Creep function of a single living cell

Abstract

We used a novel uniaxial stretching rheometer to measure the creep function J(t) of an isolated living cell. We show, for the first time at the scale of the whole cell, that J(t) behaves as a power-law J(t) = At(alpha). For N = 43 mice myoblasts (C2-7), we find alpha = 0.24 +/- 0.01 and A = (2.4 +/- 0.3) 10(-3) Pa(-1) s(-alpha). Using Laplace Transforms, we compare A and alpha to the parameters G(0) and beta of the complex modulus G*(omega) = G(0)omega(beta) measured by other authors using magnetic twisting cytometry and atomic force microscopy. Excellent agreement between A and G(0) on the one hand, and between alpha and beta on the other hand, indicated that the power-law is an intrinsic feature of cell mechanics and not the signature of a particular technique. Moreover, the agreement between measurements at very different size scales, going from a few tens of nanometers to the scale of the whole cell, suggests that self-similarity could be a central feature of cell mechanical structure. Finally, we show that the power-law behavior could explain previous results first interpreted as instantaneous elasticity. Thus, we think that the living cell must definitely be thought of as a material with a large and continuous distribution of relaxation time constants which cannot be described by models with a finite number of springs and dash-pots.

FIGURE 1

The uniaxial stretching rheometer. The components of the left-hand arm are detailed in the foreground: (1) three-axis piezo stage; (2) three-axis manual stage; (3) manual rotation stage; (4) microneedle holder; and (5) manipulation chamber. The right-hand arm is circled in the background (6).

FIGURE 2

A composite microneedle with a flexible shaped tip (the tube is 1-mm in diameter).

FIGURE 3

A C2-7 cell stretched under constant force. The servo controller gradually shifts the rigid (thick) microplate to compensate for cell deformation, and thus maintains a fixed deflection of the flexible (thin) plate tip. Pictures correspond to t = 0 s and t = 30 s.

FIGURE 4

Apparent contact diameters (D_flexible, D_rigid) and cell-length L perpendicular to the microplates.

FIGURE 5

Strain data illustrating the typical behavior of a living cell submitted to a constant force. (a) The lin-lin plot showing the cell rupture at ∼50 min. (b) The Ln-Ln plot emphasizing the existence of two different regimes.

FIGURE 6

In the short time regime, strain data are well fitted by a power-law ɛ(t) = kt^α over three time-decades (r² = 0.9997). The first measured strain values ranged from <1% to ∼30%, with one-third of these values <7%. Remarkably, power-law behavior was thus observed over a strain range going from <1% to values as high as 100%.

FIGURE 7

Shapes of a stretched cell at ɛ ∼ 1 and ɛ ∼ 6. Whereas apparent contact diameters (dots) decrease slightly at high strains, the mean cell diameter parallel to the microplates (arrows) is nearly divided by a factor of 2.

FIGURE 8

CDF of the exponent α. For a given value α₀, the CDF gives the percentage of α-values <α₀. Measured data (solid steps) are well described by an error function (solid line), CDF of a Gaussian density of probability (dashed bell-curve). Classical histogram representation of the data (inset) leads to the same conclusion, but with an unavoidable arbitrariness in data binning.

FIGURE 9

CDF of the prefactor A. Measured data (solid steps) are well described by the CDF (solid line) of a Gaussian density of probability (dashed bell-curve).

FIGURE 10

The values α (solid circles) and Ln(A) (squares) are independent of the applied stress magnitude σ₀.

FIGURE 11

Strain data of Fig. 6 are here fitted by: a power-law in a lin-lin plot (a), the creep function of the four-elements' model (c), in a lin-lin plot (b) and in a Ln-Ln plot (d).

FIGURE 12

Schematic representation of a constant-rate-of-charge experiment. (a) Initial state with a cell of length L₀. (b) The rigid plate is moved at a constant rate (the displacement D is proportional to the time t); both the cell length L(t) and the flexible plate deflection δ(t) are continuously varying and their values are geometrically related.

FIGURE 13

Data calculated using the mathematical analysis of Appendix 2 and representing the stress versus the strain for a cell characterized by a creep function J(t) = At^α and submitted to a constant-rate-of-charge (Fig. 12). Fitting this data by a linear relationship may appear acceptable in a lin-lin plot (a), whereas discrepancies are revealed by a Ln-Ln representation over more than one strain decade (b). Thus, an apparent linear (elastic) regime may hide a power-law behavior.