The Role of Topology and Topological Changes in the Mechanical Properties of Epithelia by Meryl Spencer A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Physics) in The University of Michigan 2019 Doctoral Committee: Associate Professor David K Lubensky, Chair Professor Mark Newman Professor Jennifer Ogilvie Assistant Professor Kevin Wood Meryl Spencer [email protected] ORCID iD: 0000-0003-1994-4100 c Meryl Spencer 2019 All Rights Reserved ACKNOWLEDGEMENTS This work would have not have been possible without the mentorship of my adviser David. Much of my thesis work relies on analysis of thousands of images taken by Jesus Lopez-Gay. I sustained a major spinal cord injury just as I started to write my thesis. I would not have been physically capable of finishing it, if not for the amazing medical care I received from my entire team at the neuro-ICU and 6A. I could not have afforded the care without medical benefits from my grad union GEO. Thank you from the bottom of my heart everyone who cooked me meals, cleaned my house, and gave me moral support throughout my long recovery. Special thanks to Patricia Klein for organizing my care, and my parents for their constant support. This work probably would have been possible without the support of my fellow graduate students, but it wouldn't have been nearly as fun. I'd like to thank my amazing first-year cohort especially, Paige, Rutu, Karishima, Chrisy and Kevin for working through Jackson with me, as well as Jessie, Joe, Glenn, Peter, Ansel, and Anthony for joining in second-year. John Ware has supported me from start to finish, from working through old qualifying exams through proof-reading thesis chapters. My entire family has given me moral and finantial support throughout college and graduate school, even if they often didn't understand what I was doing. Thanks for sticking with me. ii TABLE OF CONTENTS ACKNOWLEDGEMENTS :::::::::::::::::::::::::: ii LIST OF FIGURES ::::::::::::::::::::::::::::::: vi LIST OF TABLES :::::::::::::::::::::::::::::::: xvii LIST OF APPENDICES :::::::::::::::::::::::::::: xviii ABSTRACT ::::::::::::::::::::::::::::::::::: xix CHAPTER I. Introduction ..............................1 II. Vertex stability and topological transitions in vertex models of foams and epithelia ........................ 10 2.1 Introduction . 10 2.2 The vertex model . 14 2.2.1 Definition . 14 2.2.2 Dynamics . 16 2.3 Fourfold vertex stability . 18 2.3.1 Dynamics of a small edge . 19 2.3.2 Defining fourfold vertex stability . 20 2.3.3 Stability conditions . 21 2.4 No stable, stationary fourfold vertices exist in Plateau's model 23 2.5 No stable, stationary fourfold vertices exist in the equal tension vertex model . 25 2.5.1 Streamlining notation . 27 2.5.2 Exploiting symmetries . 27 2.5.3 Bounds on the angle between non-adjacent edges . 28 2.5.4 Finding a contradiction when non-adjacent edges have 180◦ separation . 31 2.6 Examples of modifications that allow for stable fourfold vertices 32 iii 2.6.1 Vertices not in mechanical equilibrium . 32 2.6.2 Anisotropic tension . 32 2.7 Implications for computational models . 35 2.8 Discussion . 37 III. Implementing the vertex model .................. 44 3.1 Introduction . 44 3.2 Overview . 45 3.3 Standard assumptions and parameter set . 46 3.4 Input-output . 47 3.5 Defining the cell, edge, and vertex classes . 48 3.6 Anatomy of the tissue class . 50 3.7 Evolution in time . 51 3.7.1 Implementing external stress . 53 3.8 Extending the base code through the use of polymorphism . 57 IV. Multicellular actomyosin cables in epithelia under external anisotropic stress ........................... 59 4.1 Introduction . 59 4.2 Materials and methods . 63 4.2.1 Theoretical framework: vertex model . 63 4.2.2 Stretching procedure . 65 4.3 Results . 68 4.3.1 Defining measures of cableness . 68 4.3.2 Oriented cell divisions promote cableness . 74 4.3.3 Example: Drosophila epithelium . 74 4.4 Conclusion . 77 V. Mechanics of actomyosin fibers in the Drosophila pupal notum 84 5.1 Introduction . 84 5.2 The toy model . 87 5.3 Model predictions . 89 5.4 Model predictions: relations between area elongation and fiber number .............................. 91 5.5 Model prediction: fiber alignment and cell orientation . 92 5.5.1 Physical intuition: why are fibers more helpful in the NCFO? . 92 5.5.2 Data analysis: fiber alignment . 93 5.5.3 Data analysis: cell orientation . 95 5.6 Conclusion . 96 VI. Image analysis and machine learning ............... 98 iv 6.1 Introduction . 98 6.2 Background: image segmentation . 98 6.2.1 Classical algorithms . 99 6.2.2 Machine learning based classification . 102 6.3 Automated fiber detection . 104 6.3.1 Stage one: segment potential fibers . 106 6.3.2 Stage two: remove misidentified fibers through ma- chine learning classification . 106 6.4 Discussion . 108 VII. Honeycomb Lattices with Defects ................. 111 7.1 Introduction . 111 7.2 Defective Honeycomb (DHC) Lattice . 113 7.2.1 Definition . 113 7.2.2 Example . 115 7.2.3 Properties . 116 7.3 Generation of Lattices . 116 7.4 Relationship between flips and defects . 117 7.5 Determination of pc ....................... 120 7.6 Discussion . 122 VIII. Conclusion ............................... 125 APPENDICES :::::::::::::::::::::::::::::::::: 127 v LIST OF FIGURES Figure 1.1 A cartoon of an epithelium. Adherens junctions keep cells in contact with each other. An apical band of actomyosin provides a contractile force in the cell. B Physicists model of the epithelium as a two dimen- sional network of contractile edge tensions balanced by cell pressures.1 1.2 A Cartoon of a T1 transition. An edge shrinks to form a fourfold ver- tex and then elongates in the perpendicular direction causing neigh- bor exchange. B Cartoon of convergent extension. Vertical edges undergo T1 neighbor exchange causing the tissue to elongate. .3 2.1 A: Cartoon of cells in an epithelial sheet. A single cell is shaded blue. The interface between two cells forms an edge (one edge is highlighted by the bold green line). The red dot indicates a vertex, defined as a point at which three or more cells touch. We treat the epithelium as a two-dimensional sheet, focusing on the level of the adherens junctions near the apical (top) surface. B: Cartoon of epithelial cells undergoing a T1 topological transition (viewed from above). An edge shrinks down until a fourfold vertex is formed, then a new edge elongates in a roughly perpendicular direction. As a result, the cells exchange neighbors, altering the topology of the cell packing. The middle panel shows the moment at which a fourfold vertex (light green dot) appears. The fourfold vertex has four neighboring cells and four neighboring edges and could in principle either be stable or resolve into either of the two different topologies shown to the left and right. 11 2.2 Cartoon of a cell with vertices at positions r0, r1 and r2. Movement of vertex r0 affects the lengths of the adjacent edges l1 and l2 and the area of the shaded triangle bounded by these edges. We assume that the face of the cell is in the x-y plane of a standard right-handed coordinate system with the z^ axis projecting out of the plane . The 1 area of the shaded region is then 2 z^ · (l1 × l2). 13 vi 2.3 A : Cartoon of a fourfold vertex with neighboring cells L, M, N, O and edges l1, l2, l3, l4. Note that the direction of the edges is outward from the vertex, and the edges are numbered clockwise. B : Cartoon showing eight different forces acting on the fourfold vertex, two asso- ciated with each edge (eq. 2.15). The four edges produce a tension ^ Γili. The effect of the pressures from the four cells can be written in terms of the pressure differences across the edges; if pi is the pressure difference across edge i, we can view the pressures as exerting a force pi 2 (z^ × li) perpendicular to each edge. C : Cartoon of two threefold vertices which share an edge lδ. As its length lδ shrinks to zero, the vertices ra and rb will merge to form a single fourfold vertex. D : Cartoon of the resolution of a fourfold vertex. The fourfold vertex (center) can break apart into two threefold vertices in either of two topologies (left, right). In each case, we can associate a told force fi with each edge i that includes tension and pressure jump contri- butions. In Topology 1 (left), forces f3 and f4 act one of the new vertices and forces f1 and f2 act on the other new vertex, so that the net force trying to extend the new edge lδ is f1 + f2 − f3 − f4; this is counteracted by the tension Γδ on the new edge (eq. 2.18). The situation is the same in Topology 2 (right), but with the edges paired differently. 17 2.4 A : The two angles \(l1l3)l2 and \(l1l3)l4 between the non-adjacent edges l1 and l3 are shown. The quantities \(lilj)lk are defined as the unsigned magnitudes of the angles, so \(lilj)lk = \(ljli)lk . The angles \(l1l3)l2 and \(l1l3)l4 together make a full circle, implying \(l1l3)l2 + \(l1l3)l4 = 2π. B : The angles θ1 and θ3 are defined in the usual manner as the signed angles between the positive x axis (which here coincides with l2) and, respectively, l1 and l3. Hence, as drawn, θ1 > 0 and θ3 < 0. ........................... 23 2.5 Cartoon of the procedure in section 2.5.2 to exploit the symmetries of the problem in order to reduce the number of free variables. An arbitrary angle \(lilj)lk between non adjacent edges can be trans- formed under rotations such that lk lies on the positive x axis; if ^ needed, a reflection through the x axis ensures that ρk(z^× lk) lies on the positive y axis and thus that ρk ≥ 0.