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Red Blood Cell Membrane Elasticity

Etirement Globule Rouge
Cell Microrheology

Optical Tweezers

Active forces in living cells

Dynamics of adhesive contacts



Optical trapping is achieved by focusing a powerful laser beam through a microscope objective of large numerical aperture. The gain in electromagnetic energy for a small dielectric particle of typical size d, located in the beam, is directly related to the light intensity, and thus restoring forces holds it in the focusing region.  More precisely, in the Rayleigh regime (d<<
) the force F exerted on a spherical silica or latex bead of optical index n is proportional to the intensity gradient, to the bead volume and to the index difference n²-n0² (n0 is the index of surrounding medium, usually water). The large numerical aperture increases the intensity gradient and ensures stable trapping, by making the gradient force in the z direction larger than the scattering force due to radiation pressure. Near infrared laser are usually used to minimize heating related to light absorption. Current trapping force on micron sized objects are in the range 1 - 100 pN for typical incident powers 10mW - 1W.  This is an adequate range to apply forces to biological objects, from single molecules to larger molecular assemblies, or to the whole cell.

Principle of optical tweezers

On the right figure, a high power infrared laser beam (here a Nd:YAG laser,
=1.064 µm, total power 600 mW) is focused through the immersion objective (NA>1.25) of an inverted optical microscope. The trap is located in the same plane as the observation plane of the microscope. One can use either galvanometric mirrors or acousto-optic modulators to control the trap position or to create multiple traps by rapidly commuting the focusing point between different positions.  The role of the afocal telescope is to enlarge the beam and to conjugate the mirrors and the objective pupil.

To lowest order, the restoring force F exerted on a trapped bead in the observation plane can be written F = -kr, where r represents the distance of the bead from the trap center. The trap stiffness k, proportional to the laser power, must be determined through an independant calibration, for instance by applying a known viscous drag force to the bead.  In steady state regime, F'=F, which allows to determine the stiffness k.

Scheme of the optical tweezers set up
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(Coll. Sylvie HENON, PhD Guillaume LENORMAND)

The human red blood cell (RBC) membrane is made of a lipid bilayer reinforced on its inner face by a flexible two-dimensional protein network. This skeleton is made of spectrin dimers associated to form mainly tetramers, approximately 200 nm long. They are linked together by complex junctions (primarily composed of actin filaments and protein 4.1) and attached to the lipid bilayer via transmembrane proteins (glycophorin C and band 3). In a simplified description, they form a triangular network in which each actin filament is connected to six spectrin tetramers.
In the classical elastic model, the membrane response to a given stress is assumed to be linear for small deformations and is characterized by three elastic moduli: the area expansion modulus K, the shear modulus µ and the binding stiffness B. The binding stiffness ranges from a few kBT to a few tens of kBT, and is too small to have a noticeable influence on the membrane response to an in-plane stress. The area expansion modulus of the RBC membrane is determined by the total amount of lipids in the bilayer, and the membrane extension under stretching is usually small. By comparison, the deformation under a shear stress is quite important and is mainly controlled by the elastic response of the spectrin network.

Schematic view of the Red Blood Cell membrane

In order to measure the membrane shear modulus µ, a red blood cell is seized and deformed by means of two small silica beads (2.1 µm in diameter) non-specifically bound to the membrane. The beads are trapped in split optical tweezers, and used like handles to pull on the membrane The measurements are made on selected RBCs with two silica beads in diametrical position.  For a given laser power P, one applies an increasing stress to the membrane by slowly incrementing the distance between the two trapped beads. The cell becomes elongated and its diameter D decreases in the direction perpendicular to the applied force F.
In the small deformation limit (linear regime), it can be shown that :
D = Do -

The shear stress µ is infered from the slope of the curve D vs F.

Stretching of a RBC between two silica beads trapped in two optical traps

The histogram on the right shows measurements of the shear stress µ for RBCs in different buffers (red : discotic RBC in isotonic conditions; blue : swollen RBCs in hypotonic buffer. In both cases, as far as the force remains small enough (typically F<15 pN), D presents linear variations with F. The shear modulus is deduced from the slope : on average one finds µ = 2.5 ± 0.4 mN/m for discotic and µ = 2.35 ± 0.5 mN/m for spherical cells. At higher forces, one observes deviations from linear regime.
Our value µ = 2.5 ± 0.4 µN/m is comparable but definitely smaller than other measurements. A widely applied method consists in measuring µ from the length of RBCs aspirated in a micropipette ; the values obtained by this technique lie between 4 and 10 µN/m, the most commonly accepted value being around 6 µN/m.  However, the different techniques may not operate on the same RBCs population. The optical tweezers technique concerns cells which are tightly bound to glass beads. On the contrary, the micropipette technique supposes that the cells do not adhere to the glass pipette wall, because this would make friction forces play an important role in the force balance. Thus one can suppose that these two techniques sort out the cells according to their affinity to glass, which depends on the membrane surface state, and which may be associated to different elastic properties from one cell to another.
Histogram of shear modulus values
for discotic and spherical RBCs

We also measured the elastic behavior of the isolated spectrin skeleton, after extraction of the lipid bilayer. This bidimensional network is characterized by its own area expansion modulus KC and shear modulus µC. For a triangular network constrained to in-plane deformations, analytical and numerical works have predicted that the ratio KCC should be equal to 2 in the small stress regime. We measured both  KC and µC by means of three silica beads bound to the extrated network and manipulated with three optical traps. In a low osmolarity buffer, we find <KC>=4.8 ± 2.7 µN/m, <µC>=2.4 ± 0.7 µN/m and <KCC>=1.9 ± 1.0. The skeleton shear modulus µC has the same order of magnitude as the membrane shear modulus µ. The ratio <KCC> is close to values obtained by theoretical and numerical studies. We also measure KC and µC in an isotonic buffer in order to evaluate the influence of the ionic strength.

Fluorescently labelled spectrin network, manipulated by three optical traps after extraction of the lipid bilayer
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(Coll. Sylvie HENON, Atef ASNACIOS, PhD Martial BALLAND, Nicolas DESPRAT, Delphine ICARD-ARCIZET)
To probe the dynamic properties of the actin network in a single cell, silica microbeads are specifically bound to transmembrane receptors and manipulated as handles  to apply a mechanical stress to the cell cytoskeleton. The specific binding (here to integrins) is achieved by coating the beads with RGD tripeptide. A bead is seized with the optical trap, which remains at a fixed position, while the experimentalchamber is mounted on a piezoelectric stage, allowing the application of either constant or variable stresses to the cell. The force/displacement relationship in static or quasi-static regime leads to the Young modulus E of the cytoskeletal network at zero frequency. Applying a sinusoidal stress (here in the range 0.1-100 Hz) allows to measure the frequency dependance of its viscoelastic modulus Ge(f) :


Scheme of the microrheological experiment

The measured values of the modulus |Ge| and of the phase of Ge(f) are shown on the right side for a single myoblast (C2 cell). |Ge| behaves as a power law of f over three frequency decades. For this particular cell, the exponent  is found equal to 0.30, and the value G0 of |Ge| at 1Hz equal to 155 Pa. Moreover, the measured phase shift  remains constant, within a good approximation.
For a whole set of C2 cells tested with optical tweezers, Ge(f) present similar power law behaviors. The distribution of exponents  follows a normal law, with <> = 0.208 ± 0.021, while the distribution of G0 appears log-normal, with G0M = 310 (+130/-100) Pa (median value).

Treating cell with blebbistatin, an inhibitor of actomyosin molecular motor, lowers both the value of the exponent down to <> = 0.10, and of the prefactor G0.  This indicates that the myosin activity takes a large part in the mechanical dissipation inside the cell.

Power-law variations of the viscoelastic modulus G(f)

We have performed microrheological experiments on several other cell types, using two complementary techniques : optical tweezers or uniaxial stretching (which leads to the creep function J(t)), and in various coating conditions. The individual cell behavior appears independant of the cell type and of the nature of the receptor transmitting the mechanical stress (integrin, cadherin, ICAM). When stretching the whole cell, the creep function
J(t) is accurately adjusted by a power law function of time t. Similarly, in oscillating force experiments, the viscoelastic modulus Ge(f) behaves as a power law of the frequency, with the same exponent . The average value <>  always remains in the range 0.15-0.25, whatever the cell type and function: this holds for premuscular cells (C2 myoblasts), epithelial cells (alveolar A549 and MDCK) , fibroblasts (primary and L929), and macrophages (primary). The prefactor G0 differs from one cell type to the other.

Results for different cell types and different techniques
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(Coll. Sylvie HENON, Atef ASNACIOS, Julien BROWAEYS)

The power law rheology indicates that there is no characteristic dissipation time in the cellular response, or more exactly that the relaxation times are broadly distributed over a wide time interval. This behavior is characteristic of structural damping, where the mechanisms responsible for the storage of elastic energy and its dissipation are strongly correlated.
Our model considers that the cytoskeleton is made of many interconnected units of different length scales (from actin individual filaments to actin bundles and stress fibers). Their size continuously spread from the nanometer scale to the scale of the whole cell. We describe the mechanical response of each unit, labelled by the index i, by a simple Kelvin-Voigt model with a set of response time i widely and densely distributed.
Assuming that the cytoskeletal structure is close to a fractal (as supported by fluorescent images of the actin stress fibers, bundles and filaments), the relaxation times i are naturally distributed according to a power law. As a consequence, the creep function J(t) (or the viscolelastic modulus Ge(f)) is found to behave as a power law of time t (or of frequency f). 

Actin labelled epithelial cell

The mechanical answer is modelled as an assembly of units with power law distributed response time

By summing the contributions of 10
5 elementary units, we numerically calculated the creep function dJ()/d as a function of the reduced time /m, where m is the largest relaxation time in the cell. The result (in black on the right figure) is quite well adjusted by a power law.
To build a more realistic picture of the cytoskeleton dynamics, taking into account the dispersion observed from one cell to the other, we assumed that only a proportion p of the elementary Kelvin-Voigt units are actually present in a given cell. The actual distribution of relaxation times is constructed by selecting a random set of relaxation times i from the ideal power law distribution. A set of numerical calculations of dJ()/d are shown in red on the right figure.
They roughly behave as power laws of 
, and exhibit approximately the same exponent as the ideal response.  The exponents and prefactors are found respectively normally and log-normally distributed, in agreement with experiments. Precise adjustment with experimetnal data leads to a reasonnable value for the largest relaxation time in the cell m,  about 103s.

The simulated creep function for the above model
behaves as a power law of time
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(Coll. Claire WILHELM, PhD Pierre BOHEC and Damien ROBERT)

Living cells are constitutively out-of-equilibrium systems. Biological activity is characterized by constant exchanges of matter and energy with the environment, and biochemical reactions provide the power supply necessary to the metabolism. Non-equilibrium activity allows cell motion and deformation when needed for function or survival, and cargo transport within the dense microenvironment of the cell. These mechanical effects are directly related to the dynamics of polymerization/ depolymerization of the cytoskeleton and to the activity of molecular motors, which convert the chemical energy stored in ATP into mechanical energy. 

Consequently, the fluctuation-dissipation theorem (FDT) is violated within a living cell. At thermal equilibrium, this theorem relates the mean square displacement (MSD) of a micron-sized probe diffusing in a viscoelastic medium to its mobility µ and to the temperature T. If the medium is not at equilibrium, one can nevertheless relate the MSD and the mobility to the spectrum of the non-thermal forces acting on the probe. A Generalized Einstein Relation is sometimes writen, replacing T by a so-called "effective temperature" Teff, which depends on frequency.
For instance, we recently brought evidence of such deviations from equilibirum FDT in an aging colloidal glass made of Laponite, by
combining the results of  active and passive microrhelogy experiments, simultaneously performed on a bead embedded in the glass (collaboration B. Abou, P. Monceau, N. Pottier, MSC)

Relations between mean square displacement and mobility,
at thermal equilibrium (FDT) or out of equilibrium

Due to biological activity and especially to molecular motors, the intracellular medium is not at thermal equilibrium and the fluctuation-dissipation theorem no longer applies. We have performed active and passive microrheology experiments on the same probe linked to the actin cytoskeleton, namely a bead specifically bound to integrins, which are transmembrane adhesive receptors linked to the actin cortex.  

The experiment consists in (i) tracking the free diffusion of the bead and calculating its mean square displacement (passive rheology), and (ii) measuring its creep function, in response to a step force applied with an optical tweezers (active rheology). The MSD curve exhibits two regimes of diffusion: a subdiffusive regime at short time scales (t<2s), and a superdiffusive one at large time scales (t>2s).  The creep function J(t) = x(t)/F exhibits the same weak power law behavior over the full time range.

By combining both results one obtains the Laplace transform of the autocorrelation function <F(t)F(t+t')> of the forces exerted on the bead. This force spectrum is found larger than the spectrum calculated at equilibrium. The departure from equilibrium corresponds to more than one order of magnitude for cells in control conditions, and is reduced for ATP depleted cells. Very similar observations are made on magnetic endosomes embedded in the cell body, and animated by non-equilibrium
active forces which drive them along the microtubules network.

We interpret the force spectra as generated by molecular motors acting cooperatively, and in a processive way, to inprint a directed motion to the probe. A model taking into account the distribution of step force durations accurately accounts for the observations.

A probe bound to the membrane is driven by active forces
generated by molecular motors


Effective temperature for the probe attached to the cortex
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(Coll. Sylvie DUFOUR, Institut Curie, Post-Doc ALIA AL-KILANI)

In a living tissue, a cell builds several types of contacts with the extracellular matrix (ECM) and with the adjacent cells, especially focal adhesions (involving integrins) and adherens junctions (involving cadherins). These complex act as mechanosensors and their structure is sensitive to the mechanical environment. For instance, it has been observed that the contacts grow with the substrate rigidity, and that they strengthen under the application of an external mechanical stress.

Here the goal is to investigate the coupling between the cell-cell and cell-matrix adhesion complexes, and the mechanisms of communication (crosstalk) which ensure the modulation of the adhesive and mechanical properties, and the cohesion of the tissues.
The work focuses on the structural changes and rigidity variations of intercellular contacts, induced by a controlled modulation of the density and of the spatial distribution of cell-ECM contacts.

The cell adhesiveness to the ECM is modulated by using micropatterned substrates of various shape and size, stamped with fibronectin, constraining the cell to spread on a controlled area and geometry. A bead coated with cadherin fragments is bound to the cell membrane wirth an optical tweezers, to simulate an intercellular contact. The mechanical rigidity, and the force necessary to disrupt this contact is measured as a function of the time elapsed from contact initiation, while visualizing by fluorescence microscopy the remodeling of its structure. Carcinoma S180 cells are used as a biological model, since they can be transfected to express the different adhesive receptors (cadherins, integrins) in a controlled way.

As a preliminary result, we observe that the rigidity of the intercellular contact, five minuts after initiation, is a decreasing function of the area allowed for  cell-matrix adhesion. 

E-cadherin  GFP labelled S180 cell
 with a cadherin coated bead
simulating a cell-cell contact


Adhesive pattern stamped with fibronectin

Scheme of crosstalk experiment
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