Mechanical properties of pleural membrane

The pleural membrane is modeled as a planar collection of interconnected randomly oriented line elements. By assuming that the line elements follow the strain field of a continuum, a strain-energy function is formulated. From the strain-energy function, an explicit stress-strain equation for large deformations is derived. In the linear approximation of the stress-strain equation the shear modulus and the area modulus of the membrane are respectively found to be 2.4 and 2.8 times the tension at the reference state. The stress-strain equation for large deformations is used to predict the displacement field around a circular hole in pleura. Good agreement is found between these predictions and measurements made on ablated pleura from dog lungs. From these theoretical and experimental results the conclusion is drawn that the pleura has a significant role in carrying shear forces and maintaining the lung's shape.     

Softness, strength and self-repair in intermediate filament networks

One cellular function of intermediate filaments is to provide cells with compliance to small deformations while strengthening them when large stresses are applied. How IFs accomplish this mechanical role is revealed by recent studies of the elastic properties of single IF protein polymers and by viscoelastic characterization of the networks they form. IFs are unique among cytoskeletal filaments in withstanding large deformations. Single filaments can stretch to more than 3 times their initial length before breaking, and gels of IF withstand strains greater than 100% without damage. Even after mechanical disruption of gels formed by crossbridged neurofilaments, the elastic modulus of these gels rapidly recovers under conditions where gels formed by actin filaments are irreversibly ruptured. The polyelectrolyte properties of IFs may enable crossbridging by multivalent counterions, but identifying the mechanisms by which IFs link into bundles and networks in vivo remains a challenge.     

Does subcutaneous adipose tissue behave as an (anti-)thixotropic material?

Although subcutaneous adipose tissue undergoes large deformations on a daily basis, there is no adequate mechanical model to describe the transfer of mechanical load from the skin throughout the tissue to deeper layers. In order to develop such a non-linear model, a set of experimental data is required. Accordingly, this study examines the long term behavior of adipose tissue under small strain and its response to various large strain profiles. The results show that the shear modulus dramatically increases to about an order of magnitude after a loading period between 250 and 1250 s, but returns to its initial value within 3 h of recovery from loading. In addition, it was observed that the stress–strain responses for various large strain history sequences are reproducible up to a strain of 0.15. For increasing strains, the stress decreases for subsequent loading cycles and, above 0.3 strain, tissue structure changes such that the stress becomes independent of the applied strain. From the results, it can be concluded that adipose tissue likely behaves as an (anti-) thixotropic material and that a Mooney–Rivlin model might be appropriate to simulate behavior at physiologically relevant high strains. However, before the model is developed more fully, further experimental research is needed to ratify that the material is (anti-)thixotropic.     

An elastic network model based on the structure of the red blood cell membrane skeleton.

A finite element network model has been developed to predict the macroscopic elastic shear modulus and the area expansion modulus of the red blood cell (RBC) membrane skeleton on the basis of its microstructure. The topological organization of connections between spectrin molecules is represented by the edges of a random Delaunay triangulation, and the elasticity of an individual spectrin molecule is represented by the spring constant, K, for a linear spring element. The model network is subjected to deformations by prescribing nodal displacements on the boundary. The positions of internal nodes are computed by the finite element program. The average response of the network is used to compute the shear modulus (mu) and area expansion modulus (kappa) for the corresponding effective continuum. For networks with a moderate degree of randomness, this model predicts mu/K = 0.45 and kappa/K = 0.90 in small deformations. These results are consistent with previous computational models and experimental estimates of the ratio mu/kappa. This model also predicts that the elastic moduli vary by 20% or more in networks with varying degrees of randomness. In large deformations, mu increases as a cubic function of the extension ratio lambda 1, with mu/K = 0.62 when lambda 1 = 1.5.     

The optimal density of cellular solids in axial tension

For cellular bodies with uniform cell size, wall thickness, and shape, an important question is whether the same volume of material has the same effect when arranged as many small cells or as fewer large cells. To answer this question, for finite element models of periodic structures of Mooney-type material with different structural geometry and subject to large strain deformations, we identify a nonlinear elastic modulus as the ratio between the mean effective stress and the mean effective strain in the solid cell walls, and show that this modulus increases when the thickness of the walls increases, as well as when the number of cells increases while the volume of solid material remains fixed. Since, under the specified conditions, this nonlinear elastic modulus increases also as the corresponding mean stress increases, either the mean modulus or the mean stress can be employed as indicator when the optimum wall thickness or number of cells is sought.     

Alterations of the apparent area expansivity modulus of red blood cell membrane by electric fields.

Red blood cell membrane exhibits a large resistance to changes in surface area. This resistance is characterized by the area expansivity modulus K, which relates the isotropic membrane force resultant, T, to the fractional change in membrane surface area delta A/Ao. The experimental technique commonly used to determine K is micropipette aspiration. Using this method, E. A. Evans and R. Waugh (1977, Biophys. J. 20:307-313) obtained a value of 450 dyn/cm for the modulus. In the present report, it is shown that the value of K, as determined using this method, is affected by electric potential differences applied across the tip of the pipette. Using Ag-AgCl electrodes and current clamping electronics, we obtained values for K ranging from 150 dyn/cm with -1.0 V applied, to 1,500 dyn/cm with 1.0 V applied. At 0.0 V the modulus obtained was approximately 500 dyn/cm. A reversible, voltage- and pressure-dependent change in the cell volume probably accounts for the effect of the voltage on the calculated value of the modulus. The use of lanthanum chloride or increasing the extra- and intracellular solute concentrations reduced the voltage dependence of the measurements. It was also found that when dissimilar metals were used to "ground" the pipette to the chamber to prevent lysis of cells by static charge, values for K ranged from 121 to 608 dyn/cm. Based on measurements made at zero applied volts, in the presence of 0.4 mM lanthanum and at high solute concentration, we conclude that the true value of the modulus is approximately 500 dyn/cm.     

Cell-to-cell variability in deformations across compressed myoblasts

Many biological consequences of external mechanical loads applied to cells depend on localized cell deformations rather than on average whole-cell-body deformations. Such localized intracellular deformations are likely to depend, in turn, on the individual geometrical features of each cell, e.g., the local surface curvatures or the size of the nucleus, which always vary from one cell to another, even within the same culture. Our goal here was to characterize cell-to-cell variabilities in magnitudes and distribution patterns of localized tensile strains that develop in the plasma membrane (PM) and nuclear surface area (NSA) of compressed myoblasts, in order to identify resemblance or differences in mechanical performances across the cells. For this purpose, we utilized our previously developed confocal microscopy-based three-dimensional cell-specific finite element modeling methodology. Five different C2C12 undifferentiated cells belonging to the same culture were scanned confocally and modeled, and were then subjected to compression in the simulation setting. We calculated the average and peak tensile strains in the PM and NSA, the percentage of PM area subjected to tensile strains above certain thresholds and the coefficient of variation (COV) in average and peak strains. We found considerable COV values in tensile strains developing at the PM and NSA (up to ~35%) but small external compressive deformations induced greater variabilities in intracellular strains across cells compared to large deformations. Interestingly, the external deformations needed to cause localized PM or NSA strains exceeding each threshold were very close across the different cells. Better understanding of variabilities in mechanical performances of cells-either of the same type or of different types-is important for interpreting experimental data in any experiments involving delivery of mechanical loads to cells.     

A model of non-uniform lung parenchyma distortion

A finite element model of mammalian lung parenchyma is used to study the effect of large non-uniform distortions on lung elastic behaviour. The non-uniform distortion is a uni-axial stretch from an initial state of uniform pressure expansion. For small distortions, the parenchymal properties are linearly isotropic and described by two elastic moduli. However, for large distortions, the parenchyma has anisotropic non-linear elastic properties described by five independent elastic moduli dependent on the degree of distortion; they are computed for a range of distortions and initial pressures. Ez, the Young's modulus in the direction of stretch, increases significantly with distortion (epsilon(z)) while Ex, the Young's modulus in the plane perpendicular to the stretch, is approximately constant. The greater the initial pressure, the bigger the difference between the two moduli at larger distortion strains. The shear modulus G(xz) is approximately independent of degree of distortion except at the highest initial pressure. The Poisson's ratio, nu(xz) is approximately constant with distortion strain for lower initial pressures, but increases significantly with epsilon(z) at higher pressures. Model predictions of the relation between G(xz) and initial uniform inflation pressure show a good correlation with reported experimental data for small distortion strains in a range of species. The model also exhibits similar behaviour to the experimentally measured uni-axial large deformations of a tri-axially pre-loaded block of parenchyma (Hoppin et al., 1975, Journal of Applied Physiology 39, 742-751).     

Cell Visco-Elasticity Measured with AFM and Optical Trapping at Sub-Micrometer Deformations

The measurement of the elastic properties of cells is widely used as an indicator for cellular changes during differentiation, upon drug treatment, or resulting from the interaction with the supporting matrix. Elasticity is routinely quantified by indenting the cell with a probe of an AFM while applying nano-Newton forces. Because the resulting deformations are in the micrometer range, the measurements will be affected by the finite thickness of the cell, viscous effects and even cell damage induced by the experiment itself. Here, we have analyzed the response of single 3T3 fibroblasts that were indented with a micrometer-sized bead attached to an AFM cantilever at forces from 30–600 pN, resulting in indentations ranging from 0.2 to 1.2 micrometer. To investigate the cellular response at lower forces up to 10 pN, we developed an optical trap to indent the cell in vertical direction, normal to the plane of the coverslip. Deformations of up to two hundred nanometers achieved at forces of up to 30 pN showed a reversible, thus truly elastic response that was independent on the rate of deformation. We found that at such small deformations, the elastic modulus of 100 Pa is largely determined by the presence of the actin cortex. At higher indentations, viscous effects led to an increase of the apparent elastic modulus. This viscous contribution that followed a weak power law, increased at larger cell indentations. Both AFM and optical trapping indentation experiments give consistent results for the cell elasticity. Optical trapping has the benefit of a lower force noise, which allows a more accurate determination of the absolute indentation. The combination of both techniques allows the investigation of single cells at small and large indentations and enables the separation of their viscous and elastic components.     

Mechanical deformation of spherical viruses with icosahedral symmetry

Virus capsids and crystalline surfactant vesicles are two examples of self-assembled shells in the nano- to micrometer size range. Virus capsids are particularly interesting since they have to sustain large internal pressures while encapsulating and protecting the viral DNA. We therefore study the mechanical properties of crystalline shells of icosahedral symmetry on a substrate under a uniaxial applied force by computer simulations. We predict the elastic response for small deformations, and the buckling transitions at large deformations. Both are found to depend strongly on the number of elementary building blocks N (the capsomers in the case of viral shells), the F?ppl-von Kármán number gamma (which characterizes the relative importance of shear and bending elasticity), and the confining geometry. In particular, we show that whereas large shells are well described by continuum elasticity-theory, small shells of the size of typical viral capsids behave differently already for small deformations. Our results are essential to extract quantitative information about the elastic properties of viruses and vesicles from deformation experiments.     

Axisymmetric optical-trap measurement of red blood cell membrane elasticity

The elastic properties of the cell membrane play a crucial role in determining the equilibrium shape of the cell, as well as its response to the external forces it experiences in its physiological environment. Red blood cells are a favored system for studying membrane properties because of their simple structure: a lipid bilayer coupled to a membrane cytoskeleton and no cytoplasmic cytoskeleton. An optical trap is used to stretch a red blood cell, fixed to a glass surface, along its symmetry axis by pulling on a micron-sized latex bead that is bound at the center of the exposed cell dimple. The system, at equilibrium, shows Hookean behavior with a spring constant of 1.5×10(-6)?N/m over a 1-2 μm range of extension. This choice of simple experimental geometry preserves the axial symmetry of the native cell throughout the stretch, probes membrane deformations in the small-extension regime, and facilitates theoretical analysis. The axisymmetry makes the experiment amenable to simulation using a simple model that makes no a priori assumption on the relative importance of shear and bending in membrane deformations. We use an iterative relaxation algorithm to solve for the geometrical configuration of the membrane at mechanical equilibrium for a range of applied forces. We obtain estimates for the out-of-plane membrane bending modulus B≈1×10(-19)?Nm and an upper limit to the in-plane shear modulus H<2×10(-6)?N/m. The partial agreement of these results with other published values may serve to highlight the dependence of the cell's resistance to deformation on the scale and geometry of the deformation.     

Bone poroelasticity

Poroelasticity is a well-developed theory for the interaction of fluid and solid phases of a fluid-saturated porous medium. It is widely used in geomechanics and has been applied to bone by many authors in the last 30 years. The purpose of this work is, first, to review the literature related to the application of poroelasticity to the interstitial bone fluid and, second, to describe the specific physical and modeling considerations that establish poroelasticity as an effective and useful model for deformation-driven bone fluid movement in bone tissue. The application of poroelasticity to bone differs from its application to soft tissues in two important ways. First, the deformations of bone are small while those of soft tissues are generally large. Second, the bulk modulus of the mineralized bone matrix is about six times stiffer than that of the fluid in the pores while the bulk moduli of the soft tissue matrix and the pore water are almost the same. Poroelasticity and electrokinetics can be used to explain strain-generated potentials in wet bone. It is noted that strain-generated potentials can be used as an effective tool in the experimental study of local bone fluid flow, and that the knowledge of this technique will contribute to the answers of a number of questions concerning bone mineralization, osteocyte nutrition and the bone mechanosensory system.     

MECHANICAL BEHAVIOR OF PLANT TISSUES AS INFERRED FROM THE THEORY OF PRESSURIZED CELLULAR SOLIDS

The mechanical behavior of plant tissues and its dependency on tissue geometry and turgor pressure are analytically dealt with in terms of the theory of cellular solids. A cellular solid is any material whose matter is distributed in the form of beamlike struts or complete “cell” walls. Therefore, its relative density is less than one and typically less than 0.3. Relative density is the ratio of the density of the cellular solid to the density of its constitutive (“cell wall”) material. Relative density depends upon cell shape and the density of cell wall material. It largely influences the mechanical behavior of cellular solids. Additional important parameters to mechanical behavior are the elastic modulus of “cell walls” and the magnitude of internal “cell” pressure. Analyses indicate that two “stiffening” agents operate in natural cellular solids (plant tissues): 1) cell wall infrastructure and 2) the hydrostatic influence of the protoplasm within each cellular compartment. The elastic modulus measured from a living tissue sample is the consequence of both agents. Therefore, the mechanical properties of living tissues are dependent upon the magnitude of turgor pressure. High turgor pressure places cell walls into axial tension, reduces the magnitude of cell wall deformations under an applied stress, and hence increases the apparent elastic modulus of the tissue. In the absence of turgid protoplasts or in the case of dead tissues, the cell wall infrastructure will respond as a linear elastic, nonlinear elastic, or “densifying” material (under compression) dependent upon the magnitude of externally applied stress. Accordingly, it is proposed that no single tangent (elastic) modulus from a stress-strain curve of a plant tissue is sufficient to characterize the material properties of a sample. It is also suggested that when a modulus is calculated that it be referred to as the tissue composite modulus to distinguish it from the elastic modulus of a noncellular solid material.     

Ion permeability of isolated chromaffin granules.

The passive ion permeability, regulation of volume, and internal pH of isolated bovine chromaffin granules were studied by radiochemical, potentiometric, gravimetric, and spectrophotometric techniques. Chromaffin granules behave as perfect osmometers between 340 and 1,000 mosM in choline chloride, NaCl, and KCl as measured by changes in absorbance at 430 nm or from intragranular water measurements using 3H2O and [14C]polydextran. By suspending chromaffin granules in iso-osmotic media of various metal ions and selectively increasing the permeability to either the cation or the anion by intrinsically permeable ions or specific ionophores, it was possible to determine by turbidity and potentiometric measurements the permeability to the counterion. These measurements indicate that the chromaffin granule is impermeable to the cations tested (Na+, K+, and H+). Limited H+ permeability across the chromaffin granule membrane was also shown by means of the time course of pH re-equilibration after pulsed pH changes in the surrounding media. The measurement of [14C]methylamine distribution indicates that a significant deltapH exists across the membrane, inside acidic, which at an external value of 6.85 has a value of 1.16. The deltapH is relatively insensitive to changes in the composition of the external media and can be enhanced or collapsed by the addition of ionophores and uncouplers. Measurement at various values of external pH indicates an internal pH of 5.5. Use of the ionophore A23187 indicates that Ca++ and Mg++ can be accumulated against an apparent concentration gradient with calcium uptake exceeding 50 nmol/mg of protein at saturation. These measurements also show that Ca++ and Mg++ are impermeable. Measurement of catecholamine release under conditions where intravesicular calcium accumulation is maximal indicates that catecholamine release does not occur. The physiological significance of the high impermeability to ions and the existence of a large deltapH are discussed in terms of regulation of uptake, storage, and release of catecholamines in chromaffin granules.     

Undulation Contributions to the Area Compressibility in Lipid Bilayer Simulations

It is here shown that there is a considerable system size-dependence in the area compressibility calculated from area fluctuations in lipid bilayers. This is caused by the contributions to the area fluctuations from undulations. This is also the case in experiments. At present, such a contribution, in most cases, is subtracted from the experimental values to obtain a true area compressibility. This should also be done with the simulation values. Here, this is done by extrapolating area compressibility versus system size, down to very small (zero) system size, where undulations no longer exist. The area compressibility moduli obtained from such simulations do not agree with experimental true area compressibility moduli as well as the uncorrected ones from contemporary or earlier simulations, but tend, instead, to be ∼50% too large. As a byproduct, the bending modulus can be calculated from the slope of the compressibility modulus versus system-size. The values obtained in this way for the bending modulus are then in good agreement with experiment.     

Self-diffusion of water into silk fibers from magnetic field pulse gradient data

Self-diffusion of water was studied in fibers of natural silk (Bombyx mori) with a water content of 0.18 g H2O/g dried material. Self-diffusion measurements were conducted by pulsed gradient of magnetic field (stimulated echo) at diffusion times from 10 to 200 mc. The dependence of experimental diffusion coefficients Dexp = f(delta) (observed decrease when delta increased) was determined to be responsible for the restricted diffusion. A model of planar and regularly spaced permeable barriers to diffusion of water molecules was applied to estimate the barrier spacing a and the permeability constant p. The maximal value of Dexp (at short diffusion time) in B. mori silk fibres was about 0.06 of the value of Dexp in bulk free water. The results obtained are compared to literature data on self-diffusion of water in hydrated biopolymer fibers and are discussed in connection with molecular mobility in natural macromolecular systems with low water content.     

Internal mobility in a double-stranded B DNA hexamer and undecamer. A time-dependent proton-proton nuclear Overhauser enhancement study

The internal mobility of the deoxyribose H2'-H2" and base C(H5)-C(H6) and T(CH3)-T(H6) vectors has been investigated by means of time-dependent nuclear Overhauser enhancement (NOE) measurements in a B DNA hexamer and undecamer. Cross-relaxation rates between these proton pairs are determined from the initial slopes of the time development of the NOEs, and, as the interproton distances between these proton pairs are fixed, apparent correlation times for the 3 interproton vectors are calculated from the cross-relaxation rate data. It is shown that there is little residue to residue variation in the cross-relaxation rates of the interproton vectors within each oligonucleotide, that the mean apparent correlation times of the C(H5)-C(H6) and T(CH3)-T(H6) vectors are approximately equal and significantly greater than that of the H2'-H2" vectors, and that the data for the H2'-H2" vectors of both oligonucleotides and the C(H5)-C(H6) and T(CH3)-T(H6) vectors of the undecamer cannot be accounted for by isotropic tumbling alone. The data are analysed in terms of a two motion model with isotropic tumbling and a single internal motion. The relaxation time of the internal motion at 23 degrees C is less than or equal to 1 ns for the H2'-H2" vectors of both oligonucleotides and less than or equal to 3 ns for the C(H5)-C(H6) and T(CH3)-T(H6) vectors of the undecamer. In the case of the H2'-H2" vectors, however, the amplitude of the internal motion is found to be too large to be compatible with the known stereochemistry of DNA. This finding can only be explained by invoking additional degrees of internal freedom with a larger number of internal motions of small amplitude of the type deduced from the analysis of crystallographic thermal factors [(1984) J. Mol. Biol. 173, 361-388].     

Equilibrium mechanics of monolayered epithelium

In order to fully understand the epithelial mechanics it is essential to integrate different levels of epithelial organization. In this work, we propose a theoretical approach for connecting the macroscopic mechanical properties of a monolayered epithelium to the mechanical properties at the cellular level. The analysis is based on the established mechanical models—at the macroscopic scale the epithelium is described within the mechanics of thin layers, while the cellular level is modeled in terms of the cellular surface (cortical) tension and the intercellular adhesion. The macroscopic elastic energy of the epithelium is linked to the energy of an average epithelial cell. The epithelial equilibrium state is determined by energy minimization and the macroscopic elastic moduli are calculated from deformations around the equilibrium. The results indicate that the epithelial equilibrium state is defined by the ratio between the adhesion strength and the cellular surface tension. The lower and the upper bounds for this ratio are estimated. If the ratio is small, the epithelium is cuboidal, if it is large, the epithelium becomes columnar. Importantly, it is found that the cellular cortical tension and the intercellular adhesion alone cannot produce the flattened squamous epithelium. Any difference in the surface tension between the apical and basal cellular sides bends the epithelium towards the side with the larger surface tension. Interestingly, the analysis shows that the epithelial area expansivity modulus and the shear modulus depend only on the cellular surface tension and not on the intercellular adhesion. The results are presented in a general analytical form, and are thus applicable to a variety of monolayered epithelia, without relying on the specifics of numerical finite-element methods. In addition, by using the standard theoretical tools for multi-laminar systems, the results can be applied to epithelia consisting of layers with different mechanical properties.     

Cell wall elasticity: I. A critique of the bulk elastic modulus approach and an analysis using polymer elastic principles

The traditional bulk elastic modulus approach to plant cell pressure-volume relations is inconsistent with its definition. The relationship between the bulk modulus and Young's modulus that forms the basis of their usual application to cell pressure-volume properties is demonstrated to be physically meaningless. The bulk modulus describes stress/strain relations of solid, homogeneous bodies undergoing small deformations, whereas the plant cell is best described as a thin-shelled, fluid-filled structure with a polymer base. Because cell walls possess a polymer structure, an alternative method of mechanical analysis is presented using polymer elasticity principles. This initial study presents the groundwork of polymer mechanics as would be applied to cell walls and discusses how the matrix and microfibrillar network induce nonlinear stress/strain relationships in the cell wall in response to turgor pressure. In subsequent studies, these concepts will be expanded to include anisotropic expansion as regulated by the microfibrillar network.     

Simulation of the superelastic response of SMA orthodontic wires.

Shape-memory alloys have properties that make them well suited to a variety of applications. One application for which their unique combination of properties (large elastic range, low modulus of elasticity, ability to deliver nearly constant forces over a wide range of deformations) seems ideally suited is for orthodontic retraction appliances where these properties are very desirable. The mechanical response of shape-memory alloys is modeled by a simple constitutive model that captures the essential superelastic behavior of the shape-memory wires. An initial value approach that iteratively converges to the appropriate boundary conditions is utilized to deliver numerical solutions. Qualitative agreement is shown with previous experimental works. The possible benefits of using such wires in an orthodontic retraction appliance are then investigated.