A stochastic description of Dictyostelium chemotaxis

Chemotaxis, the directed motion of a cell toward a chemical source, plays a key role in many essential biological processes. Here, we derive a statistical model that quantitatively describes the chemotactic motion of eukaryotic cells in a chemical gradient. Our model is based on observations of the chemotactic motion of the social ameba Dictyostelium discoideum, a model organism for eukaryotic chemotaxis. A large number of cell trajectories in stationary, linear chemoattractant gradients is measured, using microfluidic tools in combination with automated cell tracking. We describe the directional motion as the interplay between deterministic and stochastic contributions based on a Langevin equation. The functional form of this equation is directly extracted from experimental data by angle-resolved conditional averages. It contains quadratic deterministic damping and multiplicative noise. In the presence of an external gradient, the deterministic part shows a clear angular dependence that takes the form of a force pointing in gradient direction. With increasing gradient steepness, this force passes through a maximum that coincides with maxima in both speed and directionality of the cells. The stochastic part, on the other hand, does not depend on the orientation of the directional cue and remains independent of the gradient magnitude. Numerical simulations of our probabilistic model yield quantitative agreement with the experimental distribution functions. Thus our model captures well the dynamics of chemotactic cells and can serve to quantify differences and similarities of different chemotactic eukaryotes. Finally, on the basis of our model, we can characterize the heterogeneity within a population of chemotactic cells.     

A 3-D model used to explore how cell adhesion and stiffness affect cell sorting and movement in multicellular systems

A three-dimensional mathematical model is used to determine the effects of adhesion and cell signalling on cell movements during the aggregation and slug stages of Dictyostelium discoideum (Dd) and to visualize cell sorting. The building blocks of the model are individual deformable ellipsoidal cells, where movement depends on internal parameter state (cell size and stiffness) and on external cues from the neighboring cells, extracellular matrix, and chemical signals. Cell movement and deformation are calculated from equations of motion using the total force acting on each cell, ensuring that forces are balanced. The simulations show that the sorting patterns of prestalk and prespore cells, emerging during the slug stage, depend critically on the type of cell adhesion and not just on chemotactic differences between cells. This occurs because cell size and stiffness can prevent the otherwise faster cells from passing the slower cells. The patterns are distinctively different when the prestalk cells are more or less adhesive than the prespore cells. These simulations suggest that sorting is not solely due to differential chemotaxis, and that differences in both adhesion strength and type between different cell types play a very significant role, both in Dictyostelium and other systems.     

Cofilin determines the migration behavior and turning frequency of metastatic cancer cells

We have investigated the effects of inhibiting the expression of cofilin to understand its role in protrusion dynamics in metastatic tumor cells, in particular. We show that the suppression of cofilin expression in MTLn3 cells (an apolar randomly moving amoeboid metastatic tumor cell) caused them to extend protrusions from only one pole, elongate, and move rectilinearly. This remarkable transformation was correlated with slower extension of fewer, more stable lamellipodia leading to a reduced turning frequency. Hence, the loss of cofilin caused an amoeboid tumor cell to assume a mesenchymal-type mode of movement. These phenotypes were correlated with the loss of uniform chemotactic sensitivity of the cell surface to EGF stimulation, demonstrating that to chemotax efficiently, a cell must be able to respond to chemotactic stimulation at any region on its surface. The changes in cell shape, directional migration, and turning frequency were related to the re-localization of Arp2/3 complex to one pole of the cell upon suppression of cofilin expression.     

Apolar and polar transitions drive the conversion between amoeboid and mesenchymal shapes in melanoma cells

Melanoma cells can adopt two functionally distinct forms, amoeboid and mesenchymal, which facilitates their ability to invade and colonize diverse environments during the metastatic process. Using quantitative imaging of single living tumor cells invading three-dimensional collagen matrices, in tandem with unsupervised computational analysis, we found that melanoma cells can switch between amoeboid and mesenchymal forms via two different routes in shape space—an apolar and polar route. We show that whereas particular Rho-family GTPases are required for the morphogenesis of amoeboid and mesenchymal forms, others are required for transitions via the apolar or polar route and not amoeboid or mesenchymal morphogenesis per se. Altering the transition rates between particular routes by depleting Rho-family GTPases can change the morphological heterogeneity of cell populations. The apolar and polar routes may have evolved in order to facilitate conversion between amoeboid and mesenchymal forms, as cells are either searching for, or attracted to, particular migratory cues, respectively.     

Multiscale modelling of vascular tumour growth in 3D: the roles of domain size and boundary conditions

We investigate a three-dimensional multiscale model of vascular tumour growth, which couples blood flow, angiogenesis, vascular remodelling, nutrient/growth factor transport, movement of, and interactions between, normal and tumour cells, and nutrient-dependent cell cycle dynamics within each cell. In particular, we determine how the domain size, aspect ratio and initial vascular network influence the tumour's growth dynamics and its long-time composition. We establish whether it is possible to extrapolate simulation results obtained for small domains to larger ones, by constructing a large simulation domain from a number of identical subdomains, each subsystem initially comprising two parallel parent vessels, with associated cells and diffusible substances. We find that the subsystem is not representative of the full domain and conclude that, for this initial vessel geometry, interactions between adjacent subsystems contribute to the overall growth dynamics. We then show that extrapolation of results from a small subdomain to a larger domain can only be made if the subdomain is sufficiently large and is initialised with a sufficiently complex vascular network. Motivated by these results, we perform simulations to investigate the tumour's response to therapy and show that the probability of tumour elimination in a larger domain can be extrapolated from simulation results on a smaller domain. Finally, we demonstrate how our model may be combined with experimental data, to predict the spatio-temporal evolution of a vascular tumour.     

Cytoplasm dynamics and cell motion: two-phase flow models

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

Combined effects of pulsatile flow and dynamic curvature on wall shear stress in a coronary artery bifurcation model

A three-dimensional model with simplified geometry for the branched coronary artery is presented. The bifurcation is defined by an analytical intersection of two cylindrical tubes lying on a sphere that represents an idealized heart surface. The model takes into account the repetitive variation of curvature and motion to which the vessel is subject during each cardiac cycle, and also includes the phase difference between arterial motion and blood flowrate, which may be nonzero for patients with pathologies such as aortic regurgitation. An arbitrary Lagrangian Eulerian (ALE) formulation of the unsteady, incompressible, three-dimensional Navier-Stokes equations is employed to solve for the flow field, and numerical simulations are performed using the spectral/hp element method. The results indicate that the combined effect of pulsatile inflow and dynamic geometry depends strongly on the aforementioned phase difference. Specifically, the main findings of this work show that the time-variation of flowrate ratio between the two branches is minimal (less than 5%) for the simulation with phase difference angle equal to 90 degrees, and maximal (51%) for 270 degrees. In two flow pulsatile simulation cases for fixed geometry and dynamic geometry with phase angle 270 degrees, there is a local minimum of the normalized wall shear rate amplitude in the vicinity of the bifurcation, while in other simulations a local maximum is observed.     

Activated membrane patches guide chemotactic cell motility

Many eukaryotic cells are able to crawl on surfaces and guide their motility based on environmental cues. These cues are interpreted by signaling systems which couple to cell mechanics; indeed membrane protrusions in crawling cells are often accompanied by activated membrane patches, which are localized areas of increased concentration of one or more signaling components. To determine how these patches are related to cell motion, we examine the spatial localization of RasGTP in chemotaxing Dictyostelium discoideum cells under conditions where the vertical extent of the cell was restricted. Quantitative analyses of the data reveal a high degree of spatial correlation between patches of activated Ras and membrane protrusions. Based on these findings, we formulate a model for amoeboid cell motion that consists of two coupled modules. The first module utilizes a recently developed two-component reaction diffusion model that generates transient and localized areas of elevated concentration of one of the components along the membrane. The activated patches determine the location of membrane protrusions (and overall cell motion) that are computed in the second module, which also takes into account the cortical tension and the availability of protrusion resources. We show that our model is able to produce realistic amoeboid-like motion and that our numerical results are consistent with experimentally observed pseudopod dynamics. Specifically, we show that the commonly observed splitting of pseudopods can result directly from the dynamics of the signaling patches.     

A 'chemotactic dipole' mechanism for large-scale vortex motion during primitive streak formation in the chick embryo

Primitive streak formation in the chick embryo involves significant coordinated cell movement lateral to the streak, in addition to the posterior-anterior movement of cells in the streak proper. Cells lateral to the streak are observed to undergo 'polonaise movements', i.e. two large counter-rotating vortices, reminiscent of eddies in a fluid. In this paper, we propose a mechanism for these movement patterns which relies on chemotactic signals emitted by a dipolar configuration of cells in the posterior region of the epiblast. The 'chemotactic dipole' consists of adjacent regions of cells emitting chemo-attractants and chemo-repellents. We motivate this idea using a mathematical analogy between chemotaxis and electrostatics, and test this idea using large-scale computer simulations. We implement active cell response to both neighboring mechanical interactions and chemotactic gradients using the Subcellular Element Model. Simulations show the emergence of large-scale vortices of cell movement. The length and time scales of vortex formation are in reasonable agreement with experimental data. We also provide quantitative estimates for the robustness of the chemotaxis dipole mechanism, which indicate that the mechanism has an error tolerance of about 10% to variation in chemotactic parameters, assuming that only 1% of the cell population is involved in emitting signals. This tolerance increases for larger populations of cells emitting signals.     

Up-regulation of Rho/ROCK signaling in sarcoma cells drives invasion and increased generation of protrusive forces

Tumor cell invasion is the most critical step of metastasis. Determination of the mode of invasion within the particular tumor is critical for effective cancer treatment. Protease-independent amoeboid mode of invasion has been described in carcinoma cells and more recently in sarcoma cells on treatment with protease inhibitors. To analyze invasive behavior, we compared highly metastatic sarcoma cells with parental nonmetastatic cells. The metastatic cells exhibited a functional up-regulation of Rho/ROCK signaling and, similarly to carcinoma cells, an amoeboid mode of invasion. Using confocal and traction force microscopy, we showed that an up-regulation of Rho/ROCK signaling leads to increased cytoskeletal dynamics, myosin light chain localization, and increased tractions at the leading edge of the cells and that all of these contributed to increased cell invasiveness in a three-dimensional collagen matrix. We conclude that cells of mesenchymal origin can use the amoeboid nonmesenchymal mode of invasion as their primary invading mechanism and show the dependence of ROCK-mediated amoeboid mode of invasion on the increased capacity of cells to generate force.     

Actin polymerization and pseudopod extension during amoeboid chemotaxis

Amoebae of the cellular slime mold Dictyostelium discoideum are an excellent model system for the study of amoeboid chemotaxis. These cells can be studied as a homogeneous population whose response to chemotactic stimulation is sufficiently synchronous to permit the correlation of the changes in cell shape and biochemical events during chemotaxis. Having demonstrated this synchrony of response, we show that actin polymerization occurs in two stages during stimulation with chemoattractants. The assembly of F-actin that peaks between 40 and 60 sec after the onset of stimulation is temporally correlated with the growth of new pseudopods. F-actin, which is assembled by 60 sec after stimulation begins, is localized in the new pseudopods that are extended at this time. Both stages of actin polymerization during chemotactic stimulation involve polymerization at the barbed ends of actin filaments based on the cytochalasin sensitivity of this response. We present a hypothesis in which actin polymerization is one of the major driving forces for pseudopod extension during chemotaxis. The predictions of this model, that localized regulation of actin nucleation activity and actin filament cross-linking must occur, are discussed in the context of current models for signal transduction and of recent information regarding the types of actin-binding proteins that are present in the cell cortex.     

Three-Dimensional Balance of Cortical Tension and Axial Contractility Enables Fast Amoeboid Migration

Fast amoeboid migration requires cells to apply mechanical forces on their surroundings via transient adhesions. However, the role these forces play in controlling cell migration speed remains largely unknown. We used three-dimensional force microscopy to measure the three-dimensional forces exerted by chemotaxing Dictyostelium cells, and examined wild-type cells as well as mutants with defects in contractility, internal F-actin crosslinking, and cortical integrity. We showed that cells pull on their substrate adhesions using two distinct, yet interconnected mechanisms: axial actomyosin contractility and cortical tension. We found that the migration speed increases when axial contractility overcomes cortical tension to produce the cell shape changes needed for locomotion. We demonstrated that the three-dimensional pulling forces generated by both mechanisms are internally balanced by an increase in cytoplasmic pressure that allows cells to push on their substrate without adhering to it, and which may be relevant for amoeboid migration in complex three-dimensional environments.     

DOCK10-mediated Cdc42 activation is necessary for amoeboid invasion of melanoma cells

BACKGROUND: Tumor cells can move in a three-dimensional (3D) environment in either mesenchymal-type or amoeboid modes. In mesenchymal-type movement, cells have an elongated morphology with Rac-induced protrusions at the leading edge. Amoeboid cells have high levels of actomyosin contractility, and movement is associated with deformation of the cell body through the matrix without proteolysis. Because signaling pathways that control the activation of GTPases for amoeboid movement are poorly understood, we sought to identify regulators of amoeboid movement by screening an siRNA library targeting guanine nucleotide exchange factors (GEFs) for Rho-family GTPases. RESULTS: We identified DOCK10, a Cdc42 GEF, as a key player in amoeboid migration; accordingly, we find that expression of activated Cdc42 induces a mesenchymal-amoeboid transition and increases cell invasion. Silencing DOCK10 expression promotes conversion to mesenchymal migration and is associated with decreased MLC2 phosphorylation and increased Rac1 activation. Consequently, abrogating DOCK10 and Rac1 expression suppresses both amoeboid and mesenchymal migration and results in decreased invasion. We show that the Cdc42 effectors N-WASP and Pak2 are required for the maintenance of the rounded-amoeboid phenotype. Blocking Cdc42 results in loss of mesenchymal morphology, arguing that Cdc42 is also involved in mesenchymal morphology through different activation and effector pathways. CONCLUSIONS: Previous work has identified roles of Rho and Rac signaling in tumor cell movement, and we now elucidate novel roles of Cdc42 signaling in amoeboid and mesenchymal movement and tumor cell invasion.     

Contact response of cells can mediate morphogenetic pattern formation

Theories of morphogenetic pattern formation have included Turing's chemical prepatterns, mechanochemical interactions, cell sorting, and other mechanisms involving guided motion or signalling of cells. Many of these theories presuppose long-range cellular communication or other controls such as chemical concentration fields. However, the possibility that direct interactions between cells can lead to order and structure has not been seriously investigated in mathematical models. In this paper we consider this possibility, with emphasis on cells that reorient and align with each other when they come into contact. We show that such contact responses can account for the formation of multicellular patterns called parallel arrays. These patterns typically occur in tissue cultures of fibroblasts, and consist of clusters of cells sharing a common axis of orientation. Using predictions of a mathematical model and computer simulations of cell motion and interactions we show that contact responses alone, in the absence of other global controls, can promote the formation of these patterns. We suggest other situations in which patterns may result from direct cellular communication. Previous theories of morphogenesis are briefly reviewed and compared with this proposed mechanism.     

Computational fluid dynamics of aggregating red blood cells in postcapillary venules

Aggregate formation of red blood cells (RBCs) in a postcapillary venular bifurcation is investigated with three-dimensional computer simulations using the Chimera grid method. Interaction energy between the RBCs is modelled by a depletion interaction theory; RBCs are modelled as rigid oblate ellipsoids. The cell–cell interactions of RBCs are strongly dependent on vessel geometry and shear rates. The experimental data on vessel geometry, pseudoshear rates, and Dextran concentration obtained in our previous in vivo RBC aggregation study in postcapillary venules of the rat spinotrapezius muscle were used to simulate RBC aggregation. The computational results were compared to the experimental results from the in vivo study. The results show that cells have a larger tendency to form an aggregate under reduced flows. Aggregate formation also depends on the angle and location of the cells before they enter the bifurcation region. Comparisons with experimental data are discussed.     

A new interpretation of the Keller-Segel model based on multiphase modelling

In this paper an alternative derivation and interpretation are presented of the classical Keller-Segel model of cell migration due to random motion and chemotaxis. A multiphase modelling approach is used to describe how a population of cells moves through a fluid containing a diffusible chemical to which the cells are attracted. The cells and fluid are viewed as distinct components of a two-phase mixture. The principles of mass and momentum balance are applied to each phase, and appropriate constitutive laws imposed to close the resulting equations. A key assumption here is that the stress in the cell phase is influenced by the concentration of the diffusible chemical. By restricting attention to one-dimensional cartesian geometry we show how the model reduces to a pair of nonlinear coupled partial differential equations for the cell density and the chemical concentration. These equations may be written in the form of the Patlak-Keller-Segel model, naturally including density-dependent nonlinearities in the cell motility coefficients. There is a direct relationship between the random motility and chemotaxis coefficients, both depending in an inter-related manner on the chemical concentration. We suggest that this may explain why many chemicals appear to stimulate both chemotactic and chemokinetic responses in cell populations. After specialising our model to describe slime mold we then show how the functional form of the chemical potential that drives cell locomotion influences the ability of the system to generate spatial patterns. The paper concludes with a summary of the key results and a discussion of avenues for future research.     

A cell-based framework for modeling cardiac mechanics

Cardiomyocytes are the functional building blocks of the heart—yet most models developed to simulate cardiac mechanics do not represent the individual cells and their surrounding matrix. Instead, they work on a homogenized tissue level, assuming that cellular and subcellular structures and processes scale uniformly. Here we present a mathematical and numerical framework for exploring tissue-level cardiac mechanics on a microscale given an explicit three-dimensional geometrical representation of cells embedded in a matrix. We defined a mathematical model over such a geometry and parametrized our model using publicly available data from tissue stretching and shearing experiments. We then used the model to explore mechanical differences between the extracellular and the intracellular space. Through sensitivity analysis, we found the stiffness in the extracellular matrix to be most important for the intracellular stress values under contraction. Strain and stress values were observed to follow a normal-tangential pattern concentrated along the membrane, with substantial spatial variations both under contraction and stretching. We also examined how it scales to larger size simulations, considering multicellular domains. Our work extends existing continuum models, providing a new geometrical-based framework for exploring complex cell–cell and cell–matrix interactions.

     

Binding and diffusion of CheR molecules within a cluster of membrane receptors

Adaptation of the attractant response in Escherichia coli is attributable to the methylation of its transmembrane chemotactic receptors by the methyltransferase CheR. This protein contains two binding domains, one for the sites of methylation themselves and the other for a flexible tether at the C terminus of the receptor. We have explored the theoretical consequences of this binding geometry for a CheR molecule associated with a cluster of chemotactic receptors. Calculations show that the CheR molecule will bind with high net affinity to the receptor lattice, having a high probability of being attached by one or both of its domains at any instant of time. Because of the relatively low affinity of its individual domains and the close proximity of neighboring receptors, it is likely that when one domain unbinds it will reattach to the array before the other domain unbinds. Stochastic simulations show that the enzyme will move through the receptor cluster in a hand-over-hand fashion, like a gibbon swinging through the branches of a tree. We explore the possible consequences of this motion, which we term "molecular brachiation", for chemotactic adaptation and suggest that a similar mechanism may be operative in other large assemblies of protein molecules.     

CompuCell3D Simulations Reproduce Mesenchymal Cell Migration on Flat Substrates

Mesenchymal cell crawling is a critical process in normal development, in tissue function, and in many diseases. Quantitatively predictive numerical simulations of cell crawling thus have multiple scientific, medical, and technological applications. However, we still lack a low-computational-cost approach to simulate mesenchymal three-dimensional (3D) cell crawling. Here, we develop a computationally tractable 3D model (implemented as a simulation in the CompuCell3D simulation environment) of mesenchymal cells crawling on a two-dimensional substrate. The Fürth equation, the usual characterization of mean-squared displacement (MSD) curves for migrating cells, describes a motion in which, for increasing time intervals, cell movement transitions from a ballistic to a diffusive regime. Recent experiments have shown that for very short time intervals, cells exhibit an additional fast diffusive regime. Our simulations’ MSD curves reproduce the three experimentally observed temporal regimes, with fast diffusion for short time intervals, slow diffusion for long time intervals, and intermediate time -interval-ballistic motion. The resulting parameterization of the trajectories for both experiments and simulations allows the definition of time- and length scales that translate between computational and laboratory units. Rescaling by these scales allows direct quantitative comparisons among MSD curves and between velocity autocorrelation functions from experiments and simulations. Although our simulations replicate experimentally observed spontaneous symmetry breaking, short-timescale diffusive motion, and spontaneous cell-motion reorientation, their computational cost is low, allowing their use in multiscale virtual-tissue simulations. Comparisons between experimental and simulated cell motion support the hypothesis that short-time actomyosin dynamics affects longer-time cell motility. The success of the base cell-migration simulation model suggests its future application in more complex situations, including chemotaxis, migration through complex 3D matrices, and collective cell motion.     

Simulating T-cell motility in the lymph node paracortex with a packed lattice geometry

Agent-based simulation modelling of T-cell trafficking, activation and proliferation in the lymph node paracortex requires a model for cell motility. Such a model must be able to reproduce the observed random-walk behaviour of T cells, while accommodating large numbers of tightly packed cells, and must be computationally efficient. We report the development of a motility model, based on a three-dimensional lattice geometry, that meets these objectives. Cells make discrete jumps between neighbouring lattice sites in directions that are randomly determined from specified discrete probability distributions, which are defined by a small number of parameters. It is shown that the main characteristics of the random motion of T cells as typically observed in vivo can be reproduced by suitable specification of model parameters. The model is computationally highly efficient and provides a suitable engine for a model capable of simulating the full T-cell population of the paracortex.