Biaxial Nematic Order in Liver Tissue

Understanding how biological cells organize to form complex functional tissues is a question of key interest at the interface between biology and physics. The liver is a model system for a complex three-dimensional epithelial tissue, which performs many vital functions. Recent advances in imaging methods provide access to experimental data at the subcellular level. Structural details of individual cells in bulk tissues can be resolved, which prompts for new analysis methods. In this thesis, we use concepts from soft matter physics to elucidate and quantify structural properties of mouse liver tissue. Epithelial cells are structurally anisotropic and possess a distinct apico-basal cell polarity that can be characterized, in most cases, by a vector. For the parenchymal cells of the liver (hepatocytes), however, this is not possible. We therefore develop a general method to characterize the distribution of membrane-bound proteins in cells using a multipole decomposition. We first verify that simple epithelial cells of the kidney are of vectorial cell polarity type and then show that hepatocytes are of second order (nematic) cell polarity type. We propose a method to quantify orientational order in curved geometries and reveal lobule-level patterns of aligned cell polarity axes in the liver. These lobule-level patterns follow, on average, streamlines defined by the locations of larger vessels running through the tissue. We show that this characterizes the liver as a nematic liquid crystal with biaxial order. We use the quantification of orientational order to investigate the effect of specific knock-down of the adhesion protein Integrin ß-1. Building upon these observations, we study a model of nematic interactions. We find that interactions among neighboring cells alone cannot account for the observed ordering patterns. Instead, coupling to an external field yields cell polarity fields that closely resemble the experimental data. Furthermore, we analyze the structural properties of the two transport networks present in the liver (sinusoids and bile canaliculi) and identify a nematic alignment between the anisotropy of the sinusoid network and the nematic cell polarity of hepatocytes. We propose a minimal lattice-based model that captures essential characteristics of network organization in the liver by local rules. In conclusion, using data analysis and minimal theoretical models, we found that the liver constitutes an example of a living biaxial liquid crystal.:1. Introduction 1 1.1. From molecules to cells, tissues and organisms: multi-scale hierarchical organization in animals 1 1.2. The liver as a model system of complex three-dimensional tissue 2 1.3. Biology of tissues 5 1.4. Physics of tissues 9 1.4.1. Continuum descriptions 11 1.4.2. Discrete models 11 1.4.3. Two-dimensional case study: planar cell polarity in the fly wing 15 1.4.4. Challenges of three-dimensional models for liver tissue 16 1.5. Liquids, crystals and liquid crystals 16 1.5.1. The uniaxial nematic order parameter 19 1.5.2. The biaxial nematic ordering tensor 21 1.5.3. Continuum theory of nematic order 23 1.5.4. Smectic order 25 1.6. Three-dimensional imaging of liver tissue 26 1.7. Overview of the thesis 28 2. Characterizing cellular anisotropy 31 2.1. Classifying protein distributions on cell surfaces 31 2.1.1. Mode expansion to characterize distributions on the unit sphere 31 2.1.2. Vectorial and nematic classes of surface distributions 33 2.1.3. Cell polarity on non-spherical surfaces 34 2.2. Cell polarity in kidney and liver tissues 36 2.2.1. Kidney cells exhibit vectorial polarity 36 2.2.2. Hepatocytes exhibit nematic polarity 37 2.3. Local network anisotropy 40 2.4. Summary 41 3. Order parameters for tissue organization 43 3.1. Orientational order: quantifying biaxial phases 43 3.1.1. Biaxial nematic order parameters 45 3.1.2. Co-orientational order parameters 51 3.1.3. Invariants of moment tensors 52 3.1.4. Relation between these three schemes 53 3.1.5. Example: nematic coupling to an external field 55 3.2. A tissue-level reference field 59 3.3. Orientational order in inhomogeneous systems 62 3.4. Positional order: identifying signatures of smectic and columnar order 64 3.5. Summary 67 4. The liver lobule exhibits biaxial liquid-crystal order 69 4.1. Coarse-graining reveals nematic cell polarity patterns on the lobulelevel 69 4.2. Coarse-grained patterns match tissue-level reference field 73 4.3. Apical and basal nematic cell polarity are anti-correlated 74 4.4. Co-orientational order: nematic cell polarity is aligned with network anisotropy 76 4.5. RNAi knock-down perturbs orientational order in liver tissue 78 4.6. Signatures of smectic order in liver tissue 81 4.7. Summary 86 5. Effective models for cell and network polarity coordination 89 5.1. Discretization of a uniaxial nematic free energy 89 5.2. Discretization of a biaxial nematic free energy 91 5.3. Application to cell polarity organization in liver tissue 92 5.3.1. Spatial profile of orientational order in liver tissue 93 5.3.2. Orientational order from neighbor-interactions and boundary conditions 94 5.3.3. Orientational order from coupling to an external field 99 5.4. Biaxial interaction model 101 5.5. Summary 105 6. Network self-organization in a liver-inspired lattice model 107 6.1. Cubic lattice geometry motivated by liver tissue 107 6.2. Effective energy for local network segment interactions 110 6.3. Characterizing network structures in the cubic lattice geometry 113 6.4. Local interaction rules generate macroscopic network structures 115 6.5. Effect of mutual repulsion between unlike segment types on network structure 118 6.6. Summary 121 7. Discussion and Outlook 123 A. Appendix 127 A.1. Mean field theory fo the isotropic-uniaxial nematic transition 127 A.2. Distortions of the Mollweide projection 129 A.3. Shape parameters for basal membrane around hepatocytes 130 A.4. Randomized control for network segment anisotropies 130 A.5. The dihedral symmetry group D2h 131 A.6. Relation between orientational order parameters and elements of the super-tensor 134 A.7. Formal separation of molecular asymmetry and orientation 134 A.8. Order parameters under action of axes permutation 137 A.9. Minimal integrity basis for symmetric traceless tensors 139 A.10. Discretization of distortion free energy on cubic lattice 141 A.11. Metropolis Algorithm for uniaxial cell polarity coordination 142 A.12. States in the zero-noise limit of the nearest-neighbor interaction model 143 A.13. Metropolis Algorithm for network self-organization 144 A.14. Structural quantifications for varying values of mutual network segment repulsion 146 A.15. Structural quantifications for varying values of self-attraction of network segments 148 A.16. Structural quantifications for varying values of cell demand 150 Bibliography 152 Acknowledgements 17

Quantification of Nematic Cell Polarity in Three-dimensional Tissues

How epithelial cells coordinate their polarity to form functional tissues is an open question in cell biology. Here, we characterize a unique type of polarity found in liver tissue, nematic cell polarity, which is different from vectorial cell polarity in simple, sheet-like epithelia. We propose a conceptual and algorithmic framework to characterize complex patterns of polarity proteins on the surface of a cell in terms of a multipole expansion. To rigorously quantify previously observed tissue-level patterns of nematic cell polarity (Morales-Navarette et al., eLife 8:e44860, 2019), we introduce the concept of co-orientational order parameters, which generalize the known biaxial order parameters of the theory of liquid crystals. Applying these concepts to three-dimensional reconstructions of single cells from high-resolution imaging data of mouse liver tissue, we show that the axes of nematic cell polarity of hepatocytes exhibit local coordination and are aligned with the biaxially anisotropic sinusoidal network for blood transport. Our study characterizes liver tissue as a biological example of a biaxial liquid crystal. The general methodology developed here could be applied to other tissues or in-vitro organoids.Comment: 27 pages, 9 color figure

Biaxial Nematic Order in Liver Tissue

Understanding how biological cells organize to form complex functional tissues is a question of key interest at the interface between biology and physics. The liver is a model system for a complex three-dimensional epithelial tissue, which performs many vital functions. Recent advances in imaging methods provide access to experimental data at the subcellular level. Structural details of individual cells in bulk tissues can be resolved, which prompts for new analysis methods. In this thesis, we use concepts from soft matter physics to elucidate and quantify structural properties of mouse liver tissue. Epithelial cells are structurally anisotropic and possess a distinct apico-basal cell polarity that can be characterized, in most cases, by a vector. For the parenchymal cells of the liver (hepatocytes), however, this is not possible. We therefore develop a general method to characterize the distribution of membrane-bound proteins in cells using a multipole decomposition. We first verify that simple epithelial cells of the kidney are of vectorial cell polarity type and then show that hepatocytes are of second order (nematic) cell polarity type. We propose a method to quantify orientational order in curved geometries and reveal lobule-level patterns of aligned cell polarity axes in the liver. These lobule-level patterns follow, on average, streamlines defined by the locations of larger vessels running through the tissue. We show that this characterizes the liver as a nematic liquid crystal with biaxial order. We use the quantification of orientational order to investigate the effect of specific knock-down of the adhesion protein Integrin ß-1. Building upon these observations, we study a model of nematic interactions. We find that interactions among neighboring cells alone cannot account for the observed ordering patterns. Instead, coupling to an external field yields cell polarity fields that closely resemble the experimental data. Furthermore, we analyze the structural properties of the two transport networks present in the liver (sinusoids and bile canaliculi) and identify a nematic alignment between the anisotropy of the sinusoid network and the nematic cell polarity of hepatocytes. We propose a minimal lattice-based model that captures essential characteristics of network organization in the liver by local rules. In conclusion, using data analysis and minimal theoretical models, we found that the liver constitutes an example of a living biaxial liquid crystal.:1. Introduction 1 1.1. From molecules to cells, tissues and organisms: multi-scale hierarchical organization in animals 1 1.2. The liver as a model system of complex three-dimensional tissue 2 1.3. Biology of tissues 5 1.4. Physics of tissues 9 1.4.1. Continuum descriptions 11 1.4.2. Discrete models 11 1.4.3. Two-dimensional case study: planar cell polarity in the fly wing 15 1.4.4. Challenges of three-dimensional models for liver tissue 16 1.5. Liquids, crystals and liquid crystals 16 1.5.1. The uniaxial nematic order parameter 19 1.5.2. The biaxial nematic ordering tensor 21 1.5.3. Continuum theory of nematic order 23 1.5.4. Smectic order 25 1.6. Three-dimensional imaging of liver tissue 26 1.7. Overview of the thesis 28 2. Characterizing cellular anisotropy 31 2.1. Classifying protein distributions on cell surfaces 31 2.1.1. Mode expansion to characterize distributions on the unit sphere 31 2.1.2. Vectorial and nematic classes of surface distributions 33 2.1.3. Cell polarity on non-spherical surfaces 34 2.2. Cell polarity in kidney and liver tissues 36 2.2.1. Kidney cells exhibit vectorial polarity 36 2.2.2. Hepatocytes exhibit nematic polarity 37 2.3. Local network anisotropy 40 2.4. Summary 41 3. Order parameters for tissue organization 43 3.1. Orientational order: quantifying biaxial phases 43 3.1.1. Biaxial nematic order parameters 45 3.1.2. Co-orientational order parameters 51 3.1.3. Invariants of moment tensors 52 3.1.4. Relation between these three schemes 53 3.1.5. Example: nematic coupling to an external field 55 3.2. A tissue-level reference field 59 3.3. Orientational order in inhomogeneous systems 62 3.4. Positional order: identifying signatures of smectic and columnar order 64 3.5. Summary 67 4. The liver lobule exhibits biaxial liquid-crystal order 69 4.1. Coarse-graining reveals nematic cell polarity patterns on the lobulelevel 69 4.2. Coarse-grained patterns match tissue-level reference field 73 4.3. Apical and basal nematic cell polarity are anti-correlated 74 4.4. Co-orientational order: nematic cell polarity is aligned with network anisotropy 76 4.5. RNAi knock-down perturbs orientational order in liver tissue 78 4.6. Signatures of smectic order in liver tissue 81 4.7. Summary 86 5. Effective models for cell and network polarity coordination 89 5.1. Discretization of a uniaxial nematic free energy 89 5.2. Discretization of a biaxial nematic free energy 91 5.3. Application to cell polarity organization in liver tissue 92 5.3.1. Spatial profile of orientational order in liver tissue 93 5.3.2. Orientational order from neighbor-interactions and boundary conditions 94 5.3.3. Orientational order from coupling to an external field 99 5.4. Biaxial interaction model 101 5.5. Summary 105 6. Network self-organization in a liver-inspired lattice model 107 6.1. Cubic lattice geometry motivated by liver tissue 107 6.2. Effective energy for local network segment interactions 110 6.3. Characterizing network structures in the cubic lattice geometry 113 6.4. Local interaction rules generate macroscopic network structures 115 6.5. Effect of mutual repulsion between unlike segment types on network structure 118 6.6. Summary 121 7. Discussion and Outlook 123 A. Appendix 127 A.1. Mean field theory fo the isotropic-uniaxial nematic transition 127 A.2. Distortions of the Mollweide projection 129 A.3. Shape parameters for basal membrane around hepatocytes 130 A.4. Randomized control for network segment anisotropies 130 A.5. The dihedral symmetry group D2h 131 A.6. Relation between orientational order parameters and elements of the super-tensor 134 A.7. Formal separation of molecular asymmetry and orientation 134 A.8. Order parameters under action of axes permutation 137 A.9. Minimal integrity basis for symmetric traceless tensors 139 A.10. Discretization of distortion free energy on cubic lattice 141 A.11. Metropolis Algorithm for uniaxial cell polarity coordination 142 A.12. States in the zero-noise limit of the nearest-neighbor interaction model 143 A.13. Metropolis Algorithm for network self-organization 144 A.14. Structural quantifications for varying values of mutual network segment repulsion 146 A.15. Structural quantifications for varying values of self-attraction of network segments 148 A.16. Structural quantifications for varying values of cell demand 150 Bibliography 152 Acknowledgements 17

Biaxial Nematic Order in Liver Tissue

Understanding how biological cells organize to form complex functional tissues is a question of key interest at the interface between biology and physics. The liver is a model system for a complex three-dimensional epithelial tissue, which performs many vital functions. Recent advances in imaging methods provide access to experimental data at the subcellular level. Structural details of individual cells in bulk tissues can be resolved, which prompts for new analysis methods. In this thesis, we use concepts from soft matter physics to elucidate and quantify structural properties of mouse liver tissue. Epithelial cells are structurally anisotropic and possess a distinct apico-basal cell polarity that can be characterized, in most cases, by a vector. For the parenchymal cells of the liver (hepatocytes), however, this is not possible. We therefore develop a general method to characterize the distribution of membrane-bound proteins in cells using a multipole decomposition. We first verify that simple epithelial cells of the kidney are of vectorial cell polarity type and then show that hepatocytes are of second order (nematic) cell polarity type. We propose a method to quantify orientational order in curved geometries and reveal lobule-level patterns of aligned cell polarity axes in the liver. These lobule-level patterns follow, on average, streamlines defined by the locations of larger vessels running through the tissue. We show that this characterizes the liver as a nematic liquid crystal with biaxial order. We use the quantification of orientational order to investigate the effect of specific knock-down of the adhesion protein Integrin ß-1. Building upon these observations, we study a model of nematic interactions. We find that interactions among neighboring cells alone cannot account for the observed ordering patterns. Instead, coupling to an external field yields cell polarity fields that closely resemble the experimental data. Furthermore, we analyze the structural properties of the two transport networks present in the liver (sinusoids and bile canaliculi) and identify a nematic alignment between the anisotropy of the sinusoid network and the nematic cell polarity of hepatocytes. We propose a minimal lattice-based model that captures essential characteristics of network organization in the liver by local rules. In conclusion, using data analysis and minimal theoretical models, we found that the liver constitutes an example of a living biaxial liquid crystal.:1. Introduction 1 1.1. From molecules to cells, tissues and organisms: multi-scale hierarchical organization in animals 1 1.2. The liver as a model system of complex three-dimensional tissue 2 1.3. Biology of tissues 5 1.4. Physics of tissues 9 1.4.1. Continuum descriptions 11 1.4.2. Discrete models 11 1.4.3. Two-dimensional case study: planar cell polarity in the fly wing 15 1.4.4. Challenges of three-dimensional models for liver tissue 16 1.5. Liquids, crystals and liquid crystals 16 1.5.1. The uniaxial nematic order parameter 19 1.5.2. The biaxial nematic ordering tensor 21 1.5.3. Continuum theory of nematic order 23 1.5.4. Smectic order 25 1.6. Three-dimensional imaging of liver tissue 26 1.7. Overview of the thesis 28 2. Characterizing cellular anisotropy 31 2.1. Classifying protein distributions on cell surfaces 31 2.1.1. Mode expansion to characterize distributions on the unit sphere 31 2.1.2. Vectorial and nematic classes of surface distributions 33 2.1.3. Cell polarity on non-spherical surfaces 34 2.2. Cell polarity in kidney and liver tissues 36 2.2.1. Kidney cells exhibit vectorial polarity 36 2.2.2. Hepatocytes exhibit nematic polarity 37 2.3. Local network anisotropy 40 2.4. Summary 41 3. Order parameters for tissue organization 43 3.1. Orientational order: quantifying biaxial phases 43 3.1.1. Biaxial nematic order parameters 45 3.1.2. Co-orientational order parameters 51 3.1.3. Invariants of moment tensors 52 3.1.4. Relation between these three schemes 53 3.1.5. Example: nematic coupling to an external field 55 3.2. A tissue-level reference field 59 3.3. Orientational order in inhomogeneous systems 62 3.4. Positional order: identifying signatures of smectic and columnar order 64 3.5. Summary 67 4. The liver lobule exhibits biaxial liquid-crystal order 69 4.1. Coarse-graining reveals nematic cell polarity patterns on the lobulelevel 69 4.2. Coarse-grained patterns match tissue-level reference field 73 4.3. Apical and basal nematic cell polarity are anti-correlated 74 4.4. Co-orientational order: nematic cell polarity is aligned with network anisotropy 76 4.5. RNAi knock-down perturbs orientational order in liver tissue 78 4.6. Signatures of smectic order in liver tissue 81 4.7. Summary 86 5. Effective models for cell and network polarity coordination 89 5.1. Discretization of a uniaxial nematic free energy 89 5.2. Discretization of a biaxial nematic free energy 91 5.3. Application to cell polarity organization in liver tissue 92 5.3.1. Spatial profile of orientational order in liver tissue 93 5.3.2. Orientational order from neighbor-interactions and boundary conditions 94 5.3.3. Orientational order from coupling to an external field 99 5.4. Biaxial interaction model 101 5.5. Summary 105 6. Network self-organization in a liver-inspired lattice model 107 6.1. Cubic lattice geometry motivated by liver tissue 107 6.2. Effective energy for local network segment interactions 110 6.3. Characterizing network structures in the cubic lattice geometry 113 6.4. Local interaction rules generate macroscopic network structures 115 6.5. Effect of mutual repulsion between unlike segment types on network structure 118 6.6. Summary 121 7. Discussion and Outlook 123 A. Appendix 127 A.1. Mean field theory fo the isotropic-uniaxial nematic transition 127 A.2. Distortions of the Mollweide projection 129 A.3. Shape parameters for basal membrane around hepatocytes 130 A.4. Randomized control for network segment anisotropies 130 A.5. The dihedral symmetry group D2h 131 A.6. Relation between orientational order parameters and elements of the super-tensor 134 A.7. Formal separation of molecular asymmetry and orientation 134 A.8. Order parameters under action of axes permutation 137 A.9. Minimal integrity basis for symmetric traceless tensors 139 A.10. Discretization of distortion free energy on cubic lattice 141 A.11. Metropolis Algorithm for uniaxial cell polarity coordination 142 A.12. States in the zero-noise limit of the nearest-neighbor interaction model 143 A.13. Metropolis Algorithm for network self-organization 144 A.14. Structural quantifications for varying values of mutual network segment repulsion 146 A.15. Structural quantifications for varying values of self-attraction of network segments 148 A.16. Structural quantifications for varying values of cell demand 150 Bibliography 152 Acknowledgements 17