Cytoplasmic rheology of passive neutrophils.

The rheological properties of leukocytes are important to their effectiveness in the microcirculation. Previous studies based on in vitro data from micropipette experiments suggest that a Maxwell fluid bounded by a cortical shell with persistent tension is a realistic model for non-activated neutrophils in both the rapid and slow deformation phases. However, various viscoelastic coefficients have been obtained depending on the degree of cell deformation. In the present paper it is demonstrated that the cytoplasmic apparent viscosity and elasticity vary continuously, depending on the degree of deformation. These apparent variations are due to the inhomogeneous nature of the neutrophil internal structure. It is shown that the nucleus is much stiffer than the cytoplasm. The composite structure of the cell results in the deformation-dependent properties.     

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.     

Baseline mechanical characterization of J774 macrophages

Macrophage cell lines like J774 cells are ideal model systems for establishing the biophysical foundations of autonomous deformation and motility of immune cells. To aid comparative studies on these and other types of motile cells, we report measurements of the cortical tension and cytoplasmic viscosity of J774 macrophages using micropipette aspiration. Passive J774 cells cultured in suspension exhibited a cortical resting tension of ∼0.14 mN/m and a viscosity (at room temperature) of 0.93 kPa·s. Both values are about one order of magnitude higher than the respective values obtained for human neutrophils, lending support to the hypothesis that a tight balance between cortical tension and cytoplasmic viscosity is a physical prerequisite for eukaryotic cell motility. The relatively large stiffness of passive J774 cells contrasts with their capacity for a more than fivefold increase in apparent surface area during active deformation in phagocytosis. Scanning electron micrographs show how microscopic membrane wrinkles are smoothed out and recruited into the apparent surface area during phagocytosis of large targets.     

Leukocyte relaxation properties.

Study of the mechanical properties of leukocytes is useful to understand their passage through narrow capillaries and interaction with other cells. Leukocytes are known to be viscoelastic and their properties have been established by micropipette aspiration techniques. Here, the recovery of leukocytes to their normal spherical form is studied after prolonged deformation in a pipette which is large enough to permit complete entry of the leukocyte. The recovery history is characterized by the time history of the major diameter (d1) and minor diameter (d2). When the cell is removed from the pipette, it shows initially a small rapid recoil followed by a slower asymptotic recovery to the spherical shape. In the presence of cell activation and formation of pseudopods, the time history for recovery is prolonged compared with passive cell recovery. If a protopod pre-existed during the holding period, the recovery only begins when the protopod starts to retract.     

The mechanics of neutrophils: synthetic modeling of three experiments

Much experimental data exist on the mechanical properties of neutrophils, but so far, they have mostly been approached within the framework of liquid droplet models. This has two main drawbacks: 1), It treats the cytoplasm as a single phase when in reality, it is a composite of cytosol and cytoskeleton; and 2), It does not address the problem of active neutrophil deformation and force generation. To fill these lacunae, we develop here a comprehensive continuum-mechanical paradigm of the neutrophil that includes proper treatment of the membrane, cytosol, and cytoskeleton components. We further introduce two models of active force production: a cytoskeletal swelling force and a polymerization force. Armed with these tools, we present computer simulations of three classic experiments: the passive aspiration of a neutrophil into a micropipette, the active extension of a pseudopod by a neutrophil exposed to a local stimulus, and the crawling of a neutrophil inside a micropipette toward a chemoattractant against a varying counterpressure. Principal results include: 1), Membrane cortical tension is a global property of the neutrophil that is affected by local area-increasing shape changes. We argue that there exists an area dilation viscosity caused by the work of unfurling membrane-storing wrinkles and that this viscosity is responsible for much of the regulation of neutrophil deformation. 2), If there is no swelling force of the cytoskeleton, then it must be endowed with a strong cohesive elasticity to prevent phase separation from the cytosol during vigorous suction into a capillary tube. 3), We find that both swelling and polymerization force models are able to provide a unifying fit to the experimental data for the three experiments. However, force production required in the polymerization model is beyond what is expected from a simple short-range Brownian ratchet model. 4), It appears that, in the crawling of neutrophils or other amoeboid cells inside a micropipette, measurement of velocity versus counterpressure curves could provide a determination of whether cytoskeleton-to-cytoskeleton interactions (such as swelling) or cytoskeleton-to-membrane interactions (such as polymerization force) are predominantly responsible for active protrusion.     

An application of the micropipette technique to the measurement of the mechanical properties of cultured bovine aortic endothelial cells

The mechanical properties of endothelial cells were measured using the micropipette technique. The cells employed were collected from bovine aortic endothelium and cultured in our laboratory. Endothelial cells from confluent monolayers under no-flow conditions were detached from their substrate by trypsin or by a mechanical method and suspended in modified Dulbecco medium (MDM). In the micropipette technique, a part of the cell is aspirated into the tip of the micropipette under a microscope, and the deformation measured from a photograph. In this study, the data obtained were analyzed using a model where the cytoskeletal elements, which are considered to be the primary stress bearing components, are assumed to reside in a submembranous, cortical layer. Detached cells were found to have almost homogeneous mechanical properties based on measurements from different regions of the surface of a single cell. However, a hysteresis loop was observed in the relation between pressure and cell deformation during the loading and unloading processes. The calculated elastic shear moduli obtained for the trypsin-detached cells were as much as 10-20 times larger than those of a red blood cell. Mechanically-detached cells had moduli approximately twice that of the trypsin detached cells. Passage time, i.e., cell culture age, had no influence on the mechanical properties of the trypsin-detached cells, but did have an effect on the mechanically-detached cells, with both the younger and older cells being somewhat stiffer.     

A sensitive measure of surface stress in the resting neutrophil.

The simplest parameterized model of the "passive" or "resting receptive" neutrophil views the cell as being composed of an outer cortex surrounding an essentially liquid-like highly viscous cytoplasm. This cortex has been measured to maintain a small persistent tension of approximately 0.035 dyn/cm (Evans and Yeung. 1989. Biophys. J. 56:151-160) and is responsible for recovering the spherical shape of the cell after large deformation. The origin of the cortical tension is at present unknown, but speculations are that it may be an active process related to the sensitivity of a given cell to external stimulation and the "passive-active" transition. In order to characterize further this feature of the neutrophil we have used a new micropipet manipulation method to give a sensitive measure of the surface stress as a function of the surface area dilation of the highly ruffled cellular membrane. In the experiment, a single cell is driven down a tapered pipet in a series equilibrium deformation positions. Each equilibrium position represents a balance between the stress in the membrane and the pressure drop across the cell. For most cells that seemed to be "passive," as judged by their spherical appearance and lack of pseudopod activity, area dilations of approximately 30% were accompanied by only a small increase in the membrane tension, indicative of a very small apparent elastic area expansion modulus (approximately 0.04 dyn/cm). Extrapolations back to zero area dilation gave a value for the tension in the resting membrane of 0.024 +/- 0.003 dyn/cm, in close agreement with earlier measures.(ABSTRACT TRUNCATED AT 250 WORDS)     

Passive mechanical behavior of human neutrophils: power-law fluid.

The mechanical behavior of the neutrophil plays an important role in both the microcirculation and the immune system. Several laboratories in the past have developed mechanical models to describe different aspects of neutrophil deformability. In this study, the passive mechanical properties of normal human neutrophils have been further characterized. The cellular mechanical properties were assessed by single cell micropipette aspiration at fixed aspiration pressures. A numerical simulation was developed to interpret the experiments in terms of cell mechanical properties based on the Newtonian liquid drop model (Yeung and Evans, Biophys. J., 56: 139-149, 1989). The cytoplasmic viscosity was determined as a function of the ratio of the initial cell size to the pipette radius, the cortical tension, aspiration pressure, and the whole cell aspiration time. The cortical tension of passive neutrophils was measured to be about 2.7 x 10(-5) N/m. The apparent viscosity of neutrophil cytoplasm was found to depend on aspiration pressure, and ranged from approximately 500 Pa.s at an aspiration pressure of 98 Pa (1.0 cm H2O) to approximately 50 Pa.s at 882 Pa (9.0 cm H2O) when tested with a 4.0-micron pipette. These data provide the first documentation that the neutrophil cytoplasm exhibits non-Newtonian behavior. To further characterize the non-Newtonian behavior of human neutrophils, a mean shear rate gamma m was estimated based on the numerical simulation. The apparent cytoplasmic viscosity appears to decrease as the mean shear rate increases. The dependence of cytoplasmic viscosity on the mean shear rate can be approximated as a power-law relationship described by mu = mu c(gamma m/gamma c)-b, where mu is the cytoplasmic viscosity, gamma m is the mean shear rate, mu c is the characteristic viscosity at characteristic shear rate gamma c, and b is a material coefficient. When gamma c was set to 1 s-1, the material coefficients for passive neutrophils were determined to be mu c = 130 +/- 23 Pa.s and b = 0.52 +/- 0.09 for normal neutrophils. The power-law approximation has a remarkable ability to reconcile discrepancies among published values of the cytoplasmic viscosity measured using different techniques, even though these values differ by nearly two orders of magnitude. Thus, the power-law fluid model is a promising candidate for describing the passive mechanical behavior of human neutrophils in large deformation. It can also account for some discrepancies between cellular behavior in single-cell micromechanical experiments and predictions based on the assumption that the cytoplasm is a simple Newtonian fluid.     

Lipid-assisted microinjection: introducing material into the cytosol and membranes of small cells.

The microinjection of synthetic molecules, proteins, and nucleic acids into the cytosol of living cells is a powerful technique in cell biology. However, the insertion of a glass micropipette into the cell is a potentially damaging event, which presents significant problems, especially for small mammalian cells (spherical diameter = 2-15 micron), especially if they are only loosely adherent. The current technique is therefore limited to cells that are both sufficiently large or robust and firmly attached to a substrate. We describe here a modification of the standard technique that overcomes some of the problems associated with conventional microinjection but that does not involve the insertion of a micropipette deep into the cell cytoplasm. Instead, this method depends on lipid fusion at the micropipette tip to form a continuous but temporary conductance pathway between the interiors of the micropipette and cell. This technique thus also provides a novel method of transferring lipids and lipid-associated molecules to the plasma membrane of cells.     

Application of the micropipette technique to the measurement of cultured porcine aortic endothelial cell viscoelastic properties

The viscoelastic deformation of porcine aortic endothelial cells grown under static culture conditions was measured using the micropipette technique. Experiments were conducted both for control cells (mechanically or trypsin detached from the substrate) and for cells in which cytoskeletal elements were disrupted by cytochalasin B or colchicine. The time course of the aspirated length into the pipette was measured after applying a stepwise increase in aspiration pressure. To analyze the data, a standard linear viscoelastic half-space model of the endothelial cell was used. The aspirated length was expressed as an exponential function of time. The actin microfilaments were found to be the major cytoskeletal component determining the viscoelastic response of endothelial cells grown in static culture.     

Passive mechanical behavior of human neutrophils: effect of cytochalasin B.

Actin is a ubiquitous protein in eukaryotic cells. It plays a major role in cell motility and in the maintenance and control of cell shape. In this article, we intend to address the contribution of actin to the passive mechanical properties of human neutrophils. As a framework for assessing this contribution, the neutrophil is modeled as a simple viscous fluid drop with a constant cortical ("surface") tension. The reagent cytochalasin B (CTB) was used to disrupt the F-actin structure, and the neutrophil cortical tension and cytoplasmic viscosity were evaluated by single-cell micropipette aspiration. The cortical tension was calculated by simple force balance, and the viscosity was calculated according to a numerical analysis of the cell entry into the micropipette. CTB reduced the cell cortical tension in a dose-dependent fashion: by 19% at a concentration of 3 microM and by 49% at 30 microM. CTB also reduced the cytoplasmic viscosity by approximately -25% at a concentration of 3 microM and by approximately 65% at a concentration of 30 microM when compared at the same aspiration pressures. All three groups of neutrophils, normal cells, and cells treated with either 3 or 30 microM CTB, exhibited non-Newtonian behavior, in that the apparent viscosity decreased with increasing shear rate. The dependence of the cytoplasmic viscosity on deformation rate can be described empirically by mu = mu c(gamma m/gamma c)-b, where mu is cytoplasmic viscosity, gamma m is mean shear rate, mu c is the characteristic viscosity at the characteristic shear rate gamma c, and b is a material coefficient.(ABSTRACT TRUNCATED AT 250 WORDS)     

A modified micropipette aspiration technique and its application to tether formation from human neutrophils

Tether formation, which is mechanically characterized by its threshold force and effective viscosity, is involved in neutrophil emigration from blood circulation. Using the micropipette aspiration technique, which was improved by quantitative contact control and computerized data analysis, we extracted tethers from human neutrophils treated with IL-8, PMA, or cytochalasin D. We found that both IL-8 and PMA elevated the threshold force to about twice as large as the value for passive neutrophils. All these treatments decreased the effective viscosity dramatically (approximately 80%). With a novel method, the residual cortical tension of the cytochalasin-D-treated non-spherical neutrophils was measured to be approximately 8.8 pN/microm.     

Viscosity of passive human neutrophils undergoing small deformations.

At issue is the type of constitutive equation that can be used to describe all possible types of deformation of the neutrophil. Here a neutrophil undergoing small deformations is studied by aspirating it into a glass pipet with a diameter that is only slightly smaller than the diameter of the spherically shaped cell. After being held in the pipet for at least seven seconds, the cell is rapidly expelled and allowed to recover its undeformed, spherical shape. The recovery takes approximately 15 s. An analysis of the recovery process that treats the cell as a simple Newtonian liquid drop with a constant cortical (surface) tension gives a value of 3.3 x 10(-5) cm/s for the ratio of the cortical tension to cytoplasmic viscosity. This value is about twice as large as a previously published value obtained with the same model from studies of large deformations of neutrophils. This discrepancy indicates that the cytoplasmic viscosity decreases as the amount of deformation decreases. An extrapolated value for the cytoplasmic viscosity at zero deformation is approximately 600 poise when a value for the cortical tension of 0.024 dyn/cm is assumed. Clearly the neutrophil does not behave like a simple Newtonian liquid drop in that small deformations are inherently different from large deformations. More complex models consisting either of two or more fluids or multiple shells must be developed. The complex structure inside the neutrophil is shown in scanning electron micrographs of osmotically burst cells and cells whose membrane has been dissolved away.     

A power-law rheology-based finite element model for single cell deformation

Physical forces can elicit complex time- and space-dependent deformations in living cells. These deformations at the subcellular level are difficult to measure but can be estimated using computational approaches such as finite element (FE) simulation. Existing FE models predominantly treat cells as spring-dashpot viscoelastic materials, while broad experimental data are now lending support to the power-law rheology (PLR) model. Here, we developed a large deformation FE model that incorporated PLR and experimentally verified this model by performing micropipette aspiration on fibroblasts under various mechanical loadings. With a single set of rheological properties, this model recapitulated the diverse micropipette aspiration data obtained using three protocols and with a range of micropipette sizes. More intriguingly, our analysis revealed that decreased pipette size leads to increased pressure gradient, potentially explaining our previous counterintuitive finding that decreased pipette size leads to increased incidence of cell blebbing and injury. Taken together, our work leads to more accurate rheological interpretation of micropipette aspiration experiments than previous models and suggests pressure gradient as a potential determinant of cell injury.     

Cortical shell-liquid core model for passive flow of liquid-like spherical cells into micropipets.

Many nonadherent cells exist as spheres in suspension and when sucked into pipets, deform continuously like liquids within the fixed surface area limitation of a plasma membrane envelope. After release, these cells eventually recover their spherical form. Consequently, pipet aspiration test provides a useful method to assay the apparent viscosity of such cells. For this purpose, we have analyzed the inertialess flow of a liquid-like model cell into a tube at constant suction pressure. The cell is modeled as a uniform liquid core encapsulated by a distinct cortical shell. The method of analysis employs a variational approach that minimizes errors in boundary conditions defined by the equations of motion for the cortical shell where the trial functions are exact solutions for the flow field inside the liquid core. For the particular case of an anisotropic liquid cortex with persistent tension, we have determined universal predictions for flow rate scaled by the ratio of excess pressure (above the threshold established by the cortical tension) and core viscosity which is the reciprocal of the dynamic resistance to entry. The results depend on pipet to cell size ratio and a parameter that characterizes the ratio of viscous flow resistance in the cortex to that inside the cytoplasmic core. The rate of entry increases markedly as the pipet size approaches the outer segment diameter of the cell. Viscous dissipation in the cortex strongly influences the entry flow resistance for small tube sizes but has little effect for large tubes. This indicates that with sufficient experimental resolution, measurement of cell entry flow with different-size pipets could establish both the cortex to cell dissipation ratio as well as the apparent viscosity of the cytoplasmic core.     

Quantifying the contribution of actin networks to the elastic strength of fibroblasts

The structural models created to understand the cytoskeletal mechanics of cells in suspension are described here. Suspended cells can be deformed by well-defined surface stresses in an Optical Stretcher [Guck, J., Ananthakrishnan, R., Mahmood, H., Moon, T.J., Cunningham, C.C., K?s, J., 2001. The optical stretcher: a novel laser tool to micromanipulate cells. Biophys. J. 81(2), 767-784], a two-beam optical trap designed for the contact-free deformation of cells. Suspended cells have a well-defined cytoskeleton, displaying a radially symmetric actin cortical network underlying the cell membrane with no actin stress fibers, and microtubules and intermediate filaments in the interior. Based on experimental data using suspended fibroblasts, we create two structural models: a thick shell actin cortex model that describes cell deformation for a localized stress distribution on these cells and a three-layered model that considers the entire cytoskeleton when a broad stress distribution is applied. Applying the models to data, we obtain a (actin) cortical shear moduli G of approximately 220 Pa for normal fibroblasts and approximately 185 Pa for malignantly transformed fibroblasts. Additionally, modeling the cortex as a transiently crosslinked isotropic actin network, we show that actin and its crosslinkers must be co-localized into a tight shell to achieve these cortical strengths. The similar moduli values and cortical actin and crosslinker densities but different deformabilities of the normal and cancerous cells suggest that a cell's structural strength is not solely determined by cytoskeletal composition but equally importantly by (actin) cytoskeletal architecture via differing cortical thicknesses. We also find that although the interior structural elements (microtubules, nucleus) contribute to the deformed cell's exact shape via their loose coupling to the cortex, it is the outer actin cortical shell (and its thickness) that mainly determines the cell's structural response.     

An Eulerian/XFEM formulation for the large deformation of cortical cell membrane

Most animal cells are surrounded by a thin layer of actin meshwork below their membrane, commonly known as the actin cortex (or cortical membrane). An increasing number of studies have highlighted the role of this structure in many cell functions including contraction and locomotion, but modelling has been limited by the fact that the membrane thickness (about 1?μm) is usually much smaller than the typical size of a cell (10-100?μm). To overcome theoretical and numerical issues resulting from this observation, we introduce in this paper a continuum formulation, based on surface elasticity, that views the cortex as an infinitely thin membrane that can resists tangential deformation. To accurately model the large deformations of cells, we introduced equilibrium equations and constitutive relations within the Eulerian viewpoint such that all quantities (stress, rate of deformation) lie in the current configuration. A solution procedure is then introduced based on a coupled extended finite element approach that enables a continuum solution to the boundary value problem in which discontinuities in both strain and displacement (due to cortical elasticity) are easily handled. We validate the approach by studying the effect of cortical elasticity on the deformation of a cell adhering on a stiff substrate and undergoing internal contraction. Results show very good prediction of the proposed method when compared with experimental observations and analytical solutions for simple cases. In particular, the model can be used to study how cell properties such as stiffness and contraction of both cytoskeleton and cortical membrane lead to variations in cell's surface curvature. These numerical results show that the proposed method can be used to gain critical insights into how the cortical membrane affects cell deformation and how it may be used as a means to determine a cell's mechanical properties by measuring curvatures of its membrane.     

One-dimensional steady continuum model of retraction of pseudopod in leukocytes

A one-dimensional steady state continuum mechanics model of retraction of pseudopod in leukocytes is developed. The retracting pseudopod is assumed to move bodily toward the main cell body, the bulk motion of which can be represented by cytoplasmic flow within a typical stream tube through the leukocyte. The stream tube is approximated by a frictionless tube with prescribed geometry. The passive rheological properties of cytoplasm in the main cell body and in the pseudopod are modeled, respectively, by Maxwell fluid and Hookean solid. The two regions are assumed to be separated by a sharp interface at which actin gel solates and thereby changes its rheological properties as it flows from the pseudopod to the main cell body. The driving mechanism responsible for the active retraction motion is hypothesized to be a spontaneous deformation of the actin gel, analogous but not necessarily equal to the well known actin-myosin interaction. This results in an active contractile stress being developed in the pseudopod as well as in the cell cortex. The transverse traction pulls against the inclined wall of the stream tube and is transduced into an axial stress gradient, which in turn drives the flow. The tension on the tube wall is picked up by the prestressed cortical shell. Governing equations and boundary conditions are derived. A solution is obtained. Sample data are computed. Comparison of the theory with experiments shows that the model is compatible to the observations.     

Deformation properties of articular chondrocytes: a critique of three separate techniques

This paper presents a series of techniques, which examine the deformation characteristics of bovine articular chondrocytes. The direct contact approach employs well established methodology, involving AFM and micropipette aspiration, to yield structural properties of local regions of isolated chondrocytes. The former technique yields a non-linear response with increased structural stiffness in a central location on a projected image of the chondrocyte. A simple viscoelastic model can be used with data from the micropipette aspiration technique to yield a mean value of Young's modulus, which is similar to that recently reported (Jones et al., 1999). An indirect approach is also described, involving the response of chondrocytes seeded within compressed agarose constructs. For 1% agarose constructs, the resulting cell strain, yields a gross cell modulus of 2.7 kPa. The study highlights the difficulties in establishing unique mechanical parameters, which reflect the deformation behaviour of articular chondrocytes.