A master relation defines the nonlinear viscoelasticity of single fibroblasts

Cell mechanical functions like 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.Comment: 12 pages, 11 figures. Accepted for publication in Biophysical Journal, scheduled to appear in May 200

A one-step procedure to probe the viscoelastic properties of cells by Atomic Force Microscopy

The increasingly recognised importance of viscoelastic properties of cells in pathological conditions requires rapid development of advanced cell microrheology technologies. Here, we present a novel Atomic Force Microscopy (AFM)-microrheology (AFM2) method for measuring the viscoelastic properties in living cells, over a wide range of continuous frequencies (0.005 Hz ~ 200 Hz), from a simple stress-relaxation nanoindentation. Experimental data were directly analysed without the need for pre-conceived viscoelastic models. We show the method had an excellent agreement with conventional oscillatory bulk-rheology measurements in gels, opening a new avenue for viscoelastic characterisation of soft matter using minute quantity of materials (or cells). Using this capability, we investigate the viscoelastic responses of cells in association with cancer cell invasive activity modulated by two important molecular regulators (i.e. mutation of the p53 gene and Rho kinase activity). The analysis of elastic (G′(ω)) and viscous (G″(ω)) moduli of living cells has led to the discovery of a characteristic transitions of the loss tangent (G″(ω)/G′(ω)) in the low frequency range (0.005 Hz ~ 0.1 Hz) that is indicative of the capability for cell restructuring of F-actin network. Our method is ready to be implemented in conventional AFMs, providing a simple yet powerful tool for measuring the viscoelastic properties of living cells

Modeling problems in mucus viscoelasticity and mucociliary clearance

From the common cold and allergies to severe chronic and acute respiratory impairments, the function of the body\u27s mucociliary clearance mechanism plays a primary defense role. Mucus demonstrates numerous non-Newtonian behaviors which set it apart from viscous fluids. Among them: Bingham plastic behavior, shear-thinning, and elasticity on short time scales due to relaxation time. Experimental evidence suggests that certain rheologies promote effective transport. In an effort to reveal the mechanisms controlling transport, models are developed. Firstly, a steady state model which idealizes the mucus as a rigid body is created in order to bring together disparate bodies of experimental work from the literature. The force balance reveals that the force cilia are capable of exerting cannot be related, simply, to the velocity of mucus. That is, only a fraction of the force cilia are capable of exerting is required to steadily transport mucus at the velocities observed experimentally. Likewise, the velocities estimated by this model when cilia force is the input are overestimated by one to two orders of magnitude. This incongruity formally motivates the inclusion of one of mucus\u27s rheological behaviors, stress relaxation. The first viscoelastic problem considered is the response of the linear Maxwell fluid to an oscillating plate. Though a problem commonly discussed in textbooks on theoretical viscoelasticity, the complete analytical solutions are not provided. Here, solutions are found and graphed in terms of the phase and amplitude of the velocity field resultant from the oscillations of the plate; all derivations are shown in their entirety. The effects of stress relaxation (sometimes referred to as memory) and inertia on phase and amplitude are shown to have frequency dependence. Furthermore, it is shown that oscillatory shear perturbations to a viscoelastic Maxwell fluid always travel further and faster away from the source as Deborah number (a dimensionless parameter governing the importance of viscoelastic forces, De=0 corresponds to a Newtonian fluid) is increased. The limitation of the linear Maxwell fluid is illustrated by attempting to apply the constitutive equation to a thin film flow problem. It is found that the stress field of the solution only differs from the viscous case if the boundary conditions are transient; that is, the constitutive equation cannot account for the changes in stress that occur over space. The time derivative must be replaced by a Convected Derivative to achieve the proper Lagrangian to Eulerian coordinate transformation and is considered in a final set of problems. Three problems were completed using the Upper Convected Maxwell model for viscoelasticity. The first considers a purely unidirectional shear flow which, unlike a viscous fluid, possesses tensile stresses along streamlines. The model posits that these additional stresses are essential for transport by allowing regions which are actively sheared, to hold up inactive regions. A novel relationship between applied stress and relaxation time is developed; the model shows that increasing the relaxation time of mucus decreases the amount of stress that must be imparted by cilia. In the second two problems, the UCM equations are simplified via a perturbation series expansion for small Weissenberg number (also a dimensionless group governing the importance of viscoelastic forces). This technique allows the analytically solvable viscous (also referred to as the unperturbed or order one) solutions to be used to estimate the impact of small amounts of stress memory. It is found that elasticity increases the developing region of a viscous flow; all stress components are convected downstream due to flow memory. Likewise, in the sinusoidally varying stress case, the velocity field is always shifted further away from the phase of the applied stress as viscoelastic forces are increased. It is also found that the departure from the viscous solution is dramatically reduced if the stress distribution is moving at the same velocity as the mucal flow. This shows the benefit of an antiplectic wave speed (the physiologically relevant case in which the phase of the cilial beat is moving opposite to transport) as there is no danger that these two can be in phase with one another. Model restrictions prevent quantitative gauges of transport efficiency as a function of metachronal wave parameters and relaxation time to be made. Several additional problems are proposed to address unanswered modeling questions and experimental solutions for the lack of rheological data on tracheal mucus are suggested

Strain-Rate Frequency Superposition in Large-Amplitude Oscillatory Shear

In a recent work, Wyss, {\it et.al.} [Phys. Rev. Lett., {\bf 98}, 238303 (2007)] have noted a property of `soft solids' under oscillatory shear, the so-called strain-rate frequency superposition (SRFS). We extend this study to the case of soft solids under large-amplitude oscillatory shear (LAOS). We show results from LAOS studies in a monodisperse hydrogel suspension, an aqueous gel, and a biopolymer suspension, and show that constant strain-rate frequency sweep measurements with soft solids can be superimposed onto master curves for higher harmonic moduli, with the {\it same} shift factors as for the linear viscoelastic moduli. We show that the behavior of higher harmonic moduli at low frequencies in constant strain-rate frequency sweep measurements is similar to that at large strain amplitudes in strain-amplitude sweep tests. We show surface plots of the harmonic moduli and the energy dissipation rate per unit volume in LAOS for soft solids, and show experimentally that the energy dissipated per unit volume depends on the first harmonic loss modulus alone, in both the linear and the nonlinear viscoelastic regime.Comment: 10 pages, 25 figures, accepted for publication in Physical Review E. Incorporates referee comment

Dissipative Dynamics of Polymer Phononic Materials

Phononic materials are artificial composites with unprecedented abilities to control acoustic waves in solids. Their performance is mainly governed by their architecture, determining frequency ranges in which wave propagation is inhibited. However, the dynamics of phononic materials also depends on the mechanical and material properties of their constituents. In the case of viscoelastic constituents, such as most polymers, it is challenging to correctly predict the actual dynamic behavior of real phononic structures. Existing studies on this topic either lack experimental evidence or are limited to specific materials and architectures in restricted frequency ranges. A general framework is developed and employed to characterize the dynamics of polymer phononic materials with different architectures made of both thermoset and thermoplastic polymers, presenting qualitatively different viscoelastic behaviors. Through a comparison of experimental results with numerical predictions, the reliability of commonly used elastic and viscoelastic material models is evaluated in broad frequency ranges. Correlations between viscous effects and the two main band-gap formation mechanisms in phononic materials are revealed, and experimentally verified guidelines on how to correctly predict their dissipative response are proposed in a computationally efficient way. Overall, this work provides comprehensive guidelines for the extension of phononics modeling to applications involving dissipative viscoelastic materials.</p

A Laplace Transform Method for Molecular Mass Distribution Calculation from Rheometric Data

Polydisperse linear polymer melts can be microscopically described by the tube model and fractal reptation dynamics, while on the macroscopic side the generalized Maxwell model is capable of correctly displaying most of the rheological behavior. In this paper, a Laplace transform method is derived and different macroscopic starting points for molecular mass distribution calculation are compared to a classical light scattering evaluation. The underlying assumptions comprise the modern understanding on polymer dynamics in entangled systems but can be stated in a mathematically generalized way. The resulting method is very easy to use due to its mathematical structure and it is capable of calculating multimodal molecular mass distributions of linear polymer melts