Deformation analyses in cell and developmental biology. Part I--Formal methodology

This study presents a computational approach for the deformation analyses of problems in cell and developmental biology. Cells and embryos are viewed mechanically as axisymmetric shell-like bodies containing a body of incompressible material. The analysis approach is based on the finite element method. It is comprised of three finite element ingredients: an axisymmetric shell/membrane element valid for modeling finite bending, shearing and stretching; a volume constraint algorithm for modeling the membrane-bound incompressible material; and a contact algorithm for modeling the mechanical interactions between deformable bodies. Part II of this study will demonstrate how these three ingredients can be applied to analyze mechanical experiments on cells. This same method is also useful for simulating embryonic shape changes during development.     

A method to fabricate mesoscopic freestanding polydimethylsiloxane membranes used to probe the rheology of an epithelial sheet

Details are presented for the formulation, fabrication, and mechanical characterization of mesoscopic freestanding polydimethylsiloxane (PDMS) elastomer membranes, 10.0 μm thick and 5.0 mm in diameter, used to probe the rheology of a living epithelial sheet. In what is described as a composite diaphragm inflation (CDI) experiment, freestanding PDMS membranes are utilized as substrates for the culture of a sheet of epithelial cells. Together, the cell layer and the PDMS elastomer form a composite diaphragm (CD) that is suitable for mechanical testing in an axisymmetric membrane inflation experiment. In order to distinguish the rheological behavior of the epithelial sheet from the mechanical response of the elastomer using inflation test data, freestanding PDMS membranes should exhibit a highly compliant yet mechanically invariant finite load-deformation response when subjected to multiple inflation cycles following intermittent periods of cell culture. Given these considerations, we describe a method for preparing freestanding PDMS elastomer membrane specimens that are optically transparent, tensed, and wrinkle-free. Surface modifications intended to facilitate cell culture, namely water vapor plasma and ultraviolet light treatments, were shown to dramatically stiffen the mechanical response of the membranes, rendering them unusable as CD substrates. In this study, only PDMS membranes with physiosorbed collagen demonstrated the mechanical compliance, fatigue resistance, and environmental stability necessary for reliable use in CDI experiments.     

Detachment of agglutinin-bonded red blood cells. II. Mechanical energies to separate large contact areas.

As detailed in a companion paper (Berk, D., and E. Evans. 1991. Biophys. J. 59:861-872), a method was developed to quantitate the strength of adhesion between agglutinin-bonded membranes without ambiguity due to mechanical compliance of the cell body. The experimental method and analysis were formulated around controlled assembly and detachment of a pair of macroscopically smooth red blood cell surfaces. The approach provides precise measurement of the membrane tension applied at the perimeter of an adhesive contact and the contact angle theta c between membrane surfaces which defines the mechanical leverage factor (1-cos theta c) important in the definition of the work to separate a unit area of contact. Here, the method was applied to adhesion and detachment of red cells bound together by different monoclonal antibodies to red cell membrane glycophorin and the snail-helix pomatia-lectin. For these tests, one of the two red cells was chemically prefixed in the form of a smooth sphere then equilibrated with the agglutinin before the adhesion-detachment procedure. The other cell was not exposed to the agglutinin until it was forced into contact with the rigid cell surface by mechanical impingement. Large regions of agglutinin bonding were produced by impingement but no spontaneous spreading was observed beyond the forced contact. Measurements of suction force to detach the deformable cell yielded consistent behavior for all of the agglutinins: i.e., the strength of adhesion increased progressively with reduction in contact diameter throughout detachment. This tension-contact diameter behavior was not altered over a ten-fold range of separation rates. In special cases, contacts separated smoothly after critical tensions were reached; these were the highest values attained for tension. Based on measurements reported in another paper (Evans et al. 1991. Biophys. J. 59:838-848) of the forces required to rupture molecular-point attachments, the density of cross-bridges was estimated with the assumption that the tension was proportional to the discrete rupture force x the number of attachments per unit length. These estimates showed that only a small fraction of agglutinin formed cross-bridges at initial assembly and increased progressively with separation. When critical tension levels were reached, it appeared that nearly all local agglutinin was involved as cross-bridges. Because one cell surface was chemically fixed, receptor accumulation was unlikely; thus, microscopic "roughness" and steric repulsion probably modulated formation of cross-bridges on initial contact.(ABSTRACT TRUNCATED AT 400 WORDS)     

Rapid flow of passive neutrophils into a 4 microns pipet and measurement of cytoplasmic viscosity

Neutrophils from five different individuals are isolated with a density separation technique. A total of 151 unactivated (passive) cells are rapidly aspirated at constant suction pressure and at room temperature into a pipet with a diameter of 4 microns. The suction pressures in excess of an initial yield threshold are 0.5, 1 and 2 kPa and are comparable to those encountered in the microcirculation. These pressures are well in excess of the small suction pressure of approximately 20 Pa that is required to form a static hemispherical bump on the cell. At a given aspiration pressure, the leading edge of an individual cell is "tracked" as it flows into the pipet. A theory based on the flow of a Newtonian liquid from either a hemisphere or a spherical segment into a cylinder is used to model the entry process. Both theory and experiment show that during most of the entry process the leading edge of the cell moves at a nearly constant velocity with a rapid acceleration at the end. For cells from five different individuals at the three different excess aspiration pressures, Newtonian theory gives a cytoplasmic viscosity of 135 +/- 54 Pa.s and overall entry times of 3.3s (0.5 kPa), 1.6s (1 kPa) and 0.82s (2 kPa). These results and those of Evans and Yeung at lower aspiration pressures indicate that the complex cytoplasm inside unactivated neutrophils behaves as a nearly Newtonian fluid with a viscosity on the order of 10(2) Pa.s over almost a two order of magnitude range in aspiration pressure and, thus, rate of deformation.     

Leukocyte deformability: finite element modeling of large viscoelastic deformation.

An axisymmetric deformation of a viscoelastic sphere bounded by a prestressed elastic thin shell in response to external pressure is studied by a finite element method. The research is motivated by the need for understanding the passive behavior of human leukocytes (white blood cells) and interpreting extensive experimental data in terms of the mechanical properties. The cell at rest is modeled as a sphere consisting of a cortical prestressed shell with incompressible Maxwell fluid interior. A large-strain deformation theory is developed based on the proposed model. General non-linear, large strain constitutive relations for the cortical shell are derived by neglecting the bending stiffness. A representation of the constitutive equations in the form of an integral of strain history for the incompressible Maxwell interior is used in the formulation of numerical scheme. A finite element program is developed, in which a sliding boundary condition is imposed on all contact surfaces. The mathematical model developed is applied to evaluate experimental data of pipette tests and observations of blood flow.     

Changes in proteoglycan synthesis of chondrocytes in articular cartilage are associated with the time-dependent changes in their mechanical environment

Explant loading experiments were conducted to investigate the effect of load duration on proteoglycan synthesis. A compressive load of 0.1 MPa applied for 10 min was found to stimulate proteoglycan synthesis, while the same load applied for 20 h suppressed synthesis. This bimodal response suggests that the cells are responding to different mechanical stimuli as time progresses. A theoretical model has therefore been developed to describe the mechanical environment perceived by cells within soft hydrated tissues (e.g. articular cartilage) while the tissue is being loaded. The cells are modeled, using the biphasic theory, as fluid-solid inclusions embedded in and attached to a biphasic extracellular matrix of distinct material properties. A method of solution is developed which is valid for any axisymmetric loading configuration, provided that the cell radius, a, is small relative to the tissue height, h (i.e. h/a 1). A closed-form analytical solution for this inclusion problem is then presented for the confined compression configuration. Results from this model show that the mechanical environment in and around the cells is time dependent and inhomogeneous, and can be significantly influenced by differences in properties between the cell and the extracellular matrix.     

CELL LOSS FROM THREE ESTABLISHED LINES OF THE C3H MOUSE MAMMARY TUMOR: A COMPARISON OF THE 125I-UdR AND THE 3H-TdR-AUTORADIOGRAPHIC METHODS

The¹²⁵I-UdR method for measuring cell loss from solid tumors has been re-evaluated. The rate of tumor cell loss from three established lines (S102F, S102S and Slow) of the C3H mouse mammary tumor was determined by the ¹²⁵I-UdRmethod and the results were compared to the estimates for cell loss as determined by the combined approach of cellular ³H-TdR autoradiography and volumetric growth-rate determinations. This detailed comparison shows that the two methods complement each other but cannot substitute for one another because they give different quantitative information. The combined approach measures the flow of viable cells, as determined morphologically, from the proliferating compartment to the quiescent compartment, the quiescent compartment out of the tumor, etc., but does not evaluate the flow of degenerate cells or acellular (necrotic) debris. In contrast, the ¹²⁵I-UdR method indicates the net flow of intact cells and/or dead cells as well as debris from the tumor as the ^I25l-labeled material is lost from the tumor, but gives limited internal information. Thus, depending on the specific experiment, an investigator could choose one or the other of the methods to answer the question. Perhaps both would be desirable at times; however, in most cases, one could not substitute one method for the other. The data from the Slow tumors also indicate that in certain tumors, the quantitative information from the ¹²⁵I-UdR method may be quite limited, i.e. the confidence limits within an experiment as well as the replication error between experiments may be high.     

Separation and properties of prestalk and prespore cells of Dictyostelium discoideum

We describe a method of separating prestalk and prespore cells of Dictyostelium discoideum slugs using a self-generating Percoll gradient. This method gives quantitative recovery of cells and good purity. Separated prestalk and prespore cells possess different levels of the enzymes UDP galactose :polysaccharide transferase, cAMP phosphodiesterase and glycogen phosphorylase. We have used this method, as well as mechanical dissection of slugs, to examine the fate of separated prestalk and prespore cells in Dictyostelium strains that are able to give rise to mature stalk and spore cells in cell monolayers. The results from such experiments provide direct evidence that prestalk and prespore cells from the migrating slug stage are programmed to differentiate into stalk and spore cells respectively.     

Indentation and adhesive probing of a cell membrane with AFM: theoretical model and experiments

In probing adhesion and cell mechanics by atomic force microscopy (AFM), the mechanical properties of the membrane have an important if neglected role. Here we theoretically model the contact of an AFM tip with a cell membrane, where direct motivation and data are derived from a prototypical ligand-receptor adhesion experiment. An AFM tip is functionalized with a prototypical ligand, SIRPalpha, and then used to probe its native receptor on red cells, CD47. The interactions prove specific and typical in force, and also show in detachment, a sawtooth-shaped disruption process that can extend over hundreds of nm. The theoretical model here that accounts for both membrane indentation as well as membrane extension in tip retraction incorporates membrane tension and elasticity as well as AFM tip geometry and stochastic disruption. Importantly, indentation depth proves initially proportional to membrane tension and does not follow the standard Hertz model. Computations of detachment confirm nonperiodic disruption with membrane extensions of hundreds of nm set by membrane tension. Membrane mechanical properties thus clearly influence AFM probing of cells, including single molecule adhesion experiments.     

Cell signals allow the expression of a pre-existent neural pattern in C. elegans.

In C. elegans, a simple pattern develops within a row of epidermal precursor cells, V1-V6. One cell, V5, gives rise to a neuroblast called the postdeirid neuroblast, while the other V cells produce epidermal cells instead. Here we describe experiments suggesting that in order for V5 to produce the postdeirid it must be in close or direct contact with neighboring V cells. Signaling between V cells is required for the formation of the neuroblast; however, which of the V cells can make a postdeirid is not determined by these signals but rather by the action of the homeotic lin-22 and pal-1 genes. These genes prevent V cells in specific body regions from responding to intercellular signals and producing postdeirids. This is a clear example of cell signals playing a permissive rather than an instructive role in neuroblast induction.     

A biphasic multilayer computational model of human skin

The present study investigates the layer-specific mechanical behavior of human skin. Motivated by skin’s histology, a biphasic model is proposed which differentiates between epidermis, papillary and reticular dermis, and hypodermis. Inverse analysis of ex vivo tensile and in vivo suction experiments yields mechanical parameters for each layer and predicts a stiff reticular dermis and successively softer papillary dermis, epidermis and hypodermis. Layer-specific analysis of simulations underlines the dominating role of the reticular dermis in tensile loading. Furthermore, it shows that the observed out-of-plane deflection in ex vivo tensile tests is a direct consequence of the layered structure of skin. In in vivo suction experiments, the softer upper layers strongly influence the mechanical response, whose dissipative part is determined by interstitial fluid redistribution within the tissue. Magnetic resonance imaging-based visualization of skin deformation in suction experiments confirms the deformation pattern predicted by the multilayer model, showing a consistent decrease in dermal thickness for large probe opening diameters.

     

Mechanical and failure properties of single attached cells under compression

Eukaryotic cells are continuously subjected to mechanical forces under normal physiological conditions. These forces and associated cellular deformations induce a variety of biological processes. The degree of deformation depends on the mechanical properties of the cell. As most cells are anchorage dependent for normal functioning, it is important to study the mechanical properties of cells in their attached configuration. The goal of the present study was to obtain the mechanical and failure properties of attached cells. Individual, attached C2C12 mouse myoblasts were subjected to unconfined compression experiments using a recently developed loading device. The device allows global compression of the cell until cell rupture and simultaneously measures the associated forces. Cell bursting was characterized by a typical reduction in the force, referred to as the bursting force. Mean bursting forces were calculated as 8.7+/-2.5 microN at an axial strain of 72+/-4%. Visualization of the cell using confocal microscopy revealed that cell bursting was preceded by the formation of bulges at the cell membrane, which eventually led to rupturing of the cell membrane. Finite element calculations were performed to simulate the obtained force-deformation curves. A finite element mesh was built for each cell to account for its specific geometrical features. Using an axisymmetric approximation of the cell geometry, and a Neo-Hookean constitutive model, excellent agreement between predicted and measured force-deformation curves was obtained, yielding an average Young's modulus of 1.14+/-0.32 kPa.     

Continuum model of collective cell migration in wound healing and colony expansion

Collective cell migration plays an important role during wound healing and embryo development. Although the exact mechanisms that coordinate such migration are still unknown, experimental studies of moving cell layers have shown that the primary interactions governing the motion of the layer are the force of lamellipodia, the adhesion of cells to the substrate, and the adhesion of cells to each other. Here, we derive a two-dimensional continuum mechanical model of cell-layer migration that is based on a novel assumption of elastic deformation of the layer and incorporates basic mechanical interactions of cells as well as cell proliferation and apoptosis. The evolution equations are solved numerically using a level set method. The model successfully reproduces data from two types of experiments: 1), the contraction of an enterocyte cell layer during wound healing; and 2), the expansion of a radially symmetric colony of MDCK cells, both in the edge migration velocity and in cell-layer density. In accord with experimental observations, and in contrast to reaction-diffusion models, this model predicts a partial wound closure if lamellipod formation is inhibited at the wound edge and gives implications of the effect of spatially restricted proliferation.     

Modelling the mechanical properties of single suspension-cultured tomato cells

BACKGROUND AND AIMS: The relationship between composition and structure of plant primary cell walls, and cell mechanical properties is not fully understood, partly because intrinsic properties of walls such as Young's modulus cannot be obtained readily. The aim of this work is to show that Young's modulus of walls of single suspension-cultured tomato cells can be determined by modelling force-deformation data. METHODS: The model simulates the compression of a cell between two flat surfaces, with the cell treated as a liquid-filled sphere with thin compressible walls. The cell wall and membrane were taken to be permeable, but the compression was so fast that water loss could be neglected in the simulations. Force-deformation data were obtained by compressing the cells in micromanipulation experiments. RESULTS:Good fits were obtained between the model and low-strain experimental data, using the modulus and initial inflation of the cell as adjustable parameters. The mean Young's modulus for 2-week-old cells was found to be 2.3 +/- 0.2 GPa at pH 5. This corresponds to an instantaneous bulk modulus of elasticity of approx. 7 MPa, similar to a value found by the pressure probe method. However, Young's modulus is a better parameter, as it should depend only on the composition and structure of the cell wall, not on bulk cell behaviour. This new method has been used to show that Young's modulus of cultured tomato cell walls is at its lowest at pH 4.5, the pH optimum for expansin activity. CONCLUSIONS:The linear elastic model is very suitable for estimating wall Young's modulus from micromanipulation experiments on single tomato cells. This is a powerful method for determining cell wall material properties.     

A novel technique for the assessment of mechanical properties of vascular tissue

Accurate estimation of mechanical properties of the different atherosclerotic plaque constituents is important in assessing plaque rupture risk. The aim of this study was to develop an experimental set-up to assess material properties of vascular tissue, while applying physiological loading and being able to capture heterogeneity. To do so, a ring-inflation experimental set-up was developed in which a transverse slice of an artery was loaded in the radial direction, while the displacement was estimated from images recorded by a high-speed video camera. The performance of the set-up was evaluated using seven rubber samples and validated with uniaxial tensile tests. For four healthy porcine carotid arteries, material properties were estimated using ultrasound strain imaging in whole-vessel-inflation experiments and compared to the properties estimated with the ring-inflation experiment. A 1D axisymmetric finite element model was used to estimate the material parameters from the measured pressures and diameters, using a neo-Hookean and Holzapfel–Gasser–Ogden material model for the rubber and porcine samples, respectively. Reproducible results were obtained with the ring-inflation experiment for both rubber and porcine samples. Similar mean stiffness values were found in the ring-inflation and tensile tests for the rubber samples as 202 kPa and 206 kPa, respectively. Comparable results were obtained in vessel-inflation experiments using ultrasound and the proposed ring-inflation experiment. This inflation set-up is suitable for the assessment of material properties of healthy vascular tissue in vitro. It could also be used as part of a method for the assessment of heterogeneous material properties, such as in atherosclerotic plaques.

     

An immersed boundary method for simulating a single axisymmetric cell growth and division

For an animal cell, cytokinesis is the process by which a cell divides its cytoplasm to produce two daughter cells. We propose a new mathematical model for simulating cytokinesis. The proposed model is robust and realistic in deciding the position of the cleavage furrow and in defining the contractile force leading to cell division. We use an immersed boundary method to track the morphology of cell membrane during cytokinesis. For accurate calculation, we adaptively add and delete the immersed boundary points. We perform numerical simulations on the axisymmetric domain to have sufficient resolution and to incorporate three-dimensional effects such as anisotropic surface tension. Finally, we investigate the effects of each model parameter and compare a numerical result with the experimental data to demonstrate the efficiency and accuracy of our proposed method.     

A novel approach for extracting viscoelastic parameters of living cells through combination of inverse finite element simulation and Atomic Force Microscopy

Dynamic mechanical behaviour of living cells has been described by viscoelasticity. However, quantitation of the viscoelastic parameters for living cells is far from sophisticated. In this paper, combining inverse finite element (FE) simulation with Atomic Force Microscope characterization, we attempt to develop a new method to evaluate and acquire trustworthy viscoelastic index of living cells. First, influence of the experiment parameters on stress relaxation process is assessed using FE simulation. As suggested by the simulations, cell height has negligible impact on shape of the force–time curve, i.e. the characteristic relaxation time; and the effect originates from substrate can be totally eliminated when stiff substrate (Young’s modulus larger than 3 GPa) is used. Then, so as to develop an effective optimization strategy for the inverse FE simulation, the parameters sensitivity evaluation is performed for Young’s modulus, Poisson’s ratio, and characteristic relaxation time. With the experiment data obtained through typical stress relaxation measurement, viscoelastic parameters are extracted through the inverse FE simulation by comparing the simulation results and experimental measurements. Finally, reliability of the acquired mechanical parameters is verified with different load experiments performed on the same cell.     

Lipid-glass adhesion in giga-sealed patch-clamped membranes.

Adhesion between patch-clamped lipid membranes and glass micropipettes is measured by high contrast video imaging of the mechanical response to the application of suction pressure across the patch. The free patch of membrane reversibly alters both its contact angle and radius of curvature on pressure changes. The assumption that an adhesive force between the membrane and the pipette can sustain normal tension up to a maximum Ta at the edge of the free patch accounts for the observed mechanical responses. When the normal component of the pressure-induced membrane tension exceeds Ta membrane at the contact point between the free patch and the lipid-glass interface is pulled away from the pipette wall, resulting in a decreased radius of curvature for the patch and an increased contact angle. Measurements of the membrane radius of curvature as a function of the suction pressure and pipette radius determine line adhesion tensions Ta which range from 0.5 to 4.0 dyn/cm. Similar behavior of patch-clamped cell membranes implies similar adhesion mechanics.     

An Insight into Cell Elasticity and Load-Bearing Ability. Measurement and Theory

We have studied the elasticity and load bearing ability of plant tissue at the cellular level, using onion (Allium cepa) epidermal cells. The Young's modulus and Poisson's ratio of the cells were obtained by loading a tensile force on onion epidermal peels of different turgor pressures, and measuring the elongation and the lateral contraction of the peels. The Young's moduli and the Poisson's ratios ranged from 3.5 to 8.0 MPa and 0.18 to 0.30, respectively. To determine the effects of cell elasticity and turgor pressure on the cell's ability to bear load, we loaded a small glass ball onto a cell and measured the projected contact area between the ball and the cell. Unlike previous studies, we considered the cell as a whole entity, and utilized the Boussinesq's solution to derive the relevant equations that related the elastic parameters and cell deformation. For cells with a turgor pressure > or = 0.34 MPa, the predicted contact area agreed well with the measured area. The equations could also predict cell turgor pressure with a deviation from the measured value of 0.01 MPa. This study gives strong support to ball tonometry, a new method of measuring cell turgor pressure.