On the mechanics of the first cleavage division of the sea urchin egg

Biomechanics of cells and subcellular components: A comprehensive review of computational models and applications

Cells are a fundamental structural, functional and biological unit for all living organisms. Up till now, considerable efforts have been made to study the responses of single cells and subcellular components to an external load, and understand the biophysics underlying cell rheology, mechanotransduction and cell functions using experimental and in silico approaches. In the last decade, computational simulation has become increasingly attractive due to its critical role in interpreting experimental data, analysing complex cellular/subcellular structures, facilitating diagnostic designs and therapeutic techniques, and developing biomimetic materials. Despite the significant progress, developing comprehensive and accurate models of living cells remains a grand challenge in the 21st century. To understand current state of the art, this review summarises and classifies the vast array of computational biomechanical models for cells. The article covers the cellular components at multi-spatial levels, that is, protein polymers, subcellular components, whole cells and the systems with scale beyond a cell. In addition to the comprehensive review of the topic, this article also provides new insights into the future prospects of developing integrated, active and high-fidelity cell models that are multiscale, multi-physics and multi-disciplinary in nature. This review will be beneficial for the researchers in modelling the biomechanics of subcellular components, cells and multiple cell systems and understanding the cell functions and biological processes from the perspective of cell mechanics.

Classical and Emerging Regulatory Mechanisms of Cytokinesis in Animal Cells

The primary goal of cytokinesis is to produce two daughter cells, each having a full set of chromosomes. To achieve this, cells assemble a dynamic structure between segregated sister chromatids called the contractile ring, which is made up of filamentous actin, myosin-II, and other regulatory proteins. Constriction of the actomyosin ring generates a cleavage furrow that divides the cytoplasm to produce two daughter cells. Decades of research have identified key regulators and underlying molecular mechanisms; however, many fundamental questions remain unanswered and are still being actively investigated. This review summarizes the key findings, computational modeling, and recent advances in understanding of the molecular mechanisms that control the formation of the cleavage furrow and cytokinesis.

Unite to divide - how models and biological experimentation have come together to reveal mechanisms of cytokinesis

Cytokinesis is the fundamental and ancient cellular process by which one cell physically divides into two. Cytokinesis in animal and fungal cells is achieved by contraction of an actomyosin cytoskeletal ring assembled in the cell cortex, typically at the cell equator. Cytokinesis is essential for the development of fertilized eggs into multicellular organisms and for homeostatic replenishment of cells. Correct execution of cytokinesis is also necessary for genome stability and the evasion of diseases including cancer. Cytokinesis has fascinated scientists for well over a century, but its speed and dynamics make experiments challenging to perform and interpret. The presence of redundant mechanisms is also a challenge to understand cytokinesis, leaving many fundamental questions unresolved. For example, how does a disordered cytoskeletal network transform into a coherent ring? What are the long-distance effects of localized contractility? Here, we provide a general introduction to 'modeling for biologists', and review how agent-based modeling and continuum mechanics modeling have helped to address these questions.

Agent-based modeling: case study in cleavage furrow models

The number of studies in cell biology in which quantitative models accompany experiments has been growing steadily. Roughly, mathematical and computational techniques of these models can be classified as "differential equation based" (DE) or "agent based" (AB). Recently AB models have started to outnumber DE models, but understanding of AB philosophy and methodology is much less widespread than familiarity with DE techniques. Here we use the history of modeling a fundamental biological problem-positioning of the cleavage furrow in dividing cells-to explain how and why DE and AB models are used. We discuss differences, advantages, and shortcomings of these two approaches.

Prediction of traction forces of motile cells

When crawling on a flat substrate, living cells exert forces on it via adhesive contacts, enabling them to build up tension within their cytoskeleton and to change shape. The measurement of these forces has been made possible by traction force microscopy (TFM), a technique which has allowed us to obtain time-resolved traction force maps during cell migration. This cell 'footprint' is, however, not sufficient to understand the details of the mechanics of migration, that is how cytoskeletal elements (respectively, adhesion complexes) are put under tension and reinforce or deform (respectively, mature and/or unbind) as a result. In a recent paper, we have validated a rheological model of actomyosin linking tension, deformation and myosin activity. Here, we complement this model with tentative models of the mechanics of adhesion and explore how closely these models can predict the traction forces that we recover from experimental measurements during cell migration. The resulting mathematical problem is a PDE set on the experimentally observed domain, which we solve using a finite-element approach. The four parameters of the model can then be adjusted by comparison with experimental results on a single frame of an experiment, and then used to test the predictive power of the model for following frames and other experiments. It is found that the basic pattern of traction forces is robustly predicted by the model and fixed parameters as a function of current geometry only.

On the embryonic cell division beyond the contractile ring mechanism: experimental and computational investigation of effects of vitelline confinement, temperature and egg size

Embryonic cell division is a mechanical process which is predominantly driven by contraction of the cleavage furrow and response of the remaining cellular matter. While most previous studies focused on contractile ring mechanisms of cytokinesis, effects of environmental factors such as pericellular vitelline membrane and temperature on the mechanics of dividing cells were rarely studied. Here, we apply a model-based analysis to the time-lapse imaging data of two species (Saccoglossus kowalevskii and Xenopus laevis) with relatively large eggs, with the goal of revealing the effects of temperature and vitelline envelope on the mechanics of the first embryonic cell division. We constructed a numerical model of cytokinesis to estimate the effects of vitelline confinement on cellular deformation and to predict deformation of cellular contours. We used the deviations of our computational predictions from experimentally observed cell elongation to adjust variable parameters of the contractile ring model and to quantify the contribution of other factors (constitutive cell properties, spindle polarization) that may influence the mechanics and shape of dividing cells. We find that temperature affects the size and rate of dilatation of the vitelline membrane surrounding fertilized eggs and show that in native (not artificially devitellinized) egg cells the effects of temperature and vitelline envelope on mechanics of cell division are tightly interlinked. In particular, our results support the view that vitelline membrane fulfills an important role of micromechanical environment around the early embryo the absence or improper function of which under moderately elevated temperature impairs normal development. Furthermore, our findings suggest the existence of scale-dependent mechanisms that contribute to cytokinesis in species with different egg size, and challenge the view of mechanics of embryonic cell division as a scale-independent phenomenon.

An integrative toy model of cell flattening, spreading, and ruffling

BACKGROUND

The processes of cell spreading and crawling are frequently associated with mysterious waves and ruffling cycles of the leading edge.

OBJECTIVE

To develop a physical model that can account for these phenomena based on a few simple and plausible rules governing adhesion, contractility, polymerization of cytoskeleton, and membrane tension.

METHODS

Extension of a continuum mechanical model of phagocytosis [J Cell Sci. (2006);119(Pt 9):1903-13] adding a simple coupling between membrane curvature and cytoskeletal polymerization.

RESULTS

We show that our generalized model has just the right nonlinearity needed for triggering of stochastic/chaotic cycles of ruffling similar to those that are observed in real cells.

CONCLUSIONS

The cycles are caused by a branching instability at the leading edge that leads to bifurcations of protrusion into forward moving lamellipodium and upward and rearward folding ruffles. The amplitude of the instability is modulated by the surface tension, with higher tension stabilizing against ruffling (but inhibiting protrusion) and lower tension promoting ruffling and protrusion.

Modeling large-scale dynamic processes in the cell: polarization, waves, and division

The past decade has witnessed significant developments in molecular biology techniques, fluorescent labeling, and super-resolution microscopy, and together these advances have vastly increased our quantitative understanding of the cell. This detailed knowledge has concomitantly opened the door for biophysical modeling on a cellular scale. There have been comprehensive models produced describing many processes such as motility, transport, gene regulation, and chemotaxis. However, in this review we focus on a specific set of phenomena, namely cell polarization, F-actin waves, and cytokinesis. In each case, we compare and contrast various published models, highlight the relevant aspects of the biology, and provide a sense of the direction in which the field is moving.

Cells as liquid motors: mechanosensitivity emerges from collective dynamics of actomyosin cortex

Living cells adapt and respond actively to the mechanical properties of their environment. In addition to biochemical mechanotransduction, evidence exists for a myosin-dependent purely mechanical sensitivity to the stiffness of the surroundings at the scale of the whole cell. Using a minimal model of the dynamics of actomyosin cortex, we show that the interplay of myosin power strokes with the rapidly remodeling actin network results in a regulation of force and cell shape that adapts to the stiffness of the environment. Instantaneous changes of the environment stiffness are found to trigger an intrinsic mechanical response of the actomyosin cortex. Cortical retrograde flow resulting from actin polymerization at the edges is shown to be modulated by the stress resulting from myosin contractility, which in turn, regulates the cell length in a force-dependent manner. The model describes the maximum force that cells can exert and the maximum speed at which they can contract, which are measured experimentally. These limiting cases are found to be associated with energy dissipation phenomena, which are of the same nature as those taking place during the contraction of a whole muscle. This similarity explains the fact that single nonmuscle cell and whole-muscle contraction both follow a Hill-like force-velocity relationship.

The value of mechanistic biophysical information for systems-level understanding of complex biological processes such as cytokinesis

This review illustrates the value of quantitative information including concentrations, kinetic constants and equilibrium constants in modeling and simulating complex biological processes. Although much has been learned about some biological systems without these parameter values, they greatly strengthen mechanistic accounts of dynamical systems. The analysis of muscle contraction is a classic example of the value of combining an inventory of the molecules, atomic structures of the molecules, kinetic constants for the reactions, reconstitutions with purified proteins and theoretical modeling to account for the contraction of whole muscles. A similar strategy is now being used to understand the mechanism of cytokinesis using fission yeast as a favorable model system.

Furrow constriction in animal cell cytokinesis

Cytokinesis is the process of physical cleavage at the end of cell division; it proceeds by ingression of an acto-myosin furrow at the equator of the cell. Its failure leads to multinucleated cells and is a possible cause of tumorigenesis. Here, we calculate the full dynamics of furrow ingression and predict cytokinesis completion above a well-defined threshold of equatorial contractility. The cortical acto-myosin is identified as the main source of mechanical dissipation and active forces. Thereupon, we propose a viscous active nonlinear membrane theory of the cortex that explicitly includes actin turnover and where the active RhoA signal leads to an equatorial band of myosin overactivity. The resulting cortex deformation is calculated numerically, and reproduces well the features of cytokinesis such as cell shape and cortical flows toward the equator. Our theory gives a physical explanation of the independence of cytokinesis duration on cell size in embryos. It also predicts a critical role of turnover on the rate and success of furrow constriction. Scaling arguments allow for a simple interpretation of the numerical results and unveil the key mechanism that generates the threshold for cytokinesis completion: cytoplasmic incompressibility results in a competition between the furrow line tension and the cell poles' surface tension.

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.

Influence of the standard free energy on swelling kinetics of gels

Classical theories of gel swelling employ the mixing free energy, thereby ignoring any effects of the free energy of the pure phases,i.e., the polymer standard free energy. In this paper we present a model for the swelling kinetics of gels that incorporates the free energy, including the polymer standard free energy. We provide a complete analysis of how the swelling kinetics and stable states and sizes of the swelled gel depends on the free energy parameters and show that theories that use only the mixing free energy cannot correctly describe equilibrium states or the swelling kinetics.

Cytopede: a three-dimensional tool for modeling cell motility on a flat surface

When cultured on flat surfaces, fibroblasts and many other cells spread to form thin lamellar sheets. Motion then occurs by extension of the sheet at the leading edge and retraction at the trailing edge. Comprehensive quantitative models of these phenomena have so far been lacking and to address this need, we have designed a three-dimensional code called Cytopede specialized for the simulation of the mechanical and signaling behavior of plated cells. Under assumptions by which the cytosol and the cytoskeleton are treated from a continuum mechanical perspective, Cytopede uses the finite element method to solve mass and momentum equations for each phase, and thus determine the time evolution of cellular models. We present the physical concepts that underlie Cytopede together with the algorithms used for their implementation. We then validate the approach by a computation of the spread of a viscous sessile droplet. Finally, to exemplify how Cytopede enables the testing of ideas about cell mechanics, we simulate a simple fibroblast model. We show how Cytopede allows computation, not only of basic characteristics of shape and velocity, but also of maps of cell thickness, cytoskeletal density, cytoskeletal flow, and substratum tractions that are readily compared with experimental data.

Form and function in cell motility: from fibroblasts to keratocytes

It is plain enough that a horse is made for running, but similar statements about motile cells are not so obvious. Here the basis for structure-function relations in cell motility is explored by application of a new computational technique that allows realistic three-dimensional simulations of cells migrating on flat substrata. With this approach, some cyber cells spontaneously display the classic irregular protrusion cycles and handmirror morphology of a crawling fibroblast, and others the steady gliding motility and crescent morphology of a fish keratocyte. The keratocyte motif is caused by optimal recycling of the cytoskeleton from the back to the front so that more of the periphery can be devoted to protrusion. These calculations are a step toward bridging the gap between the integrated mechanics and biophysics of whole cells and the microscopic molecular biology of cytoskeletal components.

Analysis of actin FLAP dynamics in the leading lamella

Background

The transport of labeled G-actin from the mid-lamella region to the leading edge in a highly motile malignant rat fibroblast line has been studied using fluorescence localization after photobleaching or FLAP, and the transit times recorded in these experiments were so fast that simple diffusion was deemed an insufficient explanation (see Zicha et al., Science, v. 300, pp. 142–145 [1]).

Methodology/Principal Findings

We re-examine the Zicha FLAP experiments using a two-phase reactive interpenetrating flow formalism to model the cytoplasm and the transport dynamics of bleached and unbleached actin. By allowing an improved treatment of effects related to the retrograde flow of the cytoskeleton and of the geometry and finite thickness of the lamella, this new analysis reveals a mechanism that can realistically explain the timing and the amplitude of all the FLAP signals observed in [1] without invoking special transport modalities.

Conclusions/Significance

We conclude that simple diffusion is sufficent to explain the observed transport rates, and that variations in the transport of labeled actin through the lamella are minor and not likely to be the cause of the observed physiological variations among different segments of the leading edge. We find that such variations in labeling can easily arise from differences and changes in the microscopic actin dynamics inside the edge compartment, and that the key dynamical parameter in this regard is the so-called “dilatation rate” (the velocity of cytoskeletal retrograde flow divided by a characteristic dimension of the edge compartment where rapid polymerization occurs). If our dilatation hypothesis is correct, the transient kinetics of bleached actin relocalization constitute a novel and very sensitive method for probing the cytoskeletal dynamics in leading edge micro-environments which are otherwise very difficult to directly interrogate.

Multiphase flow models of biogels from crawling cells to bacterial biofilms

This article reviews multiphase descriptions of the fluid mechanics of cytoplasm in crawling cells and growing bacterial biofilms. These two systems involve gels, which are mixtures composed of a polymer network permeated by water. The fluid mechanics of these systems is essential to their biological function and structure. Their mathematical descriptions must account for the mechanics of the polymer, the water, and the interaction between these two phases. This review focuses on multiphase flow models because this framework is natural for including the relative motion between the phases, the exchange of material between phases, and the additional stresses within the network that arise from nonspecific chemical interactions and the action of molecular motors. These models have been successful in accounting for how different forces are generated and transmitted to achieve cell motion and biofilm growth and they have demonstrated how emergent structures develop though the interactions of the two phases. A short description of multiphase flow models of tumor growth is included to highlight the flexibility of the model in describing diverse biological applications.

Structural memory in the contractile ring makes the duration of cytokinesis independent of cell size

Cytokinesis is accomplished by constriction of a cortical contractile ring. We show that during the early embryonic divisions in C. elegans, ring constriction occurs in two phases--an initial phase at a constant rate followed by a second phase during which the constriction rate decreases in proportion to ring perimeter. Cytokinesis completes in the same amount of time, despite the reduction in cell size during successive divisions, due to a strict proportionality between initial ring size and the constant constriction rate. During closure, the myosin motor in the ring decreases in proportion to perimeter without turning over. We propose a "contractile unit" model to explain how the ring retains a structural memory of its initial size as it disassembles. The scalability of constriction may facilitate coordination of mitotic events and cytokinesis when cell size, and hence the distance traversed by the ring, varies during embryogenesis and in other contexts.

Size and speed go hand in hand in cytokinesis

In animal cells, cytokinesis is mediated by the constriction of a cortical ring. In this issue, Carvalho et al. (2009) show in embryos of the worm Caenorhabditis elegans that the rate of ring constriction during cytokinesis is proportional to the initial cell perimeter, ensuring that the duration of cytokinesis is cell-size independent.

Intracellular microrheology of motile Amoeba proteus

The motility of Amoeba proteus was examined using the technique of passive particle tracking microrheology, with the aid of newly developed particle tracking software, a fast digital camera, and an optical microscope. We tracked large numbers of endogeneous particles in the amoebae, which displayed subdiffusive motion at short timescales, corresponding to thermal motion in a viscoelastic medium, and superdiffusive motion at long timescales due to the convection of the cytoplasm. Subdiffusive motion was characterized by a rheological scaling exponent of 3/4 in the cortex, indicative of the semiflexible dynamics of the actin fibers. We observed shear-thinning in the flowing endoplasm, where exponents increased with increasing flow rate; i.e., the endoplasm became more fluid-like. The rheology of the cortex is found to be isotropic, reflecting an isotropic actin gel. A clear difference was seen between cortical and endoplasmic layers in terms of both viscoelasticity and flow velocity, where the profile of the latter is close to a Poiseuille flow for a Newtonian fluid.

Modeling cell interactions under flow

In this review, we summarize the current state of understanding of the processes by which leukocytes, and other cells, such as tumor cells interact with the endothelium under various blood flow conditions. It is shown that the interactions are influenced by cell-cell adhesion properties, shear stresses due to the flow field and can also be modified by the cells microrheological properties. Different adhesion proteins are known to be involved leading to particular mechanisms by which interactions take place during inflammation or metastasis. Cell rolling, spreading, migration are discussed, as well as the effect of flow conditions on these mechanisms, including microfluidic effects. Several mathematical models proposed in recent years capturing the essential features of such interaction mechanisms are reviewed. Finally, we present a recent model in which the adhesion is given by a kinetics theory based model and the cell itself is modeled as a viscoelastic drop. Qualitative agreement is found between the predictions of this model and in vitro experiments.

Distinct pathways for the early recruitment of myosin II and actin to the cytokinetic furrow

Equatorial organization of myosin II and actin has been recognized as a universal event in cytokinesis of animal cells. Current models for the formation of equatorial cortex favor either directional cortical transport toward the equator or localized de novo assembly. However, this process has never been analyzed directly in dividing mammalian cells at a high resolution. Here we applied total internal reflection fluorescence microscope (TIRF-M), coupled with spatial temporal image correlation spectroscopy (STICS) and a new analytical approach termed temporal differential microscopy (TDM), to image the dynamics of myosin II and actin during the assembly of equatorial cortex. Our results indicated distinct and at least partially independent mechanisms for the early equatorial recruitment of myosin and actin filaments. Cortical myosin showed no detectable directional flow during early cytokinesis. In addition to equatorial assembly, we showed that localized inhibition of disassembly contributed to the formation of the equatorial myosin band. In contrast to myosin, actin filaments underwent a striking flux toward the equator. Myosin motor activity was required for the actin flux, but not for actin concentration in the furrow, suggesting that there was a flux-independent, de novo mechanism for actin recruitment along the equator. Our results indicate that cytokinesis involves signals that regulate both assembly and disassembly activities and argue against mechanisms that are coupled to global cortical movements.

Mechanics of mesenchymal contribution to clefting force in branching morphogenesis

Branching morphogenesis is ubiquitous and may involve several different mechanisms. Glandular morphogenesis is affected by growth, cell rearrangements, changes in the basal lamina, changes in the stromal ECM, changes in cell-cell and cell-ECM adhesions, mesenchymal contractility, and possibly other mechanisms. We have developed a 3D model of the mechanics of clefting, focusing in this paper solely on the potential role of mesenchyme-generated traction forces. The tissue mechanics are assumed to be those of fluids, and the hypothesized traction forces are modeled as advected by the deformations which they generate. We find that mesenchymal traction forces are sufficient to generate a cleft of the correct size and morphology, in the correct time frame. We find that viscosity of the tissues affects the time course of morphogenesis, and also affects the resulting form of the organ. Morphology is also strongly dependent on the initial distribution of contractility. We suggest an in vitro method of examining the role of mesenchyme in branching morphogenesis.

Stress generation and filament turnover during actin ring constriction

We present a physical analysis of the dynamics and mechanics of contractile actin rings. In particular, we analyze the dynamics of ring contraction during cytokinesis in the Caenorhabditis elegans embryo. We present a general analysis of force balances and material exchange and estimate the relevant parameter values. We show that on a microscopic level contractile stresses can result from both the action of motor proteins, which cross-link filaments, and from the polymerization and depolymerization of filaments in the presence of end-tracking cross-linkers.

An immersed boundary framework for modelling the growth of individual cells: an application to the early tumour development

A biomechanical approach in modelling the growth and division of a single fully deformable cell by using an immersed boundary method with distributed sources is presented, and its application to model the early tumour development is discussed. This mathematical technique couples a continuous description of a viscous incompressible cytoplasm with the dynamics of separate elastic cells, containing their own point nuclei, elastic plasma membranes with membrane receptors, and individually regulated cell processes. This model enables one to focus on the biomechanical properties of individual cells and on communication between cells and their microenvironment, simultaneously allowing for the formation of clusters or sheets of cells that act together as one complex tissue. Several examples of early tumours growing in various geometrical configurations and with distinct conditions of their initiation and progression are also presented to show the strength of our approach in modelling different topologies of the growing tissues in distinct biochemical conditions of the surrounding media.

Contractile stress generation by actomyosin gels

The tension generated by randomly distributed myosin minifilaments in an actin gel is evaluated using a rigorous theorem relating the surface forces acting on the gel to the forces exerted by the myosins. The maximum tension generated per myosin depends strongly on the lengths of the myosin minifilaments and the actin filaments. The result is used to place an upper bound on the tension that can be generated during cytokinesis. It is found that actomyosin contraction by itself generates too little force for ring contraction during cytokinesis unless the actin filaments are tightly crosslinked into inextensible units much longer than a single actin filament.

Enlazin, a natural fusion of two classes of canonical cytoskeletal proteins, contributes to cytokinesis dynamics

Cytokinesis requires a complex network of equatorial and global proteins to regulate cell shape changes. Here, using interaction genetics, we report the first characterization of a novel protein, enlazin. Enlazin is a natural fusion of two canonical classes of actin-associated proteins, the ezrin-radixin-moesin family and fimbrin, and it is localized to actin-rich structures. A fragment of enlazin, enl-tr, was isolated as a genetic suppressor of the cytokinesis defect of cortexillin-I mutants. Expression of enl-tr disrupts expression of endogenous enlazin, indicating that enl-tr functions as a dominant-negative lesion. Enlazin is distributed globally during cytokinesis and is required for cortical tension and cell adhesion. Consistent with a role in cell mechanics, inhibition of enlazin in a cortexillin-I background restores cytokinesis furrowing dynamics and suppresses the growth-in-suspension defect. However, as expected for a role in cell adhesion, inhibiting enlazin in a myosin-II background induces a synthetic cytokinesis phenotype, frequently arresting furrow ingression at the dumbbell shape and/or causing recession of the furrow. Thus, enlazin has roles in cell mechanics and adhesion, and these roles seem to be differentially significant for cytokinesis, depending on the genetic background.

The mechanism of cortical ingression during early cytokinesis: thinking beyond the contractile ring hypothesis

Owing to the rapid advances in genomic, proteomic and imaging technologies, the field of cytokinesis has seen rapid advances during the past decade. However, the basic model for the early stage of ingression, known as the contractile ring hypothesis, remains largely unchanged. From recent observations, it is becoming clear that early cytokinesis of animal cells involves a more extensive set of events, both temporally and spatially, than what is encompassed by the original contractile ring hypothesis. Activities relevant to cytokinesis, such as cortical contraction, can initiate well before onset of anaphase. Furthermore, equatorial ingression can involve multiple events in different regions of the cortex, including the establishment of anterior-posterior polarity, the modulation of cortical deformability, the expansion and compression of the cell cortex, and forces directed towards the interior of the cell or away from the equator. In this article (which is part of the Cytokinesis series), I evaluate critically key observations on when, where and how early ingression of animal cells takes place.

The stress and strain of cytokinesis

The ultimate goal of all signaling pathways in cytokinesis is to control the mechanical separation of the mother cell into two daughter cells. Because of the intrinsic mechanical nature of cytokinesis, it is essential to understand fully how cell shapes and the material properties of the cell are generated, how these shapes and material properties create force, and how motor proteins such as myosin-II modify the system to achieve successful cytokinesis. In this review (which is part of the Cytokinesis series), we discuss the relevant physical properties of cells, how these properties are measured and the basic models that are used to understand cell mechanics. Finally, we present our current understanding of how cytokinesis mechanics work.

Lamellipodial contractions during crawling and spreading

Most eukaryotic cells can crawl over surfaces. In general, this motility requires three distinct actions: polymerization at the leading edge, adhesion to the substrate, and retraction at the rear. Recent experiments with mouse embryonic fibroblasts showed that during spreading and crawling the lamellipodium undergoes periodic contractions that are substrate-dependent. Here I show that a simple model incorporating stick-slip adhesion and a simplified mechanism for the generation of contractile forces is sufficient to explain periodic lamellipodial contractions. This model also explains why treatment of cells with latrunculin modifies the period of these contractions. In addition, by coupling a diffusing chemical species that can bind actin, such as myosin light-chain kinase, with the contractile model leads to periodic rows and waves in the chemical species, similar to what is observed in experiments. This model provides a novel and simple explanation for the generation of contractile waves during cell spreading and crawling that is only dependent on stick-slip adhesion and the generation of contractile force and suggests new experiments to test this mechanism.

Balance of actively generated contractile and resistive forces controls cytokinesis dynamics

Cytokinesis, the fission of a mother cell into two daughter cells, is a simple and dramatic cell shape change. Here, we examine the dynamics of cytokinesis by using a combination of microscopy, dynamic measurements, and genetic analysis. We find that cytokinesis proceeds through a single sequence of shape changes, but the kinetics of the transformation from one shape to another differs dramatically between strains. We interpret the measurements in a simple and quantitative manner by using a previously uncharacterized analytic model. From the analysis, wild-type cytokinesis appears to proceed through an active, extremely regulated process in which globally distributed proteins generate resistive forces that slow the rate of furrow ingression. Finally, we propose that, in addition to myosin II, a Laplace pressure, resulting from material properties and the geometry of the dividing cell, generates force to help drive furrow ingression late in cytokinesis.

Dictyostelium cytokinesis: from molecules to mechanics

Cytokinesis is the mechanical process that allows the simplest unit of life, the cell, to divide, propagating itself. To divide, the cell converts chemical energy into mechanical energy to produce force. This process is thought to be active, due in large part to the mechanochemistry of the myosin-II ATPase. The cell's viscoelasticity defines the context and perhaps the magnitude of the forces that are required for cytokinesis. The viscoelasticity may also guide the force-generating apparatus, specifying the cell shape change that results. Genetic, biochemical, and mechanical measurements are providing a quantitative view of how real proteins control this essential life process.

The hydration dynamics of polyelectrolyte gels with applications to cell motility and drug delivery

We combine the physics of gels with the hydrodynamics of two-phase fluids to construct a set of equations that describe the hydration dynamics of polyelectrolyte gels. We use the model to address three problems. First, we express the effective diffusion constants for neutral and charged spherically distributed gels in terms of microscopic parameters. Second, we use the model to describe the locomotion of nematode sperm cells. Finally, we describe the swelling dynamics of polyelectrolyte gels used for drug release.

Analysis of actin dynamics at the leading edge of crawling cells: implications for the shape of keratocyte lamellipodia

Leading edge protrusion is one of the critical events in the cell motility cycle and it is believed to be driven by the assembly of the actin network. The concept of dendritic nucleation of actin filaments provides a basis for understanding the organization and dynamics of the actin network at the molecular level. At a larger scale, the dynamic geometry of the cell edge has been described in terms of the graded radial extension model, but this level of description has not yet been linked to the molecular dynamics. Here, we measure the graded distribution of actin filament density along the leading edge of fish epidermal keratocytes. We develop a mathematical model relating dendritic nucleation to the long-range actin distribution and the shape of the leading edge. In this model, a steady-state graded actin distribution evolves as a result of branching, growth and capping of actin filaments in a finite area of the leading edge. We model the shape of the leading edge as a product of the extension of the actin network, which depends on actin filament density. The feedback between the actin density and edge shape in the model results in a cell shape and an actin distribution similar to those experimentally observed. Thus, we explain the stability of the keratocyte shape in terms of the self-organization of the branching actin network.

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.

A force balance model of early spindle pole separation in Drosophila embryos

The formation and function of the mitotic spindle depends upon force generation by multiple molecular motors and by the dynamics of microtubules, but how these force-generating mechanisms relate to one another is unclear. To address this issue we have modeled the separation of spindle poles as a function of time during the early stages of spindle morphogenesis in Drosophila embryos. We propose that the outward forces that drive the separation of the spindle poles depend upon forces exerted by cortical dynein and by microtubule polymerization, and that these forces are antagonized by a C-terminal kinesin, Ncd, which generates an inward force on the poles. We computed the sum of the forces generated by dynein, microtubule polymerization, and Ncd, as a function of the extent of spindle pole separation and solved an equation relating the rate of pole separation to the net force. As a result, we obtained graphs of the time course of spindle pole separation during interphase and prophase that display a reasonable fit to the experimental data for wild-type and motor-inhibited embryos. Among the novel contributions of the model are an explanation of pole separation after simultaneous loss of Ncd and dynein function, and the prediction of a large value for the effective centrosomal drag that is needed to fit the experimental data. The results demonstrate the utility of force balance models for explaining certain mitotic movements because they explain semiquantitatively how the force generators drive a rapid initial burst of pole separation when the net force is great, how pole separation slows down as the force decreases, and how a stable separation of the spindle poles characteristic of the prophase steady state is achieved when the force reaches zero.

Quantitation of the distribution and flux of myosin-II during cytokinesis

BACKGROUND

During cytokinesis, the cell's equator contracts against the cell's global stiffness. Identifying the biochemical basis for these mechanical parameters is essential for understanding how cells divide. To achieve this goal, the distribution and flux of the cell division machinery must be quantified. Here we report the first quantitative analysis of the distribution and flux of myosin-II, an essential element of the contractile ring.

RESULTS

The fluxes of myosin-II in the furrow cortex, the polar cortex, and the cytoplasm were examined using ratio imaging of GFP fusion proteins expressed in Dictyostelium. The peak concentration of GFP-myosin-II in the furrow cortex is 1.8-fold higher than in the polar cortex and 2.0-fold higher than in the cytoplasm. The myosin-II in the furrow cortex, however, represents only 10% of the total cellular myosin-II. An estimate of the minimal amount of this motor needed to produce the required force for cell cleavage fits well with this 10% value. The cell may, therefore, regulate the amount of myosin-II sent to the furrow cortex in accordance with the amount needed there. Quantitation of the distribution and flux of a mutant myosin-II that is defective in phosphorylation-dependent thick filament disassembly confirms that heavy chain phosphorylation regulates normal recruitment to the furrow cortex.

CONCLUSION

The analysis indicates that myosin-II flux through the cleavage furrow cortex is regulated by thick filament phosphorylation. Further, the amount of myosin-II observed in the furrow cortex is in close agreement with the amount predicted to be required from a simple theoretical analysis.

Cell mechanics: mechanical response, cell adhesion, and molecular deformation

As the basic unit of life, the cell is a biologically complex system, the understanding of which requires a combination of various approaches including biomechanics. With recent progress in cell and molecular biology, the field of cell mechanics has grown rapidly over the last few years. This review synthesizes some of these recent developments to foster new concepts and approaches, and it emphasizes molecular-level understanding. The focuses are on the common themes and interconnections in three related areas: (a) the responses of cells to mechanical forces, (b) the mechanics and kinetics of cell adhesion, and (c) the deformation of biomolecules. Specific examples are also given to illustrate the quantitative modeling used in analyzing biological processes and physiological functions.

The mechanism of cytokinesis: reconsideration and reconciliation

The widely held models of cytokinesis contend that signals for cleavage are transmitted by astral microtubules, and that such signals elicit the assembly and contraction of an equatorial band of actin-myosin II filaments. However, experiments during the past decade have painted an increasingly complex picture, including strong evidence for the involvement of chromosomal passenger proteins and interzonal microtubules, and the involvement of not only cortical contraction but also cytoskeletal disintegration. The purpose of this article is to consider alternative models that might better accommodate both old and new observations. It is proposed that chromosomal passenger proteins undergo dynamic associations at centromeres during metaphase and are recruited from the cytoplasm to both astral and interzonal microtubules during anaphase. In addition, cytokinesis may be driven by global inward contractions coupled to a localized collapse of the equatorial cortex.

On the mechanism of cleavage furrow ingression in Dictyostelium

The ability of Dictyostelium cells to divide without myosin II in a cell cycle-coupled manner has opened two questions about the mechanism of cleavage furrow ingression. First, are there other possible functions for myosin II in this process except for generating contraction of the furrow by a sliding filament mechanism? Second, what could be an alternative mechanical basis for the furrowing? Using aberrant changes of the cell shape and anomalous localization of the actin-binding protein cortexillin I during asymmetric cytokinesis in myosin II-deficient cells as clues, it is proposed that myosin II filaments act as a mechanical lens in cytokinesis. The mechanical lens serves to focus the forces that induce the furrowing to the center of the midzone, a cortical region where cortexillins are enriched in dividing cells. Additionally, continual disassembly of a filamentous actin meshwork at the midzone is a prerequisite for normal ingression of the cleavage furrow and a successful cytokinesis. If this process is interrupted, as it occurs in cells that lack cortexillins, an overassembly of filamentous actin at the midzone obstructs the normal cleavage. Disassembly of the crosslinked actin network can generate entropic contractile forces in the cortex, and may be considered as an alternative mechanism for driving ingression of the cleavage furrow. Instead of invoking different types of cytokinesis that operate under attached and unattached conditions in Dictyostelium, it is anticipated that these cells use a universal multifaceted mechanism to divide, which is only moderately sensitive to elimination of its constituent mechanical processes.

Aspiration of human neutrophils: effects of shear thinning and cortical dissipation

It is generally accepted that the human neutrophil can be mechanically represented as a droplet of polymeric fluid enclosed by some sort of thin slippery viscoelastic cortex. Many questions remain however about the detailed rheology and chemistry of the interior fluid and the cortex. To address these quantitative issues, we have used a finite element method to simulate the dynamics of neutrophils during micropipet aspiration using various plausible assumptions. The results were then systematically compared with aspiration experiments conducted at eight different combinations of pipet size and pressure. Models in which the cytoplasm was represented by a simple Newtonian fluid (i.e., models without shear thinning) were grossly incapable of accounting for the effects of pressure on the general time scale of neutrophil aspiration. Likewise, models in which the cortex was purely elastic (i.e., models without surface viscosity) were unable to explain the effects of pipet size on the general aspiration rate. Such models also failed to explain the rapid acceleration of the aspiration rate during the final phase of aspiration nor could they account for the geometry of the neutrophil during various phases of aspiration. Thus, our results indicate that a minimal mechanical model of the neutrophil needs to incorporate both shear thinning and surface viscosity to remain valid over a reasonable range of conditions. At low shear rates, the surface dilatation viscosity of the neutrophil was found to be on the order of 100 poise-cm, whereas the viscosity of the interior cytoplasm was on the order of 1000 poise. Both the surface viscosity and the interior viscosity seem to decrease in a similar fashion when the shear rate exceeds approximately 0.05 s(-1). Unfortunately, even models with both surface viscosity and shear thinning studied are still not sufficient to fully explain all the features of neutrophil aspiration. In particular, the very high rate of aspiration during the initial moments after ramping of pressure remains mysterious.

Distinct roles of the equatorial and polar cortices in the cleavage of adherent cells

Over the past 100 years, many models have been proposed and tested for cytokinesis [1]. There is strong evidence that the equator represents a unique region that receives cleavage signals from the mitotic spindle [2, 3]. The nature of such a signal and the mechanism of cleavage, however, remain poorly understood. To probe the contribution of different cortical regions in the cleavage of cultured epithelial cells, we applied cytochalasin D (CD), a known inhibitor of cytokinesis [4], in a highly localized manner to different regions of dividing NRK cells. Surprisingly, equatorial application of CD not only allowed cytokinesis to complete but also appeared to facilitate the process. Conversely, local application of CD near the polar region caused inhibition of cytokinesis. Our results suggest a mechanism that involves global coordination of cortical activities, including controlled cortical disassembly along the equator and possibly global cortical contraction.

Cytokinesis without myosin II

The ability of substrate-anchored Dictyostelium cells to divide without myosin II has opened the possibility of analysing the formation of cleavage furrows in the absence of a contractile ring made of filamentous myosin and actin. Similar possibilities exist in mutants of budding yeast and, less strictly, also in drug-treated mammalian cells. Myosin-II-independent activities in Dictyostelium include the microtubule-induced programming of the cell surface into ruffling areas and regions that are converted into a concave furrow, as well as the translocation of cortexillins and cross-linked membrane proteins towards the cleavage furrow. A centripetal flow of actin filaments followed by their disassembly in the cleavage furrow is proposed to underlie the translocation.

Absence of furrowing activity following regional cortical tension reduction in sand dollar blastomere and fertilized egg fragment surfaces

The purpose of the present investigation was to test experimentally the possibility that division mechanism establishment at the equator of sand dollar eggs may be a consequence of cortical tension gradients between the equator and the poles. Cytochalasin has been shown to decrease tension at the sea urchin egg surface. The concave ends of cytochalasin D-containing agarose cylinders were held against regions of the surface of Echinarachnius parma blastomeres and enucleated fertilized egg fragments. The ability to interfere with normal furrowing activity was used as a biological indicator of the effectiveness of cytochalasin. When agarose containing 2 microg/mL cytochalasin contacted the equatorial region of the blastomeres resulting from the first cleavage, or the equatorial surfaces of nucleated fertilized egg halves, furrowing was blocked, stalled or delayed, indicating that the concentration of cytochalasin was effective. When the same concentration of cytochalasin was applied to the poles, the cells and nucleated fertilized egg fragments divided in the same way as the controls, indicating that the effectiveness of the cytochalasin did not spread from the poles to the equator and that bisection did not interfere with the division of nucleated fertilized egg fragments. When the same concentration of cytochalasin was applied to diametrically opposed surfaces of enucleated, spherical egg fragments, there was no evidence of furrowing activity between the areas that contacted the cytochalasin or in any other part of the surface. Because of the tension-reducing effect of cytochalasin, a tension gradient existed between the regions affected and unaffected by cytochalasin. The results strongly suggest that establishment of the division mechanism by simple gradients of tension at the surface is unlikely.

Cytoplasm dynamics and cell motion: two-phase flow models

The motion of amoeboid cells is characterized by cytoplasmic streaming and by membrane protrusions and retractions which occur even in the absence of interactions with a substratum. Cell translocation requires, in addition, a transmission mechanism wherein the power produced by the cytoplasmic engine is applied to the substratum in a highly controlled fashion through specific adhesion proteins. Here we present a simple mechano-chemical model that tries to capture the physical essence of these complex biomolecular processes. Our model is based on the continuum equations for a viscous and reactive two-phase fluid model with moving boundaries, and on force balance equations that average the stochastic interactions between actin polymers and membrane proteins. In this paper we present a new derivation and analysis of these equations based on minimization of a power functional. This derivation also leads to a clear formulation and classification of the kinds of boundary conditions that should be specified at free surfaces and at the sites of interaction of the cell and the substratum. Numerical simulations of a one-dimensional lamella reveal that even this extremely simplified model is capable of producing several typical features of cell motility. These include periodic 'ruffle' formation, protrusion-retraction cycles, centripetal flow and cell-substratum traction forces.