A geometrically controlled rigidity transition in a model for confluent 3D tissues

The origin of rigidity in disordered materials is an outstanding open problem in statistical physics. Previously, a class of 2D cellular models has been shown to undergo a rigidity transition controlled by a mechanical parameter that specifies cell shapes. Here, we generalize this model to 3D and find a rigidity transition that is similarly controlled by the preferred surface area: the model is solid-like below a dimensionless surface area of $s_0^\ast\approx5.413$, and fluid-like above this value. We demonstrate that, unlike jamming in soft spheres, residual stresses are necessary to create rigidity. These stresses occur precisely when cells are unable to obtain their desired geometry, and we conjecture that there is a well-defined minimal surface area possible for disordered cellular structures. We show that the behavior of this minimal surface induces a linear scaling of the shear modulus with the control parameter at the transition point, which is different from the scaling observed in particulate matter. The existence of such a minimal surface may be relevant for biological tissues and foams, and helps explain why cell shapes are a good structural order parameter for rigidity transitions in biological tissues.Comment: 6 pages main text + 13 pages appendix, 3 main text figures + 6 appendix figure

Nachweis des Eremiten (Osmoderma eremita) im Othaler Holz

Zwischen Sangerhausen und der Lutherstadt Eisleben befindet sich das FFH-Gebiet „Der Hagen und Othaler Holz nördlich Beyernaumburg“ (FFH0110LSA). Von besonderer Bedeutung sind die großflächigen und recht alten Vorkommen des Waldmeister-Buchenwaldes und des Labkraut-Eichen-Hainbuchenwaldes. In geringem Umfang finden sich zudem Bestände vom Hainsimen- Buchenwald. Im Gebiet leben auch Arten der Anhänge II und IV der FFH-Richtlinie, wie Mopsfledermaus (Barbastella barbastellus), Bechsteinfledermaus (Myotis bechsteinii) und Großes Mausohr (Myotis myotis) (Jentzsch & Katthöver 2005). Am 19.7.2011 konnte im südöstlichen Teil des Othaler Holzes ein Weibchen vom Eremiten (Osmoderma eremita) fotografiert werden. Dies ist der erste Nachweis der Art in dem FFH-Gebiet

From cells to tissues

An essential prerequisite for the existence of multi-cellular life is the organization of cells into tissues. In this thesis, we theoretically study how large-scale tissue properties can emerge from the collective behavior of individual cells. To this end, we focus on the properties of epithelial tissue, which is one of the major tissue types in animals. We study how rheological properties of epithelia emerge from cellular processes, and we develop a physical description for the dynamics of an epithelial cell polarity. We apply our theoretical studies to observations in the developing wing of the fruit fly, Drosophila melanogaster. In order to study epithelial mechanics, we first develop a geometrical framework that rigorously describes the deformation of two-dimensional cellular networks. Our framework decomposes large-scale deformation into cellular contributions. For instance, we show how large-scale tissue shear decomposes into contributions by cell shape changes and into contributions by different kinds of topological transitions. We apply this framework in order to quantify the time-dependent deformation of the fruit fly wing, and to decompose it into cellular contributions. We also use this framework as a basis to study large-scale rheological properties of epithelia and their dependence on cellular fluctuations. To this end, we represent epithelial tissues by a vertex model, which describes cells as elastic polygons. We extend the vertex model by introducing fluctuations on the cellular scale, and we develop a method to perform perpetual simple shear simulations. Analyzing the steady state of such simple shear simulations, we find that the rheological behavior of vertex model tissue depends on the fluctuation amplitude. For small fluctuation amplitude, it behaves like a plastic material, and for high fluctuation amplitude, it behaves like a visco-elastic fluid. In addition to analyzing mechanical properties, we study the reorientation of an epithelial cell polarity. To this end, we develop a simple hydrodynamic description for polarity reorientation. In particular, we account for polarity reorientation by tissue shear, by another polarity field, and by local polarity alignment. Furthermore, we develop methods to quantify polarity patterns based on microscopical images of the fly wing. We find that our hydrodynamic description does not only account for polarity reorientation in wild type fly wings. Moreover, it is for the first time possible to also account for the observed polarity patterns in a number of genetically altered flies.:1 Introduction 1.1 The development of multi-cellular organisms 1.2 Biology of epithelial tissues 1.3 The model system Drosophila melanogaster 1.4 Planar cell polarity 1.5 Physical description of biological tissues 1.6 Overview over this thesis 2 Tissue shear in cellular networks 2.1 Geometry of tissue deformation on the cellular scale 2.2 Decomposition of the large-scale flow field into cellular contributions 2.3 Cellular contributions to the flow field in the fruit fly wing 2.4 Discussion 3 Rheological behavior of vertex model tissue under external shear 3.1 A vertex model to describe epithelial mechanics 3.2 Fluctuation-induced fluidization of tissue 3.3 Discussion 4 Quantitative study of polarity reorientation in the fruit fly wing 4.1 Experimentally quantified polarity patterns 4.2 Effective hydrodynamic theory for polarity reorientation 4.3 Comparison of theory and experiment 4.4 Discussion 5 Conclusions and outlook Appendices: A Algebra of real 2 × 2 matrices B Deformation of triangle networks C Simple shear simulations using the vertex model D Coarse-graining of a cellular Core PCP model E Quantification of polarity patterns in the fruit fly wing F Theory for polarity reorientation in the fruit fly wing G Boundary conditions for the polarity field in the fruit fly wing Table of symbols BibliographyEine wesentliche Voraussetzung für die Existenz mehrzelligen Lebens ist, dass sich einzelne Zellen sinnvoll zu Geweben ergänzen können. In dieser Dissertation untersuchen wir, wie großskalige Eigenschaften von Geweben aus dem kollektiven Verhalten einzelner Zellen hervorgehen. Dazu konzentrieren wir uns auf Epitheliengewebe, welches eine der Grundgewebearten in Tieren darstellt. Wir stellen theoretische Untersuchungen zu rheologischen Eigenschaften und zu zellulärer Polarität von Epithelien an. Diese theoretischen Untersuchungen vergleichen wir mit experimentellen Beobachtungen am sich entwickelnden Flügel der schwarzbäuchigen Taufliege (Drosophila melanogaster). Um die Mechanik von Epithelien zu untersuchen, entwickeln wir zunächst eine geometrische Beschreibung für die Verformung von zweidimensionalen zellulären Netzwerken. Unsere Beschreibung zerlegt die mittlere Verformung des gesamten Netzwerks in zelluläre Beitrage. Zum Beispiel wird eine Scherverformung des gesamten Netzwerks auf der zellulären Ebene exakt repräsentiert: einerseits durch die Verformung einzelner Zellen und andererseits durch topologische Veränderungen des zellulären Netzwerks. Mit Hilfe dieser Beschreibung quantifizieren wir die Verformung des Fliegenflügels während des Puppenstadiums. Des Weiteren führen wir die Verformung des Flügels auf ihre zellulären Beiträge zurück. Wir nutzen diese Beschreibung auch als Ausgangspunkt, um effektive rheologische Eigenschaften von Epithelien in Abhängigkeit von zellulären Fluktuationen zu untersuchen. Dazu simulieren wir Epithelgewebe mittels eines Vertex Modells, welches einzelne Zellen als elastische Polygone abstrahiert. Wir erweitern dieses Vertex Modell um zelluläre Fluktuationen und um die Möglichkeit, Schersimulationen beliebiger Dauer durchzuführen. Die Analyse des stationären Zustands dieser Simulationen ergibt plastisches Verhalten bei kleiner Fluktuationsamplitude und visko-elastisches Verhalten bei großer Fluktuationsamplitude. Neben mechanischen Eigenschaften untersuchen wir auch die Umorientierung einer Zellpolarität in Epithelien. Dazu entwickeln wir eine einfache hydrodynamische Beschreibung für die Umorientierung eines Polaritätsfeldes. Wir berücksichtigen dabei insbesondere Effekte durch Scherung, durch ein anderes Polaritätsfeld und durch einen lokalen Gleichrichtungseffekt. Um unsere theoretische Beschreibung mit experimentellen Daten zu vergleichen, entwickeln wir Methoden um Polaritätsmuster im Fliegenflügel zu quantifizieren. Schließlich stellen wir fest, dass unsere hydrodynamische Beschreibung in der Tat beobachtete Polaritätsmuster reproduziert. Das gilt nicht nur im Wildtypen, sondern auch in genetisch veränderten Tieren.:1 Introduction 1.1 The development of multi-cellular organisms 1.2 Biology of epithelial tissues 1.3 The model system Drosophila melanogaster 1.4 Planar cell polarity 1.5 Physical description of biological tissues 1.6 Overview over this thesis 2 Tissue shear in cellular networks 2.1 Geometry of tissue deformation on the cellular scale 2.2 Decomposition of the large-scale flow field into cellular contributions 2.3 Cellular contributions to the flow field in the fruit fly wing 2.4 Discussion 3 Rheological behavior of vertex model tissue under external shear 3.1 A vertex model to describe epithelial mechanics 3.2 Fluctuation-induced fluidization of tissue 3.3 Discussion 4 Quantitative study of polarity reorientation in the fruit fly wing 4.1 Experimentally quantified polarity patterns 4.2 Effective hydrodynamic theory for polarity reorientation 4.3 Comparison of theory and experiment 4.4 Discussion 5 Conclusions and outlook Appendices: A Algebra of real 2 × 2 matrices B Deformation of triangle networks C Simple shear simulations using the vertex model D Coarse-graining of a cellular Core PCP model E Quantification of polarity patterns in the fruit fly wing F Theory for polarity reorientation in the fruit fly wing G Boundary conditions for the polarity field in the fruit fly wing Table of symbols Bibliograph

Stiffening of under-constrained spring networks under isotropic strain

Disordered spring networks are a useful paradigm to examine macroscopic mechanical properties of amorphous materials. Here, we study the elastic behavior of under-constrained spring networks, i.e.\ networks with more degrees of freedom than springs. While such networks are usually floppy, they can be rigidified by applying external strain. Recently, an analytical formalism has been developed to predict the mechanical network properties close to this rigidity transition. Here we numerically show that these predictions apply to many different classes of spring networks, including phantom triangular, Delaunay, Voronoi, and honeycomb networks. The analytical predictions further imply that the shear modulus $G$ scales linearly with isotropic stress $T$ close to the rigidity transition; however, this seems to be at odds with recent numerical studies suggesting an exponent between $G$ and $T$ that is smaller than one for some network classes. Using increased numerical precision and shear stabilization, we demonstrate here that close to the transition linear scaling, $G\sim T$, holds independent of the network class. Finally, we show that our results are not or only weakly affected by finite-size effects, depending on the network class.Comment: 17 pages, 10 figure

Triangles bridge the scales: Quantifying cellular contributions to tissue deformation

In this article, we propose a general framework to study the dynamics and topology of cellular networks that capture the geometry of cell packings in two-dimensional tissues. Such epithelia undergo large-scale deformation during morphogenesis of a multicellular organism. Large-scale deformations emerge from many individual cellular events such as cell shape changes, cell rearrangements, cell divisions, and cell extrusions. Using a triangle-based representation of cellular network geometry, we obtain an exact decomposition of large-scale material deformation. Interestingly, our approach reveals contributions of correlations between cellular rotations and elongation as well as cellular growth and elongation to tissue deformation. Using this Triangle Method, we discuss tissue remodeling in the developing pupal wing of the fly Drosophila melanogaster.Comment: 26 pages, 18 figure

Correlating Cell Shape and Cellular Stress in Motile Confluent Tissues

Collective cell migration is a highly regulated process involved in wound healing, cancer metastasis and morphogenesis. Mechanical interactions among cells provide an important regulatory mechanism to coordinate such collective motion. Using a Self-Propelled Voronoi (SPV) model that links cell mechanics to cell shape and cell motility, we formulate a generalized mechanical inference method to obtain the spatio-temporal distribution of cellular stresses from measured traction forces in motile tissues and show that such traction-based stresses match those calculated from instantaneous cell shapes. We additionally use stress information to characterize the rheological properties of the tissue. We identify a motility-induced swim stress that adds to the interaction stress to determine the global contractility or extensibility of epithelia. We further show that the temporal correlation of the interaction shear stress determines an effective viscosity of the tissue that diverges at the liquid-solid transition, suggesting the possibility of extracting rheological information directly from traction data.Comment: 12 pages, 9 figure

Active dynamics of tissue shear flow

We present a hydrodynamic theory to describe shear flows in developing epithelial tissues. We introduce hydrodynamic fields corresponding to state properties of constituent cells as well as a contribution to overall tissue shear flow due to rearrangements in cell network topology. We then construct a generic linear constitutive equation for the shear rate due to topological rearrangements and we investigate a novel rheological behaviour resulting from memory effects in the tissue. We identify two distinct active cellular processes: generation of active stress in the tissue, and actively driven topological rearrangements. We find that these two active processes can produce distinct cellular and tissue shape changes, depending on boundary conditions applied on the tissue. Our findings have consequences for the understanding of tissue morphogenesis during development

Lysergol monohydrate

In the title compound [systematic name: (7-methyl-4,6,6a,7,8,9-hexa­hydro­indolo[4,3,2-fg]quinoline-9-yl)methanol monohydrate], C16H18N2O·H2O, the non-aromatic ring (ring C of the ergoline skeleton) directly fused to the aromatic rings is nearly planar, with a maximum deviation of 0.659 (3) Å, and shows an envelope conformation. In the crystal, hydrogen bonds between the lysergol and water mol­ecules contribute to the formation of layers parallel to (10)

Ergometrinine

The absolute configuration of ergometrinine, C19H23N3O2 {systematic name: (6aR,9S)-N-[(S)-1-hy­droxy­propan-2-yl]-7-methyl-4,6,6a,7,8,9-hexa­hydro­indolo[4,3-fg]quinoline-9-carb­ox­amide}, was established based on epimerization reaction of ergometrine, which was followed by preparative HPLC. The non-aromatic ring (ring C of the ergoline skeleton) directly fused to the aromatic rings is nearly planar [maximum deviation = 0.271 (3) Å] and shows an envelope conformation, whereas ring D, involved in an intra­molecular N—H⋯N hydrogen bond, exibits a slightly distorted chair conformation. The structure displays undulating layers in the ac plane formed by O—H⋯O and N—H⋯O hydrogen bonds