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.     

A continuum model of protrusion of pseudopod in leukocytes.

The morphology of human leukocytes, the biochemistry of actin polymerization, and the theory of continuum mechanics are used to model the pseudopod protrusion process of leukocytes. In the proposed model, the pseudopod is considered as a porous solid of F-actin network, the pores of which are full of aqueous solution. G-actin is considered as a "solute" transported by convection and diffusion in the fluid phase. The pseudopod grows as actin filaments elongate at their barbed ends at the tip of the pseudopod. The driving force of extension is hypothesized as being provided by the actin polymerization. It is assumed that elongation of actin filaments, powered by chemical energy liberated from the polymerization reaction, does mechanical work against opposing pressure on the membrane. This also gives rise to a pressure drop in the fluid phase at the tip of the pseudopod, which is formulated by an equation relating the work done by actin polymerization to the local state of pressure. The pressure gradient along the pseudopod drives the fluid filtration through the porous pseudopod according to Darcy's Law, which in turn brings more actin monomers to the growing tip. The main cell body serves as a reservoir of G-actin. A modified first-order equation is used to describe the kinetics of polymerization. The rate of pseudopod growth is modulated by regulatory proteins. A one-dimensional moving boundary problem based on the proposed mechanism has been constructed and approximate solutions have been obtained. Comparison of the solutions with experimental data shows that the model is compatible with available observations. The model is also applicable to growth of other cellular systems such as elongation of acrosomal process in sperm cells.     

Leukocyte biophysics

The biophysical properties of leukocytes in the passive and active state are discussed. In the passive unstressed state, leukocytes are spherical with numerous membrane folds. Passive leukocytes exhibit viscoelastic properties, and the stress is carried largely by the cell cytoplasm and the nucleus. The membrane is highly deformable in shearing and bending, but resists area expansion. Membrane tension can usually be neglected but plays a role in cases of large deformation when the membrane becomes unfolded. The constant membrane area constraint is a determinant of phagocytic capacity, spreading of cells, and passage through narrow pores. In the active state, leukocytes undergo large internal cytoplasmic deformation, pseudopod projection, and granule redistribution. Several different measurements for assessment of biophysical properties and the internal cytoplasmic deformation in form of strain and strain rate tensors are presented. The current theoretical models for active cytoplasmic motion in leukocytes are discussed in terms of specific macromolecular reactions.     

Leukocyte biophysics. An invited review.

The biophysical properties of leukocytes in the passive and active state are discussed. In the passive unstressed state, leukocytes are spherical with numerous membrane folds. Passive leukocytes exhibit viscoelastic properties, and the stress is carried largely by the cell cytoplasm and the nucleus. The membrane is highly deformable in shearing and bending, but resists area expansion. Membrane tension can usually be neglected but plays a role in cases of large deformation when the membrane becomes unfolded. The constant membrane area constraint is a determinant of phagocytic capacity, spreading of cells, and passage through narrow pores. In the active state, leukocytes undergo large internal cytoplasmic deformation, pseudopod projection, and granule redistribution. Several different measurements for assessment of biophysical properties and the internal cytoplasmic deformation in form of strain and strain rate tensors are presented. The current theoretical models for active cytoplasmic motion in leukocytes are discussed in terms of specific macromolecular reactions.     

Mechanical properties of the red cell membrane skeleton: analysis of axisymmetric deformations

The mechanical properties of erythrocyte membrane composed of a membrane bilayer and membrane skeleton are considered. Two membrane models are described: the model of free boundaries (MFB) and the model of immobilized boundaries (MIB). In MFB, the skeleton is assumed to be attached to the bilayer at a finite number of points, whereas MIB allows the interaction of each spectrin filament with the bilayer along its whole length. For MFB an estimate was made of the mechanical strain generated in the membrane by sucking erythrocytes into a micropipette. The existence of the deformation threshold is demonstrated, below which no mechanical strain, except that of bending, appears in the membrane. Thus only deformations exceeding this threshold result in strain. The relationship between the applied tension and the height of erythrocyte "tongue" sucked into a micropipette was determined. The MIB characteristics correspond to the model of Evans: strains in the membrane are generated at any deformation, however small, i.e. the threshold is equal to zero. A basic feature of this model is quite a different distribution of the skeleton deformations in the membrane. A comparison of the theoretical models and experimental data demonstrated the possibility of either MFB or MIB occurring, depending on the characteristic measurement time.     

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.     

Pseudopod projection and cell spreading of passive leukocytes in response to fluid shear stress

Recent evidence suggests that circulating leukocytes respond to physiological levels of fluid shear stress. This study was designed to examine the shear stress response of individual leukocytes adhering passively to a glass surface. Human leukocytes were exposed to a step fluid shear stress with amplitude between 0.2 and 4 dyn/cm(2) and duration between 1 and 20 min. The response of the cells was determined in the form of projected cell area measurements by high-resolution observation before, during, and after fluid shear application. All cells selected initially had a round morphology. After application of fluid shear many cells projected pseudopodia and spread on the glass surface. The number of leukocytes responding with pseudopod projection and the extent of cell spreading increased with increasing amplitude and duration of fluid shear stress. Pseudopod projection after exposure to a step fluid shear occurs following a delay that is insensitive to the shear stress amplitude and duration. Leukocytes that did not project pseudopodia and spread in response to low shear stress could be shown to respond to a second shear step of higher amplitude. The spreading response requires an intact actin network and activated myosin molecules. Depleting the cell glycocalyx with protease treatment enhances the spreading response in sheared leukocytes. These results indicate that passive leukocytes respond to fluid shear stress with active pseudopod projection and cell spreading. This behavior may contribute to cell spreading on endothelium and other cells as well as to transendothelial migration of leukocytes in the microcirculation.     

A mechano-electrochemical model of radial deformation of the capillary glycocalyx

A mechano-electrochemical theory of the surface glycocalyx on capillary endothelial cells is presented that models the structure as a mixture of electrostatically charged macromolecules hydrated in an electrolytic fluid. Disturbances arising from mechanical deformation are introduced as perturbations away from a nearly electroneutral equilibrium environment. Under mechanical compression of the layer, such as might occur on the passing of stiff leukocytes through capillaries, the model predicts that gradients in the electrochemical potential of the compressed layer cause a redistribution of mobile ions within the glycocalyx and a rehydration and restoration of the layer to its equilibrium dimensions. Because of the large deformations of the glycocalyx arising from passing leukocytes, nonlinear kinematics associated with finite deformations of the layer are accounted for in the theory. A pseudo-equilibrium approximation is invoked for the transport of the mobile ions that reduces the system of coupled nonlinear integro-differential equations to a single nonlinear partial differential equation that is solved numerically for the compression and recovery of the glycocalyx using a finite difference method on a fixed grid. A linearized model for small strains is also obtained as verification of the finite difference solution. Results of the asymptotic analysis agree well with the nonlinear solution in the limit of small deformations of the layer. Using existing experimental and theoretical estimates of glycocalyx properties, the glycocalyx fixed-charge density is estimated from the analysis to be approximately 1 mEq/l, i.e., we estimate that there exists approximately one fixed charge on the glycocalyx for every 100 ions in blood. Such a charge density would result in a voltage differential between the undeformed glycocalyx and the capillary lumen of approximately 0.1 mV. In addition to providing insight into the mechano-electrochemical dynamics of the layer under deformation, the model suggests several methods for obtaining improved estimates of the glycocalyx fixed-charge density and permeability in vivo.     

Two-Dimensional Aerodynamic Models of Insect Flight for Robotic Flapping Wing Mechanisms of Maximum Efficiency

In the ＂modified quasi-steady＂ approach, two-dimensional （2D） aerodynamic models of flapping wing motions are analyzed with focus on different types of wing rotation and different positions of rotation axis to explain the force peak at the end of each half stroke. In this model, an additional velocity of the mid chord position due to rotation is superimposed on the translational relative velocity of air with respect to the wing. This modification produces augmented forces around the end of each stroke. For each case of the flapping wing motions with various combination of controlled translational and rotational velocities of the wing along inclined stroke planes with thin figure-of-eight trajectory, discussions focus on lift-drag evolution during one stroke cycle and efficiency of types of wing rotation. This ＂modified quasi-steady＂ approach provides a systematic analysis of various parameters and their effects on efficiency of flapping wing mechanism. Flapping mechanism with delayed rotation around quarter-chord axis is an efficient one and can be made simple by a passive rotation mechanism so that it can be useful for robotic application.     

A comparison of passive flexion-extension to normal gait in the ovine stifle joint

Obtaining accurate values of joint tissue loads in human subjects and animals in vivo requires exact 3D-reproduction of joint kinematics and comparisons of in vivo motions between subjects and animals, and also necessitates an accurate reference position. For the knee, passive flexion-extension of isolated joints by hand has been assumed to produce bony motions similar to those of normal gait. We hypothesized that passive flexion-extension kinematics would not accurately reproduce in vivo gait, and, further, that such kinematics would vary significantly between testers. In vivo gait motions of four ovine stifle joints were measured in six degrees of freedom, as were passive flexion-extension motions after sacrifice. Passive flexion-extension motions were performed by three testers on the same stifle joints used in vitro. Results showed statistically significant differences in all degrees of freedom, with the largest differences in the proximal-distal and internal-external directions. Differences induced by muscle loads and kinetic factors in vivo were most evident during stance and hoof-off phases of gait. The in vitro passive paths generated by hand created motions with large variability both between and within individual testers. The user dependence and "area" of motion of passive flexion-extension indicates that passive flexion-extension is contained in a volume of motion, rather than constrained to a unique path. The assumption that the passive path has relevance to precise bone positions during normal in vivo gait is not supported by these results. Thus, using passive flexion-extension as a reference between joints may introduce large motion variability in the observed outcome, and large potential errors in determining joint tissue loads.     

Evaluation of the roles of passive and active control of balance using a balance control model

At present there is a lack of consensus regarding the relative roles of passive and active control of quiet upright stance. In the current work, this issue was investigated using two simulation models based on contemporary theories. Specifically, the two models, both of which assumed active control torques to be generated from an optimal neural controller, differed with respect to whether or not passive control torques (stiffness and damping) were included. Model parameters were specified using experimental center-of-pressure (COP) time series obtained during upright stance, and comparisons then made between simulated and actual COP-based measures. Including both active and passive joint torques in the control model did not appear to lead to any improvement in the ability to simulate COP compared with only including active joint torque. Further, simulated passive control torques were typically less than 10% of the active control torques, though some exceptions were found. These results, along with existing empirical evidence, suggest that active control torque is dominant in maintaining balance during upright stance.     

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)     

Kinematics of cytoplasmic deformation in neutrophils during active motion

A procedure is proposed to measure the cytoplasmic deformation in active motile neutrophils in the form of cytoplasmic strains and strain rates. Three neighboring microspheres in a local region of the cytoplasm serve as markers for local motion. Their positions are tracked by means of a high resolution light microscope and serve to compute nonlinear measures of strains and strain rates together with the principal strains and principal directions. Active neutrophils exhibit large cytoplasmic strains both during periodic pseudopod projections and during continuous locomotion in a polarized shape. The cytoplasmic motion is often synchronized with the whole cell deformation. The local cytoplasmic strains exceed the strains estimated for the whole cell and are not reversible except in some cases of single pseudopod projections. Large strains are observed both in attached and freely suspended cells. Strain rates are relatively constant but show an increase during the pseudopod retraction phase. Local cytoplasmic strains in neutrophils are inhomogeneous and reach large values during passage of the contraction rings. Neutrophils rendered passive by treatment with cytochalasin or EDTA show a random motion of microspheres with much smaller displacements. These observations suggest that the cytoplasm of active neutrophil exhibits large cytoplasmic strains and strain rates in the absence of an external stress resulting in a high degree of intracellular mixing. The proposed technique may be applied to a wide range of problems in cell biology.     

Ligand distortion modes leading to increased chirality content of Katsuki-Jacobsen catalysts

The "chirality content" of Katsuki-Jacobsen epoxidation catalysts are computed with the Avnir continuous chirality measure (CCM). An assessment of Mn(salen) molecules from the Cambridge Structural Database shows there exist some variation in CCM and the chirality content for several triplet state complexes of these catalysts purported in the literature to be the active species show even larger CCM values. Several deformation modes were analyzed to examine how chirality content changes as catalyst distortion is induced. The deformations studied include in-plane deformations, cup-shaped puckering, ligand twisting motions, and step-like deformations. Some distortions lead to increases of chirality while others lead to a decrease in chirality content. The most influential distortion modes that can be used for ligand design are twisting and step induction.     

Effect of flexural and torsional wing flexibility on lift generation in hoverfly flight

The effect of wing flexibility in hoverflies was investigated using an at-scale mechanical model. Unlike dynamically-scaled models, an at-scale model can include all phenomena related to motion and deformation of the wing during flapping. For this purpose, an at-scale polymer wing mimicking a hoverfly was fabricated using a custom micromolding process. The wing has venation and corrugation profiles which mimic those of a hoverfly wing and the measured flexural stiffness of the artificial wing is comparable to that of the natural wing. To emulate the torsional flexibility at the wing-body joint, a discrete flexure hinge was created. A range of flexure stiffnesses was chosen to match the torsional stiffness of pronation and supination in a hoverfly wing. The polymer wing was compared with a rigid, flat, carbon-fiber wing using a flapping mechanism driven by a piezoelectric actuator. Both wings exhibited passive rotation around the wing hinge; however, these rotations were reduced in the case of the compliant polymer wing due to chordwise deformations during flapping which caused a reduced effective angle of attack. Maximum lift was achieved when the stiffness of the hinge was similar to that of a hoverfly in both wing cases and the magnitude of measured lift is sufficient for hovering; the maximum lift achieved by the single polymer and carbon-fiber wings was 5.9?×?10(2)(?)μN and 6.9?×?10(2)(?)μN, respectively. These results suggest that hoverflies could exploit intrinsic compliances to generate desired motions of the wing and that, for the same flapping motions, a rigid wing could be more suitable for producing large lift.     

Isosmotic volume reabsorption in rat proximal tubule

A theoretical model incorporation both active and passive forces has been developed for fluid reabsorption from split oil droplets in rat intermediate and late proximal tubule. Of necessity, simplifying assumptions have been introduced; we have assumed that the epithelium can be treated as a single membrane and that the membrane "effective" HCO3 permeability is near zero. Based on this model with its underlying assumptions, the following conclusions are drawn. Regardless of the presence or absence of active NaCl transport, fluid reabsorption from the split oil droplet is isosmotic. The reabsorbate osmolarity can be affected by changes in tubular permeability parameters and applied forces but is not readily altered from an osmolarity essentially equal to that of plasma. In a split droplet, isosmotic flow need not be a special consequence of active Na transport, is not the result of a particular set of permeability properties, and is not merely a trivial consequence of a very high hydraulic conductivity; isosmotic flow can be obtained with hydraulic conductivity nearly an order of magnitude lower than that previously measured in the rat proximal convoluted tubule. Isosmotic reabsorption is, in part, the result of the interdependence of salt and water flows, their changing in parallel, and thus their ratio, the reabsorbate concentration being relatively invariant. Active NaCl transport can cause osmotic water flow by reducing the luminal fluid osmolarity. In the presence of passive forces the luminal fluid can be hypertonic to plasma, and active NaCl transport can still exert its osmotic effect on volume flow. There are two passive forces for volume flow: the Cl gradient and the difference in effective osmotic pressure; they have an approximately equivalent effect on volume flow. Experimentally, we have measured volume changes in a droplet made hyperosmotic by the addition of 50 mM NaCl; the experimental results are predicted reasonably well by our theoretical model.     

Cell mechanics studied by a reconstituted model tissue

Tissue models reconstituted from cells and extracellular matrix (ECM) simulate natural tissues. Cytoskeletal and matrix proteins govern the force exerted by a tissue and its stiffness. Cells regulate cytoskeletal structure and remodel ECM to produce mechanical changes during tissue development and wound healing. Characterization and control of mechanical properties of reconstituted tissues are essential for tissue engineering applications. We have quantitatively characterized mechanical properties of connective tissue models, fibroblast-populated matrices (FPMs), via uniaxial stretch measurements. FPMs resemble natural tissues in their exponential dependence of stress on strain and linear dependence of stiffness on force at a given strain. Activating cellular contractile forces by calf serum and disrupting F-actin by cytochalasin D yield "active" and "passive" components, which respectively emphasize cellular and matrix mechanical contributions. The strain-dependent stress and elastic modulus of the active component were independent of cell density above a threshold density. The same quantities for the passive component increased with cell number due to compression and reorganization of the matrix by the cells.     

A multiaxial constitutive law for mammalian left ventricular myocardium in steady-state barium contracture or tetanus.

The constitutive law of the material comprising any structure is essential for mechanical analysis since this law enables calculation of the stresses from the deformations and vice versa. To date, there is no constitutive law for actively contracting myocardial tissue. Using 2,3-butanedione monoxime to protect the myocardium from mechanical trauma, we subjected thin midwall slices of rabbit myocardium to multiaxial stretching first in the passive state and then during steady-state barium contracture or during tetani in ryanodine-loaded tissue. Assuming transverse isotropy in both the passive and active conditions, we used our previously described methods (Humphrey et al., 1990a) to obtain both passive and active constitutive laws. The major results of this study are: (1) This is the first multiaxial constitutive law for actively contracting mammalian myocardium. (2) The functional forms of the constitutive law for barium contracture and ryanodine-induced tetani are the same but differ from those in the passive state. Hence, one cannot simply substitute differing values for the coefficients of the passive law to describe the active tissue properties. (3) There are significant stresses developed in the cross-fiber direction (more than 40 percent of those in the fiber direction) that cannot be attributed to either deformation effects or nonparallel muscle fibers. These results provide the foundation for future mechanical analyses of the heart.     

Neutrophil activation in smokers

Smoking and elevated leukocyte counts are risk factors for cardiovascular disease. Experimental studies suggest that leukocyte activation may be a requirement for certain cardiovascular complications. Clinical studies have demonstrated activated leukocytes in the peripheral blood of stroke victims. Accordingly, neutrophil activation in unseparated whole blood of smokers as well as naive neutrophils of non-smokers exposed to plasma of smokers was investigated. Both spontaneous Superoxide formation as determined by nitroblue tetrazolium reduction, as well as pseudopod formation, are significantly elevated in autologous neutrophils of smokers. The surface expression of CD 18 and L-selectin on autologous circulating neutrophils of smokers is not significantly different from non-smoker controls. In contrast, incubation of naive neutrophils with smoker plasma leads to significantly higher levels of Superoxide formation, pseudopod formation, and L-selectin shedding, compared with non-smoker plasma, suggesting that the plasma of smokers contains a transferable factor which causes leukocyte activation. The results indicate that analysis of blood samples from large peripheral veins may not accurately reflect leukocyte activation in the circulation since activated leukocytes have a higher probability to be trapped in the microcirculation.