining the flexural stiffness of neutrophil microvilli. The probes we used were 40-nm-diameter fluorescent beads, which were bound to the tips of neutrophil microvilli by anti-L-selectin antibody. The fluorescent images of the bead, which was positioned at the center of the cell bottom, were acquired with high magnification and long exposure time (3 s). Using a Gaussian function as the point spread function of our imaging system, we established a convolution equation based on Boltzmann's law, which yields an analytical expression that relates the bead image profile to the flexural stiffness of the microvillus. The flexural stiffness was then obtained by the least squares regression. On average, the flexural stiffness was determined to be 7 pN/mum for single neutrophil microvilli. With the resolution of our imaging system, this technique can be used for measuring any flexural stiffness smaller than 34 pN/mum and it has great potential in single molecule biomechanics.

FIGURE 1

Schematic diagram of bead imaging. The bead, a fluorescent nanosphere, is bound to the tip of a microvillus. The Cartesian coordinates (x, y) represents any point in the bead image. The Cartesian coordinates (ξ, η) represent the position of the bead center.

FIGURE 2

The typical image of a 40-nm fluorescent bead superposed on a human neutrophil bright-field image. The neutrophil was immobilized by being held with a micropipette affixed to the microscope stage. The fluorescent bead was at the center of the cell bottom.

FIGURE 3

The point spread function (PSF) of our microscopic imaging system. The PSF was measured by averaging 25 bead images (20). The beads were on immobilized fixed neutrophils. The images were acquired with an exposure time of 3 s. The solid wireframe is a two-dimensional Gaussian fit with a correlation coefficient of 0.995.

FIGURE 4

Dependence of the PSF parameter ${\sigma }_0^2$ on the distance between the bead and the coverslip (d). The linear regression yields ${\sigma }_0^2=19+0.71d$.

FIGURE 5

(a) The distribution of the ${\sigma }_0^2$ values of the beads on some fixed neutrophils. (b) The apparent variance distribution in one typical experiment where ten beads were imaged: seven (Nos. 1–7) were bound to live neutrophils by anti-L-selectin and three (Nos. 8–10) were affixed to the cell bodies of fixed neutrophils by anti-CD44. The apparent variance β in Eq. 16 should be equal to ${\sigma }_0^2$ when the beads were on fixed neutrophils.

FIGURE 6

The histogram of the flexural stiffness of single microvilli on human neutrophils. The solid curve represents a Poisson fit, which yields 7 ± 3 pN/μm (mean ± SD) for the flexural stiffness. The bin width is 2 pN/μm.

FIGURE 7

The mean values (circle, ± SD) of simulated flexural stiffness (K) at different diffusion coefficients (D). In the Monte Carlo simulation, the beads were allowed to diffuse for 3 seconds to obtain the convolved bead images. These simulated images were analyzed with the same procedure as used for the images obtained by fluorescence microscopy to obtain the simulated flexural stiffness.