A continuum model of protrusion of pseudopod in leukocytes

G-actin diffusion is insufficient to achieve F-actin assembly in fast-treadmilling protrusions

To generate forces that drive migration of a eukaryotic cell, arrays of actin filaments (F-actin) are assembled at the cell's leading membrane edge. To maintain cell propulsion and respond to dynamic external cues, actin filaments must be disassembled to regenerate the actin monomers (G-actin), and transport of G-actin from sites of disassembly back to the leading edge completes the treadmilling cycle and limits the flux of F-actin assembly. Whether or not molecular diffusion is sufficient for G-actin transport has been a long-standing topic of debate, in part because the dynamic nature of cell motility and migration hinders the estimation of transport parameters. In this work, we applied an experimental system in which cells adopt an approximately constant and symmetrical shape; they cannot migrate but exhibit fast, steady treadmilling in the thin region protruding from the cell. Using fluorescence recovery after photobleaching, we quantified the relative concentrations and corresponding fluxes of F- and G-actin in this system. In conjunction with mathematical modeling, constrained by measured features of each region of interest, this approach revealed that diffusion alone cannot account for the transport of G-actin to the leading edge. Although G-actin diffusion and vectorial transport might vary with position in the protruding region, good agreement with the fluorescence recovery after photobleaching measurements was achieved by a model with constant G-actin diffusivity ∼2 μm2/s and anterograde G-actin velocity less than 1 μm/s.

MagPure chip: an immunomagnetic-based microfluidic device for high purification of circulating tumor cells from liquid biopsies

The isolation of circulating tumor cells (CTCs) directly from blood, as a liquid biopsy, could lead to a paradigm shift in cancer clinical care by providing an earlier diagnosis, a more accurate prognosis, and personalized treatment. Nevertheless, CTC-specific challenges, including their rarity and heterogeneity, have hampered the wider use of CTCs in clinical studies. Microfluidic-based isolation technologies have emerged as promising tools to circumvent these limitations but still fail to meet the constraints of high purity and short processing time required to ensure compatibility with clinical follow-up. In this study, we developed an immunomagnetic-based microfluidic device, the MagPure chip, to achieve the negative selection of CTCs through the depletion of white blood cells (WBCs) and provide highly purified samples for subsequent analysis. We demonstrate that the MagPure chip depletes all magnetically labeled WBCs (85% of WBCs were successfully labeled) and ensures a CTC recovery rate of 81%. In addition, we show its compatibility with conventional biological studies, including 2D and 3D cell culture, as well as phenotypic and genotypic analyses. Finally, we successfully implemented a two-step separation workflow for whole blood processing by combining a size-based pre-enrichment system (ClearCell FX1®) with the MagPure chip as a subsequent purification step. The total workflow led to high throughput (7.5 mL blood in less than 4 h) and high purity (947 WBCs per mL remaining, 99.99% depletion rate), thus enabling us to quantify CTC heterogeneity in size and tumor marker expression level. This tumor-marker-free liquid biopsy workflow could be used in a clinical context to assess phenotype aggressiveness and the prognosis rate.

Fluid shear stress-mediated mechanotransduction in circulating leukocytes and its defect in microvascular dysfunction

Leukocytes (neutrophils, monocytes) in the active circulation exhibit multiple phenotypic indicators for a low level of cellular activity, like lack of pseudopods and minimal amounts of activated, cell-adhesive integrins on their surfaces. In contrast, before these cells enter the circulation in the bone marrow or when they recross the endothelium into extravascular tissues of peripheral organs they are fully activated. We review here a multifaceted mechanism mediated by fluid shear stress that can serve to deactivate leukocytes in the circulation. The fluid shear stress controls pseudopod formation via the FPR receptor, the same receptor responsible for pseudopod projection by localized actin polymerization. The bioactivity of macromolecular factors in the blood plasma that interfere with receptor stimulation by fluid flow, such as proteolytic cleavage in the extracellular domain of the receptor or the membrane actions of cholesterol, leads to a defective ability to respond to fluid shear stress by actin depolymerization. The cell reaction to fluid shear involves CD18 integrins, nitric oxide, cGMP and Rho GTPases, is attenuated in the presence of inflammatory mediators and modified by glucocorticoids. The mechanism is abolished in disease models (genetic hypertension and hypercholesterolemia) leading to an increased number of activated leukocytes in the circulation with enhanced microvascular resistance and cell entrapment. In addition to their role in binding to biochemical agonists/antagonists, membrane receptors appear to play a second role: to monitor local fluid shear stress levels. The fluid shear stress control of many circulating cell types such as lymphocytes, stem cells, tumor cells remains to be elucidated.

Eukaryotic Cell Dynamics from Crawlers to Swimmers

Movement requires force transmission to the environment, and motile cells are robustly, though not elegantly, designed nanomachines that often can cope with a variety of environmental conditions by altering the mode of force transmission used. As with humans, the available modes range from momentary attachment to a substrate when crawling, to shape deformations when swimming, and at the cellular level this involves sensing the mechanical properties of the environment and altering the mode appropriately. While many types of cells can adapt their mode of movement to their microenvironment (ME), our understanding of how they detect, transduce and process information from the ME to determine the optimal mode is still rudimentary. The shape and integrity of a cell is determined by its cytoskeleton (CSK), and thus the shape changes that may be required to move involve controlled remodeling of the CSK. Motion in vivo is often in response to extracellular signals, which requires the ability to detect such signals and transduce them into the shape changes and force generation needed for movement. Thus the nanomachine is complex, and while much is known about individual components involved in movement, an integrated understanding of motility in even simple cells such as bacteria is not at hand. In this review we discuss recent advances in our understanding of cell motility and some of the problems remaining to be solved.

AXIAL-SYMMETRIC MODELING AND KINEMATIC ANALYSIS OF SPREADING OF SPARSELY CULTURED FIBROBLASTS

Cell spreading plays an important role in the modulation of physiological functions such as inflammation and cancer metastasis. The Brownian ratchet model and Bell's model have been used to simulate actin dynamics and bond kinetics for focal adhesion dynamics, respectively. In the present study, these models were modified and two additional subcellular mechanisms, integrin and myosin kinetics, were incoporated. An integrin recruitment function was introduced to determine the size of a focal adhesion associated with the substrate stiffness. The relationship between myosin concentration and the actin protrusion velocity was described by a first-order differential equation. Subcellular processes, including cell protrusion, focal adhesion formation, and stress fiber formation, were integrated into an axial-symmetric biophysical model, while inputs to the model were kinematic data from time-lapse experiments. Numerical simulations of the model using the Gillespie algorithm showed that dynamics of cell spreading can be well described by the model.

Cell migration with multiple pseudopodia: temporal and spatial sensing models

Cell migration triggered by pseudopodia (or "false feet") is the most used method of locomotion. A 3D finite element model of a cell migrating over a 2D substrate is proposed, with a particular focus on the mechanical aspects of the biological phenomenon. The decomposition of the deformation gradient is used to reproduce the cyclic phases of protrusion and contraction of the cell, which are tightly synchronized with the adhesion forces at the back and at the front of the cell, respectively. First, a steady active deformation is considered to show the ability of the cell to simultaneously initiate multiple pseudopodia. Here, randomness is considered as a key aspect, which controls both the direction and the amplitude of the false feet. Second, the migration process is described through two different strategies: the temporal and the spatial sensing models. In the temporal model, the cell "sniffs" the surroundings by extending several pseudopodia and only the one that receives a positive input will become the new leading edge, while the others retract. In the spatial model instead, the cell senses the external sources at different spots of the membrane and only protrudes one pseudopod in the direction of the most attractive one.

A comparison of computational models for eukaryotic cell shape and motility

Eukaryotic cell motility involves complex interactions of signalling molecules, cytoskeleton, cell membrane, and mechanics interacting in space and time. Collectively, these components are used by the cell to interpret and respond to external stimuli, leading to polarization, protrusion, adhesion formation, and myosin-facilitated retraction. When these processes are choreographed correctly, shape change and motility results. A wealth of experimental data have identified numerous molecular constituents involved in these processes, but the complexity of their interactions and spatial organization make this a challenging problem to understand. This has motivated theoretical and computational approaches with simplified caricatures of cell structure and behaviour, each aiming to gain better understanding of certain kinds of cells and/or repertoire of behaviour. Reaction–diffusion (RD) equations as well as equations of viscoelastic flows have been used to describe the motility machinery. In this review, we describe some of the recent computational models for cell motility, concentrating on simulations of cell shape changes (mainly in two but also three dimensions). The problem is challenging not only due to the difficulty of abstracting and simplifying biological complexity but also because computing RD or fluid flow equations in deforming regions, known as a “free-boundary” problem, is an extremely challenging problem in applied mathematics. Here we describe the distinct approaches, comparing their strengths and weaknesses, and the kinds of biological questions that they have been able to address.

'Run-and-tumble' or 'look-and-run'? A mechanical model to explore the behavior of a migrating amoeboid cell

Single cell migration constitutes a fundamental phenomenon involved in many biological events. Amoeboid cells are single cell organisms that migrate in a cyclic manner like worms. In this paper, we propose a 3D finite element model of an amoeboid cell migrating over a 2D surface. In particular, we focus on the mechanical aspect of the problem. The cell is able to generate cyclic active deformations, such as protrusion and contraction, in any direction. The progression of the cell is governed by a tight synchronization between the adhesion forces, which are alternatively applied at the front and at the rear edges of the cell, and the protrusion-contraction phases of the cell body. Finally, two important aspects have been taken into account: (1) the external stimuli in response to which the cell migrates (e.g. need to feed, morphogenetic events, normal or abnormal environment cues), (2) the heterogeneity of the 2D substrate (e.g. obstacles, rugosity, slippy regions) for which two distinct approaches have been evaluated: the 'run-and-tumble' strategy and the 'look-and-run' strategy. Overall, the results show a good agreement with respect to the experimental observations and the data from the literature (e.g. velocity and strains). Therefore, the present model helps, on one hand, to better understand the intimate relationship between the deformation modes of a cell and the adhesion strength that is required by the cell to crawl over a substrate, and, on the other hand, to put in evidence the crucial role played by mechanics during the migration process.

A POROELASTIC MODEL FOR CELL CRAWLING INCLUDING MECHANICAL COUPLING BETWEEN CYTOSKELETAL CONTRACTION AND ACTIN POLYMERIZATION

Much is known about the biophysical mechanisms involved in cell crawling, but how these processes are coordinated to produce directed motion is not well understood. Here, we propose a new hypothesis whereby local cytoskeletal contraction generates fluid flow through the lamellipodium, with the pressure at the front of the cell facilitating actin polymerization which pushes the leading edge forward. The contraction, in turn, is regulated by stress in the cytoskeleton. To test this hypothesis, finite element models for a crawling cell are presented. These models are based on nonlinear poroelasticity theory, modified to include the effects of active contraction and growth, which are regulated by mechanical feedback laws. Results from the models agree reasonably well with published experimental data for cell speed, actin flow, and cytoskeletal deformation in migrating fish epidermal keratocytes. The models also suggest that oscillations can occur for certain ranges of parameter values.

Actin-myosin viscoelastic flow in the keratocyte lamellipod

The lamellipod, the locomotory region of migratory cells, is shaped by the balance of protrusion and contraction. The latter is the result of myosin-generated centripetal flow of the viscoelastic actin network. Recently, quantitative flow data was obtained, yet there is no detailed theory explaining the flow in a realistic geometry. We introduce models of viscoelastic actin mechanics and myosin transport and solve the model equations numerically for the flat, fan-shaped lamellipodial domain of keratocytes. The solutions demonstrate that in the rapidly crawling cell, myosin concentrates at the rear boundary and pulls the actin network inward, so the centripetal actin flow is very slow at the front, and faster at the rear and at the sides. The computed flow and respective traction forces compare well with the experimental data. We also calculate the graded protrusion at the cell boundary necessary to maintain the cell shape and make a number of other testable predictions. We discuss model implications for the cell shape, speed, and bi-stability.

Mathematics of cell motility: have we got its number?

Mathematical and computational modeling is rapidly becoming an essential research technique complementing traditional experimental biological methods. However, lack of standard modeling methods, difficulties of translating biological phenomena into mathematical language, and differences in biological and mathematical mentalities continue to hinder the scientific progress. Here we focus on one area-cell motility-characterized by an unusually high modeling activity, largely due to a vast amount of quantitative, biophysical data, 'modular' character of motility, and pioneering vision of the area's experimental leaders. In this review, after brief introduction to biology of cell movements, we discuss quantitative models of actin dynamics, protrusion, adhesion, contraction, and cell shape and movement that made an impact on the process of biological discovery. We also comment on modeling approaches and open questions.

Spatio-temporal analysis of eukaryotic cell motility by improved force cytometry

Cell motility plays an essential role in many biological systems, but precise quantitative knowledge of the biophysical processes involved in cell migration is limited. Better measurements are needed to ultimately build models with predictive capabilities. We present an improved force cytometry method and apply it to the analysis of the dynamics of the chemotactic migration of the amoeboid form of Dictyostelium discoideum. Our explicit calculation of the force field takes into account the finite thickness of the elastic substrate and improves the accuracy and resolution compared with previous methods. This approach enables us to quantitatively study the differences in the mechanics of the migration of wild-type (WT) and mutant cell lines. The time evolution of the strain energy exerted by the migrating cells on their substrate is quasi-periodic and can be used as a simple indicator of the stages of the cell motility cycle. We have found that the mean velocity of migration v and the period of the strain energy T cycle are related through a hyperbolic law v = L/T, where L is a constant step length that remains unchanged in mutants with adhesion or contraction defects. Furthermore, when cells adhere to the substrate, they exert opposing pole forces that are orders of magnitude higher than required to overcome the resistance from their environment.

Diffusion with attrition

This article treats the problem of the sharp front observed when a diffusing substance interacts irreversibly with binding sites within the medium. The model consists of two simultaneous partial differential equations that are nonlinear and cannot be solved in closed form. The parameters are the diffusion coefficient D in the direction under consideration (x), the interaction constant k, the binding-site concentration mu and the boundary concentration of the diffusing ion c(0). Our aim is to develop methods to enable the estimation of these parameters from the experimental data. An analytical solution for the case k --> infinity, as found by others, is given first and then a finite element analysis package is used to obtain numerical solutions for the general case. Graphs are presented to illustrate the effects of the various parameters. Simple graphical procedures are described to compute mu and c (0). The position of the advancing front xi then provides, together with mu, a way to estimate D. A mathematical identity relating D and x and a second one involving D, k and t help to reduce the complexity of the problem. A new, measurable quantity S(t) is defined as [see text] where f is the total concentration (free + bound) of the diffusing ion at time t, and detailed plots are furnished that permit the computation of k directly from S(t), mu and D. The accuracy with which such methods can be expected to determine the various parameters of the model is considered at some length. Finally, in a concluding section, we simulate typical experimental data, examine the validity of our methods, and see how their accuracy is affected by controlled amounts of various kinds of noise.

Analysis of inflammation

In the past, inflammation has been associated with infections and with the immune system. But more recent evidence suggests that a much broader range of diseases have telltale markers for inflammation. Inflammation is the basic mechanism available for repair of tissue after an injury and consists of a cascade of cellular and microvascular reactions that serve to remove damaged and generate new tissue. The cascade includes elevated permeability in microvessels, attachment of circulating cells to the vessels in the vicinity of the injury site, migration of several cell types, cell apoptosis, and growth of new tissue and blood vessels. This review provides a summary of the major microvascular, cellular, and molecular mechanisms that regulate elements of the inflammatory cascade. The analysis is largely focused on the identification of the major participants, notably signaling and adhesion molecules, and their mode of action in the inflammatory cascade. We present a new hypothesis for the generation of inflammatory mediators in plasma that are derived from the digestive pancreatic enzymes responsible for digestion. The inflammatory cascade offers a large number of opportunities for development of quantitative models that describe various aspects of human diseases.

Biochemistry and biomechanics of cell motility

Cell motility is an essential cellular process for a variety of biological events. The process of cell migration requires the integration and coordination of complex biochemical and biomechanical signals. The protrusion force at the leading edge of a cell is generated by the cytoskeleton, and this force generation is controlled by multiple signaling cascades. The formation of new adhesions at the front and the release of adhesions at the rear involve the outside-in and inside-out signaling mediated by integrins and other adhesion receptors. The traction force generated by the cell on the extracellular matrix (ECM) regulates cell-ECM adhesions, and the counter force exerted by ECM on the cell drives the migration. The polarity of cell migration can be amplified and maintained by the feedback loop between the cytoskeleton and cell-ECM adhesions. Cell migration in three-dimensional ECM has characteristics distinct from that on two-dimensional ECM. The migration of cells is initiated and modulated by external chemical and mechanical factors, such as chemoattractants and the mechanical forces acting on the cells and ECM, as well as the surface density, distribution, topography, and rigidity of the ECM.

MULTISCALE TWO-DIMENSIONAL MODELING OF A MOTILE SIMPLE-SHAPED CELL

Cell crawling is an important biological phenomenon underlying coordinated cell movement in morphogenesis, cancer, and wound healing. In recent decades the process of cell crawling has been experimentally and theoretically dissected into further subprocesses: protrusion of the cell at its leading edge, retraction of the cell body, and graded adhesion. A number of one-dimensional (1-D) models explain successfully a proximal-distal organization and movement of the motile cell. However, more adequate two-dimensional (2-D) models are lacking. We propose a multiscale 2-D computational model of the lamellipodium (motile appendage) of a simply shaped, rapidly crawling fish keratocyte cell. We couple submodels of (i) protrusion and adhesion at the leading edge, (ii) the elastic 2-D lamellipodial actin network, (iii) the actin-myosin contractile bundle at the rear edge, and (iv) the convection-reaction-diffusion actin transport on the free boundary lamellipodial domain. We simulate the combined model numerically using a finite element approach. The simulations reproduce observed cell shapes, forces, and movements and explain some experimental results on perturbations of the actin machinery. This novel 2-D model of the crawling cell makes testable predictions and posits questions to be answered by future modeling.

Traveling wave solutions for a one-dimensional crawling nematode sperm cell model

In this paper, we proved that the one-dimensional crawling nematode sperm cell model proposed by Mogilner and Verzi (2003) supports traveling wave solutions if there is no disassembly of unbundled filaments in the cell. Uniqueness of traveling wave is established under additional assumptions and numerical examples are also given in the paper. Mathematical methods used include dynamical system techniques, implicit function theorem and global bifurcation theory.

An analysis of actin delivery in the acrosomal process of thyone

The acrosomal process of the sea cucumber Thyone briareus can extend 90 microm in 10 s, but an epithelial goldfish keratocyte can only glide a few microns in the same time. Both speeds reflect the rate of extension of an actin network. The difference is in the delivery of actin monomers to the polymerization region. Diffusion supplies monomers fast enough to support the observed speed of goldfish keratocytes, but previous models have indicated that the acrosomal process of Thyone extends too rapidly for diffusion to keep up. Here we reexamine the assumptions made in earlier models and present a new model, the Actin Reconcentration Model, that includes more biological detail. Salt and water fluxes during the acrosomal reaction and the nonideality of the cytoplasm are particularly significant for actin delivery. We find that the variability of the acrosomal growth curve can be explained by the salt and water fluxes, and that nonideality magnifies the effect of actin concentration changes. We calculate the speed of process growth using biologically relevant parameters from the literature and find that the predictions of the model fall among the experimental data.

Cell motility driven by actin polymerization

Certain kinds of cellular movements are apparently driven by actin polymerization. Examples include the lamellipodia of spreading and migrating embryonic cells, and the bacterium Listeria monocytogenes, that propels itself through its host's cytoplasm by constructing behind it a polymerized tail of cross-linked actin filaments. Peskin et al. (1993) formulated a model to explain how a polymerizing filament could rectify the Brownian motion of an object so as to produce unidirectional force (Peskin, C., G. Odell, and G. Oster. 1993. Cellular motions and thermal fluctuations: the Brownian ratchet. Biophys. J. 65:316-324). Their "Brownian ratchet" model assumed that the filament was stiff and that thermal fluctuations affected only the "load," i.e., the object being pushed. However, under many conditions of biological interest, the thermal fluctuations of the load are insufficient to produce the observed motions. Here we shall show that the thermal motions of the polymerizing filaments can produce a directed force. This "elastic Brownian ratchet" can explain quantitatively the propulsion of Listeria and the protrusive mechanics of lamellipodia. The model also explains how the polymerization process nucleates the orthogonal structure of the actin network in lamellipodia.

F-actin network formation in tethers and in pseudopods stimulated by chemoattractant

Micropipets are used either to deliver a given concentration of the chemoattractant N-formyl-methionyl-leucyl-phenylalanine (fMLP) to a local region of a human neutrophil or to create a membrane tether. Pseudopods, which have a cylindrical shape and grow at a constant rate, are formed in either case. After reaching a maximum extension, they retract, even in the presence of chemoattractant. As a pseudopod grows, cell granules begin to penetrate the pseudopod region to a "boundary" that defines a distance to the pseudopod's leading edge that is almost constant. The exclusion of granules from this domain indicates that it is filled with a dense network. The formation of this network involves the plasma membrane because pseudopod growth ceases when a membrane tether is pulled away from the leading edge. The rate of pseudopod growth depends on fMLP concentration just as the number of occupied N-formyl peptide receptors depends on this concentration. The experimental data are explained by assuming that F-actin network is formed next to the plasma membrane. The newly formed network displaces the membrane and the dominant process in the network region then becomes F-actin depolymerization. The rate of pseudopod growth is determined by the rate of the process leading to network formation. This process is apparently an enzymatic type of reaction. It has a positive enthalpy change and, therefore, is endothermic.

Intracellular pressure is a motive force for cell motion in Amoeba proteus

The cortical filament layer of free-living amoebae contains concentrated actomyosin, suggesting that it can contract and produce an internal hydrostatic pressure. We report here on direct and dynamic intracellular pressure (P(ic)) measurements in Amoeba proteus made using the servo-null technique. In resting apolar A. proteus, P(ic) increased while the cells remained immobile and at apparently constant volume. P(ic) then decreased approximately coincident with pseudopod formation. There was a positive correlation between P(ic) at the onset of movement and the rate of pseudopod formation. These results are the first direct evidence that hydrostatic pressure may be a motive force for cell motion. We postulate that contractile elements in the amoeba's cortical layer contract and increase P(ic) and that this P(ic) is utilized to overcome the viscous flow resistance of the intracellular contents during pseudopod formation.

The fundamental motor of the human neutrophil is not random: evidence for local non-Markov movement in neutrophils

The search for a fundamental mechano-chemical process that results in net cell motion has led investigators to fit neutrophil tracking data to well described physical models in hopes of understanding the functional form of the driving force. The Ornstein-Uhlenbeck (OU) equation for mean square displacement describes a locally persistent and globally random process and is often used as a starting point for analysis of neutrophil displacements. Based upon the apparently close fit of neutrophil tracking data to this equation and the nature of its derivation, biologists have inferred that the motor of the neutrophil is best represented as a random process. However, 24 of 37 neutrophil paths that we investigated preferentially display programmatic rather than Markov short term correlations between displacements or turn angles. These correlations reflect a bimodal rather than a uniform distribution of subpath correlations in the two variables, and are strongly sampling rate-dependent. Significant periodic components of neutrophil shape change are also detected at the same time scale using either Fourier or elliptical Fourier transform-based descriptors of the neutrophil perimeter. Oscillations in neutrophil velocity have the same period. Taken together, these data suggest a nonstochastic, and perhaps periodic, component to the process driving neutrophil movement.

Phorbol myristate acetate induction of chemotactic migration of human polymorphonuclear neutrophils

The in vitro migration of human polymorphonuclear neutrophils (PMNs) was studied employing an enzymatic assay of cell migration with phorbol myristate acetate (PMA) as the test stimulant. Our data clearly show that PMA in concentrations between 1 and 100 ng/ml in the lower wells of blind-well chambers induced chemotactic migration. Chemokinesis (increased migration) was not induced when PMA was present in both the upper and lower chambers (i.e., in a nongradient mode). Clearly our data indicate that PMA is chemotactic for human PMNs and, coupled with published studies of the effect of PMA on PMNs, suggest activation of an intracellular gradient of membrane-associated protein kinase C as a possible new mechanism for the induction of oriented migration of PMNs. Such a mechanism may be generalized to include membrane-soluble materials (e.g., inflammatory mediators, microbial products), which establish internal gradients of activated PKC rather than via the "classic" agonist-surface receptor mechanism, providing an alternative pathway for the induction of leukocyte chemotaxis.

New physical concepts for cell amoeboid motion

Amoeboid motion of cells is an essential mechanism in the function of many biological organisms (e.g., the regiment of scavenger cells in the immune defense system of animals). This process involves rapid chemical polymerization (with numerous protein constituents) to create a musclelike contractile network that advances the cell over the surface. Significant progress has been made in the biology and biochemistry of motile cells, but the physical dynamics of cell spreading and contraction are not well understood. The reason is that general approaches are formulated from complex mass, momentum, and chemical reaction equations for multiphase-multicomponent flow with the nontrivial difficulty of moving boundaries. However, there are strong clues to the dynamics that allow bold steps to be taken in simplifying the physics of motion. First, amoeboid cells often exhibit exceptional kinematics, i.e., steady advance and retraction of local fixed-shape patterns. Second, recent evidence has shown that cell projections "grow" by polymerization along the advancing boundary of the cell. Together, these characteristics represent a local growth process pinned to the interfacial contour of a contractile network. As such, the moving boundary becomes tractable, but subtle features of the motion lead to specific requirements for the chemical nature of the boundary polymerization process. To demonstrate these features, simple examples for limiting conditions of substrate interaction (i.e., "strong" and "weak" adhesion) are compared with data from experimental studies of yeast particle engulfment by blood granulocytes and actin network dynamics in fishscale keratocytes.

Locomotion forces generated by a polymorphonuclear leukocyte

There have been very few studies which have measured the physical forces generated by cells during active movements. A special micropipette system has been designed to make it possible to observe cell motion within the pipette and to apply a pressure to counter the chemotactic migration of the cell. This provides a direct measure of the locomotion force generated by the cell. The average velocity of forward motion is 0.33 microns/s in the absence of counter-pressure. The application of a positive counter-pressure (C-P) causes a decrease in the velocity of the forward motion of the cell. At 17 cm H2O of C-P, the cell velocity drops to zero and even moves backward with a higher C-P. The results show that the decrement of velocity is linearly related to the magnitude of the C-P with a complete stoppage at a pressure of 17 cm H2O which corresponds to a force of 0.003 dyn. The maximum work rate of the cell is approximately 2.5 x 10(-8) erg/s.

Mathematical model for the effects of adhesion and mechanics on cell migration speed

Migration of mammalian blood and tissue cells over adhesive surfaces is apparently mediated by specific reversible reactions between cell membrane adhesion receptors and complementary ligands attached to the substratum. Although in a number of systems these receptors and ligand molecules have been isolated and identified, a theory capable of predicting the effects of their properties on cell migration behavior currently does not exist. We present a simple mathematical model for elucidating the dependence of cell speed on adhesion-receptor/ligand binding and cell mechanical properties. Our model can be applied to propose answers to questions such as: does an optimal adhesiveness exist for cell movement? How might changes in receptor and ligand density and/or affinity affect the rate of migration? Can cell rheological properties influence movement speed? This model incorporates cytoskeletal force generation, cell polarization, and dynamic adhesion as requirements for persistent cell movement. A critical feature is the proposed existence of an asymmetry in some cell adhesion-receptor property, correlated with cell polarity. We consider two major alternative mechanisms underlying this asymmetry: (a) a spatial distribution of adhesion-receptor number due to polarized endocytic trafficking and (b) a spatial variation in adhesion-receptor/ligand bond strength. Applying a viscoelastic-solid model for cell mechanics allows us to represent one-dimensional locomotion with a system of differential equations describing cell deformation and displacement along with adhesion-receptor dynamics. In this paper, we solve these equations under the simplifying assumption that receptor dynamics are at a quasi-steady state relative to cell locomotion. Thus, our results are strictly valid for sufficiently slow cell movement, as typically observed for tissue cells such as fibroblasts. Numerical examples relevant to experimental systems are provided. Our results predict how cell speed might vary with intracellular contractile force, cell rheology, receptor/ligand kinetics, and receptor/ligand number densities. A biphasic dependence is shown to be possible with respect to some of the system parameters, with position of the maxima essentially governed by a balance between transmitted contractile force and adhesiveness. We demonstrate that predictions for the two alternative asymmetry mechanisms can be distinguished and could be experimentally tested using cell populations possessing different adhesion-receptor numbers.

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.

Cytoplasmic strains and strain rates in motile polymorphonuclear leukocytes

A new method is presented to measure local cytoplasmic deformation and rate of deformation in motile active neutrophils. The deformation is expressed in terms of biomechanical strains and strain rates. For this purpose small phagocytosed latex microspheres were used as intracellular markers. Planar Lagrangian and Eulerian strains and the rate of strain were estimated from the positions of a triad of internalized markers. Principal strains, stretch ratios, and principal directions were computed. The intracellular strains were found to be large relative to the overall cell shape change. Principal cytoplasmic stretch ratios showed large extension in the direction of pseudopod formation and cell locomotion and contraction in perpendicular directions. Regional strain analysis showed contractile strains to predominate in the vicinity of the pseudopod or leading edge of motion. The transitional region between the pseudopod and the main cell body exhibited large shear strains. The posterior region, where the uropod is located, also revealed large extensions but small contractile strains. The rate of strains are relatively small, nonuniform in time, and largely independent of the strain. The method we propose to measure cytoplasmic strain can be applied to a variety of problems in cell mechanics.