Optical twisting to monitor the rheology of single cells

Abstract

Background:

Biological cells exhibit complex mechanical properties which determine their responses to applied force.

Objective:

We developed an optical method to probe the temporal evolution of power-law rheology of single cells.

Methods:

The method consisted in applying optically a constant mechanical torque to a birefringent microparticle bound to the cell membrane, and observing dynamics of the particle’s in-plane rotation.

Results:

The deformation dynamics of the membrane followed a power law of time, which directly relates to cytoskeletal prestress as reported in the literature. The temporal evolution of this rheological behaviour, over time scales of several minutes, showed strong variations of the exponent on single adherent cells not subject to any specific treatment.

Conclusions:

The consistent observation of variations in the exponent suggests that, in their normal activity, living cells modulate their prestress by up to three orders of magnitude within minutes.

Keywords

Adhering cells power-law rheology optically-induced torque prestress

1. Introduction

Cell mechanics has raised a growing interest from both the biophysical and the soft condensed matter standpoints. On the one hand, a growing body of work suggested its crucial role in the development of some diseases due to the mechanical regulation of cellular biochemistry and gene expression [1]. On the other hand, cells are fascinating active materials that defy the pre-existing classifications among complex materials [2]. Investigations performed over the past decade have nevertheless drawn several universal trends, beyond the diversity of living cells [2,3]. A robust common feature of adherent cells, regardless of the cell type or line considered, the drug treatment applied, or the measurement technique used, is their power-law rheological behaviour. When loaded by a constant stress, cells deform following a weak power law of time – equivalently, under oscillating load the elastic modulus scales with the excitation frequency at the same power [4–6]. This scaling law behaviour suggests that stress relaxation involves a large number of characteristic times rather than a specific one [7], a feature shared with many complex viscoelastic systems belonging to the generic class of ‘soft glassy materials’ [8]. These systems can be described as an energy landscape comprising a succession of potential wells which reflect the disordered structural elements of the material. The exponent, α, reflects the level of non-thermal (i.e., chemical, mechanical) noise allowing the elements to hop between adjacent potential wells, leading the material to yield. The value of the exponent therefore quantifies the material’s ability to deform and flow, the extreme cases $α = 0$ and $α = 1$ corresponding to a purely elastic (Hookean) solid and a purely viscous (Newtonian) liquid, respectively. Cells feature rather low exponents, typically 0.1–0.4.

Besides rheological studies, extensive investigations of the cytoskeleton have evidenced the pre-existence of contractile forces, even when the cell is not forced [9,10]. Tensile forces exerted by actin filaments, resisted by focal adhesions and microtubules, result in a tensile stress called prestress. Cells can modulate the prestress passively, in response to an external load, or actively, so as to allow movements (crawling, migration, division) that require more deformability, or some biochemical mechanisms (through the opening of ion channels) [11]. The physiological activity of a cell should therefore involve a modulation of the cytoskeletal prestress in time scales of minutes. Surprisingly, however, the evolution of cell prestress over time is, to our knowledge, scarcely documented.

Previous investigations [12–14] have established the intimate link between the rheological exponent α and the cytoskletal prestress [2]. In this paper, we propose a simple approach to investigate this evolution, by monitoring the power-law rheology of single cells during physiologically-relevant time scales. To this end, we developed a new technique that exploits the constant torque exerted by the circularly-polarised laser beam of an optical trap to a birefringent micron-sized particle [15]. By measuring the resulting rotation of such particles when bound to cell membranes, we characterised their power-law rheology. Consecutive measurements on individual cells showed that the exponent α may strongly vary within time scales of 5–10 min, and consistently follow specific temporal patterns. The measured variations suggest that the cytoskeletal prestress intrinsically modulates by up to three orders of magnitude without specific external intervention.

2. Experimental procedure

The experiment consists in exerting optically a constant torque on microscopic particles bound to membranes and measuring their resulting in-plane rotation. When transmitted through a birefringent particle, a circularly-polarised laser beam transfers spin angular momentum, thereby exerting on the particle a constant torque τ, oriented along the propagation axis, which makes it rotate. When bound to a cell membrane, the particle’s rotation is limited by the ability of the cell to deform, which overrides the purely viscous contribution of the culture medium.

This constant torque procedure allows to apply a step stress forcing to living cells, in a way similar to previously reported creep function measurements [5–7,16]. In fact, the creep function experiments reported in the literature mainly consist in twisting laterally (rocking) [5] or pulling away [16] a particle bound on the cell membrane, or stretching axially the whole cell [6,7]. In our case, the probe twists in the plane of the membrane. Although in-plane twisting has been used within the (three-dimensional) intracellular medium [17], we note that this deformation mode had, to our knowledge, never been applied on the (planar) cell membrane to date. Since the deformation field radially expands all around the particle, we expect this mode to be less sensitive to the directional anisotropy within the membrane surface than the rocking mode that favours one direction.

2.1. Optical setup

The experimental setup, based on a home-built optical trap, is sketched in Fig. 1. The trapping beam, produced by a CW fibre laser (IPG Photonics GmbH, YLM-10-1070-LP, wavelength in vacuum $λ_{0} = 1070 nm$ ) operating in the TEM₀₀ Gaussian mode, is focused into the sample through a microscope objective (Olympus LUMFLN 60×, N.A. 1.1 or Nikon 60×, N.A. 0.8). A quarter-wave plate placed at the back aperture of the objective circularly-polarises the beam prior to focusing. The same objective images the sample over a fast CCD camera (Basler 640-210gm, operating at 200 frames per second) to record the twisting dynamics. All measurements and observations were carried out at room temperature, and no special treatment was applied to the sample during the experiment, typically performed within 1 h in closed sample chambers.

Fig. 1.

Sketch of the experimental setup. A circularly-polarised laser beam is tightly focused onto a vaterite particle bound to the cell membrane, exerting a constant torque τ oriented along the propagation axis. The resulting rotation of the particle is measured through the orientation angle $θ (t)$ .

2.2. Sample preparation

Chinese Hamster Ovarian (CHO-K1) cells were cultured in Minimum Essential Medium (Sigma), 10% Fetal Calf Serum (Sera Laboratories International) and L-Glutamine, Penicillin and Streptomycin (Sigma). The cells were plated on 23-mm glass-bottom sample dishes (Fluorodish, World Precision Instruments) and incubated for at least 24 hours until adherent on the bottom surface.

Birefringent (vaterite) microparticles were prepared following the protocol described in Ref. [18]. Briefly, calcium carbonate was precipitated from CaCl₂ and K₂CO₃ solutions in the presence of MgSO₄ under vigorous pipetting, leading to the formation of 4–6 µm spheroidal vaterite crystals. Particles, stabilised by using small amounts of Tween 20, a water-soluble surfactant, can be stored up to several months in isopropanol. Prior to any experiment, vaterite particles were dried, rinsed and coated with Bovine Serum Albumin (BSA) to avoid sticking at the glass surface of the chamber [19], and then resuspended in the culture medium solution. Finally, ∼30 µl were dripped into the sample chamber, and the sample was incubated for 30 min. The particles randomly sank into the sample, some of them getting attached to cells (see Fig. 1), as can be checked by the inability of moving them optically (free particles were easily trapped in the laser focus, and could be moved throughout the sample by displacing the beam).

2.3. Particle adhesion

Under exposure, vaterite particles twist by an angle θ, as shown in Fig. 1, in response to the torque exerted by the laser beam. When the laser is turned off, the particle reorients backwards, but keeps a remanent angular shift compared to its initial orientation. A part of the deformation has thus been stored, which confirms that the particle has actually probed a viscoelastic medium. We also visualised the distortion of the membrane in the vicinity of the particle, through the collective displacement of rough spots on the cell membrane as illustrated in Fig. 2. The two images on the left are snapshots taken from an experimental movie, at the beginning and the end of a twist, respectively. Six bright spots on the cell membrane, identified by automated edge detection, are circled. The third picture on the right is the average of the first two, showing the displacement of these spots along the direction given by the solid arrows. In addition, out-of-focus objects, not well resolved enough to be tracked by image analysis, could also be seen moving in the movie. This displacement is schematically represented by the dashed arrows. All these displacements are rather weak, typically in the range 0.1–1 µm; the arrows are plotted edge-to-edge and therefore exaggerate the actual displacements. This figure thus shows the membrane strains induced by the particle rotation. Although the strain field looks rather complex due to the cellular organisation, it clearly follows a gear-like pattern around the rotating particle, showing its distortion when the particle twists. This observation demonstrates a posteriori that the particle adhesion to the cell membrane, yet unknown in nature, is sufficient to distort it. Note also that the strain observed is, at first sight, of amplitude comparable to that reported in the literature [4,12] for mechanical stresses in the range of tens of Pa, which is the typical order of magnitude expected from optical forces.

Fig. 2.

Images showing the deformation of the membrane by the particle twist. Left: still images extracted from the movie, before the laser was turned on ( $t = 0 s$ ) and at the end of the twist ( $t = 7.950 s$ ); the particle has twisted counterclockwise by 52°. Six bright spots, corresponding to rough spots on the membrane, are circled for reference. Right: average of the two images. The solid arrows depict the displacement of the bright spots; the dashed arrows depict the displacement of out-of-focus objects. Beam power is 90 mW (in the sample).

Finally, the literature suggests that, if the binding mechanism influences the absolute stress–strain relationship (i.e., the scaling viscoelastic modulus), it does not influence the dynamics of the cell membrane’s response to the solicitation (i.e., the exponent α) [7], inherently dominated by the cytoskeleton. We checked this by also using particles coated with poly-D-lysine (PDL), a peptide used to enhance the adhesion of cells to glass surfaces, and changed the incubation time between 0 and 2 h. As expected, none of these parameters seemed to affect the dynamics of the cell response.

2.4. Angular deviation measurements

The twist angles were measured from the recorded movies, by image pair correlation. Setting the initial image (the last before the laser was turned on) as a reference, we rotated each image of the movie (more precisely, a selection of frames spanning a logarithmic scale) up to the angle which best matches it with the reference. This correlation was performed both manually and with a home-written Matlab routine, with comparable results yet automated detection sometimes missed the particle rotation. We thus systematically kept the manual measurement.

The accuracy of this measurement was estimated about 0.5° for small angles (typically below 5°, although the limit is not well defined). For larger deviations, the membrane deformation also induced an in-plane distortion which slightly changed the apparent shape of the particle, making the correlation more difficult. The uncertainty is thus more pronounced, up to a few degrees, and depends on the distortion.

3. Results and discussion

3.1. Power-law rheology

When shining a bound vaterite particle, its orientation progressively changes, following a dynamics initially fast then gradually slowing down. Measuring the twisting dynamics $θ (t)$ then gives access to the dynamics of cell deformation. We applied consecutive step stresses of about 1–10 s each, during time intervals of 5 to 20 min on each single cell, waiting at least 30 s between two consecutive measurements.

Typical twisting dynamics are represented in Fig. 3. The three curves correspond to measurements performed on the same cell, at three instants from 38 to 43 min after the sample was removed from the incubator. In all cases, the twisting dynamics follows a weak power law over at least two decades in time. Discrepancies from this behaviour were sometimes seen at very short (<20 ms) or very long (>2 s) times. We identified the short-time dynamics as the faster $\sim t^{3 / 4}$ regime governed by the viscoelasticity of actin filaments [20,21]. This regime, of entropic origin, is attributed to thermal fluctuations of semiflexible polymers. At the other end, several investigations [6,16,22] over a wide range of frequencies (10⁻³ to 10³ Hz) showed that a distinct regime, whose origin remains unclear to our knowledge, occurs at long times (after a few seconds). Thus, the short-time dynamics is clearly not related to the cytoskeletal prestress, and whether the long-time dynamics is or not related to the cytoskeletal prestress is an open debate. We therefore chose to ignore the corresponding data and focus on the intermediate regime, unambiguously related to the prestress.

Fig. 3.

Time-dependent twisting angle, $θ (t)$ , measured on a single cell at three instants, about 40 min (from left to right graphs, 38, 39, and 43 min) after the sample was removed from the incubator. The twisting dynamics follows a power law, $θ \sim t^{α}$ . Solid lines are the best power-law fits to each dynamics, dashed lines depict extreme acceptable fits according to the experimental uncertainty. Beam power is 175 mW (in the sample) in all cases. Left inset: same data in linear scale. Right inset: evolution of the exponent α with time for this cell. The experimental points correspond to the best fits obtained for each dynamics, the error bars depict extreme fits. The circle, square and triangle correspond to the three dynamics represented in the main figure and left inset.

All dynamics were fitted with three power laws: the best fit, and the two extreme acceptable fits considering the measurement uncertainty, allowing to determine the exponent and the associated error bars. The nominal value of the exponent, α, is first given by the best fit to the experimental points by a power law (solid lines in Fig. 3). To determine the experimental uncertainty on this exponent, we also computed extreme power laws (dashed lines in Fig. 3) that could reasonably fit the data points assuming the twisting angle θ was under- or overestimated, respectively, by its uncertainty. We thus obtain two extreme possible values for the exponent, $α_{+}$ and $α_{-}$ , from which we deduce $+ / -$ error bars around the exponent values, represented by the error bars in the right inset of Fig. 3. For example, the three dynamics represented in Fig. 3 yield, from left to right, $α = 0.371 + 0.039 / - 0.051$ , $0.132 + 0.028 / - 0.032$ , and $0.294 + 0.056 / - 0.044$ . We therefore unambiguously observed, within 5 minutes, a strong reduction of the exponent (solidification) followed by an increase (fluidisation) back to the initial value. The right inset in Fig. 3 illustrates the full evolution measured, which takes an unexpected symmetric U-shape. Such observation contrasts with the transient fluidisation followed by resolidification expected after strongly stretching cells [23], but suggests the laser may somehow affect the inegrity of the cell membrane. We thus first checked the possible influence of the laser forcing on the observed evolution.

3.2. Linearity

The laser forcing may alter the membrane structure either thermally or mechanically. We assessed these effects through order-of-magnitude estimates and experiments at variable laser power.

3.2.1. Mechanical torque

We measured the torque τ exerted by a circularly-polarised laser beam on a vaterite particle ( $2 R ≃ 5 µ m$ ) in bulk liquid according to a procedure described in Ref. [24], getting 10–100 pN µm for laser powers ranging from 0.1 to 1 W in the sample. The average stress exerted by the particle over its surrounding medium is thus in the order of $3 τ / (4 π R^{3})$ , thus a few pascals or less. Assuming a contact surface between the particle and the membrane of a fraction (a few percent to a few tens of percent) of the particle surface area, this would yield an actual stress of a few tens of pascals on the membrane. Such value lies within the range reported for other techniques routinely used in cell microrheology, involving optical or magnetic forces [2]. It is, however, very small compared to the reported values of cells elastic moduli (10²–10⁵ Pa), and still below the typical values (50–200 Pa) involved in investigations of non-linear rheological processes such as cytoskeleton ageing and rejuvenation [25,26]. This estimate therefore strongly suggests that the stresses applied here are too weak to induce non-linear rheological effects.

3.2.2. Overheating

The use of relatively high laser power in the sample raises the question of the amplitude of laser-induced overheating. We quantified this overheating by measuring the local viscosity of the culture medium when a particle is trapped at a laser intensity typical to those used in cell experiments. The viscosity η of Newtonian liquids indeed depends on temperature according to an Arrhenius law [27], $\begin{matrix} (1) & η (T) = \frac{h N_{A}}{V_{m}} exp (3.8 \frac{T_{b}}{T}), \end{matrix}$ with h, $N_{A}$ , $V_{m}$ and $T_{b}$ being Planck’s constant, Avogadro’s number, the molar volume and boiling temperature of the considered liquid, respectively.

We use optically-rotating vaterite microparticles to measure the viscosity [15,24,28]. In a Newtonian liquid, birefringent particles trapped in a circularly-polarised light beam indeed rotate at a constant rate, directly proportional to the optical torque, and inversely proportional to the viscosity of the surrounding liquid. Measuring, on the one hand, the rotation rate, and on the other hand, the torque, allows to deduce the local viscosity.

A set of viscosity measurements performed at variable power is presented in Fig. 4. The values found are close to that of water, consistently with the literature [29], and feature a clear decreasing trend that can be fitted using Eq. (1). Assuming that temperature depends linearly on laser power, we thus estimated a local temperature increase of about 3.7°C per 100 mW in the close proximity of the particle. Local heating remains therefore moderate (below 10°C) provided that the laser power is kept below ∼300 mW. In this range, the laser power density is actually comparable to that reported in a recent in-vivo investigation [30], which suggests that cell integrity should be preserved against optical damage.

Fig. 4.

Evolution of the viscosity of culture medium with the laser power in the sample. Solid line is a fit with Eq. (1).

3.2.3. Laser power effect

Finally, we experimentally assessed the possible influence of the laser power on the cell’s rheological response by performing measurements at variable laser power (gradually increasing up to 900 mW in the sample). Typical results are depicted in Fig. 5 for six cells, featuring exponents that can either remain fairly constant or vary, in some cases very significantly (up to about 0.4). The variation of the exponent, if any, is not correlated to the laser power, which confirms that the evolution of the exponent is not specifically caused by the laser forcing.

Fig. 5.

Evolution of the exponent α with the laser power in the sample, for six representative cells (distinct symbols denote distinct cells). Data corresponding to each individual cell were shifted vertically for clarity; vertical graduation unit is 0.2.

3.3. Temporal patterns

3.3.1. Exponent changes

To assess the generality of the temporal evolution reported above (Fig. 3), we reproduced the twisting dynamics measurements over $N = 31$ individual cells, taken from distinct culture dishes. While a minority (13%) of the tested cells featured roughly constant (within experimental uncertainty) exponent, we actually observed significant changes on most samples. Note that all the exponents measured fall within the distributions commonly reported when studying large cell populations [6,7,31].

More interestingly, the evolution of the exponent draws several well defined, nonmonotonic patterns, that seem quite reproducible from cell to cell. Figure 6 presents the three most typical patterns observed, superimposing several individual cells featuring them. The time origin has been set to match the individual evolutions of all considered cells; note that no correlation was found between the observed pattern and the amount of time the dish spent out of the incubator. Figure 6(a) and (b) show two patterns (U-shaped, as on Fig. 3, and Λ-shaped) characteristic of strongly variable exponents, representing about a third (32%) of the tested cells. In both cases, the exponent decreases (or increases) and symmetrically increases (decreases) by 0.2–0.3 in a matter of 4–5 min. Six cells are represented here, but several others featured similar evolutions, within slightly different time scales (7–8 min). Besides this strongly variable regime, many cells exhibited weaker fluctuations. The exponent oscillates by typically 0.05–0.1 around an average value close to 0.2 within about 10 min, as represented for 4 cells in Fig. 6(c). Quantitatively distinct patterns, featuring for example weak, apparently disordered fluctuations centred around a distinct mean exponent (0.1), were also observed. All together, these weakly-fluctuating patterns (fluctuation amplitude between 0.05 and 0.1) were observed in the majority of the considered population (55%).

Fig. 6.

Evolution of the exponent α over consecutive measurements on several representative cells. The time reference was set arbitrarily so as to superimpose data points corresponding to the distinct cells. Symbols represent distinct individual cells. The three graphs illustrate three observed variation patterns sketched by the lines on the top right corner: (a) U-shaped (2 cells), (b) Λ-shaped (4 cells), and (c) oscillating (4 cells).

These patterns suggest that the exponent may continuously evolve during cell’s life, each pattern observed here being possibly a given stage of this evolution. For example, one might expect that periods with strong modulations alternate with more stable periods. An isolated series of measurements, performed every 5 min on average over 90 min, seemed to show such alternation. However, the 5-min time step, comparable to the typical fluctuation period, is yet too long to observe the fluctuation patterns on the long term. Further long-term investigations performed with a finer time step, yet both highly time-consuming and demanding in terms of image analysis, should allow to plot the whole cell history.

3.3.2. Prestress

The changes in exponent measured here are the signature of a very significant prestress modulation. The exponent is indeed related to the cytoskeletal prestress through a logarithmic relationship [12]; the generality of this behaviour was actually shown among distinct cell types and experimental conditions, and myosin-cross-linked actin–filamin A networks – differences in contractility between these systems are accounted for by normalising the prestress values [2]. Thus, for instance, an exponent decrease from 0.3 to 0.1 corresponds to a prestress upscaling by a factor of about 300, while weaker fluctuations, of amplitude comprised between 0.05 and 0.1, would still denote a relative variation of the prestress by a factor of three to twenty. Our measurements therefore suggest that the cytoskeleton naturally modulates its prestress over time, around a mean value corresponding to $α ≃ 0.2$ . This modulation, of period typically around 4–10 min, itself varies in amplitude, being mostly quite subtle but sometimes very strong (up to three orders of magnitude), leading to the transient stiffening or softening of the cell in less than one minute.

An alternative explanation to the temporal evolution of the exponent could come from fluctuations in the probed area. Sub-cellular studies indeed showed distinct power-law responses between the cortical and intracellular media on the one hand [21], and between specific cytoskeletal areas on the other hand [32–34]. It is clear from the images we get that the particles remain outside of the cellular membrane, and we did not see them migrating along the cell membrane by more than a fraction of particle size. Invoking spatial fluctuations to explain the strong exponent changes observed therefore seems unreasonable. The geometric structure at the particle–cell contact surface may also remodel the membrane which would affect the particle’s response to the applied torque. This remodelling is very likely to happen (we actually observed variations in the deformation amplitude at given torque), but is not expected to change the frequency-dependence of the elastic modulus (i.e., the exponent α) [31]. We therefore believe that the exponent variations we observe do correspond to the actual modulation of cell prestress.

3.4. Individual cell variations

The rheological characterisation of a population of individual adherent cells would produce a distribution of power law dynamics. For example, Balland et al. performed a comprehensive investigation on distinct cell types (mouse ear fibroblasts and rat alveolar macrophages) and lines (C2 and C2-7 myogenic cells, A549 human alveolar epithelial cells, Madin–Darby Canine Kidney epithelial cells, and mouse fibroblasts L929), making one measurement per cell with two different techniques that stretch cells at distinct scales (local probe or whole cell) [7]. Strikingly, all populations yield overlapping normal distributions of exponents, all centred around a mean value $α_{0} ≃ 0.2$ , with a standard deviation σ typically in the order of 0.1, regardless of all parameters considered.

Fig. 7.

Distribution of the first exponent α measured on each individual cell (histogram, left axis). The cumulative distribution (right axis) is fitted with Eq. (2) (solid line).

For the sake of comparison, we also built the distribution of the exponents we obtained on each cell, considering only one measurement (say, the first one) to ignore the subsequent variations (Fig. 7). The cumulative distribution, superimposed to the histogram, is well adjusted by an error function, $\begin{matrix} (2) & E (α) = \frac{N}{2} (1 + \erf (\frac{α - α_{0}}{\sqrt{2} σ})), \end{matrix}$ confirming that our exponents also follow a normal distribution. In addition, the mean exponent $α_{0} = 0.180$ and the standard deviation $σ = 0.090$ are quantitatively comparable to those found by Balland et al. on distinct cells. Such broad distribution of exponents suggests that the variations found between individual cells may reflect their individual prestress state, if care is not taken to synchronise the whole cell population when performing measurements.

4. Concluding remarks

We developed a new technique to probe the power-law rheology of single cells. This technique is based on the application of an optical torque to a birefringent microparticle bound to the cell’s membrane. We observe that the power-law behaviour of the cells strongly and consistently evolves in time, without any external action on the sample other than the optical manipulation itself. We checked that this probing does not consistently affect the cell membrane’s properties, which suggests the measured fluctuations actually correspond to modulations of the cytoskeletal prestress that can span up to three orders of magnitude, within few minutes. Further confirmation of this hypothesis could arise by performing similar measurements on structurally simpler cells, such as red blood cells, in which the absence of cytoskeleton should result in a distinct, prestress-insensitive rheological behaviour.

Since the time scales involved are much shorter than the cell cycle, the prestress modulations are unlikely the signature of a specific stage of this cycle, such as cell’s division, but instead suggest a strong intrinsic activity occurring without any specific treatment. Such normal activity would make instantaneous rheological measurements on single cells meaningless, as no cell-to-cell comparison could be made unless special care is taken to take it into account. Temporal fluctuations may thus be at the origin of the broad distribution of exponents obtained on a set of single measurements [6,7].

Footnotes

Acknowledgements

We gratefully acknowledge Kishan Dholakia for hosting and funding experiments, and for comments on this work, and Alison McDonald for cell preparation and fruitful discussions. We also wish to thank Benjamin Dollet for proofreading and comments on the manuscript. This work was funded by the Engineering and Physical Sciences Research Council (EPSRC).

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