A Computational Model for Collective Cellular Motion in Three Dimensions: General Framework and Case Study for Cell Pair Dynamics

Cell migration in healthy and diseased systems is a combination of single and collective cell motion. While single cell motion has received considerable attention, our understanding of collective cell motion remains elusive. A new computational framework for the migration of groups of cells in three dimensions is presented, which focuses on the forces acting at the microscopic scale and the interactions between cells and their extracellular matrix (ECM) environment. Cell-cell adhesion, resistance due to the ECM and the factors regulating the propulsion of each cell through the matrix are considered. In particular, our approach emphasizes the role of receptors that mediate cell-cell and cell-matrix interactions, and examines how variation in their properties induces changes in cellular motion. As an important case study, we analyze two interacting cells. Our results show that the dynamics of cell pairs depends on the magnitude and the stochastic nature of the forces. Stronger intercellular stability is generally promoted by surface receptors that move. We also demonstrate that matrix resistance, cellular stiffness and intensity of adhesion contribute to migration behaviors in different ways, with memory effects present that can alter pair motility. If adhesion weakens with time, our findings show that cell pair break-up depends strongly on the way cells interact with the matrix. Finally, the motility for cells in a larger cluster (size 50 cells) is examined to illustrate the full capabilities of the model and to stress the role of cellular pairs in complex cellular structures. Overall, our framework shows how properties of cells and their environment influence the stability and motility of cellular assemblies. This is an important step in the advancement of the understanding of collective motility, and can contribute to knowledge of complex biological processes involving migration, aggregation and detachment of cells in healthy and diseased systems.     

Cell speed is independent of force in a mathematical model of amoeboidal cell motion with random switching terms

In this paper the motion of a single cell is modeled as a nucleus and multiple integrin based adhesion sites. Numerical simulations and analysis of the model indicate that when the stochastic nature of the adhesion sites is a memoryless and force independent random process, the cell speed is independent of the force these adhesion sites exert on the cell. Furthermore, understanding the dynamics of the attachment and detachment of the adhesion sites is key to predicting cell speed. We introduce a differential equation describing the cell motion and then introduce a conjecture about the expected drift of the cell, the expected average velocity relation conjecture. Using Markov chain theory, we analyze our conjecture in the context of a related (but simpler) model of cell motion, and then numerically compare the results for the simpler model and the full differential equation model. We also heuristically describe the relationship between the simplified and full models as well as provide a discussion of the biological significance of these results.     

Simulating convergent extension by way of anisotropic differential adhesion

Simulations using the Extended Potts Model suggest that anisotropic differential adhesion can account for convergent extension, as observed during embryonic development of the frog Xenopus laevis for example. During gastrulation in these frogs, convergent extension produces longitudinal tissue growth from latitudinal elongation and migration of aligned constituent cells. The Extended Potts Model employs clustered points on a grid to represent subdivided cells with probabilistic displacement of cell boundaries such that small changes in energy drive gradual tissue development. For modeling convergent extension, simulations include anisotropic differential adhesion: the degree of attachment between adjacent elongated cells depends on their relative orientation. Without considering additional mechanisms, simulations based on anisotropic differential adhesion reproduce the hallmark stages of convergent extension in the correct sequence, with random fluctuations as sufficient impetus for cell reorganization.     

Numerical investigation of the role of intercellular interactions on collective epithelial cell migration

During collective cell migration, the intercellular forces will significantly affect the collective migratory behaviors. However, the measurement of mechanical stresses exerted at cell–cell junctions is very challenging. A recent experimental observation indicated that the intercellular adhesion sites within a migrating monolayer are subjected to both normal stress exerted perpendicular to cell–cell junction surface and shear stress exerted tangent to cell–cell junction surface. In this study, an interfacial interaction model was proposed to model the intercellular interactions for the first time. The intercellular interaction model-based study of collective epithelial migration revealed that the direction of cell migration velocity has better alignment with the orientation of local principal stress at higher maximum shear stress locations in an epithelial monolayer sheet. Parametric study of the effects of adhesion strength indicated that normal adhesion strength at the cell–cell junction surface has dominated effect on local alignment between the direction of cell velocity vector and the principal stress orientation, while the shear adhesion strength has little effect, which provides compelling evidence to help explain the force transmission via cell–cell junctions between adjacent cells in collective cell motion and provides new insights into “adhesive belt” effects at cell–cell junction.     

Local actin dynamics couple speed and persistence in a cellular Potts model of cell migration

Cell migration is astoundingly diverse. Molecular signatures, cell-cell interactions, and environmental structures each play their part in shaping cell motion, yielding numerous morphologies and migration modes. Nevertheless, in recent years, a simple unifying law was found to describe cell migration across many different cell types and contexts: faster cells turn less frequently. This universal coupling between speed and persistence (UCSP) was explained by retrograde actin flow from front to back, but it remains unclear how this mechanism generalizes to cells with complex shapes and cells migrating in structured environments, which may not have a well-defined front-to-back orientation. Here, we present an in-depth characterization of an existing cellular Potts model, in which cells polarize dynamically from a combination of local actin dynamics (stimulating protrusions) and global membrane tension along the perimeter (inhibiting protrusions). We first show that the UCSP emerges spontaneously in this model through a cross talk of intracellular mechanisms, cell shape, and environmental constraints, resembling the dynamic nature of cell migration in vivo. Importantly, we find that local protrusion dynamics suffice to reproduce the UCSP—even in cases in which no clear global, front-to-back polarity exists. We then harness the spatial nature of the cellular Potts model to show how cell shape dynamics limit both the speed and persistence a cell can reach and how a rigid environment such as the skin can restrict cell motility even further. Our results broaden the range of potential mechanisms underlying the speed-persistence coupling that has emerged as a fundamental property of migrating cells.     

Cadherin-2 Controls Directional Chain Migration of Cerebellar Granule Neurons

Long distance migration of differentiating granule cells from the cerebellar upper rhombic lip has been reported in many vertebrates. However, the knowledge about the subcellular dynamics and molecular mechanisms regulating directional neuronal migration in vivo is just beginning to emerge. Here we show by time-lapse imaging in live zebrafish (Danio rerio) embryos that cerebellar granule cells migrate in chain-like structures in a homotypic glia-independent manner. Temporal rescue of zebrafish Cadherin-2 mutants reveals a direct role for this adhesion molecule in mediating chain formation and coherent migratory behavior of granule cells. In addition, Cadherin-2 maintains the orientation of cell polarization in direction of migration, whereas in Cadherin-2 mutant granule cells the site of leading edge formation and centrosome positioning is randomized. Thus, the lack of adhesion leads to impaired directional migration with a mispositioning of Cadherin-2 deficient granule cells as a consequence. Furthermore, these cells fail to differentiate properly into mature granule neurons. In vivo imaging of Cadherin-2 localization revealed the dynamics of this adhesion molecule during cell locomotion. Cadherin-2 concentrates transiently at the front of granule cells during the initiation of individual migratory steps by intramembraneous transport. The presence of Cadherin-2 in the leading edge corresponds to the observed centrosome orientation in direction of migration. Our results indicate that Cadherin-2 plays a key role during zebrafish granule cell migration by continuously coordinating cell-cell contacts and cell polarity through the remodeling of adherens junctions. As Cadherin-containing adherens junctions have been shown to be connected via microtubule fibers with the centrosome, our results offer an explanation for the mechanism of leading edge and centrosome positioning during nucleokinetic migration of many vertebrate neuronal populations.     

The coordination between actin filaments and adhesion in mesenchymal migration

Mesenchymal cell motility is characterized by a polarized distribution of actin filaments, with a network of short branched actin filaments at the leading edge, and polymers of actin filaments arranged into distinct classes of actin stress fibres behind the leading edge. Importantly, the distinct actin filaments are characteristically associated with discrete adhesion structures and both the adhesions and the actin filaments are co-ordinately regulated during cell migration. While it has long been known that these macromolecular structures are intimately linked in cells, precisely how they are co-ordinately regulated is presently unknown. Live imaging data now suggests that the focal adhesions may act as sites of actin polymerization resulting in the generation of tension-bearing actin bundles of actin filaments (stress fibres). Moreover, a picture is emerging to suggest that the tropomyosin family of proteins that can determine actin filament dynamics may also play a key role in determining the transition between adhesion states. Molecules such as the tropomyosins are therefore tantalizing candidates to orchestrate the coordination of actin and adhesion dynamics during mesenchymal cell migration.     

Fluorescence imaging of two-photon linear dichroism: cholesterol depletion disrupts molecular orientation in cell membranes

The plasma membrane of cells is an ordered environment, giving rise to anisotropic orientation and restricted motion of molecules and proteins residing in the membrane. At the same time as being an organized matrix of defined structure, the cell membrane is heterogeneous and dynamic. Here we present a method where we use fluorescence imaging of linear dichroism to measure the orientation of molecules relative to the cell membrane. By detecting linear dichroism as well as fluorescence anisotropy, the orientation parameters are separated from dynamic properties such as rotational diffusion and homo energy transfer (energy migration). The sensitivity of the technique is enhanced by using two-photon excitation for higher photo-selection compared to single photon excitation. We show here that we can accurately image lipid organization in whole cell membranes and in delicate structures such as membrane nanotubes connecting two cells. The speed of our wide-field imaging system makes it possible to image changes in orientation and anisotropy occurring on a subsecond timescale. This is demonstrated by time-lapse studies showing that cholesterol depletion rapidly disrupts the orientation of a fluorophore located within the hydrophobic region of the cell membrane but not of a surface bound probe. This is consistent with cholesterol having an important role in stabilizing and ordering the lipid tails within the plasma membrane.     

Modelling Cell Migration and Adhesion During Development

Cell?Ccell adhesion is essential for biological development: cells migrate to their target sites, where cell?Ccell adhesion enables them to aggregate and form tissues. Here, we extend analysis of the model of cell migration proposed by Anguige and Schmeiser (J. Math. Biol. 58(3):395?C427, 2009) that incorporates both cell?Ccell adhesion and volume filling. The stochastic space-jump model is compared to two deterministic counterparts (a system of stochastic mean equations and a non-linear partial differential equation), and it is shown that the results of the deterministic systems are, in general, qualitatively similar to the mean behaviour of multiple stochastic simulations. However, individual stochastic simulations can give rise to behaviour that varies significantly from that of the mean. In particular, individual simulations might admit cell clustering when the mean behaviour does not. We also investigate the potential of this model to display behaviour predicted by the differential adhesion hypothesis by incorporating a second cell species, and present a novel approach for implementing models of cell migration on a growing domain.     

Centrosome reorientation in wound-edge cells is cell type specific

The reorientation of the microtubule organizing center during cell migration into a wound in the monolayer was directly observed in living wound-edge cells expressing gamma-tubulin tagged with green fluorescent protein. Our results demonstrate that in CHO cells, the centrosome reorients to a position in front of the nucleus, toward the wound edge, whereas in PtK cells, the centrosome lags behind the nucleus during migration into the wound. In CHO cells, the average rate of centrosome motion was faster than that of the nucleus; the converse was true in PtK cells. In both cell lines, centrosome motion was stochastic, with periods of rapid motion interspersed with periods of slower motion. Centrosome reorientation in CHO cells required dynamic microtubules and cytoplasmic dynein/dynactin activity and could be prevented by altering cell-to-cell or cell-to-substrate adhesion. Microtubule marking experiments using photoactivation of caged tubulin demonstrate that microtubules are transported in the direction of cell motility in both cell lines but that in PtK cells, microtubules move individually, whereas their movement is more coherent in CHO cells. Our data demonstrate that centrosome reorientation is not required for directed migration and that diverse cells use distinct mechanisms for remodeling the microtubule array during directed migration.     

Memo-RhoA-mDia1 signaling controls microtubules, the actin network, and adhesion site formation in migrating cells

Actin assembly at the cell front drives membrane protrusion and initiates the cell migration cycle. Microtubules (MTs) extend within forward protrusions to sustain cell polarity and promote adhesion site turnover. Memo is an effector of the ErbB2 receptor tyrosine kinase involved in breast carcinoma cell migration. However, its mechanism of action remained unknown. We report in this study that Memo controls ErbB2-regulated MT dynamics by altering the transition frequency between MT growth and shortening phases. Moreover, although Memo-depleted cells can assemble the Rac1-dependent actin meshwork and form lamellipodia, they show defective localization of lamellipodial markers such as α-actinin-1 and a reduced number of short-lived adhesion sites underlying the advancing edge of migrating cells. Finally, we demonstrate that Memo is required for the localization of the RhoA guanosine triphosphatase and its effector mDia1 to the plasma membrane and that Memo–RhoA–mDia1 signaling coordinates the organization of the lamellipodial actin network, adhesion site formation, and MT outgrowth within the cell leading edge to sustain cell motility.     

Apical migration of nuclei during G2 is a prerequisite for all nuclear motion in zebrafish neuroepithelia

Nuclei in the proliferative pseudostratified epithelia of vastly different organisms exhibit a characteristic dynamics - the so-called interkinetic nuclear migration (IKNM). Although these movements are thought to be intimately tied to the cell cycle, little is known about the relationship between IKNM and distinct phases of the cell cycle and the role that this association plays in ensuring balanced proliferation and subsequent differentiation. Here, we perform a quantitative analysis of modes of nuclear migration during the cell cycle using a marker that enables the first unequivocal differentiation of all four phases in proliferating neuroepithelial cells in vivo. In zebrafish neuroepithelia, nuclei spend the majority of the cell cycle in S phase, less time in G1, with G2 and M being noticeably shorter still in comparison. Correlating cell cycle phases with nuclear movements shows that IKNM comprises rapid apical nuclear migration during G2 phase and stochastic nuclear motion during G1 and S phases. The rapid apical migration coincides with the onset of G2, during which we find basal actomyosin accumulation. Inhibiting the transition from G2 to M phase induces a complete stalling of nuclei, indicating that IKNM and cell cycle continuation cannot be uncoupled and that progression from G2 to M is a prerequisite for rapid apical migration. Taken together, these results suggest that IKNM involves an actomyosin-driven contraction of cytoplasm basal to the nucleus during G2, and that the stochastic nuclear movements observed in other phases arise passively due to apical migration in neighboring cells.     

Cell adhesion defines the topology of endocytosis and signaling

Preferred sites of endocytosis have been observed in various cell types, but whether they occur randomly or are linked to cellular cues is debated. Here, we quantified the sites of endocytosis of transferrin (Tfn) and epidermal growth factor (EGF) in cells whose adhesion geometry was defined by micropatterns. 3D probabilistic density maps revealed that Tfn was enriched in adhesive sites during uptake, whereas EGF endocytosis was restricted to the dorsal cellular surface. This spatial separation was not due to distributions of corresponding receptors but was regulated by uptake mechanisms. Asymmetric uptake of Tfn resulted from the enrichment of clathrin and adaptor protein 2 at adhesive areas. Asymmetry in EGF uptake was strongly dependent on the actin cytoskeleton and led to asymmetry in EGF receptor activation. Mild alteration of actin dynamics abolished asymmetry in EGF uptake and decreased EGF‐induced downstream signaling, suggesting that cellular adhesion cues influence signal propagation. We propose that restriction of endocytosis at distinct sites allows cells to sense their environment in an “outside‐in” mechanism.     

A stochastic model for adhesion-mediated cell random motility and haptotaxis

The active migration of blood and tissue cells is important in a number of physiological processes including inflammation, wound healing, embryogenesis, and tumor cell metastasis. These cells move by transmitting cytoplasmic force through membrane receptors which are bound specifically to adhesion ligands in the surrounding substratum. Recently, much research has focused on the influence of the composition of extracellular matrix and the distribution of its components on the speed and direction of cell migration. It is commonly believed that the magnitude of the adhesion influences cell speed and/or random turning behavior, whereas a gradient of adhesion may bias the net direction of the cell movement, a phenomenon known as haptotaxis. The mechanisms underlying these responses are presently not understood.A stochastic model is presented to provide a mechanistic understanding of how the magnitude and distribution of adhesion ligands in the substratum influence cell movement. The receptor-mediated cell migration is modeled as an interrelation of random processes on distinct time scales. Adhesion receptors undergo rapid binding and transport, resulting in a stochastic spatial distribution of bound receptors fluctuating about some mean distribution. This results in a fluctuating spatio-temporal pattern of forces on the cell, which in turn affects the speed and turning behavior on a longer time scale. The model equations are a system of nonlinear stochastic differential equations (SDE's) which govern the time evolution of the spatial distribution of bound and free receptors, and the orientation and position of the cell. These SDE's are integrated numerically to simulate the behavior of the model cell on both a uniform substratum, and on a gradient of adhesion ligand concentration.Furthermore, analysis of the governing SDE system and corresponding Fokker-Planck equation (FPE) yields analytical expressions for indices which characterize cell movement on multiple time scales in terms of cell cytomechanical, morphological, and receptor binding and transport parameters. For a uniform adhesion ligand concentration, this analysis provides expressions for traditional cell movement indices such as mean speed, directional persistence time, and random motility coefficient. In a small gradient of adhesion, a perturbation analysis of the FPE yields a constitutive cell flux expression which includes a drift term for haptotactic directional cell migration. The haptotactic drift contains terms identified as contributions from directional orientation bias (taxis).     

Finite metapopulation models with density-dependent migration and stochastic local dynamics

The effects of small density-dependent migration on the dynamics of a metapopulation are studied in a model with stochastic local dynamics. We use a diffusion approximation to study how changes in the migration rate and habitat occupancy affect the rates of local colonization and extinction. If the emigration rate increases or if the immigration rate decreases with local population size, a positive expected rate of change in habitat occupancy is found for a greater range of habitat occupancies than when the migration is density-independent. In contrast, the reverse patterns of density dependence in respective emigration and immigration reduce the range of habitat occupancies where the metapopulation will be viable. This occurs because density-dependent migration strongly influences both the establishment and rescue effects in the local dynamics of metapopulations.     

The Rho-mDia1 pathway regulates cell polarity and focal adhesion turnover in migrating cells through mobilizing Apc and c-Src

Directed cell migration requires cell polarization and adhesion turnover, in which the actin cytoskeleton and microtubules work critically. The Rho GTPases induce specific types of actin cytoskeleton and regulate microtubule dynamics. In migrating cells, Cdc42 regulates cell polarity and Rac works in membrane protrusion. However, the role of Rho in migration is little known. Rho acts on two major effectors, ROCK and mDia1, among which mDia1 produces straight actin filaments and aligns microtubules. Here we depleted mDia1 by RNA interference and found that mDia1 depletion impaired directed migration of rat C6 glioma cells by inhibiting both cell polarization and adhesion turnover. Apc and active Cdc42, which work together for cell polarization, localized in the front of migrating cells, while active c-Src, which regulates adhesion turnover, localized in focal adhesions. mDia1 depletion impaired localization of these molecules at their respective sites. Conversely, expression of active mDia1 facilitated microtubule-dependent accumulation of Apc and active Cdc42 in the polar ends of the cells and actin-dependent recruitment of c-Src in adhesions. Thus, the Rho-mDia1 pathway regulates polarization and adhesion turnover by aligning microtubules and actin filaments and delivering Apc/Cdc42 and c-Src to their respective sites of action.     

Arachidonic acid randomizes endothelial cell motion and regulates adhesion and migration

Cell adhesion and migration are essential for the evolution, organization, and repair of living organisms. An example of a combination of these processes is the formation of new blood vessels (angiogenesis), which is mediated by a directed migration and adhesion of endothelial cells (ECs). Angiogenesis is an essential part of wound healing and a prerequisite of cancerous tumor growth. We investigated the effect of the amphiphilic compound arachidonic acid (AA) on EC adhesion and migration by combining live cell imaging with biophysical analysis methods. AA significantly influenced both EC adhesion and migration, in either a stimulating or inhibiting fashion depending on AA concentration. The temporal evolution of cell adhesion area was well described by a two-phase model. In the first phase, the spreading dynamics were independent of AA concentration. In the latter phase, the spreading dynamics increased at low AA concentrations and decreased at high AA concentrations. AA also affected EC migration; though the instantaneous speed of individual cells remained independent of AA concentration, the individual cells lost their sense of direction upon addition of AA, thus giving rise to an overall decrease in the collective motion of a confluent EC monolayer into vacant space. Addition of AA also caused ECs to become more elongated, this possibly being related to incorporation of AA in the EC membrane thus mediating a change in the viscosity of the membrane. Hence, AA is a promising non-receptor specific regulator of wound healing and angiogenesis.     

Mammalian CAP (Cyclase-associated protein) in the world of cell migration: Roles in actin filament dynamics and beyond

Cell migration is essential for a variety of fundamental biological processes such as embryonic development, wound healing, and immune response. Aberrant cell migration also underlies pathological conditions such as cancer metastasis, in which morphological transformation promotes spreading of cancer to new sites. Cell migration is driven by actin dynamics, which is the repeated cycling of monomeric actin (G-actin) into and out of filamentous actin (F-actin). CAP (Cyclase-associated protein, also called Srv2) is a conserved actin-regulatory protein, which is implicated in cell motility and the invasiveness of human cancers. It cooperates with another actin regulatory protein, cofilin, to accelerate actin dynamics. Hence, knockdown of CAP1 slows down actin filament turnover, which in most cells leads to reduced cell motility. However, depletion of CAP1 in HeLa cells, while causing reduction in dynamics, actually led to increased cell motility. The increases in motility are likely through activation of cell adhesion signals through an inside-out signaling. The potential to activate adhesion signaling competes with the negative effect of CAP1 depletion on actin dynamics, which would reduce cell migration. In this commentary, we provide a brief overview of the roles of mammalian CAP1 in cell migration, and highlight a likely mechanism underlying the activation of cell adhesion signaling and elevated motility caused by depletion of CAP1.     

Mammalian CAP (Cyclase-associated protein) in the world of cell migration

Cell migration is essential for a variety of fundamental biological processes such as embryonic development, wound healing, and immune response. Aberrant cell migration also underlies pathological conditions such as cancer metastasis, in which morphological transformation promotes spreading of cancer to new sites. Cell migration is driven by actin dynamics, which is the repeated cycling of monomeric actin (G-actin) into and out of filamentous actin (F-actin). CAP (Cyclase-associated protein, also called Srv2) is a conserved actin-regulatory protein, which is implicated in cell motility and the invasiveness of human cancers. It cooperates with another actin regulatory protein, cofilin, to accelerate actin dynamics. Hence, knockdown of CAP1 slows down actin filament turnover, which in most cells leads to reduced cell motility. However, depletion of CAP1 in HeLa cells, while causing reduction in dynamics, actually led to increased cell motility. The increases in motility are likely through activation of cell adhesion signals through an inside-out signaling. The potential to activate adhesion signaling competes with the negative effect of CAP1 depletion on actin dynamics, which would reduce cell migration. In this commentary, we provide a brief overview of the roles of mammalian CAP1 in cell migration, and highlight a likely mechanism underlying the activation of cell adhesion signaling and elevated motility caused by depletion of CAP1.