2 search hits
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Mechanics of living cells: nonlinear viscoelasticity of single fibroblasts and shape instabilities in axons
(2006)
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Pablo Fernandez
- Biomechanics is a field of major biological relevance. In spite of the vast complexity of biological matter, a number of generic features are found to hold in the mechanics of soft tissues throughout all of its length scales. A major goal in biomechanics is to reduce its general features to those of the cytoskeleton, the filamentous scaffold which provides cells with mechanical integrity, architecture and contractility. The first part of this report describes single-cell uniaxial stretching experiments performed on fibroblasts. When placed between fibronectin coated microplates, fibroblasts adopt a regular, symmetrical shape and generate forces. When a constant cell length is imposed, an increase with time of the pulling force can be observed. This active behaviour can be probed in more detail by superimposing small-amplitude oscillations at frequencies in the range 0.1--1 Hz. The response to the superimposed oscillations is then characterised by the viscoelastic moduli. These are seen to be a function of the average force acting on the cell. This master-relation holds for all cells. At low forces, both moduli are constant; beyond a crossover force, power-law stress stiffening is observed, where as a function of the average force both moduli go as a power-law with exponents in the range 1-1.8. The loss factor depends only weakly on the average force. Remarkably, the moduli are a function of the average force but are independent of the cell length. Therefore this mechanical behaviour is not strain stiffening; rather, it is an example of active, intrinsic stress stiffening. The precise way of sweeping force-space is seen to be irrelevant. The stiffening relation shows a striking similarity to rheological measurements performed on purified actin gels, in an unprecedented example of quantitative agreement between living and dead matter. This mechanical response originates in the semiflexible behaviour of biopolymers. The precise mechanism is however at present not fully understood. Here, a simple explanation is proposed. It is shown that stress stiffening in fibroblasts bears a strong resemblance to the nonlinear mechanics of Euler-Bernoulli beams, which also show a linear regime at low forces and a crossover to power-law stiffening. Systematic analysis of the response of fibroblasts to large amplitude deformations reveals a striking similarity to plasticity in metals. Fibroblasts can be described as showing kinematic (or directional) hardening, a hallmark of composite materials. The second part of this report addresses experiments performed on neurites. These comprise axons --the processes extended by neurons-- as well as PC12 neurites, a model system for axons. After a sudden increase in the external osmotic pressure, axons swell and a cylindrical-peristaltic shape transformation sets in. We interprete this transition as a Rayleigh-Plateau-like instability triggered by elastic membrane tension, similar to the pearling instability known in membrane tubes. Microtubuli disruption by nocodazol strongly increases the maximum amplitude of the instability, as well as slightly increases the wavenumber of the fastest mode, showing microtubuli to be the most important cytoskeletal component in stabilising neurites. After a hypoosmotic shock the neurite volume increases, reaches a maximum, and relaxes back close to its initial value. These experiments were performed at different temperatures and initial osmotic pressure differences. The relaxation time as a function of the temperature closely follows an Arrhenius dependence, suggesting the rate-limiting factor of the relaxation to be the movement of ions through channels. Similar experiments were also performed under drug-induced perturbation of actin, myosin and microtubuli. Cytoskeleton perturbation does not have any significant effect on volume relaxation, indicating that it takes place solely by changes in osmolarity, without a significant role for hydrostatic pressures. A clear effect of drugs is seen in the initial swelling phase, especially after microtubuli disruption by nocodazol. The rate and extent of swelling are significantly higher. Taking the effect of drugs on the evolution of neurite volume together with that on the pearling instability, we suggest that hydrostatic pressure is present in the initial swelling phase and determines the swelling rate. In conclusion, reproducible, quantitative experiments at the single-cell level have been developed which address biologically relevant phenomena. Following a time-honoured tradition in physics, both the cell-pulling experiments and the shape transformations in axons address highly symmetric systems, where the geometry does not preclude the understanding. First interpretations of the observed phenomena have been found, in terms of generic behaviours common to all objects under tension.
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Modeling Pattern Formation in Biopolymer Systems induced by Reaction Kinetics and Molecular Motors
(2006)
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Falko Ziebert
- In my thesis I studied pattern formation in nonequilibrium (NE) polymer systems motivated from cell biology. Actin and microtubules (MTs) can be met in a state of continuous de-/polymerization (D/P), which is used by the cell e.g. during locomotion. This is a NE state since the polymerization is actively coupled to ATP or GTP hydrolysis. A second NE state of biological relevance is caused by motor proteins. These are mobile crosslinkers that walk on the filaments whereby creating forces and reorienting or transporting the latter. The cell displays filament-related ordered structures like aster patterns in the mitotic spindle, bundles in actin stress fibers and also oscillating structures e.g. in muscle bundles. The question is to what extent these structures inside the cell are governed by the physics of active polymers. In part I of my work, we proposed a pattern forming mechanism in a filament solution at high density that is subject to a D/P state. Since actin and MTs are rod-like objects, at high filament concentration a transition to lyotropic nematic order occurs. This transition is first order and thus accompanied by a phase separation. In the absence of D/P kinetics, the solution will thus tend to decompose into an isotropic domain with low density and a domain of high density and nematic order, i.e. the filaments preferentially aligned in one direction. To highlight that the D/P process interplays with this transition, we assumed that filaments are generated and decaying with some specific rates, implying a finite lifetime for the filaments. Accordingly the latter can only diffuse a finite length during their lifetime, which competes with the tendency of the system to phase separate and gives rise to a finite wavelength instability towards a pattern with alternating isotropic and nematic regions with a wavelength of the order of 10 microns. The model developed to describe these patterns is also interesting since it allows a feasible linear stability analysis of the homogeneous nematic state. Part II is devoted to the NE interaction of motor proteins with the filaments. As the starting point of our modeling efforts we chose a mesoscopic approach, namely a Smoluchowski equation which can be coarse-grained to obtain equations for the density and the orientation of the filaments. The main difference to a passive solution of rods are active motor-mediated currents caused by a motor density assumed sufficiently high and homogeneously distributed. These active contributions can be determined to leading order, introducing phenomenological motor transport rates containing details like active motor density, duty ratio, etc. After a thorough linear analysis of the model we obtained a rich instability diagram with an orientational finite wavelength instability which is either stationary or oscillatory and a demixing instability similar to spinodal decomposition but also motor-mediated. The finite wavelength instability has been analyzed by perturbative techniques and numerical simulations of the model equations. In the stationary case, we calculated the existence and stability regions of stripes and squares, which could be related to bundle-like structures and regular lattices of asters respectively. In the oscillatory case, there is competition between traveling and standing waves in one dimension and between traveling and alternating waves in two dimensions, the latter being a four mode solution built from two standing waves in perpendicular directions with a phase shift of 90 degrees. The long-wavelength demixing instability has also been investigated, showing coarsening aster-like structures. Experiments on MT-motor solutions display dissipative patterns in the NE state. Recent experiments on actin filaments and myosin oligomers show a rather different behavior, namely cluster patterns do not appear until ATP is nearly depleted. We proposed two mechanisms to explain these patterns: first, motors lacking ATP form rigor bonds with actin inducing small bundles, which through a combination of reduced diffusivity and enhanced interaction cross-section can be transported more efficiently, allowing the system to cross one of the instabilities discussed above. A second important feature is the presence of crosslinking proteins in the experiments. We propose that these can be interpreted as a parametric disorder. Assuming in the model a random contribution to the active current, a Ginzburg-Landau equation with multiplicative stationary noise could be derived leading to a threshold reduction. To conclude, it seems to be fruitful to apply and combine methods from statistical physics and pattern formation to NE problems in cell biology to foster the understanding of actively polymerizing filament and motor proteins in their different NE states.
