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TECHNICAL PAPERS: Cell

Estimation of Cell Young’s Modulus of Adherent Cells Probed by Optical and Magnetic Tweezers: Influence of Cell Thickness and Bead Immersion

[+] Author and Article Information
Alain Kamgoué

Laboratoire TIMC-IMAG, Equipe DynaCell, CNRS UMR 5525, Institut de l’Ingénierie de l’Information de Santé, Faculté de Médecine, 38706 La Tronche Cedex, FranceAlain.Kamgoue@imag.fr

Jacques Ohayon1

Laboratoire TIMC-IMAG, Equipe DynaCell, CNRS UMR 5525, Institut de l’Ingénierie de l’Information de Santé, Faculté de Médecine, 38706 La Tronche Cedex, FranceJacques.Ohayon@imag.fr

Philippe Tracqui1

Laboratoire TIMC-IMAG, Equipe DynaCell, CNRS UMR 5525, Institut de l’Ingénierie de l’Information de Santé, Faculté de Médecine, 38706 La Tronche Cedex, FrancePhilippe.Tracqui@imag.fr

1

Authors to whom correspondence should be addressed.

J Biomech Eng 129(4), 523-530 (Dec 07, 2006) (8 pages) doi:10.1115/1.2746374 History: Received July 25, 2006; Revised December 07, 2006

Abstract

A precise characterization of cell elastic properties is crucial for understanding the mechanisms by which cells sense mechanical stimuli and how these factors alter cellular functions. Optical and magnetic tweezers are micromanipulation techniques which are widely used for quantifying the stiffness of adherent cells from their response to an external force applied on a bead partially embedded within the cell cortex. However, the relationships between imposed external force and resulting bead translation or rotation obtained from these experimental techniques only characterize the apparent cell stiffness. Indeed, the value of the estimated apparent cell stiffness integrates the effect of different geometrical parameters, the most important being the bead embedding angle $2γ$, bead radius $R$, and cell height $h$. In this paper, a three-dimensional finite element analysis was used to compute the cell mechanical response to applied force in tweezer experiments and to explicit the correcting functions which have to be used in order to infer the intrinsic cell Young’s modulus from the apparent elasticity modulus. Our analysis, performed for an extensive set of values of $γ$, $h$, and $R$, shows that the most relevant parameters for computing the correcting functions are the embedding half angle $γ$ and the ratio $hu∕2R$, where $hu$ is the under bead cell thickness. This paper provides original analytical expressions of these correcting functions as well as the critical values of the cell thickness below which corrections of the apparent modulus are necessary to get an accurate value of cell Young’s modulus. Moreover, considering these results and taking benefit of previous results obtained on the estimation of cell Young’s modulus of adherent cells probed by magnetic twisting cytometry (MTC) (Ohayon, J., and Tracqui, P., 2005, Ann. Biomed. Eng., 33, pp. 131–141), we were able to clarify and to solve the still unexplained discrepancies reported between estimations of elasticity modulus performed on the same cell type and probed with MTC and optical tweezers (OT). More generally, this study may strengthen the applicability of optical and magnetic tweezers techniques by insuring a more precise estimation of the intrinsic cell Young’s modulus (CYM).

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Figures

Figure 1

Finite element mesh of the half cell volume of size l∕2×l×h. The arrows indicate the orientations of the bead rotation, bead translation, and force applied at the bead center. Free boundary conditions are considered except for displacement conditions explicitly indicated in the figure. Bead neighborhood has been enlarged for clarity, with the bead radius R and the bead embedding angle 2γ being specified in this enlargement.

Figure 2

Influence of cell thickness on the mechanical response of the cell to the applied force; 3D color maps show the deformed shapes of the cell and the spatial distributions of the effective strains eeff in the neighborhood of the bead (eeff=2eijeij∕3, with eij the components of the deviatoric strain tensor). Two different cell-bead geometries have been considered for fixed values γ=65deg, R=2.5μm, and l=20R. For each geometry, a force of 50 pN has been imposed, giving rise to a lateral bead translation U and rotation θ. Cell thickness values are, respectively, (A)h=6.5μm. (B)h=2μm. The bead has been removed in order to visualize more clearly the spatial strain distributions in the neighborhood of the cell-bead contact area.

Figure 3

Influence of the bead embedding half angle γ and normalized under bead thickness hu∕2R on the two correcting coefficients α and β. Solid points correspond to isovalues of bead embedding angle when hu∕2R is varied.

Figure 4

Best fits obtained for the two correcting hyperbolic functions α(hu∕2R) ((A), r2=0.990) and β(hu∕2R) ((B), r2=0.995) for the fixed value γ=80deg of the bead embedding half angle

Figure 5

Best polynomial fits of the four functions Aα(γ) ((A), r2=0.999), Bα(γ) ((B), r2=0.999), Aβ(γ) ((C), r2=0.999), and Bβ(γ) ((D), r2=0.999) given in Eqs. 9,10,11,12 and used for the estimation of the two correcting functions α and β (see Eqs. 7,8). In these functions, γ is expressed in radian.

Figure 6

Critical domain (white area) specifying the values hu∕2R and γ for which the apparent stiffness must be corrected according to a relative error on cell stiffness larger than 25%. On the contrary, the dark domain gives the values of hu∕2R and γ for which the relative error is smaller than 25%.

Figure 7

Comparison between the correcting functions α(γ,hu∕2R), defined in this study and αLa(γ) proposed by Laurent (22). The curves represent the difference of these two corrections for fixed values of hu∕2R when the bead embedding half angle γ is increased.

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