Cytoplasm dynamics and cell motion: two-phase flow models

Two-fluid dynamics and micron-thin boundary layers shape cytoplasmic flows in early Drosophila embryos

Cytoplasmic flows are widely emerging as key functional players in development. In early Drosophila embryos, flows drive the spreading of nuclei across the embryo. Here, we combine hydrodynamic modeling with quantitative imaging to develop a two-fluid model that features an active actomyosin gel and a passive viscous cytosol. Gel contractility is controlled by the cell cycle oscillator, the two fluids being coupled by friction. In addition to recapitulating experimental flow patterns, our model explains observations that remained elusive and makes a series of predictions. First, the model captures the vorticity of cytosolic flows, which highlights deviations from Stokes' flow that were observed experimentally but remained unexplained. Second, the model reveals strong differences in the gel and cytosol motion. In particular, a micron-sized boundary layer is predicted close to the cortex, where the gel slides tangentially while the cytosolic flow cannot slip. Third, the model unveils a mechanism that stabilizes the spreading of nuclei with respect to perturbations of their initial positions. This self-correcting mechanism is argued to be functionally important for proper nuclear spreading. Fourth, we use our model to analyze the effects of flows on the transport of the morphogen Bicoid and the establishment of its gradients. Finally, the model predicts that the flow strength should be reduced if the shape of the domain is more round, which is experimentally confirmed in Drosophila mutants. Thus, our two-fluid model explains flows and nuclear positioning in early Drosophila, while making predictions that suggest novel future experiments.

Phase Separation Driven by Active Flows

We extend the continuum theories of active nematohydrodynamics to model a two-fluid mixture with separate velocity fields for each fluid component, coupled through a viscous drag. The model is used to study an active nematic fluid mixed with an isotropic fluid. We find microphase separation, and argue that this results from an interplay between active anchoring and active flows driven by concentration gradients. The results may be relevant to cell sorting and the formation of lipid rafts in cell membranes.

Two-fluid dynamics and micron-thin boundary layers shape cytoplasmic flows in early Drosophila embryos

Cytoplasmic flows are widely emerging as key functional players in development. In early Drosophila embryos, flows drive the spreading of nuclei across the embryo. Here, we combine hydrodynamic modeling with quantitative imaging to develop a two-fluid model that features an active actomyosin gel and a passive viscous cytosol. Gel contractility is controlled by the cell cycle oscillator, the two fluids being coupled by friction. In addition to recapitulating experimental flow patterns, our model explains observations that remained elusive, and makes a series of new predictions. First, the model captures the vorticity of cytosolic flows, which highlights deviations from Stokes' flow that were observed experimentally but remained unexplained. Second, the model reveals strong differences in the gel and cytosol motion. In particular, a micron-sized boundary layer is predicted close to the cortex, where the gel slides tangentially whilst the cytosolic flow cannot slip. Third, the model unveils a mechanism that stabilizes the spreading of nuclei with respect to perturbations of their initial positions. This self-correcting mechanism is argued to be functionally important for proper nuclear spreading. Fourth, we use our model to analyze the effects of flows on the transport of the morphogen Bicoid, and the establishment of its gradients. Finally, the model predicts that the flow strength should be reduced if the shape of the domain is more round, which is experimentally confirmed in Drosophila mutants. Thus, our two-fluid model explains flows and nuclear positioning in early Drosophila, while making predictions that suggest novel future experiments.

Viscous shaping of the compliant cell nucleus

The cell nucleus is commonly considered to be a stiff organelle that mechanically resists changes in shape, and this resistance is thought to limit the ability of cells to migrate through pores or spread on surfaces. Generation of stresses on the cell nucleus during migration and nuclear response to these stresses is fundamental to cell migration and mechano-transduction. In this Perspective, we discuss our previous experimental and computational evidence that supports a dynamic model, in which the soft nucleus is irreversibly shaped by viscous stresses generated by the motion of cell boundaries and transmitted through the intervening cytoskeletal network. While the nucleus is commonly modeled as a stiff elastic body, we review how nuclear shape changes on the timescale of migration can be explained by simple geometric constraints of constant nuclear volume and constant surface area of the nuclear lamina. Because the lamina surface area is in excess of that of a sphere of the same volume, these constraints permit dynamic transitions between a wide range of shapes during spreading and migration. The excess surface area allows the nuclear shape changes to mirror those of the cell with little mechanical resistance. Thus, the nucleus can be easily shaped by the moving cell boundaries over a wide range of shape changes and only becomes stiff to more extreme deformations that would require the lamina to stretch or the volume to compress. This model explains how nuclei can easily flatten on surfaces during cell spreading or elongate as cells move through pores until the lamina smooths out and becomes tense.

Cell nucleus as a microrheological probe to study the rheology of the cytoskeleton

Mechanical properties of the cell are important biomarkers for probing its architectural changes caused by cellular processes and/or pathologies. The development of microfluidic technologies has enabled measuring the cell's mechanical properties at high throughput so that mechanical phenotyping can be applied to large samples in reasonable timescales. These studies typically measure the stiffness of the cell as the only mechanical biomarker and do not disentangle the rheological contributions of different structural components of the cell, including the cell cortex, the interior cytoplasm and its immersed cytoskeletal structures, and the nucleus. Recent advancements in high-speed fluorescent imaging have enabled probing the deformations of the cell cortex while also tracking different intracellular components in rates applicable to microfluidic platforms. We present a, to our knowledge, novel method to decouple the mechanics of the cell cortex and the cytoplasm by analyzing the correlation between the cortical deformations that are induced by external microfluidic flows and the nucleus displacements, induced by those cortical deformations, i.e., we use the nucleus as a high-throughput microrheological probe to study the rheology of the cytoplasm, independent of the cell cortex mechanics. To demonstrate the applicability of this method, we consider a proof-of-concept model consisting of a rigid spherical nucleus centered in a spherical cell. We obtain analytical expressions for the time-dependent nucleus velocity as a function of the cell deformations when the interior cytoplasm is modeled as a viscous, viscoelastic, porous, and poroelastic material and demonstrate how the nucleus velocity can be used to characterize the linear rheology of the cytoplasm over a wide range of forces and timescales/frequencies.

Active random forces can drive differential cellular positioning and enhance motor-driven transport

Cells are remarkable machines capable of performing an exquisite range of functions, many of which depend crucially on the activity of molecular motors that generate forces. Recent experiments have shown that intracellular random movements are not solely thermal in nature but also arise from stochasticity in the forces from these molecular motors. Here we consider the effects of these nonthermal random forces. We show that stochastic motor force not only enhances diffusion but also leads to size-dependent transport of objects that depends on the local density of the cytoskeletal filaments on which motors operate. As a consequence, we find that objects that are larger than the mesh size of the cytoskeleton should be attracted to regions of high cytoskeletal density, while objects that are smaller than the mesh size will preferentially avoid these regions. These results suggest a mechanism for size-based organelle positioning and also suggest that motor-driven random forces can additionally enhance motor-driven transport.

Active poroelastic two-phase model for the motion of physarum microplasmodia

The onset of self-organized motion is studied in a poroelastic two-phase model with free boundaries for Physarum microplasmodia (MP). In the model, an active gel phase is assumed to be interpenetrated by a passive fluid phase on small length scales. A feedback loop between calcium kinetics, mechanical deformations, and induced fluid flow gives rise to pattern formation and the establishment of an axis of polarity. Altogether, we find that the calcium kinetics that breaks the conservation of the total calcium concentration in the model and a nonlinear friction between MP and substrate are both necessary ingredients to obtain an oscillatory movement with net motion of the MP. By numerical simulations in one spatial dimension, we find two different types of oscillations with net motion as well as modes with time-periodic or irregular switching of the axis of polarity. The more frequent type of net motion is characterized by mechano-chemical waves traveling from the front towards the rear. The second type is characterized by mechano-chemical waves that appear alternating from the front and the back. While both types exhibit oscillatory forward and backward movement with net motion in each cycle, the trajectory and gel flow pattern of the second type are also similar to recent experimental measurements of peristaltic MP motion. We found moving MPs in extended regions of experimentally accessible parameters, such as length, period and substrate friction strength. Simulations of the model show that the net speed increases with the length, provided that MPs are longer than a critical length of ≈ 120 μm. Both predictions are in line with recent experimental observations.

Self-propulsion of droplets driven by an active permeating gel

We discuss the flow field and propulsion velocity of active droplets, which are driven by body forces residing on a rigid gel. The latter is modelled as a porous medium which gives rise to permeation forces. In the simplest model, the Brinkman equation, the porous medium is characterised by a single lengthscale [Formula: see text] --the square root of the permeability. We compute the flow fields inside and outside of the droplet as well as the energy dissipation as a function of [Formula: see text]. We furthermore show that there are optimal gel fractions, giving rise to maximal linear and rotational velocities. In the limit [Formula: see text], corresponding to a very dilute gel, we recover Stokes flow. The opposite limit, [Formula: see text], corresponding to a space filling gel, is singular and not equivalent to Darcy's equation, which cannot account for self-propulsion.

Electrodiffusion-Mediated Swelling of a Two-Phase Gel Model of Gastric Mucus

Gastric mucus gel is known to exhibit dramatic and unique swelling behaviors in response to the ionic composition of the hydrating solution. This swelling behavior is important in the maintenance of the mucus layer lining the stomach wall, as the layer is constantly digested by enzymes in the lumen, and must be replenished by new mucus that swells as it is secreted from the gastric wall. One hypothesis suggests that the condensed state of mucus at secretion is maintained by transient bonds with calcium that form crosslinks. These crosslinks are lost as monovalent cations from the environment displace divalent crosslinkers, leading to a dramatic change in the energy of the gel and inducing the swelling behavior. Previous modeling work has characterized the equilibrium behavior of polyelectrolyte gels that respond to calcium crosslinking. Here, we present an investigation of the dynamic swelling behavior of a polyelectrolytic gel model of mucus. In particular, we quantified the rate at which a globule of initially crosslinked gel swells when exposed to an ionic bath. The dependence of this swelling rate on several parameters was characterized. We observed that swelling rate has a non-monotone dependence on the molarity of the bath solution, with moderate concentrations of available sodium inducing the fastest swelling.

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.

Paxillin phosphorylation at serine 273 and its effects on Rac, Rho and adhesion dynamics

Focal adhesions are protein complexes that anchor cells to the extracellular matrix. During migration, the growth and disassembly of these structures are spatiotemporally regulated, with new adhesions forming at the leading edge of the cell and mature adhesions disassembling at the rear. Signalling proteins and structural cytoskeletal components tightly regulate adhesion dynamics. Paxillin, an adaptor protein within adhesions, is one of these proteins. Its phosphorylation at serine 273 (S273) is crucial for maintaining fast adhesion assembly and disassembly. Paxillin is known to bind to a GIT1-βPIX-PAK1 complex, which increases the local activation of the small GTPase Rac. To understand quantitatively the behaviour of this system and how it relates to adhesion assembly/disassembly, we developed a mathematical model describing the dynamics of the small GTPases Rac and Rho as determined by paxillin S273 phosphorylation. Our model revealed that the system possesses bistability, where switching between uninduced (active Rho) and induced (active Rac) states can occur through a change in rate of paxillin phosphorylation or PAK1 activation. The bistable switch is characterized by the presence of memory, minimal change in the levels of active Rac and Rho within the induced and uninduced states, respectively, and the limited regime of monostability associated with the uninduced state. These results were validated experimentally by showing the presence of bimodality in adhesion assembly and disassembly rates, and demonstrating that Rac activity increases after treating Chinese Hamster Ovary cells with okadaic acid (a paxillin phosphatase inhibitor), followed by a modest recovery after 20 min washout. Spatial gradients of phosphorylated paxillin in a reaction-diffusion model gave rise to distinct regions of Rac and Rho activities, resembling polarization of a cell into front and rear. Perturbing several parameters of the model also revealed important insights into how signalling components upstream and downstream of paxillin phosphorylation affect dynamics.

Cellular migration is crucial in both physiological and pathological functions. Maintenance of proper migration and development of aberrant migration are effectuated by cellular machinery involving protein complexes, called adhesions, that anchor the cell to its environment. Over time, these adhesions assemble at the leading edge, as the cell extends forward, anchoring the front of the cells to its substrate, while those at the cell rear disassemble, allowing detachment and forward movement. Their dynamics are controlled by a number of regulatory factors, occurring on both cell-wide and adhesion-level scales. The coordination of these regulatory factors is complex, but insights about their dynamics can be gained from the use of mathematical modeling techniques which integrate many of these components together. Here, we developed several molecularly explicit models to explore how local regulation of paxillin, an adhesion protein, interacts with the activities of Rac and Rho to produce cell-wide polarization associated with motility and directionality. By altering paxillin phosphorylation/dephosphorylation within such models, we have advanced our understanding of how a shift from a non-motile state to a highly motile state occurs. Deciphering these key processes quantitatively thus helped us gain insight into the subcellular factors underlying polarity and movement.

Hydrodynamic theory of active matter

We review the general hydrodynamic theory of active soft materials that is motivated in particular by biological matter. We present basic concepts of irreversible thermodynamics of spatially extended multicomponent active systems. Starting from the rate of entropy production, we identify conjugate thermodynamic fluxes and forces and present generic constitutive equations of polar active fluids and active gels. We also discuss angular momentum conservation which plays a role in the the physics of active chiral gels. The irreversible thermodynamics of active gels provides a general framework to discuss the physics that underlies a wide variety of biological processes in cells and in multicellular tissues.

A DISCRETE MATHEMATICAL MODEL FOR SINGLE AND COLLECTIVE MOVEMENT IN AMOEBOID CELLS

In this paper, we develop a new discrete mathematical model for individual and collective cell motility. We introduce a mechanical model for the movement of a cell on a two-dimensional rigid surface to describe and investigate the cell–cell and cell–substrate interactions. The cell cytoskeleton is modeled as a series of springs and dashpots connected in parallel. The cell–substrate attachments and the cell protrusions are also included. In particular, this model is used to describe the directed movement of endothelial cells on a Matrigel plate. We compare the results from our model with experimental data. We show that cell density and substrate rigidity play an important role in network formation.

Computational modeling of single-cell mechanics and cytoskeletal mechanobiology

Cellular cytoskeletal mechanics plays a major role in many aspects of human health from organ development to wound healing, tissue homeostasis and cancer metastasis. We summarize the state-of-the-art techniques for mathematically modeling cellular stiffness and mechanics and the cytoskeletal components and factors that regulate them. We highlight key experiments that have assisted model parameterization and compare the advantages of different models that have been used to recapitulate these experiments. An overview of feed-forward mechanisms from signaling to cytoskeleton remodeling is provided, followed by a discussion of the rapidly growing niche of encapsulating feedback mechanisms from cytoskeletal and cell mechanics to signaling. We discuss broad areas of advancement that could accelerate research and understanding of cellular mechanobiology. A precise understanding of the molecular mechanisms that affect cell and tissue mechanics and function will underpin innovations in medical device technologies of the future. WIREs Syst Biol Med 2018, 10:e1407. doi: 10.1002/wsbm.1407 This article is categorized under: Models of Systems Properties and Processes > Mechanistic Models Physiology > Mammalian Physiology in Health and Disease Models of Systems Properties and Processes > Cellular Models.

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.

Moving Cell Boundaries Drive Nuclear Shaping during Cell Spreading

The nucleus has a smooth, regular appearance in normal cells, and its shape is greatly altered in human pathologies. Yet, how the cell establishes nuclear shape is not well understood. We imaged the dynamics of nuclear shaping in NIH3T3 fibroblasts. Nuclei translated toward the substratum and began flattening during the early stages of cell spreading. Initially, nuclear height and width correlated with the degree of cell spreading, but over time, reached steady-state values even as the cell continued to spread. Actomyosin activity, actomyosin bundles, microtubules, and intermediate filaments, as well as the LINC complex, were all dispensable for nuclear flattening as long as the cell could spread. Inhibition of actin polymerization as well as myosin light chain kinase with the drug ML7 limited both the initial spreading of cells and flattening of nuclei, and for well-spread cells, inhibition of myosin-II ATPase with the drug blebbistatin decreased cell spreading with associated nuclear rounding. Together, these results show that cell spreading is necessary and sufficient to drive nuclear flattening under a wide range of conditions, including in the presence or absence of myosin activity. To explain this observation, we propose a computational model for nuclear and cell mechanics that shows how frictional transmission of stress from the moving cell boundaries to the nuclear surface shapes the nucleus during early cell spreading. Our results point to a surprisingly simple mechanical system in cells for establishing nuclear shapes.

An immersed boundary method for two-phase fluids and gels and the swimming of Caenorhabditis elegans through viscoelastic fluids

The swimming of microorganisms typically involves the undulation or rotation of thin, filamentary objects in a fluid or other medium. Swimming in Newtonian fluids has been examined extensively, and only recently have investigations into microorganism swimming through non-Newtonian fluids and gels been explored. The equations that govern these more complex media are often nonlinear and require computational algorithms to study moderate to large amplitude motions of the swimmer. Here, we develop an immersed boundary method for handling fluid-structure interactions in a general two-phase medium, where one phase is a Newtonian fluid and the other phase is viscoelastic (e.g., a polymer melt or network). We use this algorithm to investigate the swimming of an undulating, filamentary swimmer in 2D (i.e., a sheet). A novel aspect of our method is that it allows one to specify how forces produced by the swimmer are distributed between the two phases of the fluid. The algorithm is validated by comparing theoretical predictions for small amplitude swimming in gels and viscoelastic fluids. We show how the swimming velocity depends on material parameters of the fluid and the interaction between the fluid and swimmer. In addition, we simulate the swimming of Caenorhabditis elegans in viscoelastic fluids and find good agreement between the swimming speeds and fluid flows in our simulations and previous experimental measurements. These results suggest that our methodology provides an accurate means for exploring the physics of swimming through non-Newtonian fluids and gels.

Two-Phase Acto-Cytosolic Fluid Flow in a Moving Keratocyte: A 2D Continuum Model

The F-actin network and cytosol in the lamellipodia of crawling cells flow in a centripetal pattern and spout-like form, respectively. We have numerically studied this two-phase flow in the realistic geometry of a moving keratocyte. Cytosol has been treated as a low viscosity Newtonian fluid flowing through the high viscosity porous medium of F-actin network. Other involved phenomena including myosin activity, adhesion friction, and interphase interaction are also discussed to provide an overall view of this problem. Adopting a two-phase coupled model by myosin concentration, we have found new accurate perspectives of acto-cytosolic flow and pressure fields, myosin distribution, as well as the distribution of effective forces across the lamellipodia of a keratocyte with stationary shape. The order of magnitude method is also used to determine the contribution of forces in the internal dynamics of lamellipodia.

A mathematical model for eph/ephrin-directed segregation of intermingled cells

Eph receptors, the largest family of receptor tyrosine kinases, control cell-cell adhesion/de-adhesion, cell morphology and cell positioning through interaction with cell surface ephrin ligands. Bi-directional signalling from the Eph and ephrin complexes on interacting cells have a significant role in controlling normal tissue development and oncogenic tissue patterning. Eph-mediated tissue patterning is based on the fine-tuned balance of adhesion and de-adhesion reactions between distinct Eph- and ephrin-expressing cell populations, and adhesion within like populations (expressing either Eph or ephrin). Here we develop a stochastic, Lagrangian model that is based on Eph/ephrin biology: incorporating independent Brownian motion to describe cell movement and a deterministic term (the drift term) to represent repulsive and adhesive interactions between neighbouring cells. Comparison between the experimental and computer simulated Eph/ephrin cell patterning events shows that the model recapitulates the dynamics of cell-cell segregation and cell cluster formation. Moreover, by modulating the term for Eph/ephrin-mediated repulsion, the model can be tuned to match the actual behaviour of cells with different levels of Eph expression or activity. Together the results of our experiments and modelling suggest that the complexity of Eph/ephrin signalling mechanisms that control cell-cell interactions can be described well by a mathematical model with a single term balancing adhesion and de-adhesion between interacting cells. This model allows reliable prediction of Eph/ephrin-dependent control of cell patterning behaviour.

An active poroelastic model for mechanochemical patterns in protoplasmic droplets of Physarum polycephalum

Motivated by recent experimental studies, we derive and analyze a two-dimensional model for the contraction patterns observed in protoplasmic droplets of Physarum polycephalum. The model couples a description of an active poroelastic two-phase medium with equations describing the spatiotemporal dynamics of the intracellular free calcium concentration. The poroelastic medium is assumed to consist of an active viscoelastic solid representing the cytoskeleton and a viscous fluid describing the cytosol. The equations for the poroelastic medium are obtained from continuum force balance and include the relevant mechanical fields and an incompressibility condition for the two-phase medium. The reaction-diffusion equations for the calcium dynamics in the protoplasm of Physarum are extended by advective transport due to the flow of the cytosol generated by mechanical stress. Moreover, we assume that the active tension in the solid cytoskeleton is regulated by the calcium concentration in the fluid phase at the same location, which introduces a mechanochemical coupling. A linear stability analysis of the homogeneous state without deformation and cytosolic flows exhibits an oscillatory Turing instability for a large enough mechanochemical coupling strength. Numerical simulations of the model equations reproduce a large variety of wave patterns, including traveling and standing waves, turbulent patterns, rotating spirals and antiphase oscillations in line with experimental observations of contraction patterns in the protoplasmic droplets.

Mathematical modeling of eukaryotic cell migration: insights beyond experiments

A migrating cell is a molecular machine made of tens of thousands of short-lived and interacting parts. Understanding migration means understanding the self-organization of these parts into a system of functional units. This task is one of tackling complexity: First, the system integrates numerous chemical and mechanical component processes. Second, these processes are connected in feedback interactions and over a large range of spatial and temporal scales. Third, many processes are stochastic, which leads to heterogeneous migration behaviors. Early on in the research of cell migration it became evident that this complexity exceeds human intuition. Thus, the cell migration community has led the charge to build mathematical models that could integrate the diverse experimental observations and measurements in consistent frameworks, first in conceptual and more recently in molecularly explicit models. The main goal of this review is to sift through a series of important conceptual and explicit mathematical models of cell migration and to evaluate their contribution to the field in their ability to integrate critical experimental data.

Accurate computation of traveling wavefronts in a biological hydrodynamic model

This paper concerns a hyperbolic-elliptic system of partial differential equations for the biological cell density and cell velocity. This system appears as a mathematical model for describing the dynamics of cell motion. Traveling wavefront solutions for the system of equations are computed by using two different numerical methods. One of the methods involves the traveling wave equations and solving an initial-value problem, which leads to accurate computations of the wave profile of cell density and the distributions of the cell velocity. The second method is to solve an initial-moving boundary-value problem for the PDE system, where the traveling wave emerges as the asymptotic long time solution.

Intracellular mechanochemical waves in an active poroelastic model

Many processes in living cells are controlled by biochemical substances regulating active stresses. The cytoplasm is an active material with both viscoelastic and liquid properties. We incorporate the active stress into a two-phase model of the cytoplasm which accounts for the spatiotemporal dynamics of the cytoskeleton and the cytosol. The cytoskeleton is described as a solid matrix that together with the cytosol as an interstitial fluid constitutes a poroelastic material. We find different forms of mechanochemical waves including traveling, standing, and rotating waves by employing linear stability analysis and numerical simulations in one and two spatial dimensions.

Asymmetry between pushing and pulling for crawling cells

Eukaryotic cells possess motility mechanisms allowing them not only to self-propel but also to exert forces on obstacles (to push) and to carry cargoes (to pull). To study the inherent asymmetry between active pushing and pulling we model a crawling acto-myosin cell extract as a one-dimensional layer of active gel subjected to external forces. We show that pushing is controlled by protrusion and that the macroscopic signature of the protrusion dominated motility mechanism is concavity of the force-velocity relation. In contrast, pulling is driven by protrusion only at small values of the pulling force and it is replaced by contraction when the pulling force is sufficiently large. This leads to more complex convex-concave structure of the force-velocity relation; in particular, competition between protrusion and contraction can produce negative mobility in a biologically relevant range. The model illustrates active readjustment of the force generating machinery in response to changes in the dipole structure of external forces. The possibility of switching between complementary active mechanisms implies that if necessary "pushers" can replace "pullers" and vice versa.

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.

Traveling waves in actin dynamics and cell motility

Much of current understanding of cell motility arose from studying steady treadmilling of actin arrays. Recently, there have been a growing number of observations of a more complex, non-steady, actin behavior, including self-organized waves. It is becoming clear that these waves result from activation and inhibition feedbacks in actin dynamics acting on different scales, but the exact molecular nature of these feedbacks and the respective roles of biomechanics and biochemistry are still unclear. Here, we review recent advances achieved in experimental and theoretical studies of actin waves and discuss mechanisms and physiological significance of wavy protrusions.

Modelling cell motility and chemotaxis with evolving surface finite elements

We present a mathematical and a computational framework for the modelling of cell motility. The cell membrane is represented by an evolving surface, with the movement of the cell determined by the interaction of various forces that act normal to the surface. We consider external forces such as those that may arise owing to inhomogeneities in the medium and a pressure that constrains the enclosed volume, as well as internal forces that arise from the reaction of the cells' surface to stretching and bending. We also consider a protrusive force associated with a reaction-diffusion system (RDS) posed on the cell membrane, with cell polarization modelled by this surface RDS. The computational method is based on an evolving surface finite-element method. The general method can account for the large deformations that arise in cell motility and allows the simulation of cell migration in three dimensions. We illustrate applications of the proposed modelling framework and numerical method by reporting on numerical simulations of a model for eukaryotic chemotaxis and a model for the persistent movement of keratocytes in two and three space dimensions. Movies of the simulated cells can be obtained from http://homepages.warwick.ac.uk/∼maskae/CV_Warwick/Chemotaxis.html.

'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.

Excitable actin dynamics in lamellipodial protrusion and retraction

Many animal cells initiate crawling by protruding lamellipodia, consisting of a dense network of actin filaments, at their leading edge. We imaged XTC cells that exhibit flat lamellipodia on poly-L-lysine-coated coverslips. Using active contours, we tracked the leading edge and measured the total amount of F-actin by summing the pixel intensities within a 5-μm band. We observed protrusion and retraction with period 130-200 s and local wavelike features. Positive (negative) velocities correlated with minimum (maximum) integrated actin concentration. Approximately constant retrograde flow indicated that protrusions and retractions were driven by fluctuations of the actin polymerization rate. We present a model of these actin dynamics as an excitable system in which a diffusive, autocatalytic activator causes actin polymerization; F-actin accumulation in turn inhibits further activator accumulation. Simulations of the model reproduced the pattern of actin polymerization seen in experiments. To explore the model's assumption of an autocatalytic activation mechanism, we imaged cells expressing markers for both F-actin and the p21 subunit of the Arp2/3 complex. We found that integrated Arp2/3-complex concentrations spike several seconds before spikes of F-actin concentration. This suggests that the Arp2/3 complex participates in an activation mechanism that includes additional diffuse components. Response of cells to stimulation by fetal calf serum could be reproduced by the model, further supporting the proposed dynamical picture.

A review of models of fluctuating protrusion and retraction patterns at the leading edge of motile cells

A characteristic feature of motile cells as they undergo a change in motile behavior is the development of fluctuating exploratory motions of the leading edge, driven by actin polymerization. We review quantitative models of these protrusion and retraction phenomena. Theoretical studies have been motivated by advances in experimental and computational methods that allow controlled perturbations, single molecule imaging, and analysis of spatiotemporal correlations in microscopic images. To explain oscillations and waves of the leading edge, most theoretical models propose nonlinear interactions and feedback mechanisms among different components of the actin cytoskeleton system. These mechanisms include curvature-sensing membrane proteins, myosin contraction, and autocatalytic biochemical reaction kinetics. We discuss how the combination of experimental studies with modeling promises to quantify the relative importance of these biochemical and biophysical processes at the leading edge and to evaluate their generality across cell types and extracellular environments.

Cell motility resulting from spontaneous polymerization waves

The crawling of cells on a substrate is in many cases driven by the actin cytoskeleton. How actin filaments and associated proteins are organized to generate directed motion is still poorly understood. Recent experimental observations suggest that spontaneous cytoskeletal waves might orchestrate the actin-filament network to produce directed motion. We investigate this possibility by studying a mean-field description of treadmilling filaments interacting with nucleating proteins, a system that is known to self-organize into waves. Confining the system by a boundary that shares essential features of membranes, we find that spontaneous waves can generate directional motion. We also find that it can produce lateral waves along the confining membrane as are observed in spreading cells.

LOCOMOTION OF A TWO DIMENSIONAL KERATOCYTE MODEL

The polymerization-induced propulsion of a model cell consisting of a cell membrane enclosing mobile actin molecules and polymerizing actin filaments is studied using Monte Carlo methods. It is shown that asymmetric polymerization alone induces a rectified motion of the cell. The structural organization of the locomoting cell exhibits an anisotropic shape induced by the anisotropic distribution of actin within the cell. This nonequilibrium distribution is maintained by a constant flow of actin molecules from the rear to the front of the cell. The efficiency of the rectification process, and hence the cell velocity, depends cooperatively on the density of actin molecules. The maximum of the cell velocity is determined by the optimal interplay between the number of filaments and the fluctuation of the cell membrane.

A computational model of cell polarization and motility coupling mechanics and biochemistry

The motion of a eukaryotic cell presents a variety of interesting and challenging problems from both a modeling and a computational perspective. The processes span many spatial scales (from molecular to tissue) as well as disparate time scales, with reaction kinetics on the order of seconds, and the deformation and motion of the cell occurring on the order of minutes. The computational difficulty, even in 2D, resides in the fact that the problem is inherently one of deforming, non-stationary domains, bounded by an elastic perimeter, inside of which there is redistribution of biochemical signaling substances. Here we report the results of a computational scheme using the immersed boundary method to address this problem. We adopt a simple reaction-diffusion system that represents an internal regulatory mechanism controlling the polarization of a cell, and determining the strength of protrusion forces at the front of its elastic perimeter. Using this computational scheme we are able to study the effect of protrusive and elastic forces on cell shapes on their own, the distribution of the reaction-diffusion system in irregular domains on its own, and the coupled mechanical-chemical system. We find that this representation of cell crawling can recover important aspects of the spontaneous polarization and motion of certain types of crawling cells.

Robust organizational principles of protrusive biopolymer networks in migrating living cells

Cell migration is associated with the dynamic protrusion of a thin actin-based cytoskeletal extension at the cell front, which has been shown to consist of two different substructures, the leading lamellipodium and the subsequent lamellum. While the formation of the lamellipodium is increasingly well understood, organizational principles underlying the emergence of the lamellum are just beginning to be unraveled. We report here on a 1D mathematical model which describes the reaction-diffusion processes of a polarized actin network in steady state, and reproduces essential characteristics of the lamellipodium-lamellum system. We observe a steep gradient in filament lengths at the protruding edge, a local depolymerization maximum a few microns behind the edge, as well as a differential dominance of the network destabilizer ADF/cofilin and the stabilizer tropomyosin. We identify simple and robust organizational principles giving rise to the derived network characteristics, uncoupled from the specifics of any molecular implementation, and thus plausibly valid across cell types. An analysis of network length dependence on physico-chemical system parameters implies that to limit array treadmilling to cellular dimensions, network growth has to be truncated by mechanisms other than aging-induced depolymerization, e.g., by myosin-associated network dissociation at the transition to the cell body. Our work contributes to the analytical understanding of the cytoskeletal extension's bisection into lamellipodium and lamellum and sheds light on how cells organize their molecular machinery to achieve motility.

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.

Cell physician: reading cell motion: a mathematical diagnostic technique through analysis of single cell motion

Cell motility is an essential phenomenon in almost all living organisms. It is natural to think that behavioral or shape changes of a cell bear information about the underlying mechanisms that generate these changes. Reading cell motion, namely, understanding the underlying biophysical and mechanochemical processes, is of paramount importance. The mathematical model developed in this paper determines some physical features and material properties of the cells locally through analysis of live cell image sequences and uses this information to make further inferences about the molecular structures, dynamics, and processes within the cells, such as the actin network, microdomains, chemotaxis, adhesion, and retrograde flow. The generality of the principals used in formation of the model ensures its wide applicability to different phenomena at various levels. Based on the model outcomes, we hypothesize a novel biological model for collective biomechanical and molecular mechanism of cell motion.

Modeling of protrusion phenotypes driven by the actin-membrane interaction

We propose a mathematical model for simulating the leading-edge dynamics of a migrating cell from the interplay among elastic properties, architecture of the actin cytoskeleton, and the mechanics of the membrane. Our approach is based on the description of the length and attachment dynamics of actin filaments in the lamellipodium network. It is used to determine the total force exerted on the membrane at each position along the leading edge and at each time step. The model reproduces the marked state switches in protrusion morphodynamics found experimentally between epithelial cells in control conditions and cells expressing constitutively active Rac, a signaling molecule involved in the regulation of lamellipodium network assembly. The model also suggests a mechanistic explanation of experimental distortions in protrusion morphodynamics induced by deregulation of Arp2/3 and cofilin activity.

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.

A numerical model of cellular blebbing: a volume-conserving, fluid-structure interaction model of the entire cell

In animal cells, blebs are smooth, quasi-hemispherical protrusions of the plasma membrane that form when a section of the membrane detaches from the underlying actin cytoskeleton and is inflated by flowing cytosol. The mechanics behind this common cellular activity are not yet clear. As a first step in the development of a full computational framework, we present a numerical model of overall cell behavior based upon the interaction between a background Newtonian-fluid cytosol and elastic structures modeling the membrane and filaments. The detailed micromechanics of the cytoskeletal network are the subject of future work. Here, the myosin-driven contraction of the actin network is modeled through stressed elastic filaments. Quantitative models of cytoskeletal micromechanics and biochemistry require accurate estimates of local stress and flow conditions. The main contribution of this paper is the development of a computationally efficient fluid-structure interaction model based on operator splitting, to furnish this data. Cytosol volume conservation (as supported by experimental evidence) is enforced through an intermediate energy minimization step. Realistic bleb formation and retraction is observed from this model, offering an alternative formulation to positing complex continuum behavior of the cytoplasm (e.g. poroelastic model of Charras et al., 2008).

Modeling of the actin-cytoskeleton in symmetric lamellipodial fragments

The pushing structures of cells include laminar sheets, termed lamellipodia, made up of a meshwork of actin filaments that grow at the front and depolymerise at the rear, in a treadmilling mode.We here develop a mathematical model to describe the turnover and the mechanical properties of this network.Our basic modeling assumptions are that the lamellipodium is idealised as a two-dimensional structure, and that the actin network consists of two families of possibly bent, but locally parallel filaments. Instead of dealing with individual polymers, the filaments are assumed to be continuously distributed.The model includes (de)polymerization, of the mechanical effects of cross-linking, cell-substrate adhesion, as well as of the leading edge of the membrane.In the first version presented here, the total amount of F-actin is prescribed by assuming a constant polymerisation speed at the leading edge and a fixed total number and length distribution of filaments. We assume that cross-links at filament crossing points as well as integrin linkages with the matrix break and reform in response to incremental changes in network organization. In this first treatment, the model successfully simulates the persistence of the treadmilling network in radially spread cells.

Cortical factor feedback model for cellular locomotion and cytofission

Eukaryotic cells can move spontaneously without being guided by external cues. For such spontaneous movements, a variety of different modes have been observed, including the amoeboid-like locomotion with protrusion of multiple pseudopods, the keratocyte-like locomotion with a widely spread lamellipodium, cell division with two daughter cells crawling in opposite directions, and fragmentations of a cell to multiple pieces. Mutagenesis studies have revealed that cells exhibit these modes depending on which genes are deficient, suggesting that seemingly different modes are the manifestation of a common mechanism to regulate cell motion. In this paper, we propose a hypothesis that the positive feedback mechanism working through the inhomogeneous distribution of regulatory proteins underlies this variety of cell locomotion and cytofission. In this hypothesis, a set of regulatory proteins, which we call cortical factors, suppress actin polymerization. These suppressing factors are diluted at the extending front and accumulated at the retracting rear of cell, which establishes a cellular polarity and enhances the cell motility, leading to the further accumulation of cortical factors at the rear. Stochastic simulation of cell movement shows that the positive feedback mechanism of cortical factors stabilizes or destabilizes modes of movement and determines the cell migration pattern. The model predicts that the pattern is selected by changing the rate of formation of the actin-filament network or the threshold to initiate the network formation.