Continuum model of cell adhesion and migration

Subcellular mechano-regulation of cell migration in confined extracellular microenvironment

Cell migration is a highly coordinated cellular event that determines diverse physiological and pathological processes in which the continuous interaction of a migrating cell with neighboring cells or the extracellular matrix is regulated by the physical setting of the extracellular microenvironment. In confined spaces, cell migration occurs differently compared to unconfined open spaces owing to the additional forces that limit cell motility, which create a driving bias for cells to invade the confined space, resulting in a distinct cell motility process compared to what is expected in open spaces. Moreover, cells in confined environments can be subjected to elevated mechanical compression, which causes physical stimuli and activates the damage repair cycle in the cell, including the DNA in the nucleus. Although cells have a self-restoring system to repair damage from the cell membrane to the genetic components of the nucleus, this process may result in genetic and/or epigenetic alterations that can increase the risk of the progression of diverse diseases, such as cancer and immune disorders. Furthermore, there has been a shift in the paradigm of bioengineering from the development of new biomaterials to controlling biophysical cues and fine-tuning cell behaviors to cure damaged/diseased tissues. The external physical cues perceived by cells are transduced along the mechanosensitive machinery, which is further channeled into the nucleus through subcellular molecular linkages of the nucleoskeleton and cytoskeleton or the biochemical translocation of transcription factors. Thus, external cues can directly or indirectly regulate genetic transcriptional processes and nuclear mechanics, ultimately determining cell fate. In this review, we discuss the importance of the biophysical cues, response mechanisms, and mechanical models of cell migration in confined environments. We also discuss the effect of force-dependent deformation of subcellular components, specifically focusing on subnuclear organelles, such as nuclear membranes and chromosomal organization. This review will provide a biophysical perspective on cancer progression and metastasis as well as abnormal cellular proliferation.

Integrative experimental/computational approach establishes active cellular protrusion as the primary driving force of phagocytic spreading by immune cells

The dynamic interplay between cell adhesion and protrusion is a critical determinant of many forms of cell motility. When modeling cell spreading on adhesive surfaces, traditional mathematical treatments often consider passive cell adhesion as the primary, if not exclusive, mechanistic driving force of this cellular motion. To better assess the contribution of active cytoskeletal protrusion to immune-cell spreading during phagocytosis, we here develop a computational framework that allows us to optionally investigate both purely adhesive spreading ("Brownian zipper hypothesis") as well as protrusion-dominated spreading ("protrusive zipper hypothesis"). We model the cell as an axisymmetric body of highly viscous fluid surrounded by a cortex with uniform surface tension and incorporate as potential driving forces of cell spreading an attractive stress due to receptor-ligand binding and an outward normal stress representing cytoskeletal protrusion, both acting on the cell boundary. We leverage various model predictions against the results of a directly related experimental companion study of human neutrophil phagocytic spreading on substrates coated with different densities of antibodies. We find that the concept of adhesion-driven spreading is incompatible with experimental results such as the independence of the cell-spreading speed on the density of immobilized antibodies. In contrast, the protrusive zipper model agrees well with experimental findings and, when adapted to simulate cell spreading on discrete adhesion sites, it also reproduces the observed positive correlation between antibody density and maximum cell-substrate contact area. Together, our integrative experimental/computational approach shows that phagocytic spreading is driven by cellular protrusion, and that the extent of spreading is limited by the density of adhesion sites.

Mathematical modelling in cell migration: tackling biochemistry in changing geometries

Directed cell migration poses a rich set of theoretical challenges. Broadly, these are concerned with (1) how cells sense external signal gradients and adapt; (2) how actin polymerisation is localised to drive the leading cell edge and Myosin-II molecular motors retract the cell rear; and (3) how the combined action of cellular forces and cell adhesion results in cell shape changes and net migration. Reaction-diffusion models for biological pattern formation going back to Turing have long been used to explain generic principles of gradient sensing and cell polarisation in simple, static geometries like a circle. In this minireview, we focus on recent research which aims at coupling the biochemistry with cellular mechanics and modelling cell shape changes. In particular, we want to contrast two principal modelling approaches: (1) interface tracking where the cell membrane, interfacing cell interior and exterior, is explicitly represented by a set of moving points in 2D or 3D space and (2) interface capturing. In interface capturing, the membrane is implicitly modelled analogously to a level line in a hilly landscape whose topology changes according to forces acting on the membrane. With the increased availability of high-quality 3D microscopy data of complex cell shapes, such methods will become increasingly important in data-driven, image-based modelling to better understand the mechanochemistry underpinning cell motion.

Nonlocal and local models for taxis in cell migration: a rigorous limit procedure

A rigorous limit procedure is presented which links nonlocal models involving adhesion or nonlocal chemotaxis to their local counterparts featuring haptotaxis and classical chemotaxis, respectively. It relies on a novel reformulation of the involved nonlocalities in terms of integral operators applied directly to the gradients of signal-dependent quantities. The proposed approach handles both model types in a unified way and extends the previous mathematical framework to settings that allow for general solution-dependent coefficient functions. The previous forms of nonlocal operators are compared with the new ones introduced in this paper and the advantages of the latter are highlighted by concrete examples. Numerical simulations in 1D provide an illustration of some of the theoretical findings.

Substrate-rigidity dependent migration of an idealized twitching bacterium

Mechanical properties of the extracellular matrix are important determinants of cellular migration in diverse processes, such as immune response, wound healing, and cancer metastasis. Moreover, recent studies indicate that even bacterial surface colonization can depend on the mechanics of the substrate. Here, we focus on physical mechanisms that can give rise to substrate-rigidity dependent migration. We study a "twitcher", a cell driven by extension-retraction cycles, to idealize bacteria and perhaps eukaryotic cells that employ a slip-stick mode of motion. The twitcher is asymmetric and always pulls itself forward at its front. Analytical calculations show that the migration speed of a twitcher depends non-linearly on substrate rigidity. For soft substrates, deformations do not lead to build-up of significant force and the migration speed is therefore determined by stochastic adhesion unbinding. For rigid substrates, forced adhesion rupture determines the migration speed. Depending on the force-sensitivity of front and rear adhesions, forced bond rupture implies an increase or a decrease of the migration speed. A requirement for the occurrence of rigidity-dependent stick-slip migration is a "sticky" substrate, with binding rates being an order of magnitude larger than unbinding rates in absence of force. Computer simulations show that small stall forces of the driving machinery lead to a reduced movement on high rigidities, regardless of force-sensitivities of bonds. The simulations also confirm the occurrence of rigidity-dependent migration speed in a generic model for slip-stick migration of cells on a sticky substrate.

Simulation of Multispecies Desmoplastic Cancer Growth via a Fully Adaptive Non-linear Full Multigrid Algorithm

A fully adaptive non-linear full multigrid (FMG) algorithm is implemented to computationally simulate a model of multispecies desmoplastic tumor growth in three spatial dimensions. The algorithm solves a thermodynamic mixture model employing a diffuse interface approach with Cahn-Hilliard-type fourth-order equations that are coupled, non-linear, and numerically stiff. The tumor model includes extracellular matrix (ECM) as a major component with elastic energy contribution in its chemical potential term. Blood and lymphatic vasculatures are simulated via continuum representations. The model employs advection-reaction-diffusion partial differential equations (PDEs) for the cell, ECM, and vascular components, and reaction-diffusion PDEs for the elements diffusing from the vessels. This study provides the details of the numerical solution obtained by applying the fully adaptive non-linear FMG algorithm with finite difference method to solve this complex system of PDEs. The results indicate that this type of computational model can simulate the extracellular matrix-rich desmoplastic tumor microenvironment typical of fibrotic tumors, such as pancreatic adenocarcinoma.

A free-boundary model of a motile cell explains turning behavior

To understand shapes and movements of cells undergoing lamellipodial motility, we systematically explore minimal free-boundary models of actin-myosin contractility consisting of the force-balance and myosin transport equations. The models account for isotropic contraction proportional to myosin density, viscous stresses in the actin network, and constant-strength viscous-like adhesion. The contraction generates a spatially graded centripetal actin flow, which in turn reinforces the contraction via myosin redistribution and causes retraction of the lamellipodial boundary. Actin protrusion at the boundary counters the retraction, and the balance of the protrusion and retraction shapes the lamellipodium. The model analysis shows that initiation of motility critically depends on three dimensionless parameter combinations, which represent myosin-dependent contractility, a characteristic viscosity-adhesion length, and a rate of actin protrusion. When the contractility is sufficiently strong, cells break symmetry and move steadily along either straight or circular trajectories, and the motile behavior is sensitive to conditions at the cell boundary. Scanning of a model parameter space shows that the contractile mechanism of motility supports robust cell turning in conditions where short viscosity-adhesion lengths and fast protrusion cause an accumulation of myosin in a small region at the cell rear, destabilizing the axial symmetry of a moving cell.

To understand shapes and movements of simple motile cells, we systematically explore minimal models describing a cell as a two-dimensional actin-myosin gel with a free boundary. The models account for actin-myosin contraction balanced by viscous stresses in the actin gel and uniform adhesion. The myosin contraction causes the lamellipodial boundary to retract. Actin protrusion at the boundary counters the retraction, and the balance of protrusion and retraction shapes the cell. The models reproduce a variety of motile shapes observed experimentally. The analysis shows that the mechanical state of a cell depends on a small number of parameters. We find that when the contractility is sufficiently strong, cells break symmetry and move steadily along either straight or circular trajectory. Scanning model parameters shows that the contractile mechanism of motility supports robust cell turning behavior in conditions where deformable actin gel and fast protrusion destabilize the axial symmetry of a moving cell.

Model of vascular desmoplastic multispecies tumor growth

We present a three-dimensional nonlinear tumor growth model composed of heterogeneous cell types in a multicomponent-multispecies system, including viable, dead, healthy host, and extra-cellular matrix (ECM) tissue species. The model includes the capability for abnormal ECM dynamics noted in tumor development, as exemplified by pancreatic ductal adenocarcinoma, including dense desmoplasia typically characterized by a significant increase of interstitial connective tissue. An elastic energy is implemented to provide elasticity to the connective tissue. Cancer-associated fibroblasts (myofibroblasts) are modeled as key contributors to this ECM remodeling. The tumor growth is driven by growth factors released by these stromal cells as well as by oxygen and glucose provided by blood vasculature which along with lymphatics are stimulated to proliferate in and around the tumor based on pro-angiogenic factors released by hypoxic tissue regions. Cellular metabolic processes are simulated, including respiration and glycolysis with lactate fermentation. The bicarbonate buffering system is included for cellular pH regulation. This model system may be of use to simulate the complex interactions between tumor and stromal cells as well as the associated ECM and vascular remodeling that typically characterize malignant cancers notorious for poor therapeutic response.

A computational method for the coupled solution of reaction-diffusion equations on evolving domains and manifolds: Application to a model of cell migration and chemotaxis

In this paper, we devise a moving mesh finite element method for the approximate solution of coupled bulk-surface reaction-diffusion equations on an evolving two dimensional domain. Fundamental to the success of the method is the robust generation of bulk and surface meshes. For this purpose, we use a novel moving mesh partial differential equation (MMPDE) approach. The developed method is applied to model problems with known analytical solutions; these experiments indicate second-order spatial and temporal accuracy. Coupled bulk-surface problems occur frequently in many areas; in particular, in the modelling of eukaryotic cell migration and chemotaxis. We apply the method to a model of the two-way interaction of a migrating cell in a chemotactic field, where the bulk region corresponds to the extracellular region and the surface to the cell membrane.

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.

ONE-DIMENSIONAL MODELING AND SIMULATIONS OF MIGRATION OF CULTURED FIBROBLASTS

Cell migration is crucial for many physiological functions such as wound healing, immuno-response and carcinogenesis. In this study an one-dimensional model of migration of fibroblasts was developed by modeling and integrating five subcellular processes, namely, actin protrusion, focal adhesion formation, stress fiber formation, polarization and retraction. The direction of migration was determined by polarization, which was related to direction of the stiffness gradient of the substrate. By controlling intensity of ultraviolet exposure on type-I collagen, a substrate with a stiffness gradient could be fabricated. Kinematic analyses of positions of the cell front, the nucleus and the cell rear, were utilized as inputs to the model. Simulation results of five live NIH 3T3 fibroblasts showed that the model was capable of simulating fast moving, slow moving and back-and-forth moving of the cells on the substrate.

Analytical solutions of actin-retrograde-flow in a circular stationary cell: a mechanical point of view

The network of actin filaments in the lamellipodium (LP) of stationary and migrating cells flows in a retrograde direction, from the membrane periphery toward the cell nucleus. We have theoretically studied this phenomenon in the circular stationary (fully spread) cells. Adopting a continuum view on the LP actin network, new closed-form solutions are provided for the actin-retrograde-flow (ARF) in a polar coordinate system. Due to discrepancy in the mechanical models of the actin network in the ARF regime, solutions are provided for both assumptions of solid and fluid behavior. Other involved phenomena, including polymerizing machine at the membrane periphery, cytosol drag, adhesion friction, and membrane tension, are also discussed to provide an overall quantitative view on this problem.

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.

Effects of adhesion dynamics and substrate compliance on the shape and motility of crawling cells

Computational modeling of eukaryotic cells moving on substrates is an extraordinarily complex task: many physical processes, such as actin polymerization, action of motors, formation of adhesive contacts concomitant with both substrate deformation and recruitment of actin etc., as well as regulatory pathways are intertwined. Moreover, highly nontrivial cell responses emerge when the substrate becomes deformable and/or heterogeneous. Here we extended a computational model for motile cell fragments, based on an earlier developed phase field approach, to account for explicit dynamics of adhesion site formation, as well as for substrate compliance via an effective elastic spring. Our model displays steady motion vs. stick-slip transitions with concomitant shape oscillations as a function of the actin protrusion rate, the substrate stiffness, and the rates of adhesion. Implementing a step in the substrate's elastic modulus, as well as periodic patterned surfaces exemplified by alternating stripes of high and low adhesiveness, we were able to reproduce the correct motility modes and shape phenomenology found experimentally. We also predict the following nontrivial behavior: the direction of motion of cells can switch from parallel to perpendicular to the stripes as a function of both the adhesion strength and the width ratio of adhesive to non-adhesive stripes.

Migrating oligodendrocyte progenitor cells swell prior to soma dislocation

The migration of oligodendrocyte progenitor cells (OPCs) to the white matter is an indispensable requirement for an intact brain function. The mechanism of cell migration in general is not yet completely understood. Nevertheless, evidence is accumulating that besides the coordinated rearrangement of the cytoskeleton, a finetuned interplay of ion and water fluxes across the cell membrane is essential for cell migration. One part of a general hypothesis is that a local volume increase towards the direction of movement triggers a mechano-activated calcium influx that regulates various procedures at the rear end of a migrating cell. Here, we investigated cell volume changes of migrating OPCs using scanning ion conductance microscopy. We found that during accelerated migration OPCs undergo an increase in the frontal cell body volume. These findings are supplemented with time lapse calcium imaging data that hint an increase in calcium content the frontal part of the cell soma.

Emergent complexity of the cytoskeleton: from single filaments to tissue

Despite their overwhelming complexity, living cells display a high degree of internal mechanical and functional organization which can largely be attributed to the intracellular biopolymer scaffold, the cytoskeleton. Being a very complex system far from thermodynamic equilibrium, the cytoskeleton's ability to organize is at the same time challenging and fascinating. The extensive amounts of frequently interacting cellular building blocks and their inherent multifunctionality permits highly adaptive behavior and obstructs a purely reductionist approach. Nevertheless (and despite the field's relative novelty), the physics approach has already proved to be extremely successful in revealing very fundamental concepts of cytoskeleton organization and behavior. This review aims at introducing the physics of the cytoskeleton ranging from single biopolymer filaments to multicellular organisms. Throughout this wide range of phenomena, the focus is set on the intertwined nature of the different physical scales (levels of complexity) that give rise to numerous emergent properties by means of self-organization or self-assembly.

A comparison of computational models for eukaryotic cell shape and motility

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

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

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

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.

Quantitative mapping of averaged focal adhesion dynamics in migrating cells by shape normalization

The spatially ordered formation and disassembly of focal adhesions is a basic requirement for effective cell locomotion. Because focal adhesions couple the contractile actin-myosin network to the substrate, their distribution determines the pattern of traction forces propelling the cell in a certain direction. In the present study, we quantitatively analyzed the spatial patterning of cell-substrate adhesion in migrating cells by mapping averaged focal adhesion growth dynamics to a standardized cell coordinate system. These maps revealed distinct zones of focal adhesion assembly, disassembly and stability and were strongly interrelated with corresponding actin flow and traction force patterns. Moreover, the mapping technique enables precise detection of even minute responses of adhesion dynamics upon targeted signaling perturbations. For example, the partial inhibition of vinculin phosphorylation was followed by the reduced number of newly formed adhesions, whereas growth dynamics of existing adhesions remained unaffected.

Modeling morphodynamic phenotypes and dynamic regimes of cell motion

Many cellular processes and signaling pathways converge onto cell morphology and cell motion, which share important components. The mechanisms used for propulsion could also be responsible for shape changes, if they are capable of generating the rich observed variety of dynamic regimes. Additionally, the analysis of cell shape changes in space and time promises insight into the state of the cytoskeleton and signaling pathways controlling it. While this has been obvious for some time by now, little effort has been made to systematically and quantitatively explore this source of information. First pioneering experimental work revealed morphodynamic phenotypes which can be associated with dynamic regimes like oscillations and excitability. Here, we review the current state of modeling of morphodynamic phenotypes, the experimental results and discuss the ideas on the mechanisms driving shape changes which are suggested by modeling.

Cell-biomaterial mechanical interaction in the framework of tissue engineering: insights, computational modeling and perspectives

Tissue engineering is an emerging field of research which combines the use of cell-seeded biomaterials both in vitro and/or in vivo with the aim of promoting new tissue formation or regeneration. In this context, how cells colonize and interact with the biomaterial is critical in order to get a functional tissue engineering product. Cell-biomaterial interaction is referred to here as the phenomenon involved in adherent cells attachment to the biomaterial surface, and their related cell functions such as growth, differentiation, migration or apoptosis. This process is inherently complex in nature involving many physico-chemical events which take place at different scales ranging from molecular to cell body (organelle) levels. Moreover, it has been demonstrated that the mechanical environment at the cell-biomaterial location may play an important role in the subsequent cell function, which remains to be elucidated. In this paper, the state-of-the-art research in the physics and mechanics of cell-biomaterial interaction is reviewed with an emphasis on focal adhesions. The paper is focused on the different models developed at different scales available to simulate certain features of cell-biomaterial interaction. A proper understanding of cell-biomaterial interaction, as well as the development of predictive models in this sense, may add some light in tissue engineering and regenerative medicine fields.

Inhibition of myosin II triggers morphological transition and increased nuclear motility

We investigate the effect of myosin II inhibition on cell shape and nuclear motility in cultures of mouse radial glia-like neural progenitor and rat glioma C6 cells. Instead of reducing nucleokinesis, the myosin II inhibitor blebbistatin provokes an elongated bipolar morphology and increased nuclear motility in both cell types. When myosin II is active, time-resolved traction force measurements indicate a pulling force between the leading edge and the nucleus of C6 cells. In the absence of myosin II activity, traction forces during nucleokinesis are diminished below the sensitivity threshold of our assay. By visualizing the centrosome position in C6 cells with GFP-centrin, we show that in the presence or absence of myosin II activity, the nucleus tends to overtake or lag behind the centrosome, respectively. We interpret these findings with the help of a simple viscoelastic model of the cytoskeleton consisting active contractile and passive compressed elements.

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.

Leading-edge-gel coupling in lamellipodium motion

We present a model for actin-based motility that combines the dynamics of the semiflexible region at the leading edge of the lamellipodium with actomyosin gel properties in the bulk described by the theory of active polar gels. We calculate the velocity of the lamellipodium determined by the interaction of the gel and adhesion with forces in the semiflexible region. The stationary concave force-velocity relation of the model reproduces experimental results. We suggest that it is determined by retrograde flow at small forces and gel formation and retrograde flow at large ones. The variety of dynamic regimes of the semiflexible region reproducing experimentally observed morphodynamics is conserved when we couple the leading edge to the gel.

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.

A model of fibroblast motility on substrates with different rigidities

To function efficiently in the body, the biological cells must have the ability to sense the external environment. Mechanosensitivity toward the extracellular matrix was identified as one of the sensing mechanisms affecting cell behavior. It was shown experimentally that a fibroblast cell prefers locomoting over the stiffer substrate when given a choice between a softer and a stiffer substrate. In this article, we develop a discrete model of fibroblast motility with substrate-rigidity sensing. Our model allows us to understand the interplay between the cell-substrate sensing and the cell biomechanics. The model cell exhibits experimentally observed substrate rigidity sensing, which allows us to gain additional insights into the cell mechanosensitivity.

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.

Nonlinear modelling of cancer: bridging the gap between cells and tumours

Despite major scientific, medical and technological advances over the last few decades, a cure for cancer remains elusive. The disease initiation is complex, and including initiation and avascular growth, onset of hypoxia and acidosis due to accumulation of cells beyond normal physiological conditions, inducement of angiogenesis from the surrounding vasculature, tumour vascularization and further growth, and invasion of surrounding tissue and metastasis. Although the focus historically has been to study these events through experimental and clinical observations, mathematical modelling and simulation that enable analysis at multiple time and spatial scales have also complemented these efforts. Here, we provide an overview of this multiscale modelling focusing on the growth phase of tumours and bypassing the initial stage of tumourigenesis. While we briefly review discrete modelling, our focus is on the continuum approach. We limit the scope further by considering models of tumour progression that do not distinguish tumour cells by their age. We also do not consider immune system interactions nor do we describe models of therapy. We do discuss hybrid-modelling frameworks, where the tumour tissue is modelled using both discrete (cell-scale) and continuum (tumour-scale) elements, thus connecting the micrometre to the centimetre tumour scale. We review recent examples that incorporate experimental data into model parameters. We show that recent mathematical modelling predicts that transport limitations of cell nutrients, oxygen and growth factors may result in cell death that leads to morphological instability, providing a mechanism for invasion via tumour fingering and fragmentation. These conditions induce selection pressure for cell survivability, and may lead to additional genetic mutations. Mathematical modelling further shows that parameters that control the tumour mass shape also control its ability to invade. Thus, tumour morphology may serve as a predictor of invasiveness and treatment prognosis.

On the effect of substrate curvature on cell mechanics

Cell movement on a substrate or within the extracellular matrix is the phenomenological response to a biochemical signals' cascade transcripted into biophysical processes and viceversa. The process is complex in nature, including different length scales from the whole cell to organelle and protein levels, where substrate/ECM curvature has been shown to play an important role on cell's behavior. From a macroscopic perspective, the cytoskeleton may be modeled as a continuum body unbalanced by internal protein motors. In this work, we propose a cell constitutive model to simulate cell attachment on curved substrates, activated by contractile forces. We first analyze a single fiber bundle composed by microtubules, actin filaments and myosin machinery. Then, the model is macroscopically extended to the cytoskeletal level using homogenization. Substrate curvature has two implications in our model: (i) it forces fibers to work in a curved (bent) position and (ii) it eventually creates a pre-deformation state in the cytoskeleton. Interestingly, the model shows higher contractile force inhibition as curvature increases when implemented over different substrate morphologies, being this consistent with experimental results. The presented model may result useful in many new regenerative medicine techniques, miniaturized experimental tests, or to analyze cell behavior on manufactured nanoscaffolds for tissue engineering.