Theoretical model for cellular shapes driven by protrusive and adhesive forces

Effect of relative timescale on a system of particles sliding on a fluctuating energy landscape: Exact derivation of product measure condition

We consider a system of hardcore particles advected by a fluctuating potential energy landscape, whose dynamics is in turn affected by the particles. Earlier studies have shown that as a result of two-way coupling between the landscape and the particles, the system shows an interesting phase diagram as the coupling parameters are varied. The phase diagram consists of various different kinds of ordered phases and a disordered phase. We introduce a relative timescale ω between the particle and landscape dynamics, and study its effect on the steady state properties. We find there exists a critical value ω=ω_{c} when all configurations of the system are equally likely in the steady state. We prove this result exactly in a discrete lattice system and obtain an exact expression for ω_{c} in terms of the coupling parameters of the system. We show that ω_{c} is finite in the disordered phase, diverges at the boundary between the ordered and disordered phase, and is undefined in the ordered phase. We also derive ω_{c} from a coarse-grained level description of the system using linear hydrodynamics. We start with the assumption that there is a specific value ω^{*} of the relative timescale when correlations in the system vanish, and mean-field theory gives exact expressions for the current Jacobian matrix A and compressibility matrix K. Our exact calculations show that Onsager-type current symmetry relation AK=KA^{T} can be satisfied if and only if ω^{*}=ω_{c}. Our coarse-grained model calculations can be easily generalized to other coupled systems.

CdSe/ZnS Quantum Dots' Impact on In Vitro Actin Dynamics

Quantum dots (QDs) are a novel type of nanomaterial that has unique optical and physical characteristics. As such, QDs are highly desired because of their potential to be used in both biomedical and industrial applications. However, the mass adoption of QDs usage has raised concerns among the scientific community regarding QDs' toxicity. Although many papers have reported the negative impact of QDs on a cellular level, the exact mechanism of the QDs' toxicity is still unclear. In this investigation, we study the adverse effects of QDs by focusing on one of the most important cellular processes: actin polymerization and depolymerization. Our results showed that QDs act in a biphasic manner where lower concentrations of QDs stimulate the polymerization of actin, while high concentrations of QDs inhibit actin polymerization. Furthermore, we found that QDs can bind to filamentous actin (F-actin) and cause bundling of the filament while also promoting actin depolymerization. Through this study, we found a novel mechanism in which QDs negatively influence cellular processes and exert toxicity.

Universal properties of active membranes

We put forward a general field theory for nearly flat fluid membranes with embedded activators and analyze their critical properties using renormalization group techniques. Depending on the membrane-activator coupling, we find a crossover between acoustic and diffusive scaling regimes, with mean-field dynamical critical exponents z=1 and 2, respectively. We argue that the acoustic scaling, which is exact in all spatial dimensions, leads to an early-time behavior, which is representative of the spatiotemporal patterns observed at the leading edge of motile cells, such as oscillations superposed on the growth of the membrane width. In the case of mean-field diffusive scaling, one-loop corrections to the mean-field exponents reveal universal behavior distinct from the Kardar-Parisi-Zhang scaling of passive interfaces and signs of strong-coupling behavior.

Renormalization group study of the dynamics of active membranes: Universality classes and scaling laws

Motivated by experimental observations of patterning at the leading edge of motile eukaryotic cells, we introduce a general model for the dynamics of nearly-flat fluid membranes driven from within by an ensemble of activators. We include, in particular, a kinematic coupling between activator density and membrane slope which generically arises whenever the membrane has a nonvanishing normal speed. We unveil the phase diagram of the model by means of a perturbative field-theoretical renormalization group analysis. Due to the aforementioned kinematic coupling the natural early-time dynamical scaling is acoustic, that is the dynamical critical exponent is 1. However, as soon as the the normal velocity of the membrane is tuned to zero, the system crosses over to diffusive dynamic scaling in mean field. Distinct critical points can be reached depending on how the limit of vanishing velocity is realized: in each of them corrections to scaling due to nonlinear coupling terms must be taken into account. The detailed analysis of these critical points reveals novel scaling regimes which can be accessed with perturbative methods, together with signs of strong coupling behavior, which establishes a promising ground for further nonperturbative calculations. Our results unify several previous studies on the dynamics of active membrane, while also identifying nontrivial scaling regimes which cannot be captured by passive theories of fluctuating interfaces and are relevant for the physics of living membranes.

Cortical contraction drives the 3D patterning of epithelial cell surfaces

Cellular protrusions create complex cell surface topographies, but biomechanical mechanisms regulating their formation and arrangement are largely unknown. To study how protrusions form, we focused on the morphogenesis of microridges, elongated actin-based structures that are arranged in maze-like patterns on the apical surfaces of zebrafish skin cells. Microridges form by accreting simple finger-like precursors. Live imaging demonstrated that microridge morphogenesis is linked to apical constriction. A nonmuscle myosin II (NMII) reporter revealed pulsatile contractions of the actomyosin cortex, and inhibiting NMII blocked apical constriction and microridge formation. A biomechanical model suggested that contraction reduces surface tension to permit the fusion of precursors into microridges. Indeed, reducing surface tension with hyperosmolar media promoted microridge formation. In anisotropically stretched cells, microridges formed by precursor fusion along the stretch axis, which computational modeling explained as a consequence of stretch-induced cortical flow. Collectively, our results demonstrate how contraction within the 2D plane of the cortex can pattern 3D cell surfaces.

Cell Shape and Durotaxis Explained from Cell-Extracellular Matrix Forces and Focal Adhesion Dynamics

Many cells are small and rounded on soft extracellular matrices (ECM), elongated on stiffer ECMs, and flattened on hard ECMs. Cells also migrate up stiffness gradients (durotaxis). Using a hybrid cellular Potts and finite-element model extended with ODE-based models of focal adhesion (FA) turnover, we show that the full range of cell shape and durotaxis can be explained in unison from dynamics of FAs, in contrast to previous mathematical models. In our 2D cell-shape model, FAs grow due to cell traction forces. Forces develop faster on stiff ECMs, causing FAs to stabilize and, consequently, cells to spread on stiff ECMs. If ECM stress further stabilizes FAs, cells elongate on substrates of intermediate stiffness. We show that durotaxis follows from the same set of assumptions. Our model contributes to the understanding of the basic responses of cells to ECM stiffness, paving the way for future modeling of more complex cell-ECM interactions.

Traction Forces Control Cell-Edge Dynamics and Mediate Distance Sensitivity during Cell Polarization

Traction forces are generated by cellular actin-myosin system and transmitted to the environment through adhesions. They are believed to drive cell motion, shape changes, and extracellular matrix remodeling [1-3]. However, most of the traction force analysis has been performed on stationary cells, investigating forces at the level of individual focal adhesions or linking them to static cell parameters, such as area and edge curvature [4-10]. It is not well understood how traction forces are related to shape changes and motion, e.g., forces were reported to either increase or drop prior to cell retraction [11-15]. Here, we analyze the dynamics of traction forces during the protrusion-retraction cycle of polarizing fish epidermal keratocytes and find that forces fluctuate together with the cycle, increasing during protrusion and reaching maximum at the beginning of retraction. We relate force dynamics to the recently discovered phenomenological rule [16] that governs cell-edge behavior during keratocyte polarization: both traction forces and probability of switch from protrusion to retraction increase with the distance from the cell center. Diminishing forces with cell contractility inhibitor leads to decreased edge fluctuations and abnormal polarization, although externally applied force can induce protrusion-retraction switch. These results suggest that forces mediate distance sensitivity of the edge dynamics and organize cell-edge behavior, leading to spontaneous polarization. Actin flow rate did not exhibit the same distance dependence as traction stress, arguing against its role in organizing edge dynamics. Finally, using a simple model of actin-myosin network, we show that force-distance relationship might be an emergent feature of such networks.

Cell-Substrate Patterns Driven by Curvature-Sensitive Actin Polymerization: Waves and Podosomes

Cells adhered to an external solid substrate are observed to exhibit rich dynamics of actin structures on the basal membrane, which are distinct from those observed on the dorsal (free) membrane. Here we explore the dynamics of curved membrane proteins, or protein complexes, that recruit actin polymerization when the membrane is confined by the solid substrate. Such curved proteins can induce the spontaneous formation of membrane protrusions on the dorsal side of cells. However, on the basal side of the cells, such protrusions can only extend as far as the solid substrate and this constraint can convert such protrusions into propagating wave-like structures. We also demonstrate that adhesion molecules can stabilize localized protrusions that resemble some features of podosomes. This coupling of curvature and actin forces may underlie the differences in the observed actin-membrane dynamics between the basal and dorsal sides of adhered cells.

Modeling membrane nanotube morphology: the role of heterogeneity in composition and material properties

Membrane nanotubes are dynamic structures that may connect cells over long distances. Nanotubes are typically thin cylindrical tubes, but they may occasionally have a beaded architecture along the tube. In this paper, we study the role of membrane mechanics in governing the architecture of these tubes and show that the formation of bead-like structures along the nanotubes can result from local heterogeneities in the membrane either due to protein aggregation or due to membrane composition. We present numerical results that predict how membrane properties, protein density, and local tension compete to create a phase space that governs the morphology of a nanotube. We also find that there exists a discontinuity in the energy that impedes two beads from fusing. These results suggest that the membrane-protein interaction, membrane composition, and membrane tension closely govern the tube radius, number of beads, and the bead morphology.

How a lipid bilayer membrane responds to an oscillating nanoparticle: Promoted membrane undulation and directional wave propagation

Mechanical forces acting on a plasma membrane are of essential importance to cellular functioning via inducing delicate change of the membrane shape with the underlying mechanism yet to be elucidated. Here, we introduce an oscillating nanoparticle (NP) interaction with a lipid bilayer membrane, using the coarse-grained simulation to investigate the dynamic membrane response to constrained mechanical stimulation, which is ubiquitous in biology. Our results demonstrate that, the membrane responds to an oscillating NP by generating nanoscale undulation waves, which immediately propagate through the membrane. In dynamics, propagation of the generated membrane undulation waves always starts from flattening of the region where the NP locates, thus producing a lateral force to propel the waves away from the point of stimulation. The speed of membrane undulation wave propagation is proportional to that of NP oscillation and accelerated by increasing the integral membrane surface tension, suggesting that both the membrane bending and stretching contribute to the energy driving the unique response of membrane undulation wave propagation.

Interplay between surface and bending energy helps membrane protrusion formation

We consider a one-dimensional elastic membrane, which is pushed by growing filaments. The filaments tend to grow by creating local protrusions in the membrane and this process has surface energy and bending energy costs. Although it is expected that with increasing surface tension and bending rigidity, it should become more difficult to create a protrusion, we find that for a fixed bending rigidity, as the surface tension increases, protrusions are more easily formed. This effect also gives rise to nontrivial dependence of membrane velocity on the surface tension, characterized by a dip and a peak. We explain this unusual phenomenon by studying in detail the interplay of the surface and the bending energy and show that this interplay is responsible for a qualitative shape change of the membrane, which gives rise to the above effect.

Stick-slip dynamics of migrating cells on viscoelastic substrates

Stick-slip motion, a common phenomenon observed during crawling of cells, is found to be strongly sensitive to the substrate stiffness. Stick-slip behaviors have previously been investigated typically using purely elastic substrates. For a more realistic understanding of this phenomenon, we propose a theoretical model to study the dynamics on a viscoelastic substrate. Our model, based on a reaction-diffusion framework, incorporates known important interactions such as retrograde flow of actin, myosin contractility, force-dependent assembly, and disassembly of focal adhesions coupled with cell-substrate interaction. We show that consideration of a viscoelastic substrate not only captures the usually observed stick-slip jumps but also predicts the existence of an optimal substrate viscosity corresponding to maximum traction force and minimum retrograde flow which was hitherto unexplored. Moreover, our theory predicts the time evolution of individual bond force that characterizes the stick-slip patterns on soft versus stiff substrates. Our analysis also elucidates how the duration of the stick-slip cycles are affected by various cellular parameters.

Theoretical study of vesicle shapes driven by coupling curved proteins and active cytoskeletal forces

Eukaryote cells have a flexible shape, which dynamically changes according to the function performed by the cell. One mechanism for deforming the cell membrane into the desired shape is through the expression of curved membrane proteins. Furthermore, these curved membrane proteins are often associated with the recruitment of the cytoskeleton, which then applies active forces that deform the membrane. This coupling between curvature and activity was previously explored theoretically in the linear limit of small deformations, and low dimensionality. Here we explore the unrestricted shapes of vesicles that contain active curved membrane proteins, in three-dimensions, using Monte-Carlo numerical simulations. The activity of the proteins is in the form of protrusive forces that push the membrane outwards, as may arise from the cytoskeleton of the cell due to actin or microtubule polymerization occurring near the membrane. For proteins that have an isotropic convex shape, the additional protrusive force enhances their tendency to aggregate and form membrane protrusions (buds). In addition, we find another transition from deformed spheres with necklace type aggregates, to flat pancake-shaped vesicles, where the curved proteins line the outer rim. This second transition is driven by the active forces, coupled to the spontaneous curvature, and the resulting configurations may shed light on the formation of sheet-like protrusions and lamellipodia of adhered and motile cells.

Guided by curvature: shaping cells by coupling curved membrane proteins and cytoskeletal forces

Eukaryote cells have flexible membranes that allow them to have a variety of dynamical shapes. The shapes of the cells serve important biological functions, both for cells within an intact tissue, and during embryogenesis and cellular motility. How cells control their shapes and the structures that they form on their surface has been a subject of intensive biological research, exposing the building blocks that cells use to deform their membranes. These processes have also drawn the interest of theoretical physicists, aiming to develop models based on physics, chemistry and nonlinear dynamics. Such models explore quantitatively different possible mechanisms that the cells can employ to initiate the spontaneous formation of shapes and patterns on their membranes. We review here theoretical work where one such class of mechanisms was investigated: the coupling between curved membrane proteins, and the cytoskeletal forces that they recruit. Theory indicates that this coupling gives rise to a rich variety of membrane shapes and dynamics, while experiments indicate that this mechanism appears to drive many cellular shape changes.This article is part of the theme issue 'Self-organization in cell biology'.

Modeling Membrane Curvature Generation due to Membrane⁻Protein Interactions

To alter and adjust the shape of the plasma membrane, cells harness various mechanisms of curvature generation. Many of these curvature generation mechanisms rely on the interactions between peripheral membrane proteins, integral membrane proteins, and lipids in the bilayer membrane. Mathematical and computational modeling of membrane curvature generation has provided great insights into the physics underlying these processes. However, one of the challenges in modeling these processes is identifying the suitable constitutive relationships that describe the membrane free energy including protein distribution and curvature generation capability. Here, we review some of the commonly used continuum elastic membrane models that have been developed for this purpose and discuss their applications. Finally, we address some fundamental challenges that future theoretical methods need to overcome to push the boundaries of current model applications.

A computational model of amoeboid cell motility in the presence of obstacles

Locomotion of amoeboid cells is mediated by finger-like protrusions of the cell body, known as pseudopods, which grow, bifurcate, and retract in a dynamic fashion. Pseudopods are the primary mode of locomotion for many cells within the human body, such as leukocytes, embryonic cells, and metastatic cancer cells. Amoeboid motility is a complex and multiscale process, which involves bio-molecular reactions, cell deformation, and cytoplasmic and extracellular fluid motion. Additionally, cells within the human body are subject to a confined 3D environment known as the extra-cellular matrix (ECM), which resembles a fluid-filled porous medium. In this article, we present a 3D, multiphysics computational approach coupling fluid mechanics, solid mechanics, and a pattern formation model to simulate locomotion of amoeboid cells through a porous matrix composed of a viscous fluid and an array of finite-sized spherical obstacles. The model combines reaction-diffusion of activator/inhibitors, extreme deformation of the cell, pseudopod dynamics, cytoplasmic and extracellular fluid motion, and fully resolved extracellular matrix. A surface finite-element method is used to obtain the cell deformation and activator/inhibitor concentrations, while the fluid motion is solved using a combined finite-volume and spectral method. The immersed-boundary methods are used to couple the cell deformation, obstacles, and fluid. The model is able to recreate squeezing and weaving motion of cells through the matrix. We study the influence of matrix porosity, obstacle size, and cell deformability on the motility behavior. It is found that below certain values of these parameters, cell motion is completely inhibited. Phase diagrams are presented depicting such motility limits. Interesting dynamics seen in the presence of obstacles but absent in unconfined medium, such as freezing or cell arrest, probing, doubling-back, and tug-of-war are predicted. Furthermore, persistent unidirectional motion of cells that is often observed in an unconfined medium is shown to be lost in presence of obstacles, and is attributed to an alteration of the pseudopod dynamics. The same mechanism, however, allows the cell to find a new direction to penetrate further into the matrix without being stuck in one place. The results and analysis presented here show a strong coupling between cell deformability and ECM properties, and provide new fluid mechanical insights on amoeboid motility in confined medium.

Bleb Expansion in Migrating Cells Depends on Supply of Membrane from Cell Surface Invaginations

Cell migration is essential for morphogenesis, organ formation, and homeostasis, with relevance for clinical conditions. The migration of primordial germ cells (PGCs) is a useful model for studying this process in the context of the developing embryo. Zebrafish PGC migration depends on the formation of cellular protrusions in form of blebs, a type of protrusion found in various cell types. Here we report on the mechanisms allowing the inflation of the membrane during bleb formation. We show that the rapid expansion of the protrusion depends on membrane invaginations that are localized preferentially at the cell front. The formation of these invaginations requires the function of Cdc42, and their unfolding allows bleb inflation and dynamic cell-shape changes performed by migrating cells. Inhibiting the formation and release of the invaginations strongly interfered with bleb formation, cell motility, and the ability of the cells to reach their target.

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.

Mechanics of Cell Mechanosensing in Protrusion and Retraction of Lamellipodium

Lamellipodia (LP), a subcellular structure at cell front, plays a key role in cell spreading and migration. And its mechanosensing function is of crucial importance for cell activities. But the mechanism of the mechanosensing function remains poorly understood. Here we developed a multiscale model to consider its protrusion and retraction processes, and analyzed the forces acted on the key structural components of the LP and the effect of these forces on LP movement. Our results show that raising substrate rigidity increases the force acting on the focal adhesion (FA) and decreases the force on LP actin, thus promoting the maturation of FA while suppressing the detachment of LP actin from the cell membrane. The membrane tension also influences the LP movement, but its effect is opposite to that of the substrate rigidity. It turns out that the substrate rigidity and membrane tension together regulate the dynamics of FAs and the detachment of LP actin, which in turn determine the LP movement. Interestingly, we found that the effect of substrate rigidity and membrane tension on the LP movement both exhibit a biphasic manner. We show that our predictions agree, in general, with the experiments on cell mechanosensing behaviors at both subcellular and cellular levels.

Modeling the Excess Cell Surface Stored in a Complex Morphology of Bleb-Like Protrusions

Cells transition from spread to rounded morphologies in diverse physiological contexts including mitosis and mesenchymal-to-amoeboid transitions. When these drastic shape changes occur rapidly, cell volume and surface area are approximately conserved. Consequently, the rounded cells are suddenly presented with a several-fold excess of cell surface whose area far exceeds that of a smooth sphere enclosing the cell volume. This excess is stored in a population of bleb-like protrusions (BLiPs), whose size distribution is shown by electron micrographs to be skewed. We introduce three complementary models of rounded cell morphologies with a prescribed excess surface area. A 2D Hamiltonian model provides a mechanistic description of how discrete attachment points between the cell surface and cortex together with surface bending energy can generate a morphology that satisfies a prescribed excess area and BLiP number density. A 3D random seed-and-growth model simulates efficient packing of BLiPs over a primary rounded shape, demonstrating a pathway for skewed BLiP size distributions that recapitulate 3D morphologies. Finally, a phase field model (2D and 3D) posits energy-based constitutive laws for the cell membrane, nematic F-actin cortex, interior cytosol, and external aqueous medium. The cell surface is equipped with a spontaneous curvature function, a proxy for the cell surface-cortex couple, that is a priori unknown, which the model “learns” from the thin section transmission electron micrograph image (2D) or the “seed and growth” model image (3D). Converged phase field simulations predict self-consistent amplitudes and spatial localization of pressure and stress throughout the cell for any posited stationary morphology target and cell compartment constitutive properties. The models form a general framework for future studies of cell morphological dynamics in a variety of biological contexts.

Individual cells must have the capability for rapid morphological transformations under various physiological conditions. One of the most drastic shape transformations occurs during the transition from spread to rounded morphologies. When this transition occurs rapidly, there is insufficient time for significant changes in surface area to occur, although the final size of the rounded cell indicates a significant reduction in apparent cell surface area at light microscope resolution. By contrast, high-resolution scanning electron micrographs of rapidly rounded cells reveal that a large amount of surface area is stored in a highly convoluted surface morphology consisting of bleb-like protrusions (BLiPs) and other small structures that are unrecognizable at lower resolution. This surface reserve is an important part of the mechanism that allows rapid and efficient large scale transformations of cell shape. Remarkably, although this convoluted morphology has been observed for decades, there has been very little effort recognizing and including this surface surplus in mathematical modeling of cell morphology and physiology. In this paper, we develop three complementary models to fill this void and lay the foundation for future investigations of the mechanisms that drive cellular morphological dynamics.

Protrusive waves guide 3D cell migration along nanofibers

Reductionist approaches based on 3D fibers reveal that single-cell migration along fibers is driven by lateral actin-based waves for various cell types.

In vivo, cells migrate on complex three-dimensional (3D) fibrous matrices, which has made investigation of the key molecular and physical mechanisms that drive cell migration difficult. Using reductionist approaches based on 3D electrospun fibers, we report for various cell types that single-cell migration along fibronectin-coated nanofibers is associated with lateral actin-based waves. These cyclical waves have a fin-like shape and propagate up to several hundred micrometers from the cell body, extending the leading edge and promoting highly persistent directional movement. Cells generate these waves through balanced activation of the Rac1/N-WASP/Arp2/3 and Rho/formins pathways. The waves originate from one major adhesion site at leading end of the cell body, which is linked through actomyosin contractility to another site at the back of the cell, allowing force generation, matrix deformation and cell translocation. By combining experimental and modeling data, we demonstrate that cell migration in a fibrous environment requires the formation and propagation of dynamic, actin based fin-like protrusions.

The mechanisms of spatial and temporal patterning of cell-edge dynamics

Adherent cells migrate and change their shape by means of protrusion and retraction at their edges. When and where these activities occur defines the shape of the cell and the way it moves. Despite a great deal of knowledge about the structural organization, components, and biochemical reactions involved in protrusion and retraction, the origins of their spatial and temporal patterns are still poorly understood. Chemical signaling circuitry is believed to be an important source of patterning, but recent studies highlighted mechanisms based on physical forces, motion, and mechanical feedback.

Multiple scale model for cell migration in monolayers: Elastic mismatch between cells enhances motility

We propose a multiscale model for monolayer of motile cells that comprise normal and cancer cells. In the model, the two types of cells have identical properties except for their elasticity; cancer cells are softer and normal cells are stiffer. The goal is to isolate the role of elasticity mismatch on the migration potential of cancer cells in the absence of other contributions that are present in real cells. The methodology is based on a phase-field description where each cell is modeled as a highly-deformable self-propelled droplet. We simulated two types of nearly confluent monolayers. One contains a single cancer cell in a layer of normal cells and the other contains normal cells only. The simulation results demonstrate that elasticity mismatch alone is sufficient to increase the motility of the cancer cell significantly. Further, the trajectory of the cancer cell is decorated by several speed “bursts” where the cancer cell quickly relaxes from a largely deformed shape and consequently increases its translational motion. The increased motility and the amplitude and frequency of the bursts are in qualitative agreement with recent experiments.

Physical principles of membrane remodelling during cell mechanoadaptation

Biological processes in any physiological environment involve changes in cell shape, which must be accommodated by their physical envelope—the bilayer membrane. However, the fundamental biophysical principles by which the cell membrane allows for and responds to shape changes remain unclear. Here we show that the 3D remodelling of the membrane in response to a broad diversity of physiological perturbations can be explained by a purely mechanical process. This process is passive, local, almost instantaneous, before any active remodelling and generates different types of membrane invaginations that can repeatedly store and release large fractions of the cell membrane. We further demonstrate that the shape of those invaginations is determined by the minimum elastic and adhesive energy required to store both membrane area and liquid volume at the cell–substrate interface. Once formed, cells reabsorb the invaginations through an active process with duration of the order of minutes.

Variations in cell shape must be accommodated by the cell membrane, but how the membrane adjusts to changes in area and volume is not known. Here the authors show that the membrane responds in a nearly instantaneous, purely physical manner involving the flattening or generation of membrane invaginations.

Flightless I interacts with NMMIIA to promote cell extension formation, which enables collagen remodeling

The role of the actin-capping protein flightless I in collagen remodeling by mouse fibroblasts is examined. Flightless and nonmuscle myosin IIA cooperate to enable collagen phagocytosis.

We examined the role of the actin-capping protein flightless I (FliI) in collagen remodeling by mouse fibroblasts. FliI-overexpressing cells exhibited reduced spreading on collagen but formed elongated protrusions that stained for myosin10 and fascin and penetrated pores of collagen-coated membranes. Inhibition of Cdc42 blocked formation of cell protrusions. In FliI-knockdown cells, transfection with constitutively active Cdc42 did not enable protrusion formation. FliI-overexpressing cells displayed increased uptake and degradation of exogenous collagen and strongly compacted collagen fibrils, which was blocked by blebbistatin. Mass spectrometry analysis of FliI immunoprecipitates showed that FliI associated with nonmuscle myosin IIA (NMMIIA), which was confirmed by immunoprecipitation. GFP-FliI colocalized with NMMIIA at cell protrusions. Purified FliI containing gelsolin-like domains (GLDs) 1–6 capped actin filaments efficiently, whereas FliI GLD 2–6 did not. Binding assays showed strong interaction of purified FliI protein (GLD 1–6) with the rod domain of NMMIIA (k_D = 0.146 μM), whereas FliI GLD 2–6 showed lower binding affinity (k_D = 0.8584 μM). Cells expressing FliI GLD 2–6 exhibited fewer cell extensions, did not colocalize with NMMIIA, and showed reduced collagen uptake compared with cells expressing FliI GLD 1–6. We conclude that FliI interacts with NMMIIA to promote cell extension formation, which enables collagen remodeling in fibroblasts.

Three-dimensional numerical model of cell morphology during migration in multi-signaling substrates

Cell Migration associated with cell shape changes are of central importance in many biological processes ranging from morphogenesis to metastatic cancer cells. Cell movement is a result of cyclic changes of cell morphology due to effective forces on cell body, leading to periodic fluctuations of the cell length and cell membrane area. It is well-known that the cell can be guided by different effective stimuli such as mechanotaxis, thermotaxis, chemotaxis and/or electrotaxis. Regulation of intracellular mechanics and cell's physical interaction with its substrate rely on control of cell shape during cell migration. In this notion, it is essential to understand how each natural or external stimulus may affect the cell behavior. Therefore, a three-dimensional (3D) computational model is here developed to analyze a free mode of cell shape changes during migration in a multi-signaling micro-environment. This model is based on previous models that are presented by the same authors to study cell migration with a constant spherical cell shape in a multi-signaling substrates and mechanotaxis effect on cell morphology. Using the finite element discrete methodology, the cell is represented by a group of finite elements. The cell motion is modeled by equilibrium of effective forces on cell body such as traction, protrusion, electrostatic and drag forces, where the cell traction force is a function of the cell internal deformations. To study cell behavior in the presence of different stimuli, the model has been employed in different numerical cases. Our findings, which are qualitatively consistent with well-known related experimental observations, indicate that adding a new stimulus to the cell substrate pushes the cell to migrate more directionally in more elongated form towards the more effective stimuli. For instance, the presence of thermotaxis, chemotaxis and electrotaxis can further move the cell centroid towards the corresponding stimulus, respectively, diminishing the mechanotaxis effect. Besides, the stronger stimulus imposes a greater cell elongation and more cell membrane area. The present model not only provides new insights into cell morphology in a multi-signaling micro-environment but also enables us to investigate in more precise way the cell migration in the presence of different stimuli.

A novel mechanotactic 3D modeling of cell morphology

Cell morphology plays a critical role in many biological processes, such as cell migration, tissue development, wound healing and tumor growth. Recent investigations demonstrate that, among other stimuli, cells adapt their shapes according to their substrate stiffness. Until now, the development of this process has not been clear. Therefore, in this work, a new three-dimensional (3D) computational model for cell morphology has been developed. This model is based on a previous cell migration model presented by the same authors. The new model considers that during cell-substrate interaction, cell shape is governed by internal cell deformation, which leads to an accurate prediction of the cell shape according to the mechanical characteristic of its surrounding micro-environment. To study this phenomenon, the model has been applied to different numerical cases. The obtained results, which are qualitatively consistent with well-known related experimental works, indicate that cell morphology not only depends on substrate stiffness but also on the substrate boundary conditions. A cell located within an unconstrained soft substrate (several kPa) with uniform stiffness is unable to adhere to its substrate or to send out pseudopodia. When the substrate stiffness increases to tens of kPa (intermediate and rigid substrates), the cell can adequately adhere to its substrate. Subsequently, as the traction forces exerted by the cell increase, the cell elongates and its shape changes. Within very stiff (hard) substrates, the cell cannot penetrate into its substrate or send out pseudopodia. On the other hand, a cell is found to be more elongated within substrates with a constrained surface. However, this elongation decreases when the cell approaches it. It can be concluded that the higher the net traction force, the greater the cell elongation, the larger the cell membrane area, and the less random the cell alignment.

Dynamics and instabilities of lipid bilayer membrane shapes

Biological membranes undergo constant shape remodeling involving the formation of highly curved structures. The lipid bilayer represents the fundamental architecture of the cellular membrane with its shapes determined by the Helfrich curvature bending energy. However, the dynamics of bilayer shape transitions, especially their modulation by membrane proteins, and the resulting shape instabilities, are still not well understood. Here, we review in a unifying manner several theories that describe the fluctuations (i.e. undulations) of bilayer shapes as well as their local coupling with lipid or protein density variation. The coupling between local membrane curvature and lipid density gives rise to a 'slipping mode' in addition to the conventional 'bending mode' for damping the membrane fluctuation. This leads to a number of interesting experimental phenomena regarding bilayer shape dynamics. More importantly, curvature-inducing proteins can couple with membrane shape and eventually render the membrane unstable. A criterion for membrane shape instability is derived from a linear stability analysis. The instability criterion reemphasizes the importance of membrane tension in regulating the stability and dynamics of membrane geometry. Recent progresses in understanding the role of membrane tension in regulating dynamical cellular processes are also reviewed. Protein density is emphasized as a key factor in regulating membrane shape transitions: a threshold density of curvature coupling proteins is required for inducing membrane morphology transitions.

Exploring the elasticity and adhesion behavior of cardiac fibroblasts by atomic force microscopy indentation

AFM was used to collect the whole force-deformation cell curves. They provide both the elasticity and adhesion behavior of mouse primary cardiac fibroblasts. To confirm the hypothesis that a link exists between the membrane receptors and the cytoskeletal filaments causing therefore changing in both elasticity and adhesion behavior, actin-destabilizing Cytochalsin D was administrated to the fibroblasts. From immunofluorescence observation and AFM loading/unloading curves, cytoskeletal reorganization as well as a change in the elasticity and adhesion was indeed observed. Elasticity of control fibroblasts is three times higher than that for fibroblasts treated with 0.5 μM Cytochalasin. Moreover, AFM loading-unloading curves clearly show the different mechanical behavior of the two different cells analyzed: (i) for control cells the AFM cantilever rises during the dwell time while cells with Cytochalasin fail to show such an active resistance; (ii) the maximum force to deform control cells is quite higher and as far as adhesion is concern (iii) the maximum separation force, detachment area and the detachment process time are much larger for control compared to the Cytochalasin treated cells. Therefore, alterations in the cytoskeleton suggest that a link must exist between the membrane receptors and the cytoskeletal filaments beneath the cellular surface and inhibition of actin polymerization has effects on the whole cell mechanical behavior as well as adhesion.

Phospho-NHE3 forms membrane patches and interacts with beta-actin to sense and maintain constant direction during cell migration

The Na(+)/H(+) exchanger NHE3 colocalizes with beta-actin at the leading edge of directionally migrating cells. Using human osteosarcoma cells (SaOS-2), rat osteoblasts (calvaria), and human embryonic kidney (HEK) cells, we identified a novel role for NHE3 via beta-actin in anode and cathode directed motility, during electrotaxis. NHE3 knockdown by RNAi revealed that NHE3 expression is required to achieve constant directionality and polarity in migrating cells. Phosphorylated NHE3 (pNHE3) and beta-actin complex formation was impaired by the NHE3 inhibitor S3226 (IC50 0.02µM). Fluorescence cross-correlation spectroscopy (FCCS) revealed that the molecular interactions between NHE3 and beta-actin in membrane protrusions increased 1.7-fold in the presence of a directional cue and decreased 3.3-fold in the presence of cytochalasin D. Data from flow cytometric analysis showed that membrane potential of cells (Vmem) decreases in directionally migrating, NHE3-deficient osteoblasts and osteosarcoma cells whereas only Vmem of wild type osteoblasts is affected during directional migration. These findings suggest that pNHE3 has a mechanical function via beta-actin that is dependent on its physiological activity and Vmem. Furthermore, phosphatidylinositol 3,4,5-trisphosphate (PIP3) levels increase while PIP2 remains stable when cells have persistent directionality. Both PI3 kinase (PI3K) and Akt expression levels change proportionally to NHE3 levels. Interestingly, however, the content of pNHE3 level does not change when PI3K/Akt is inhibited. Therefore, we conclude that NHE3 can act as a direction sensor for cells and that NHE3 phosphorylation in persistent directional cell migration does not involve PI3K/Akt during electrotaxis.

Morphological alterations of T24 cells on flat and nanotubular TiO2 surfaces

Aim

To investigate morphological alterations of malignant cancer cells (T24) of urothelial origin seeded on flat titanium (Ti) and nanotubular titanium dioxide (TiO₂) nanostructures.

Methods

Using anodization method, TiO₂ surfaces composed of vertically aligned nanotubes of 50-100 nm diameters were produced. The flat Ti surface was used as a reference. The alteration in the morphology of cancer cells was evaluated using scanning electron microscopy (SEM). A computational model, based on the theory of membrane elasticity, was constructed to shed light on the biophysical mechanisms responsible for the observed changes in the contact area of adhesion.

Results

Large diameter TiO₂ nanotubes exhibited a significantly smaller contact area of adhesion (P < 0.0001) and had more membrane protrusions (eg, microvilli and intercellular membrane nanotubes) than on flat Ti surface. Numerical membrane dynamics simulations revealed that the low adhesion energy per unit area would hinder the cell spreading on the large diameter TiO₂ nanotubular surface, thus explaining the small contact area.

Conclusion

The reduction in the cell contact area in the case of large diameter TiO₂ nanotube surface, which does not enable formation of the large enough number of the focal adhesion points, prevents spreading of urothelial cells.

Envisioning migration: mathematics in both experimental analysis and modeling of cell behavior

The complex nature of cell migration highlights the power and challenges of applying mathematics to biological studies. Mathematics may be used to create model equations that recapitulate migration, which can predict phenomena not easily uncovered by experiments or intuition alone. Alternatively, mathematics may be applied to interpreting complex data sets with better resolution--potentially empowering scientists to discern subtle patterns amid the noise and heterogeneity typical of migrating cells. Iteration between these two methods is necessary in order to reveal connections within the cell migration signaling network, as well as to understand the behavior that arises from those connections. Here, we review recent quantitative analysis and mathematical modeling approaches to the cell migration problem.

Dynamic modeling of cell migration and spreading behaviors on fibronectin coated planar substrates and micropatterned geometries

An integrative cell migration model incorporating focal adhesion (FA) dynamics, cytoskeleton and nucleus remodeling, actin motor activity, and lamellipodia protrusion is developed for predicting cell spreading and migration behaviors. This work is motivated by two experimental works: (1) cell migration on 2-D substrates under various fibronectin concentrations and (2) cell spreading on 2-D micropatterned geometries. These works suggest (1) cell migration speed takes a maximum at a particular ligand density (∼1140 molecules/µm(2)) and (2) that strong traction forces at the corners of the patterns may exist due to combined effects exerted by actin stress fibers (SFs). The integrative model of this paper successfully reproduced these experimental results and indicates the mechanism of cell migration and spreading. In this paper, the mechanical structure of the cell is modeled as having two elastic membranes: an outer cell membrane and an inner nuclear membrane. The two elastic membranes are connected by SFs, which are extended from focal adhesions on the cortical surface to the nuclear membrane. In addition, the model also includes ventral SFs bridging two focal adhesions on the cell surface. The cell deforms and gains traction as transmembrane integrins distributed over the outer cell membrane bond to ligands on the ECM surface, activate SFs, and form focal adhesions. The relationship between the cell migration speed and fibronectin concentration agrees with existing experimental data for Chinese hamster ovary (CHO) cell migrations on fibronectin coated surfaces. In addition, the integrated model is validated by showing persistent high stress concentrations at sharp geometrically patterned edges. This model will be used as a predictive model to assist in design and data processing of upcoming microfluidic cell migration assays.

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.

Cell polarity: quantitative modeling as a tool in cell biology

Among a number of innovative approaches that have modernized cell biology, modeling has a prominent yet unusual place. One popular view is that we progress linearly, from conceptual to ever more detailed models. We review recent discoveries of cell polarity mechanisms, in which modeling played an important role, to demonstrate that the experiment-theory feedback loop requires diverse models characterized by varying levels of biological detail and mathematical complexity. We argue that a quantitative model is a tool that has to fit an experimental study, and the model's value should be judged not by how complex and detailed it is, but by what could be learned from it.

Deconvolution of the cellular force-generating subsystems that govern cytokinesis furrow ingression

Cytokinesis occurs through the coordinated action of several biochemically-mediated stresses acting on the cytoskeleton. Here, we develop a computational model of cellular mechanics, and using a large number of experimentally measured biophysical parameters, we simulate cell division under a number of different scenarios. We demonstrate that traction-mediated protrusive forces or contractile forces due to myosin II are sufficient to initiate furrow ingression. Furthermore, we show that passive forces due to the cell's cortical tension and surface curvature allow the furrow to complete ingression. We compare quantitatively the furrow thinning trajectories obtained from simulation with those observed experimentally in both wild-type and myosin II null Dictyostelium cells. Our simulations highlight the relative contributions of different biomechanical subsystems to cell shape progression during cell division.

Cytokinesis, the physical separation of a mother cell into two daughter cells, requires force to deform the cell. Though there is ample evidence in many systems that myosin II provides some of this force, it is also well known that some cell types can divide in the absence of myosin II. To elucidate the mechanisms by which cells control furrow ingression, we developed a computational model of cellular dynamics during cytokinesis in the social amoeba, Dictyostelium discoideum. We took advantage of a large number of experimentally measured parameters and well-characterized furrow ingression dynamics for a number of different strains. Our simulations demonstrate that there are distinct phases of cytokinesis. Myosin II plays a role providing the stress that initiates furrow ingression. In its absence, however, this force can be supplied by a combination of adhesion and protrusion-mediated stresses. Thereafter, Laplace-like pressures take over and provide stresses that enable the cell to divide. Overall, we show how various mechanical parameters quantitatively impact furrow ingression kinetics, accounting for the cytokinesis dynamics of wild type and mutant cell-lines.

The role of cholesterol-sphingomyelin membrane nanodomains in the stability of intercellular membrane nanotubes

Intercellular membrane nanotubes (ICNs) are highly curved tubular structures that connect neighboring cells. The stability of these structures depends on the inner cytoskeleton and the cell membrane composition. Yet, due to the difficulty in the extraction of ICNs, the cell membrane composition remains elusive. In the present study, a raft marker, ostreolysin, revealed the enrichment of cholesterol-sphingomyelin membrane nanodomains along ICNs in a T24 (malignant) urothelial cancer cell line. Cholesterol depletion, due to the addition of methyl-β-cyclodextrin, caused the dispersion of cholesterol-sphingomyelin membrane nanodomains and the retraction of ICNs. The depletion of cholesterol also led to cytoskeleton reorganization and to formation of actin stress fibers. Live cell imaging data revealed the possible functional coupling between the change from polygonal to spherical shape, cell separation, and the disconnection of ICNs. The ICN was modeled as an axisymmetric tubular structure, enabling us to investigate the effects of cholesterol content on the ICN curvature. The removal of cholesterol was predicted to reduce the positive spontaneous curvature of the remaining membrane components, increasing their curvature mismatch with the tube curvature. The mechanisms by which the increased curvature mismatch could contribute to the disconnection of ICNs are discussed.

The transport along membrane nanotubes driven by the spontaneous curvature of membrane components

Intercellular membrane nanotubes (ICNs) serve as a very specific transport system between neighboring cells. The underlying mechanisms responsible for the transport of membrane components and vesicular dilations along the ICNs are not clearly understood. The present study investigated the spatial distribution of anisotropic membrane components of tubular shapes and isotropic membrane components of spherical shapes. Experimental results revealed the preferential distribution of CTB (cholera toxin B)-GM1 complexes mainly on the spherical cell membrane, and cholesterol-sphingomyelin at the membrane leading edge and ICNs. In agreement with previous studies, we here propose that the spatial distribution of CTB-GM1 complexes and cholesterol-sphingomyelin rafts were due to their isotropic and anisotropic shapes, respectively. To elucidate the relationship between a membrane component shape and its spatial distribution, a two-component computational model was constructed. The minimization of the membrane bending (free) energy revealed the enrichment of the anisotropic component along the ICN and the isotropic component in the parent cell membrane, which was due to the curvature mismatch between the ICN curvature and the spontaneous curvature of the isotropic component. The equations of motion, derived from the differentiation of the membrane free energy, revealed a curvature-dependent flux of the isotropic component and a curvature-dependent force exerted on a vesicular dilation along the ICN. Finally, the effects of possible changes in the orientational ordering of the anisotropic component attendant to the transport of the vesicular dilation were discussed with connection to the propagation of electrical and chemical signals.

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.

Fusion pore diameter regulation by cations modulating local membrane anisotropy

The fusion pore is an aqueous channel that is formed upon the fusion of the vesicle membrane with the plasma membrane. Once the pore is open, it may close again (transient fusion) or widen completely (full fusion) to permit vesicle cargo discharge. While repetitive transient fusion pore openings of the vesicle with the plasma membrane have been observed in the absence of stimulation, their frequency can be further increased using a cAMP-increasing agent that drives the opening of nonspecific cation channels. Our model hypothesis is that the openings and closings of the fusion pore are driven by changes in the local concentration of cations in the connected vesicle. The proposed mechanism of fusion pore dynamics is considered as follows: when the fusion pore is closed or is extremely narrow, the accumulation of cations in the vesicle (increased cation concentration) likely leads to lipid demixing at the fusion pore. This process may affect local membrane anisotropy, which reduces the spontaneous curvature and thus leads to the opening of the fusion pore. Based on the theory of membrane elasticity, we used a continuum model to explain the rhythmic opening and closing of the fusion pore.

On the role of membrane anisotropy and BAR proteins in the stability of tubular membrane structures

Recent studies have demonstrated that actin filaments are not crucial for the short-term stability of tubular membrane protrusions originating from the cell surface. It has also been demonstrated that prominin nanodomains and curvature inducing I-BAR proteins could account for the stability of the membrane protrusion. Here we constructed an axisymmetric model of a membrane protrusion that excludes actin filaments in order to investigate the contributions of prominin nanodomains (rafts) and I-BAR proteins to the membrane protrusion stability. It was demonstrated that prominin nanodomains and I-BAR proteins can stabilize the membrane protrusion only over a specific range of spontaneous curvature. On the other hand, high spontaneous curvature and/or high density of I-BAR proteins could lead to system instability and to non-uniform contraction in the radial direction of the membrane protrusion. In agreement with previous studies, it was also shown that the isotropic bending energy of lipids is not sufficient to explain the stability of the observed tubular membrane protrusion without actin filaments.