A master relation defines the nonlinear viscoelasticity of single fibroblasts

Abstract

Cell mechanical functions such as locomotion, contraction, and division are controlled by the cytoskeleton, a dynamic biopolymer network whose mechanical properties remain poorly understood. We perform single-cell uniaxial stretching experiments on 3T3 fibroblasts. By superimposing small amplitude oscillations on a mechanically prestressed cell, we find a transition from linear viscoelastic behavior to power law stress stiffening. Data from different cells over several stress decades can be uniquely scaled to obtain a master relation between the viscoelastic moduli and the average force. Remarkably, this relation holds independently of deformation history, adhesion biochemistry, and intensity of active contraction. In particular, it is irrelevant whether force is actively generated by the cell or externally imposed by stretching. We propose that the master relation reflects the mechanical behavior of the force-bearing actin cytoskeleton, in agreement with stress stiffening known from semiflexible filament networks.

FIGURE 1

Image of an actively contracting fibroblast between two fibronectin coated microplates. The distance between plates is ≃5 μm.

FIGURE 2

Schematic of the micromanipulation setup. A fibroblast is held between two coated microplates. The deformation of the flexible microplate gives the force F acting on the cell. The position of the laser beam emerging from the optical fiber, which is in contact with the tip of the flexible plate, is detected using a position sensitive detector. A personal computer reads the signal from the detector and controls the piezoelectric translator.

FIGURE 3

Force as a function of time at constant cell extension, recorded immediately after incorporation of the fibroblast into the rheometer. No significant cell shape alterations are seen throughout. The experiment is performed using fibronectin mediated adhesion.

FIGURE 4

Amplitude sweep. Modulus |Θ| as a function of the strain amplitude Δ_ℓ/〈ℓ〉 for an arbitrary selection of cells, at a frequency of 0.2 Hz. Each curve is a different experiment; some at negative, compressive forces, some at positive tensions. No systematic change is seen in the shape of the amplitude sweeps as a function of force.

FIGURE 5

Lissajoux figures for different amplitudes. (Top) Typical response at forces above crossover. Force F as a function of relative cell length ℓ/〈ℓ〉 for strain amplitudes 3.5%, 6%, 13%, and 20%. The oscillation frequency is 0.2 Hz. (Bottom) Typical response at compressive forces. Force F as a function of relative cell length ℓ/〈ℓ〉 for strain amplitudes 3%, 5%, 12%, and 20%. The oscillation frequency is 0.2 Hz.

FIGURE 6

Frequency sweep (preliminary data). Modulus |Θ| and loss angle δ as a function of frequency. Each curve is obtained using a different cell.

FIGURE 7

Stress stiffening due to active contraction at constant length. The average cell length is kept constant throughout at 〈ℓ〉 = 9 μm. Sinusoidal oscillations are superimposed with a strain amplitude Δ_ℓ/〈ℓ〉 = 0.03. The frequency of the oscillations is cyclically changed from 0.1 to 1.0 Hz. (a) The average force 〈F〉 is seen to increase in time. (b) The modulus |Θ| increases in time for all frequencies. (c) Stiffness |Θ| as a function of average force 〈F〉, for different frequencies. The line shows a power law function y ∼ x^1.7 for comparison. (d) Loss angle δ as a function of average force 〈F〉, for different frequencies.

FIGURE 8

Force response as result of imposed length changes We step-strain the cell by ∼50% at a rate of 1.5 μm/s, and apply length oscillations at an amplitude of 0.5 μm and a frequency of 0.2 Hz. The experiment was performed using fibronectin mediated adhesion.

FIGURE 9

Inset shows the modulus |Θ| as a function of average stress for 13 cells, measured using length steps plus oscillation experiments such as in Fig. 8. The main plot shows the data scaled using two factors, as described in the text, which gives an exponent . All experiments are performed at 26°C and using fibronectin mediated adhesion.

FIGURE 10

(a) Ramp experiment with glutaraldehyde coating. Average stress (top), loss angle (middle), and average length (bottom) as a function of time. Oscillations at 1 Hz, 5% amplitude are superposed throughout, also during the ramps (shaded areas). The loss angle δ increases with the deformation rate. (b) Stiffness as a function of average stress. The relationship between |Θ| and 〈σ〉 depends on the deformation rate. The curves correspond to the shaded areas 1, 2, and 3 in a.

FIGURE 11

(a) Single-filament nonlinear elasticity: filaments are regarded as inextensible, characterized by a contour length L and a bending modulus κ. One end is clamped, the other one free. The force F bends the filament by an amount x. The dimensionless force is defined as f = FL²/κ. (b) Differential stiffness df/d(x/L) as a function of the dimensionless force f. (c) Euler-Bernoulli elastica: filament shapes for different forces.