Cell migration is an essential feature of, either physiologic or pathologic, phenomena in biology, such as embryonic development, wound healing or tumor invasion. According to the local microenvironment and the cell function, the characteristics of the migration may vary considerably.
Here we look closer at the influence of the cell density on the migration dynamics, and we assume two different regimes: when cells are isolated, the corresponding motion is essentially characterized by a sequence of "runs" separated by random reorientations of the velocity; in denser areas, migrating cells interact with other cells and "collision effects" become relevant.
Additionally a "resting" regime is included in the migration modelling. This can either result from environmental conditions or relates to a strategy of cells to fulfil efficiently their function. As an example, cells undergo mitosis only under favourable environmental conditions, and an immotile state is then required. A second illustration is the "Go or Grow" hypothesis currently accepted in the biology of brain tumor invasion.
A kinetic (mesoscopic) model is first derived and a continuous (macroscopic) model is deduced as its diffusive limit. This so-called "Go or Rest" model provides anomalous diffusion which is furthermore analyzed.
The model is then extended to include proliferating phenomena. The study of the invasive front will aim to understand heterogeneous patterns observed in tumor invasion.
Pollen is the male gametophyte of plants. From a dehydrated, quiescent organ in the atmosphere, pollen grains take up water from the female tissues in a matter of seconds, and in minutes develop a unique form of cellular outgrowth, the pollen tube. Pollen tubes are among the fastest growing cells in Nature, they achieve growth rates up to 4 um/sec, and in a matter of few hours can increase their volume up to 50 fold without division (1). Despite copious amounts of detailed physiological and molecular data, the mechanistic regulation of growth in pollen tubes still lacks an integrative model. While transcriptomics reveals the presence of about 7.000 genes, theoretical modeling shows that cooperation of all of these into two processes- wall surface and cytoplamic volume growth- is condition enough to generate apical growth as we know it. Spatial and temporal integration of extended biochemical and biophysical processes is mandatory, and in the past we have propose and demonstrated that "ion dynamics"- which we define specifically as the regulation of ion membrane fluxes and cytosolic free ion concentration- is maybe a common denominator of this integrative processes, and follow the behavior of a dynamical attractor on the space phase (2). In pollen tubes, ion fluxes are polarized, implying that carriers for protons, calcium, chloride and potassium shows non-linear patterns in space (polarized distribution of carriers) and also in time (oscillatory and chaotic behaviors) (3). We thus are systematically cloning and GFP expressing the putative genes involved in ion dynamics. Recently we cloned the first proton pump P-ATPase from tobacco pollen (Nt AHA- Nicotiana H+ ATPase), which shows clear correlation with the flux patterns: NHA is excluded from the apex membrane by mechanisms involving polarized translation, actin cytoskeleton and membrane recycling (4). Nt AHA gets polarized soon after hydration, and its absence is predictive of the germination pore. Information about fluxes and molecular underlying entities are now being integrated on intracellular diffusion models which should allow to test hypothesis about the adequacy of the stipulated molecular mechanism.
(1)- Int.J.Dev.Biol. 49:595-614, 2005; Int.J.Dev.Biol. 49:615-632, 2005; (2) BioEssays, 23:86-94, 2001 (3 ) Int.J.Dev.Biol., doi:10.1387/ijdb.072296em, 2008; Sex.Plant Reprod., doi:10.1007/s00497-008-0076-x, 2008 (4) Plant Cell, 20:614-634, 2008.
The chemotactic response in Escherichia coli bacteria is a well-characterized signal transduction mechanism, which has been cited as a useful model system for information processing in higher-order organisms. In this lecture I will briefly describe the molecular mechanism underlying chemotaxis and the run-and-tumble swimming behavior that it controls. The motivation behind our experimental work has been to build on this fundamental understanding of the underlying mechanisms to develop predictive models for bacterial migration in complex natural systems. Our focus has been on migration of chemotactic bacteria in porous media with application to remediation of polluted groundwater systems where factors such as fluid flow and restricted pore spaces alter the swimming trajectories. I will present a series of experimental approaches that range from tracking individual cells to imaging chemotactic bands in microfluidic devices to monitoring bacterial migration in a natural groundwater aquifer.
The capacity to organize cells into epithelial sheets is a defining feature of all metazoans, and the ability of cells to adhere and polarize is in turn central to nearly every aspect of organ morphogenesis and physiology. While the many benefits of epitheliality are clear, the corresponding constraints imposed on spatial relationships between cells remain poorly understood. In this talk I will describe a combination of experimental and theoretical approaches to understanding the effect of cell proliferation on the dynamic spatial relationships between cells in Drosophila epithelia.
During this workshop we will encounter various models in the form of partial differential equations (PDEs). These models include reaction-advection-diffusion equations, transport equations, and continuum mechanics equations.
All of these models can be motivated from stochastic processes, and this is what I plan to do in my tutorial lecture. I start from simple random walk equations and show how to derive diffusion models, transport models and continuum mechanics models. This should allow participants to put models from following lectures into context and to see relations between different approaches.
Many biological systems have the ability to sense the direction of external chemical sources and respond by polarizing and migrating toward chemoattractants or away from chemorepellants. This phenomenon, referred to as chemotaxis, is crucial for proper functioning of single cell organisms, such as bacteria and amoebae, as well as multi-cellular systems as complex as the immune and nervous systems. Chemotaxis also appears to be important in wound healing and tumor metastasis.
I will discuss our groups efforts at elucidating the mechanisms underlying chemotaxis. Using known biochemical data, we have developed mathematical models that can account for many of the observed chemotactic behavior of the model organism Dictyostelium. I will discuss experiments used to test these models. Finally, I will describe how information-theoretic methods can be used to evaluate the optimality of the gradient sensing mechanisms.
Cell polarization is a process in which various proteins are recruited to the plasma membrane and segregate at an emergent front or back of the cell in response to external signals. Many polarity proteins such as Rho and Rap GTPases cycle between active membrane-bound forms and inactive cytosolic forms. A simple biochemical model for a single active/inactive protein pair with positive feedback to its own activation can account for establishment and maintenance of polarity, as well as sensitivity to new stimuli. The model's capability for polarizing crucially depends on unequal rates of diffusion between the active and inactive forms of the chemicals and overall conservation of the protein. A stable homogeneous chemical distribution, (typical of a "resting cell") coexists with a stable asymmetric stationary profile (as in a "polarized cell"). Using singular perturbation theory we can explain the mathematical basis for this wave-based pattern-formation mechanism and suggest its implications to cell behaviour.
Cell intercalation has emerged as a major mechanism of transducing local cell behavior into massive changes in tissue shape, and though its inner workings were once unseen, advances in imaging have brought to light many aspects of the cell motility, cell shape, and cell boundary exchanges that occur during cell intercalation. Coupled with molecular and genetic studies, these studies have identified a number of players and processes essential for cell intercalation and the resulting tissue shape changes to occur. However, we still to not understand how cell intercalation-driven tissue shape change works. A number of unsolved issues about the cellular, molecular and biomechanical aspects of cell intercalation will be discussed with the hope of stimulating further modeling and biophysical studies.
During tissue invasion the movement of tumour cells can be supported or constrained by the structure of the tissue network. The fibre density as well as the fibre orientations play an important role for the velocity and direction of the cell movement. Besides, the cells can remodel the network by mechanical and chemical processes. On the one hand, cells adhere to the fibres and thus impose traction forces on the network during the contraction of the cell skeleton. On the other hand, tumour cells can degrade the fibres by the production of protease. The resulting digested fibronectin acts as chemoattractant for the cells.
In this talk, I will present a macroscopic model that comprises the complex interactions of cells and tissue as well as the chemosensitive cell dynamics during the invasion process.
Ventral furrow formation in Drosophila is the first large-scale morphogenetic movement during the life of the embryo, and is driven by co-ordinated changes in the shape of individual epithelial cells. Although many of the genes involved have been identified, the mechanical processes that convert local changes in gene expression into changes in embryonic form remain unknown. In this work we used a computer model to analyse the mechanics of invagination, and to investigate the ability of different combinations of independant active cell shape changes to bring about invagination. We systematically screened for combinations of active cell shape changes that are able to induce furrow formation. This identified a set of distinct combinations of active forces able to generate a furrow. An examination of this set reveals two important general features of the mechanics of the system: that no single active shape change of ventral cells can explain furrow formation in this system, and that no single force is indispensable for invagination. Instead, quantitatively similar morphogenetic changes can be brought about by different combinations of active cell shape changes. Thus, different combinations of force generating mechanisms could underlie epithelial infolding in different biological systems. We compared the results of our model with the previously described effects of mutations in the morphogenetic regulators, Twist and Snail. This comparison shows that ventral furrow formation in Drosophila is likely to be a consequence of a mesodermally localised mechanism consisting of active apical constriction, downstream of Snail, combined with active apical-basal shortening in the ventral domain downstream of Twist. We also show that there is an ecto-pushing class of invagination mechanisms which are very robust to mutation and mechanical pertubation, suggesting that ectodermal pushing could well play an important role in gastrulation movements in other systems.
Cell migration plays an essential role during both embryonic development (e.g. gastrulation, neural crest migration) and in the normal physiological responses of the adult (e.g. immune response, wound healing). The ECM plays a key role in migration, providing cells with a scaffold for migration and providing guidance information to the cells through matrix fibre following (contact guidance).
Individual cell migration in the ECM can be classified into two main groups: amoeboid and mesenchymal. In the former, cells move quickly and have negligible effect on the structure of the surrounding ECM. Mesenchymal migration, however, is much slower and extensive matrix degradation takes place through the focussed expression of specific matrix degrading proteins by the cells (pericellular proteolysis).
In this talk, I will describe both discrete and continuous models for amoeboid and mesenchymal cell migration. Numerical investigations will be used to demonstrate a potential role of contact guidance and matrix degradation in directing the macroscopic organisation of cells and the matrix.
Epithelial tissues that form highly selective barriers between different body compartments are composed from tightly packed cells which life processes, such as cell proliferation, survival, differentiation and enzyme secretion, need to be coordinated to ensure tissue integrity. To do so, epithelial cells develop specialised cell-cell connections with their neighbours and with the surrounding extracellular matrix that provide a mechanical attachment and mediate exchange of various extracellular signals. All epithelial cells maintain also an apical-basal polarity that is synchronised among all cells in the epithelium and must conform to the overlying architecture of the tissue. In contrast, disruption of such a well-organised epithelial architecture leads to various forms of epithelial tumours (carcinomas).
To enable in vitro investigation of genotypic and molecular abnormalities associated with epithelial cancers, the 3-dimensional experimental models of epithelial acini have been developed. In their mature form the acini are composed of a layer of polarised epithelial cells enclosing the hollow lumen. The self-assembly of these multi-cellular structures from a single cell is a result of cell growth and tissue movement driven by changes in cell shape and cell-cell interactions.
I will address several stages of the development of epithelial acini using a biomechanical model of the cross section through a typical acinus (IBcell model) and compare computational results to experimental data. I will also discuss different rules of cell collaboration or competition that lead either to the self-arrangement of normal acini or to the emergence of degenerated structures resembling acinar mutants, such as ductal carcinoma in situ or invasive tumours. In particular, I will focus on the dynamics of cell membrane receptors that drive interactions between neighbouring cells and between cells and their immediate microenvironment.
Work done in collaboration with Sandy Anderson and Vito Quaranta.
Ascidians begin to gastrulate at only 64 cells, each large relative to embryo size, providing an unparalleled window into the cellular basis of morphogenesis. On the experimental side, detailed descriptions of shape change are facilitated both by the global deformations being spread over a small number of cells, and by the ability to finely localize cytoskeletal players such as actin and myosin (each endoderm cell is the size of an entire C. elegans embryo). On the modeling side, a low cell count allows simulations to be mechanically and geometrically detailed at the level individual cells, while encompassing the entire embryo. Our evidence from kinematics, localization of actively contractile myosin, and inhibition of apical contraction suggests the invagination occurs in two steps: (1) apical contraction forms a placode and (2) apically constrained basolateral contraction shortens the endoderm causing it to sink inwards.
To test this hypothesis, we created a 2D computational simulation to explore which combinations of localized cortical contractility generated placodes and invaginations. We model each cell in the embryo cross-section as a connected belt of cortical segments, endowed with contractile and viscous properties (representing the inevitable relaxation of the cortex under stress due to rapid and continual assembly and disassembly of the actomyosin network) and tightly linked at cell vertices. Two cell types give rise to seven boundary types whose contractile tension could vary: two apical, two homo-lateral, one heterolateral, and two basal. Because viscosity is included in the force-balance equations solved at each timestep, the simulations exhibit true kinematics and solutions need not reach equilibrium. Crucially, many desirable geometries that do not reach a mechanical steady state are nevertheless sufficiently stable to be useful to real embryos. Wide-ranging searches of tension parameter space largely confirmed our hypothesized invagination mechanism: most placodes formed with conditions of strong apical contraction, and invaginations invariably required elevated basolateral contractility on the invaginating endoderm cells. Additionally, reducing apical tension in either step mimicked our experimental results. Simulations further suggested a strong role in step 1 for active rounding of ectoderm cells generating a form of epiboly, and showed that (if limited to differential cortical tension mechanisms) only passive stretching could generate the squamous-shaped ectoderm cells of step 2.
Our rather surprising finding that apical contraction was insufficient to drive invagination while basolateral contractility was essential, has implications for invaginations with similar kinematics such as Drosophila ventral furrow ingression. Further studies explicitly linking action at the level of individual cell boundaries, which comprise the functional units of morphogenesis, to global embryonic deformations, and which take advantage of modern molecular biology and imaging tools on the one hand, and astounding computational power on the other, should greatly improve our understanding of fundamental morphogenetic processes.
Cells move and divide by dynamically assembling and disassembling their cytoskeleton. In order to unveil generic mechanisms of cell movements, we developed simplified stripped-down systems that reconstitute cellular behaviours.
In the past, we have reconstituted Listeria movement by replacing the bacteria with beads that move by the same biochemical mechanism as the lamellipodium of cells. The advantage of such systems is that physical and mechanical properties of the load can be changed for a thorough study of the movement. Recently, we have reconstituted actin dynamics and assembly at an artificial membrane that mimics Golgi membranes, and show that actin dynamics have a role in the movement and reorganization of Golgi membranes.
We have looked into the dynamics of the acto-myosin cortex in cells in conditions where cortex contractility is enhanced, and shown that symmetry breaking occurs for cell polarization and movement. We mimick the actin cortex of cells by triggering actin polymerization at the inside liflet of a liposome, thus providing an experimental system for a controlled mechanical characterization.
Moreover, we study how membrane dynamics is changed in the presence of a cytoskeletal cortex. We use red blood cells and an optical tweezers inspired system, and show that the membrane is softened or stiffened depending on the structure of the underlying cytoskeleton. The aim of this study is to adapt this sytem to cells, and liposomes filled with an artificial actin cortex.
Works done in collaboration with L.-L. Pontani and T. Betz.
Mesenchymal motion describes the movement of cells in biological tissues formed by fiber networks, which can be observed for certain cancer metastasis. Cells move in a field of fibers (e.g. collagen) and change their velocities according to the local orientation of the fibers and the cells remodel the fibers simultaneously. Here we present the models proposed by T. Hillen and provide a comprehensive analysis including: steady states, existence of solutions, macroscopic limits, traveling waves, numerical simulations (by K. Painter).
Semi-rigid polysaccharide walls generally prevent plant cells from rapid movements. With the exception of pollen tubes and the motile sperm of certain taxa, plant cells do not migrate. Instead, plant cell movement occurs by expansion of the cell wall, which, driven by turgor pressure either occurs by tip growth or, more commonly, by diffuse expansion, which accounts for the massive enlargement that enables roots to penetrate and mine the soil, stems to grow into the canopy, and leaves to achieve appropriate dimensions to optimize the harnessing of solar energy. The multicellular nature of plants underlies the importance of precise coordination of cell growth between adjacent cells, across tissue layers and within organs. Rather than expanding equally in all directions, cells within elongating plant organs expand along one major axis, which is generally perpendicular to the orientation of highly tensile cellulose microfibrils in the cell wall. The mechanical properties of cellulose microfibrils, in turn, are governed by dynamic microtubules, which are self-organized into parallel arrays at the plant cell cortex. How spatial organization of the cortical array is achieved is one of the most enduring questions in plant cell biology.
The research in my laboratory explores how the dynamic properties of cortical microtubules, which we quantify in live cells using fluorescent reporter proteins, determine the spatial organization of microtubule arrays. We are also exploring how these arrays control the growth, morphology and performance of plants. I will outline how genetic approaches to identify key proteins like MOR1, CLASP and ARK1 that modulate microtubule dynamics are generating experimental tools for understanding the mechanisms that drive organization of microtubule arrays. Comparing the dynamic behaviour of microtubules in mutant lines that have defective versions of MOR1, CLASP or ARK proteins enables us to test models of the molecular mechanisms that drive microtubule organization. We are currently using this knowledge to explore the role microtubules play in the mechanical properties of the cellulosic cell wall, the helical handedness of elongating organs, and even the polar transport of the hormone auxin.
Tumor cell migration through extracellular matrix involves proteolytic cell-matrix interactions for matrix barrier removal, yet the molecular topography and its relevance to invasion patterning are unclear. Using dynamic bright-field and confocal imaging of highly proteolytic HT1080 fibrosarcoma cells invading 3D fibrillar collagen lattices, we quantified the topology of cell-matrix-interactions, structural matrix break-down, collagen neoepitope generation and cooperating molecules, and related shape change and nuclear deformation. During migration, major focalized degradation of those fibers occurred that caused "belt-like" constrictions towards the cell body. Beta1 integrins and F-actin colocalized with cleavage events only at these constrictions, indicating cooperation of adhesion and proteolysis. Cleaved collagen fibers then became dislocated and aligned in parallel along the forward-moving cell body. The resulting tube-like matrix defects were further widened by following cells to arrange in chains (Indian file) and invading multicellular strands. Proteolysis, path generation and widening as well as transition to collective invasion were abrogated by protease inhibition, not however single-cell movement which was maintained by dynamic shape change and constriction of cell body and nucleus (minimum width: 3\mu m). These findings directly demonstrate, how step-wise focalized matrix breakdown results in cell and tissue patterning towards invading cell chains and collectives. The more complex multicellular invasion critically depends on proteolytic trail formation and widening, whereas single cell migration can persist via nonproteolytic, physical mechanisms.
Work done in collaboration with Peter Friedl.