Schedule for: 21w5225 - Modeling and Computational Approaches to Individual and Collective Cell Movement in Complex Environments (Online)

Dr. Othmer and Dr. Hillen will welcome the participants and set the stage for the rest of the conference. Common issues about cell motility will be discussed and methods that are used during the conference will be outlined.

Over the past decade, in an interdisciplinary collaboration between Philip Maini and colleagues in the Wolfson Centre for Mathematical Biology and the Kulesa lab at The Stowers Institute for Medical Research, we have developed a suite of cell-based models in parallel with experiments to investigate the underlying dynamics of collective cell migration. We focus on the highly invasive neural crest cells that in the vertebrate embryo migrate in discrete streams along stereotypical pathways. In this talk, we will review this work and show how it has led to new insights into the underlying biology. Specifically, it will be shown how modelling, combined with experiments, led us to the identification of different cell phenotypes and phenotypic switching, as well as generating hypotheses on how cells may be assembling and deforming the extracellular matrix through which they migrate, and how they may be signalling to each other.

Over the past decade, in an interdisciplinary collaboration between Philip Maini and colleagues in the Wolfson Centre for Mathematical Biology and the Kulesa lab at The Stowers Institute for Medical Research, we have developed a suite of cell-based models in parallel with experiments to investigate the underlying dynamics of collective cell migration. We focus on the highly invasive neural crest cells that in the vertebrate embryo migrate in discrete streams along stereotypical pathways. In this talk, we will review this work and show how it has led to new insights into the underlying biology. Specifically, it will be shown how modelling, combined with experiments, led us to the identification of different cell phenotypes and phenotypic switching, as well as generating hypotheses on how cells may be assembling and deforming the extracellular matrix through which they migrate, and how they may be signalling to each other.

Cells can be structurally anisotropic, and they can be heterogeneous due to either genetic or environment clues. Cellular anisotropy and heterogeneity might lead to interesting behaviors in individual or collective cell movement. In this talk we will discuss the roles of cellular anisotropy and heterogeneity in two systems. In the first part, we will discuss how anisotropic flagella bending rigidity affects the flagellar beating dynamics. Flagellar beating is controlled by molecular motors that exert forces along the length of the flagellum and are regulated by a feedback mechanism coupled to the flagellar motion. We build on previous work on sliding-controlled motor feedback to develop a fully three-dimensional description of flagellar beating, accounting for both bending and twist. We show that with isotropic bending, three-dimensional spiral modes are spontaneously generated beyond a critical molecular activity. On the other hand, when a difference is introduced into the bending rigidity along orthogonal directions, a preferential bending plane is established, and we find that the generic three-dimensional spiral modes give way to planar beating along the soft axis as the difference in bending rigidity increases. In the second part, we will discuss how hair follicle heterogeneous responses to signals affect the cell flows which then regulate the follicle temporal growth dynamics. Hair follicles are mini skin organs rich of stem cells, and they undergo cyclic growth. The growing phase – anagen of a hair follicle is tightly controlled by a group of epithelial transient amplifying (TA) cells. Using an interdisciplinary approach combined of multi-scale modeling and lineage tracing experiments, we show that cellular heterogeneity based on cell division generations drive the upward cell flows, which guarantees the refill of the follicle TA cells that prolongs the anagen.

During embryonic development, cells often migrate collectively. The cardiogenic progenitors of the ascidian Ciona provide one of the simplest examples of collective migration: two cells migrate with defined leader-trailer polarity, squeezed between stiff epidermis and deformable endoderm. The cells are also capable of migrating individually, in a less persistent way. Two cells with upregulated protrusion migrate side by side. When more than two cells migrate together, they form a single file. We developed three computational models (Cellular Potts, vortex and 'bubble' models) that capture various aspects of the cell collective movements and shed light on mechanical constraints of the migrating cell pair.

In this talk I will discuss two models of cell motion. One assumes cells are ellipsoid and the other model makes no assumption about cell shape and focuses on cell adhesions. In the second model random switching terms are used to model the attachment and detachment of adhesions. In the first model formation the focus is on force and force transmission. In the second model the focus shifts to the dynamics of the adhesions and how they affect the cell speed.

The active response of cells to mechanical cues due to their interaction with the environment has been of increasing interest, since it is involved in many physiological phenomena, pathologies, and in tissue engineering. In particular, several experiments have shown that, if a substrate with overlying cells is cyclically stretched, they will reorient with their main axis either perpendicular or at an oblique angle with respect to the main stretching direction. In the first part of the seminar, held by Luigi Preziosi, the phenomenon is studied from the deterministic point of view working in the framework of continuum mechanics. First a nonlinear elastic energy for a quite general orthotropic material is used and a complete bifurcation analysis is performed to explain the dependence of the reorientation angle on the applied strain. Then, a linear viscoelastic model is proposed to describe the dependence on the applied frequency. In the second part of the seminar, held by Nadia Loy, stochastic effects are considered. In fact, in many cases results are given in terms of the percentage of cells having an orientation in certain intervals. With the aim of describing both the evolution and the stationary state of the probability density function over cell orientations, Fokker-Planck equations are deduced starting from microscopic rules. connected with the continuum mechanics models previously introduced. In addition, we introduce a way of describing the microscopic re-orientation rule as a result of an optimal control internally activated by the cell. The results of both models compare very well with experimental results.

The active response of cells to mechanical cues due to their interaction with the environment has been of increasing interest, since it is involved in many physiological phenomena, pathologies, and in tissue engineering. In particular, several experiments have shown that, if a substrate with overlying cells is cyclically stretched, they will reorient with their main axis either perpendicular or at an oblique angle with respect to the main stretching direction. In the first part of the seminar, held by Luigi Preziosi, the phenomenon is studied from the deterministic point of view working in the framework of continuum mechanics. First a nonlinear elastic energy for a quite general orthotropic material is used and a complete bifurcation analysis is performed to explain the dependence of the reorientation angle on the applied strain. Then, a linear viscoelastic model is proposed to describe the dependence on the applied frequency. In the second part of the seminar, held by Nadia Loy, stochastic effects are considered. In fact, in many cases results are given in terms of the percentage of cells having an orientation in certain intervals. With the aim of describing both the evolution and the stationary state of the probability density function over cell orientations, Fokker-Planck equations are deduced starting from microscopic rules. connected with the continuum mechanics models previously introduced. In addition, we introduce a way of describing the microscopic re-orientation rule as a result of an optimal control internally activated by the cell. The results of both models compare very well with experimental results.

The submembranous actin cortex is the main determinant of cell shape. During mitosis and migration, spatiotemporal changes in cortex mechanics give rise to shape changes. These result from tightly orchestrated global and local changes in RhoGTPase activity regulated by recruitment of RhoGEFs and RhoGAPs to the cortex. Yet, little is known about how signalling controls cell mechanics to drive shape change. I will present work investigating how signalling controls cell mechanics. We use optogenetics to control the activity of RhoGTPases by relocalising a RhoGEF to the cortex and investigate the resulting temporal changes in surface tension using AFM. I will discuss how to coarse-grain signalling downstream of RhoGTPases to link signalling to mechanics and shape change.

Beneath the membrane of many cells lies the cell cortex, a composite layer of actin, myosin, and various cross-linking proteins. The cell cortex is believed to have a strong influence on the ability of a cell to move about in its enviroment, and in turn, cell motility plays an important role in cancer metastasis and in many developmental processes. Cells can employ various strategies to move, such as crawling or swimming, and here I will discuss recent modeling and computational results on how the forces, applied externally or generated internally by the cellular cortex, in conjunction with the mechanical properties of the cell can lead to various shape changes and cell motion.

In the presence of sparse attractants, the movement of both cells and large organisms has been shown to be governed by long distance runs, according to an approximate Levy distribution. In this talk we clarify the form of biologically relevant PDE descriptions for such movements. Motivated by experiments we consider a microscopic velocity-jump model in which the motion of the individuals is characterized by long runs and long waiting times, according to a heavy-tailed distribution. Furthermore, this nonlocal movement of individuals has been observed in more complex geometries, e.g., the brain. We propose to study the (nonlocal) diffusion using a network of subdomains, corresponding to the nodes of a graph. I will introduce metaplex networks which are networks with internal structure, and we will extend our analysis to two real world examples: a brain and a landscape network.

We introduce a hybrid two-dimensional multiscale model of angiogenesis, the process by which endothelial cells (ECs) migrate from a pre-existing vascular bed in response to local environmental cues and cell-cell interactions, to create a new vascular network. Recent experimental studies have highlighted the central role of cell rearrangements in the formation of angiogenic networks. Our model accounts for this phenomenon via the heterogeneous response of ECs to their microenvironment. These cell rearrangements, in turn, dynamically remodel the local environment. The model reproduces characteristic features of angiogenic sprouting that include branching, chemotactic sensitivity, the brush border effect, and cell mixing. These properties, rather than being hardwired into the model, emerge naturally from the gene expression patterns of individual cells. After calibrating and validating our model against experimental data, we use it to predict how the structure of the vascular network changes as the baseline gene expression levels of the VEGF-Delta-Notch pathway, and the composition of the extracellular environment, vary. In order to investigate the impact of cell rearrangements on the vascular network structure, we introduce the mixing measure, a scalar metric that quantifies cell mixing as the vascular network grows. We calculate the mixing measure for the simulated vascular networks generated by ECs of different lineages (wild-type cells and mutant cells with impaired expression of a specific receptor). Our results show that the time evolution of the mixing measure is directly correlated to the generic features of the vascular branching pattern, thus, supporting the hypothesis that cell rearrangements play an essential role in sprouting angiogenesis. Furthermore, we predict that lower cell rearrangement leads to an imbalance between branching and sprout elongation. Since the computation of this statistic requires only individual cell trajectories, it can be computed for networks generated in biological experiments, making it a potential biomarker for pathological angiogenesis.

We introduce a hybrid two-dimensional multiscale model of angiogenesis, the process by which endothelial cells (ECs) migrate from a pre-existing vascular bed in response to local environmental cues and cell-cell interactions, to create a new vascular network. Recent experimental studies have highlighted the central role of cell rearrangements in the formation of angiogenic networks. Our model accounts for this phenomenon via the heterogeneous response of ECs to their microenvironment. These cell rearrangements, in turn, dynamically remodel the local environment. The model reproduces characteristic features of angiogenic sprouting that include branching, chemotactic sensitivity, the brush border effect, and cell mixing. These properties, rather than being hardwired into the model, emerge naturally from the gene expression patterns of individual cells. After calibrating and validating our model against experimental data, we use it to predict how the structure of the vascular network changes as the baseline gene expression levels of the VEGF-Delta-Notch pathway, and the composition of the extracellular environment, vary. In order to investigate the impact of cell rearrangements on the vascular network structure, we introduce the mixing measure, a scalar metric that quantifies cell mixing as the vascular network grows. We calculate the mixing measure for the simulated vascular networks generated by ECs of different lineages (wild-type cells and mutant cells with impaired expression of a specific receptor). Our results show that the time evolution of the mixing measure is directly correlated to the generic features of the vascular branching pattern, thus, supporting the hypothesis that cell rearrangements play an essential role in sprouting angiogenesis. Furthermore, we predict that lower cell rearrangement leads to an imbalance between branching and sprout elongation. Since the computation of this statistic requires only individual cell trajectories, it can be computed for networks generated in biological experiments, making it a potential biomarker for pathological angiogenesis.

For cells to cooperate in healing a wound or work together to follow a signal, they must coordinate their motion. One stereotyped behavior found in many cell types is "contact inhibition of locomotion" (CIL), in which cells that collide with one another repolarize away from contact. Experiments studying CIL are often performed on flat rigid two-dimensional substrates, unlike the natural fibrous environment of many cells in vivo. How does extracellular matrix geometry and adhesivity affect CIL? First, I will talk about recent experiments by our collaborators in the Nain group, which show that when cells are attached to single suspended nanofibers, the outcomes of CIL can be radically different, with cells walking past each other. Our modeling shows that this likely arises from the additional degrees of freedom that cells have to rotate around the fiber, and can be abolished by forcing cells to attach to two fibers. I will also discuss more recent modeling on how cell-cell collisions can be moderated by the geometry of the cell-substrate contact angle.

Border cell migration in the Drosophila ovary is a relatively simple model for the study of collective, cooperative, cell-on cell migration in vivo that is amenable to live imaging, genetic and optogenetic approaches. Decades of work have revealed the secreted signals that govern which 6 of the 850 epithelial cells acquire the ability to migrate, when during development they do so, and where they go. In addition to biochemical signals, moving cells also sense and respond to physical features of the microenvironment; however, the significance of tissue topography was unknown. We used Drosophila border cells to study how chemical and physical information influences path selection. Although chemical cues were thought to be sufficient, live imaging, genetics, modeling, and simulations show that microtopography is also important. Chemoattractants promote predominantly posterior movement, whereas tissue architecture presents orthogonal information, a path of least resistance concentrated near the center of the egg chamber. E-cadherin supplies a permissive haptotactic cue. Our results provide insight into how cells integrate and prioritize topographical, adhesive, and chemoattractant cues to choose one path among many. New findings on the role of septin proteins in border cell morphology and migration will also be presented.

In collective navigation, a population travels as a group from an origin to a destination. Famous examples include the migrations of birds, between their winter and summer grounds, but collective movements also extend down to microorganisms and cell populations. Collective navigation is believed to improve the efficiency of migration, for example through the presence of more knowledgeable individuals that guide naive members ("leader-follower behaviour") or through the averaging out of individual undertainty ("many wrongs"). In this talk I will describe individual and continuous approaches for modelling collective navigation. The individual based model is predicated on a random walk model, where individuals supplement their own inherent guidance information with information acquired from other group members. The continuous model is based on a nonlocal hyperbolic PDE system. We investigate the point at which group information becomes beneficial to migration and how it can help a population navigate through "information voids", i.e. areas with negligible guidance information. We also explore the effectiveness of different modes through which a leader can herd a group of naïve followers.

Mechanical and mechanochemical models of pattern formation in biological tissues have been used to study a variety of biomedical systems, particularly in developmental biology, and describe the physical interactions between cells and their local surroundings. These models in their original form consist of a balance equation for the cell density, a balance equation for the density of the extracellular matrix (ECM), and a force-balance equation describing the mechanical equilibrium of the cell-ECM system. In these models, the stress-strain relation of the ECM is often described using the Kelvin-Voigt model of linear viscoelasticity. However, due to the multifaceted bio-physical nature of the ECM constituents, there are rheological aspects that cannot be effectively captured by this model and, therefore, depending on the pattern formation process and the type of biological tissue considered, other constitutive models of linear viscoelasticity may be better Suited. (Co-authors: Chiara Villa and Tommaso Lorenzi.)

Mechanical and mechanochemical models of pattern formation in biological tissues have been used to study a variety of biomedical systems, particularly in developmental biology, and describe the physical interactions between cells and their local surroundings. These models in their original form consist of a balance equation for the cell density, a balance equation for the density of the extracellular matrix (ECM), and a force-balance equation describing the mechanical equilibrium of the cell-ECM system. In these models, the stress-strain relation of the ECM is often described using the Kelvin-Voigt model of linear viscoelasticity. However, due to the multifaceted bio-physical nature of the ECM constituents, there are rheological aspects that cannot be effectively captured by this model and, therefore, depending on the pattern formation process and the type of biological tissue considered, other constitutive models of linear viscoelasticity may be better suited. Co-authors: Chiara Villa and Tommaso Lorenzi.

Cells migrating in vivo can encounter microenvironments with varying physical properties. One such physical variable is the viscosity of the fluid surrounding the cell. Increased fluid viscosity is expected to increase the hydraulic resistance experienced by the migrating cell and therefore decrease the cell speed. We demonstrate that contrary to this expected result, cells migrate faster in high viscosity media on 2D substrates. To reveal the molecular mechanism, we examined both actin dynamics and water dynamics driven by ion channel activity. Results show that cells increased in area in high viscosity and actomyosin dynamics remained similar, except that actin retrograde flow speed is reduced. Inhibiting ion channel fluxes in high viscosity media results in a large reduction in cell speed, suggesting that water flux contributes to the observed speed increase. Moreover, inhibiting actin-dependent vesicular trafficking that transports ion channels from the ER to the cell boundary changes ion channel spatial positioning and reduces cell speed in high viscosity media. Cells also displayed altered Ca2+-activity in high viscosity media, and when cytoplasmic Ca2+ is sequestered, cell speed reduction and altered ion channel positioning were observed. Taken together, we find that the cell cytoplasmic actin-phase and water-phase are coupled during cell migration in high viscosity media. Directional water fluxes are mediated by ion channels whose position depend on actin-based vesicular trafficking. These results, together with observed cell migration behavior in micro channels, suggest that hydraulic resistance and local hydraulic pressure are important mechanical variables governing cell polarization and influence cell migration speed. A physical 2-phase model of cell migration incorporating actin and water dynamics is presented to explain the experimental results.

Collective cell migration plays an important role in many processes including in the cohesion of epithelial cell monolayers, in wound healing, and in embryonic development. In tissues with multiple cell types, such as differentiating stem cells that spread in a single layer or epithelial and mesenchymal cells that spread in stratified layers, continuum mechanical models may be used to understand the mechanisms involved. We develop PDE models to examine the spread and maturation of astrocytes in retinal development and the spread of multi-layer embryonic tissue explants in gastrulation, and we compare numerical simulations to experimental data to decipher the spatiotemporal distribution of cell types.

The origins of disordered actomyosin network contraction such as in the cellular cortex remain an active topic of research. We derive a mathematical model for the evolution of two-dimensional networks. A major advantage of our approach is that it enables direct calculation of the network stress tensor, which provides a quantitative measure of contractility. Exploiting this, we use repeated simulations of disordered networks to confirm that both protein friction and actin filament bending are required for contraction. We also show that actin filament turnover is necessary to sustain contraction and prevent pattern formation. We then consider a toy-model version of the model for only two filaments immersed in an actomyosin network. Using asymptotic analysis numerical simulation of the resulting PDE and numerical solutions, we find that bending facilitates contraction by inducing a geometric asymmetry that enables motors to move faster close to filament plus-ends, inhibiting expansion.

In fast moving cells such as amoeba and immune cells, dendritic actin filaments are spatio-temporally regulated to shape large-scale plasma membrane protrusions. Through quantitative image analysis of Dictyostelium on micro-fabricated surfaces, we show that there is a distinct mode of topographical guidance directed by the macropinocytic membrane cup. Unlike other topographic guidance known to date that depends on nanometer-scale curvature sensing protein or stress fibers, the macropinocytic membrane cup is driven by the Ras/PI3K/F-actin signaling patch and its dependency on the micrometer-scale topographic features; namely PI3K/F-actin-independent accumulation of Ras-GTP at the convex curved surface, PI3K-dependent patch propagation along the convex edge and its actomyosin-dependent constriction at the concave edge. We will introduce a basic mathematical model of macropinocytic cup formation and closure and apply it to study this newly discovered mode of directed cell migration. Our simulations demonstrate that the topographically-dependent initiation in combination with the mutually-defining patch patterning and the membrane deformation gives rise to the topographical guidance. The results suggest that macropinocytic cup serves as a global surveyor of substrate topology. It is a self-enclosing structure that can support liquid ingestion by default, however in the presence of structured surfaces, it is directed to faithfully trace bent and bifurcating ridges for particle ingestion and cell guidance.

In this talk, I will present a hybrid individual cell-based mathematical and computational model, incorporating single-cell based intracellular dynamics, the cell microenvironment and cell-cell interactions to study the growth and progression of cancer cell mass. The modelling framework will then be used to study the effects of various anticancer therapies such as radiotherapy and chemotherapy, illustrating its adaptability and usefulness. Furthermore, we will see how this modelling framework can be used to understand the role of known mechanisms in driving treatment effects and while in some cases, how it can provide testable hypotheses on poorly understood concepts such as “radiation-induced bystander effects”.

Small GTPases, such as Rac and Rho, are well known central regulators of cell morphology and motility, whose dynamics also play a role in coordinating collective cell migration. Experiments have shown GTPase dynamics to be affected by both chemical and mechanical cues, but also to be spatially and temporally heterogeneous. This heterogeneity is found both within a single cell, and between cells in a tissue. For example, sometimes the leader and follower cells display an inverted GTPase configuration. While progress on understanding GTPase dynamics in single cells has been made, a major remaining challenge is to understand the role of GTPase heterogeneity in collective cell migration. Motivated by recent one-dimensional experiments (e.g. micro-channels) we introduce a one-dimensional modelling framework allowing us to integrate cell bio-mechanics, changes in cell size, and detailed intra-cellular signalling circuits (reaction-diffusion equations). Using this framework, we build cell migration models of both loose (mesenchymal) and cohering (epithelial) tissues. We use numerical simulations, and analysis tools, such as bifurcation analysis, to provide insights into the regulatory mechanisms coordinating collective cell migration. We show how local perturbations to GTPase signalling due to cell-cell interactions or tension lead to a variety of dynamics, resembling the behavior of small cell groups.

We discuss a mathematical model for the growth of cellulolytic biofilms. Cellulolytic biofilms play an important role in the production of cellulosic ethanol, a biofuel with large economic potential. Different from traditional models where the biofilm grows into the aqueous phase and nutrients are transported by diffusion, bacteria colonize, consume and degrade a cellulosic substratum that supports them. Hence, the nutrients are immobilized and modelled by an ODE. The ODE is coupled to a two-fold degenerate reaction diffusion equation for the biomass density that exhibits a polynomial degeneracy (as known from the porous medium equation) and a singularity as the biomass density approaches its maximum value (fast diffusion effect). We show the well-posedness of the model and prove the existence of travelling wave solutions. Invading fronts had been observed in biological experiments as well as in numerical simulations of the model. This is joint work with Hermann Eberl, Jack Hughes and Koondanibha Mitra.