Thesis
Reference
Numerical Modelling of Confluent Cell Monolayers : Study of Tissue Mechanics and Morphogenesis
MERZOUKI, Fatma Aziza
Abstract
In this thesis, we present the numerical model of confluent cell monolayers that we developed in order to study the interplay between cell biophysical properties and tissue mechanics and morphology. Our cell based model combines both cell physics and biology; it includes the representation of cell mechanics, the application of external constraints, as well as the simulation of cell proliferation and signalling. This model was inspired from the vertex model of Farhadifar et al. 2007, which we adapted and progressively extended within our EpiCells framework. Our model allowed us to investigate (i) how cell mechanical properties affect the response of tissues to stretching and how cells may adapt to the resulting strain, (ii) the development of the spine follicles covering the lower back of Acomys Dimidiatus, and (iii) tissue buckling. Finally, we extended our 2D model to the 3D space and presented its application for further studies of tissue folding.
MERZOUKI, Fatma Aziza. Numerical Modelling of Confluent Cell Monolayers : Study of Tissue Mechanics and Morphogenesis . Thèse de doctorat : Univ. Genève, 2018, no. Sc.
5210
URN : urn:nbn:ch:unige-1064341
DOI : 10.13097/archive-ouverte/unige:106434
Available at:
http://archive-ouverte.unige.ch/unige:106434
Disclaimer: layout of this document may differ from the published version.
UNIVERSITÉ DE GENÈVE FACULTÉ DES SCIENCES
Département d’informatique Professeur B. Chopard
Numerical Modelling of Confluent Cell Monolayers : Study of Tissue Mechanics and
Morphogenesis
THÈSE
présentée à la Faculté des sciences de l’Université de Genève pour obtenir le grade de Docteur ès sciences, mention informatique
par
Fatma Aziza MERZOUKI
de Alger (Algérie)
Thèse N◦5210
GENÈVE
Repro-Mail - Université de Genève 2018
To my family...
Acknowledgements
This thesis is dedicated to my late and very missed grandfather Taoufik Zouaï who truly believed in me and who has supported me during my early years of studies.
This thesis would not have been possible without the support of many people.
I would like to acknowledge the precious guidance and advices of my supervisors, Prof. Bastien Chopard and Dr. Orestis Malaspinas. Thank you Bastien for giving me the opportunity to join the Scientific and Parallel Computing (SPC) group and for offering me a PhD position. Thank you Orestis for your time and patience. Thanks to both of you for enabling me to benefit from your experience and for sharing with me your skills. I am deeply grateful for your help and trust.
Thank you Prof. Jaap Kaadorp, Prof. Michel Milinkovitch and Prof. Orélien Roux for allowing me to defend my thesis and for your stimulating feedback. I am honoured that you have accepted to be part of my PhD committee.
A sincere acknowledgment goes to our biologist and biophysicist collaborators of the EpiPhysX project, Prof. Michel Milinkovitch, Dr. Athanasia Tzika, Antonio Martins, Prof. Orélien Roux, Dr. Anastasiya Trushko, Ilaria Di Meglio, Prof. Marcos Gonzalez-Gaetan, Prof. Andreas Wagner and Dr. Charles De Santana. Our work would not have been possible without your laboratory studies, our stimulating discussions and your precious feedback.
Of course, thank you also to all my past and present colleagues of the SPC group, Jean-Luc, Pierre, Paul, Gregor, Federico, Raphael, Jonathan, Mohamed, Reto, Anthony, Alex, Jayro, Jonas, Christophe, Christos, Xavier, Yann, Sha Li and Francesco, for your sympathy and friendship. I enjoyed our discussions during the lunch and coffee breaks, with their wide variety of subjects, from the most geeky and scientific ones to the most unbelievably random ones :D
I would also like to thank all the administrative and IT teams of the CUI (Centre Universitaire d’Informatique). Thanks a lot Anne-Isabelle Giuntini, Elie Zagury, Nicolas Mayencourt and Daniel Agulleiro for your help during these years.
I can not forget to acknowledge all the professors of the CUI since my Bachelor studies at the University of Geneva for their impact and inspiration. Thank you Prof. Didier Buchs, Prof. Alexandros Kalousis, Prof. Pierre Leone, Prof. Stéphane Marchand-Maillet, Prof. Thierry Pun, Prof. José Rolim and Prof. Sviatoslav Voloshynovskiy for sharing your knowledge and experience with generosity.
Acknowledgements
Finally, a special thanks goes to my family, particularly to my parents, Assia and Mohamed, and my brothers, Sid-Ali and Toufik, for always supporting and encouraging me. I am very fortunate and grateful to have you. Thank you also to my husband’s family, Azza, Adel and Nesma. And last but certainly not least, thanks a lot to my husband, Hisham, for your continuous support, patience and precious help.
Geneva, May 2018 F. A. M.
Abstract
In this thesis, we present the numerical model of confluent cell monolayers that we developed and implemented in order to study the interplay between cell biophysical properties on the one hand and tissue mechanics and morphology on the other hand. Our model is cell-based and combines both cell physics and biology; it includes the representation of cell mechanical properties, the application of external mechanical constraints, as well as the simulation of cell proliferation and signalling. This model was initially inspired from the vertex model of Farhadifar et al. 2007, which we adapted and progressively extended within ourEpiCells framework (www.epicells.unige.ch) along with our different studies and collaborations.
Our model allowed us to investigate how cell mechanical properties, namely the cellular apical contractility and the intercellular adhesion, affect the response of tissues to stretching.
We were able to compare our simulations of tissue stretching to the experiments of Harris et al. 2012 and to integrate their experimental measurements to calibrate our model. The calibration results suggested how cell mechanical properties should adjust to tissue stretching in order to match the mechanics of cultured epithelium.
Moreover, we used our EpiCells framework to study the development of the spine follicles covering the lower back ofAcomys Dimidiatus(also called spiny mouse). We focused on how the forces generated by the signal-based proliferation of cells shape the spine follicles. We investigated the factors that drive the Dermal Papilla (DP) cells, located at the center of the follicle, to flatten and slightly off-center between the embryonic stages E32 and E36 (32 and 36 days after fertilisation), while Matrix cells at the periphery of the follicle proliferate yielding an enlarged follicle.
We also used the vertex model to study tissue buckling. We simulated cross-sections of circular cell-monolayers and showed how cell mechanics affect the geometry and the relaxation times of cell monolayers, which are characteristic of buckling tissues. We showed that it is the competition between the cell monolayer relaxation and the cell proliferation that controls the buckling of unconstrained tissues. Moreover, in the context of our collaboration with the Roux Lab where epithelial cell monolayers are cultured inside hydrogel microcapsules, we also investigated the folding of simulated tissues growing under the constraints of an elastic environment.
Finally, we extended our initial 2D model to the 3D space and presented its application for further studies of tissue folding.
Acknowledgements
Key words: vertex models, numerical simulations, confluent cell monolayers, mechanics, external constraints, stretching, signalling, proliferation, buckling, 3D space model
Résumé
Cette thèse présente le modèle numérique que nous avons développé et implémenté afin de simuler des mono-couches de cellules et d’étudier la relation entre la biophysique des cellules d’une part, et les propriétés mécaniques et morphologiques des tissus d’autre part. Notre modèle combine la physique et la biologie des cellules ; il permet de représenter la mécanique des cellules, l’application de contraintes externes, la déformation, ainsi que la prolifération et la signalisation cellulaire. Notre modèle cellulaire a été initialement inspiré par le modèle à sommets de Farhadifar et al. 2007, que nous avons adapté et progressivement étendu au sein de notre frameworkEpiCells(www.epicells.unige.ch) au cours de nos différentes études et collaborations.
Ce modèle nous a permis d’examiner la façon dont les propriétés mécaniques des cellules, i.e. leur contractilité apicale et l’adhésion intercellulaire, affectent la réponse des tissus à l’étirement. Nous avons pu comparer nos simulations d’étirement de tissus aux expériences effectuées par Harris et al. 2012 et avons pu calibrer notre modèle de sorte que nos résultats numériques correspondent aux mesures expérimentales. Les résultats de la calibration ont suggéré la façon dont les cellules devraient ajuster leurs propriétés mécaniques en réponse à l’étirement du tissu afin de reproduire la mécanique de l’épithélium cultivé in vitro.
De plus, nous avons utilisé le framework EpiCells afin d’étudier le développement des follicules épineux qui recouvrent le bas du dos d’Acomys Dimidiatus(aussi connue sous le nom de souris épineuse). Nous nous sommes concentrés sur la façon dont les forces générées par la prolifération cellulaire controlée par la diffusion d’un signal permettent de façonner les follicules épineux. Nous avons examiné les facteurs qui entrainent la papille dermique, située au centre du follicule, à s’aplatir et à se décentrer légèrement entre les phases embryonaires E32 et E36 (32 et 36 jours après la fertilisation), alors que les cellules de la Matrice prolifèrent en périphérie et mènent à l’élargissement du follicule.
Nous avons également utilisé le modèle à sommets afin d’étudier le flambement des tissus.
Nous avons simulé des coupes transversales de mono-couches cellulaires et nous avons montré comment la mécanique des cellules affecte la géométrie et le temps de relaxation des couches cellulaires, caractérisant tous deux le flambement des tissus. Nous avons montré que c’est la compétition entre la relaxation de la couche de cellules et la prolifération cellulaire qui contrôle le flambement des tissus en l’absence de contraintes extérieures. De plus, dans le contexte de notre collaboration avec le Laboratoire du Professeur Roux, où des mono-couches cellulaires sont cultivées à l’intérieur de microcapsules d’hydrogel, nous avons également examiné la déformation des tissus simulés lorsque ces derniers se développent sous les
Acknowledgements
contraintes d’un environnement élastique.
Enfin, nous avons étendu notre modèle initialement 2D à l’espace 3D et nous avons présenté certaines de ses applications pour l’étude plus poussée du flambement des tissus.
Mots clefs : Modèles à sommets, simulations numériques, mono-couches cellulaires, méca- nique, contraintes externes, étirement, signalisation, prolifération, flambement, modèle 3D
Contents
Acknowledgements i
Abstract (English/Français) iii
List of figures xi
1 Introduction 1
1.1 State of the art . . . 3
1.1.1 Epithelial cell packing . . . 4
1.1.2 Single cell and tissue mechanics . . . 4
1.1.3 Buckling . . . 6
1.2 Numerical models . . . 6
1.2.1 Cellular Potts Models . . . 6
1.2.2 Boundary Cell Models . . . 7
1.2.3 Vertex Models . . . 7
1.2.4 3D models . . . 8
1.3 Thesis outline . . . 8
2 2D vertex-model of cell monolayers 11 2.1 Cells and tissue representation . . . 11
2.2 Model dynamics . . . 12
2.3 Cell proliferation . . . 15
2.4 Cell signalling . . . 16
2.5 External mechanical constraints . . . 17
2.5.1 Elastic environments . . . 18
2.5.2 Stretching constraints . . . 19
2.6 Topological changes . . . 20
2.6.1 Neighbour cells exchange - T1 . . . 20
2.6.2 Cell extrusion - T2 . . . 20
2.7 Boundary management . . . 21
2.8 Summary . . . 21
Contents
3 EpiCells: Our framework for modelling confluent cell monolayers 23
3.1 Cell monolayer description and dynamics . . . 24
3.1.1 Computational complexity . . . 25
3.2 External force application and environments . . . 27
3.3 Cell types management . . . 28
3.4 Signalling . . . 28
3.5 Cell proliferation . . . 29
3.6 Topological transitions . . . 31
3.7 Perspectives . . . 31
3.7.1 Parallelisation . . . 31
3.7.2 Extension to 3D space . . . 36
4 Cell physical properties and tissue mechanics 39 4.1 Cell monolayers model . . . 39
4.2 Stretching simulations and elastic properties computation . . . 42
4.3 Impact of cell mechanical properties on tissues elasticity . . . 44
4.3.1 Comparison to experimental observations . . . 45
4.4 Tissue response to different stress loading intensities . . . 46
4.4.1 Creep responses . . . 46
4.4.2 Stress-strain curves . . . 47
4.5 Parameter estimator algorithm . . . 49
4.5.1 Hypotheses . . . 49
4.5.2 Iterative process . . . 50
4.5.3 Example . . . 51
4.6 Strain-dependent model parameters . . . 52
4.7 Discussion . . . 54
5 Acomysspine morphogenesis 57 5.1 Experimental observations . . . 57
5.2 Simulation ofAcomysspine follicle development . . . 59
5.2.1 Modelling the follicle cross-section . . . 59
5.2.2 Different cell types . . . 60
5.2.3 Cell signalling . . . 61
5.2.4 Signal-based proliferation . . . 62
5.3 Effect of signal-based proliferation on DP flattening . . . 62
5.4 Effect of signal profile on DP off-centring . . . 63
5.5 Summary and discussion . . . 67
6 Buckling of growing cell monolayers 69 6.1 Simulation of a spherical cell monolayer cross-section . . . 70
6.2 Influence of cell mechanical properties on buckling . . . 71
6.3 Effect of proliferation rate and tissue size on buckling . . . 74
6.4 Discussion . . . 76
Contents
7 Buckling under spherical confinement constraints 79
7.1 In vitro experiments and observations . . . 79
7.2 Model . . . 81
7.2.1 Cell monolayer cross-section . . . 81
7.2.2 Heterogeneous elastic capsules . . . 82
7.3 Folding quantification . . . 83
7.4 Simulations and results . . . 85
7.4.1 Effect of cell mechanics on buckling . . . 85
7.4.2 Effect of confining environment . . . 90
7.5 Conclusion . . . 91
8 3D model of cell monolayers and applications 93 8.1 Model transition from the 2D to the 3D space . . . 93
8.1.1 Cell monolayer representation . . . 93
8.1.2 Energy function . . . 94
8.1.3 Force function . . . 95
8.1.4 Cell proliferation . . . 96
8.1.5 Topological transitions . . . 97
8.1.6 External environments and constraints . . . 97
8.2 Simulations . . . 97
8.2.1 Spherical cell monolayers and cell packing . . . 97
8.2.2 Buckling under compressive forces . . . 98
8.2.3 Buckling due to confined proliferation . . . 99
8.2.4 Cylindrical cell monolayer . . . 99
8.3 Summary and discussion . . . 100
9 Conclusion 103 A Model dynamics 107 A.1 Verlet integration method . . . 107
A.2 Dimensionless dynamics . . . 108
B The EpiCells framework 111 B.1 Cell cluster and dynamics . . . 111
B.2 External environments . . . 112
C Buckling of growing epithelium under confinement 113 C.1 Effect of cell mechanical properties on constrained buckling . . . 113
C.2 Pressure estimation . . . 115
C.3 Cell elasticity . . . 116
C.4 Bending rigidity . . . 117 C.5 Effect of non-slipping force between the capsule’s inner surface and the cells . 120
Contents
D Cell fronts roughness 123
D.1 Experiments . . . 123 D.2 Simulations . . . 124
Bibliography 136
List of Figures
2.1 (a) Apical view of a spherical epithelium monolayer. (b) Cross-section view of
the same cell monolayer. (c) Schema of an epithelial cell monolayer. . . 12
2.2 Polygonal representation of cells with common edges shared between neighbour cells. (a) Apical model. (b) Transversal model. . . 13
2.3 Open boundary conditions. . . 14
2.4 In the transversal model, all cells have two boundary edges, apical and basal, and two bulk edges that are at the interface with the neighbour cells. . . 14
2.5 Cell proliferation when modelling the apical view of a cell monolayer. . . 16
2.6 Cell proliferation when modelling a cell monolayer cross-section. . . 16
2.7 Signal propagation. . . 17
2.8 External mechanical constraint generated by the elastic shell encapsulating the tissue.The red ring displayed in dashed-lines represents the resting elastic capsule. 19 2.9 External mechanical constraints generated by the elastic walls surrounding the tissue. . . 19
2.10 External stretching forces applied on the tissue along the X axis. . . 20
2.11 T1 transitions. (a) T1 around bulk (top) and boundary (bottom) edges. (b) Succession of T1 underlying cell migration through the tissue. The vertices of the red cell are subjected to an external horizontal forceFext=2e−7N. . . 21
2.12 T2 transition on a 3-sided cell. . . 21
2.13 Boundary management to avoid cell overlapping. . . 22
3.1 EpiCells code structure. . . 23
3.2 Thecoremodule describes a cell cluster and its dynamics. It defines the notions of cells, edges, vertices, cell cluster and dynamics. C++ templates are used to parametrise the dimensionality of the model. For 2D space models,d=2. . . . 24
3.3 Sequence diagram showing how the forces applied on vertices are computed and how vertices move consequently over time. . . 27
3.4 Abstract environment classes. . . 27
3.5 ThecellTypesmodule defines the different cell types and their relation to each others. . . 28
3.6 Thesignalingmodule allows the propagation of a signal through the tissue. . . 29
3.7 Thedivisionmodule allows the selection of cells entering mitosis, their growth and their division. . . 30
List of Figures
3.8 Thetransitionmodule allows the occurrence of topological transitions. T1 transi- tions rearrange cells around edges whose lengths drop bellow a threshold length.
T2 transitions remove three-sided cells whose areas drop bellow a threshold area. 31
3.9 Cell distribution over 4 processors. . . 32
3.10 T1 transition in parallel. . . 36
4.1 Counterclockwise ordering of cell vertices. . . 41
4.2 Relaxed tissue and stretching forces applied through boundary vertices. . . 42
4.3 Resulting tissue after stretching along theX-axis. . . 43
4.4 Mechanical properties of a simulated tissue as a function of the normalised cell perimeter contractility ¯Γand normalised line tension ¯Λ, for a stress ¯σ=0.0738. 45 4.5 Strain versus time of a simulated cell monolayer subject to different stress loadingσ, and comparison with experimental data[1]. The parameters of the simulation areK =2.67·109 N/m3, A0=70·10−12 m2, ( ¯Γ=0.16, ¯Λ= −1.10), δt=0.005sec,η=10sec−1, andm=10−5kg. . . 47
4.6 Stress-strain curves for different cell monolayers simulated with our model and compared to the experimental average stress-strain curve of cultured cell monolayers[1]. . . 48
4.7 Organigram of the iterative process for model parameter estimation. . . 50
4.8 Ratio between the steady area of cells and their preferred area as a function of the normalised line tension ¯Λat cell interfaces and the normalized contractility Γ¯of cells. . . 51
4.9 Experimental values of applied stress and resulting strain[1]. . . 52
4.10 Change of the cell mechanical properties with an increasing tissue strain. . . . 53
4.11 Stress-Strain curves. Black dots: experimental results. Red curve: model with constant parameters (K=2.256·109N/m3,A0=70·10−12m2,Λ= −1.7·10−6N andΓ=0.0298N/m). Green line: model with strain-dependent values forKand Γ, andA0=70·10−12m2,Λ= −1.7·10−6N. . . 54
5.1 Acomysspines versus laboratory mouse hairs [2]. . . 57
5.2 Transverse section ofAcomys Dimidiatusat E32, E34 and E36 embryonic skin showing the changing shape of the Dermal Papilla [2]. . . 58
5.3 Diagram illustrating the hypothesis made on the mechanisms underlying the symmetry break of the spiny hairs follicles morphology. . . 58
5.4 Initial circular aggregate of cells. At this point, all the cells are from the same type and are assigned with the same mechanical properties. . . 60
5.5 Numerical modelling of a follicle’s cross-section. Two types of cells are repre- sented, the DP (coloured in red) at the centre and the Matrix cells (coloured in blue) at the periphery. . . 60
5.6 Signal propagation. Signal gradient over the follicle cross-section. . . 61
5.7 Signal-based proliferation. Probabilitypof a cell to enter mitosis as a sigmoid function of the signalsit senses. The sigmoid is parametrised by (st hr esh,λs). . 62
List of Figures
5.8 DP axes. The longest and shortest axes are represented by arrows coloured in white and yellow, respectively. The estimated lengths of DP along the longest and shortest axes,Landl, are indicated. Clickhereto watch the video of a simulated spine follicle development or follow the link:https://youtu.be/WA6IVQS4ANY. 63
5.9 Signal-based proliferation and its effect on DP elongation. . . 64
5.10 Simulated transverse sections of hairs follicles. . . 64
5.11 Signal intensity profiles. Signal intensity as a function of the absolute and relative distance of cells to the signal source. . . 65
5.12 DP elongation and off-centring within different signal-based proliferation pa- rameter regions (st hr esh,λs). . . 66
5.13 Examples of DP off-centring obtained with different signal profiles scaling and non-scaling with the follicle size. . . 66
6.1 Confocal scan of an epithelium monolayer, which was cultured inside a hydrogel microcapsule, and that starts buckling. . . 69
6.2 Numerical and experimental models of an epithelium monolayer. . . 70
6.3 Representation of forces applied by one cellαon its vertices. . . 71
6.4 Cell proliferation when modelling a cell monolayer cross-section. . . 72
6.5 Diagrams showing how cell contractility ¯Γand intercellular adhesion ¯Λaffect the tissue’s cross-section buckling . . . 73
6.6 Diagrams showing how cell contractility ¯Γand intercellular adhesion ¯Λaffect the tissue’s cross-section thickness and circumference. . . 74
6.7 Influence of cell contractility ¯Γand intercellular adhesion ¯Λon the time needed by the tissue’s cross-section to relax after one cell division, which correlates with tissue buckling. . . 75
6.8 Diagram showing how the probability of a cell to enter mitosispmi t osi sand the number of cell divisions control whether the tissue end up smooth or folded. All these simulations started from a circular monolayer made up of 20 cells. . . 75
6.9 Diagram showing how the probability of a cell to enter mitosispmi t osi sand the number of cell divisions control whether the tissue end up smooth or folded. All these simulations started from a circular monolayer made up of 60 cells. . . 76
7.1 Cross-section of a cell monolayer cultured inside a hydrogel micrcapsule. . . . 80
7.2 Illustration of a capsule cross-section showing its inner radiusRi nc ap sand outer radiusRc ap sout . . . 80
7.3 Cross-section of a cultured cell monolayer pulling on the capsule while folding. 81 7.4 Confining environment constraints. (a) Elastic constraints. (b) Non-slipping forces. . . 83
7.5 (a) Dynamics of a cell monolayer contour growing under spherical confinement constraints. (b) Corresponding contour profiles. . . 84
7.6 Identification of cell monolayer detachment. . . 85
7.7 Tissue buckling within different parameter regions. . . 87
7.8 Fold width and positions within different parameter regions. . . 88
List of Figures
7.9 Early cell monolayer detachment from the capsule, before later actual folding.
Video:https://youtu.be/6bjZnPKrB6o. . . 89
7.10 Evolution over time of the external force applied by the elastic capsule on the growing tissue cross-section. . . 89
7.11 Effect of non-slipping forces on cell monolayer folding. Video:https://youtu.be/ C73R6kLNfQk. . . 90
8.1 Growth of a spherical cell monolayer from an initial cubical configuration. Video: https://youtu.be/ghqohXO17jE. . . 98
8.2 Compressed cell monolayer along the X axis. Video:https://youtu.be/1wgZ8-kh9a8. 99 8.3 Confined cell proliferation yields buckling. Video:https://youtu.be/cyZBR7g95D0.100 8.4 Simulation of a growing tube after 2000 cell divisions. Video:https://youtu.be/ ACtmOZ9FvKk. . . 101
B.1 (a) Specification of theCellClusterclass. (b) Specification of the abstract class Dynamics. . . 111
B.2 Inheritance diagram of the abstract classElasticEnvironment. . . 112
B.3 Inheritance diagram of the abstract classStickyEnvironment. . . 112
C.1 Average fold width for different cell and tissue mechanical properties. . . 113
C.2 Average cell aspect ratio when the first tissue buckling is detected, for different cell and tissue mechanical properties. . . 114
C.3 Number of cells when the first tissuedetachmentfrom the capsule is detected, for different cell and tissue mechanical properties. . . 114
C.4 (a) Number of cells when the first fold is detected withFN S=10−3. (b) Number of cells when the first fold is detected with increased non-slipping forcesFN S= 10−2. Larger friction enhances and accelerates folding. . . 120
C.5 (a) Distribution of fold widths withFN S=10−3. (b) Same figure with increased non-slipping forcesFN S =10−2. The fold’s width measure used here is the distance (in meters) between the two points delimiting the fold. Larger friction increases folds’ width. . . 121
C.6 (a) Number of folds withFN S=10−3. (b) Number of folds with increased non- slipping forcesFN S=10−2. Larger friction enhances folding. . . 121
D.1 Cultured rat epithelial cell fronts. . . 123
D.2 Experimental measurements of epithelial cell front roughnessBas a function of the length scaler[3]. . . 124
D.3 (a) Experimental (top) and simulated (bottom) proliferating cell fronts. (b) Roughness measureB(r) of simulated cell fronts [3]. Clickhereandhereto watch videos of simulated proliferating cell fronts, or follow the links: https: //youtu.be/9FL1Weq8udMandhttps://youtu.be/w6prbMsXZ2w. . . 124
D.4 Effect of cell mechanical properties on cell front roughness. . . 125
1 Introduction
Animals are complex multicellular organisms. They undergo metabolism1, maintain home- ostasis2, respond to stimuli, grow and develop, reproduce and adapt[5, 6]. However, before they become composed of trillions of cells3and being characterised by a complex anatomy and physiology, almost all animals originate from a single eukaryotic cell, the zygote4. The zygote results from the fertilisation between two gametes5, the egg and the sperm cells.
Successive cycles of mitotic divisions6, the cleavage process, form a multicellular embryo. It initially consists of a compact ball of cells, the morula. Further cell divisions and a cavitation process transform the morula into a hollow sphere of cells surrounding a fluid-filled cavity called the blastula. An important phase of early embryonic development is the gastrulation.
It reorganises the blastula from a continuous single layer of epithelial cells to a multilayered structure called the gastrula. It often starts with an invagination of the cell monolayer and results in the creation of the primary germ layers, namely the ectoderm, the mesoderm and the endoderm. From these germ layers, the internal organs develop during the following organogenesis process. Successive specialisation of the germ layers and their folding form the organs constituting the foetus. For example, rudiments of the central nervous system develop from the ectoderm in the process of neurulation. Specialised tissues thicken and fold forming the neural tube. Like the central nervous system, the heart also begins its development in the embryo as a tube-like structure derived from mesodermal germ-layer cells[7]. Through cell proliferation, specialisation and embryonic folding, the foetus begins to take shape.
Cells are the most basic structural and functional units of all known living organisms. They
1Metabolism is the life-sustaining chemical transformations of nutrients to produce energy and synthesise the compounds needed by cells.
2Homeostasis is the process in which an organism works to maintain a stable internal environment, which requires constant adjustments[4].
3The human reaching adulthood is composed of 100 trillions (1014) cells[4].
4Rare animals reproduce asexually.
5Gametes are haploid cells, which fuse during the conception of organisms that reproduce sexually. The resulting cell’s genome is the combination of the DNA of each gamete.
6Each cycle of divisions doubles the number of cells, while maintaining the size of the overall embryo equal to that of the original zygote.
Chapter 1. Introduction
are the smallest units of life, which can replicate independently. Many cells are specialised in shape and function, such as neurones, which have long projections that allow them to receive and transmit electrical and chemical signals, and muscle cells, which are packed full of mitochondria to provide the energy needed to muscle contraction.
An eukaryotic cell consists of a flexible membrane that encloses the cytoplasm, a water-based cellular fluid, and a variety of tiny membrane-bound functioning units called organelles. The nucleus is the central organelle of eukaryotic cells, as it contains the DNA carrying the genetic program of life. The cell membrane (also known as the plasma membrane) and intracellular membranes surrounding sub-cellular structures consist of lipid bilayers. They separate the inner aqueous compartment contents from their surroundings. The plasma membrane protects the cell and plays the role of a barrier regulating the movement of substances in and out of the cell. It is also involved in many cellular processes, such as cell adhesion and cell signalling7. The cell shape and the cell mechanical resistance to deformation are given by the cytoskeleton underlying the plasma membrane. The cytoskeleton is a complex network of interlinking filaments and tubules that extend throughout the cytoplasm. It is made of three kinds of filaments: the microfilaments (7nm diameter), the intermediate filaments (10nmdiameter) and the microtubules (25nmdiameter)8. Microfilaments, also known as actin filaments, act as tracks for the movement of myosin molecules. The association between the actin filaments and the myosin motors allows most contractile activity in cells. Like actin filaments, intermediate filaments, such as keratin9, function in the maintenance of the cell- shape by bearing tension. They are more stable than actin filaments. They can be stretched several times their initial length. Microtubules are fine tubular structures. They help resist compression and play an important role to set the paths along which the genetic material can be pulled during cell division, so that each new daughter cell receives the appropriate set of chromosomes[9].
Cells with similar functions are organised into tissues. There are four basic types of animal tissues: epithelial, connective, muscle and nervous tissues. Epithelium line cavities, glands and surfaces throughout the body. It protects the body and its internal organs, secretes substances such as hormones and absorbs substances such as nutrients. The skin and the intestinal system are major examples of epithelium. Connective tissue binds the cells and organs of the body together. It protects, supports and integrates all parts of the body. Muscle tissue is excitable, respond to stimulation and contract to provide movement. Nervous tissue is also excitable and allows the propagation of electrochemical signals in the form of nerve impulses that communicate between different regions of the body[10]. It is the combination of two or more tissues that form the working structures called organs, such as the lung, the heart and the brain. Functionally related organs often cooperate to form whole organ systems,
7Cell signalling is the way cell communicate with one another to regulate tissue and organ development, to control their growth and division and to coordinate their functions[8].
8In addition to determining the shape of cells, these filaments play an important role in the movement of organelles and cytoplasmic vesicles. They also allow the movement of entire cells.
9Keratin is present in epithelial cells.
1.1. State of the art
such as the respiratory system, the cardiovascular system and the nervous system.
Epithelial tissues consist of large sheets of tightly packed cells, with little or no extracellular material present between them. Epithelial cells exhibit polarity with differences in structure and function between the apical facing surface of the cell and the basal surface close to the underlying body structures. Three types of connections allow varying degrees of interaction between the cells, namely the tight junctions, the anchoring junctions and the gap junctions.
The tight junctions separate the cells into apical and basal compartments. When two adjacent epithelial cells form a tight junction, there is no extracellular space between them and the movement of substances through the extracellular space between the cells is blocked. The anchoring junctions include several types of cell junctions, namely the adherens junction, the desmosomes and the hemidesmosomes. They are common on the lateral and basal surfaces of cells where they provide strong and flexible connections. Adherens junctions connect the actin microfilaments of adjacent cells, through cadherin10molecules. Desmosomes occur in patches on the membranes of cells and attach to keratin intermediate filaments. The cadherin embedded in these patches project through the cell membrane to link with the cadherin molecules of adjacent cells. Hemidesmosomes, which look like half a desmosome, link cells to the extracellular matrix. They include adhesion proteins called integrins rather than cadherins. In contrast with the tight and anchoring junctions, gap junctions form intercellular passageways between the membranes of adjacent cells to facilitate the movement of small molecules and ions between the cytoplasm of adjacent cells[10].
Many epithelial tissues are capable of rapidly replacing damaged and dead cells. Epithelia are renewed by mitosis. Cells grow and replicate their chromosomes (during interphase), con- dense the chromosomes and initiate the formation of the mitotic spindle (during prophase), disintegrate the nuclear envelop and attach the mitotic spindle to the chromosomes (during prometaphase), align the chromosomes at the equatorial plane (during metaphase), separate and move the sister chromatids along opposite poles (during anaphase), depolymerise the spindle, form a nuclear membrane around each set of daughter chromosomes and develop a contractile ring in the peripheral cytoplasm at the cells’ equator (during telophase). The cell division completes by the constriction of the ring (cleavage furrow) during cytokinesis until the cytoplasm and its organelles are divided into two daughter cells, each with a nucleus[8].
1.1 State of the art
An important work is done to study epithelial tissues and understand the effect of cell bio- physics on the emergence of higher scale properties, such as tissue’s mechanics, shape and size. This state of the art does not aim at being exhaustive. Instead, it aims at presenting important research related to epithelial tissues, give an overview of the hot topics addressed and presents the hypothesis considered to explain epithelial behaviour and characteristics across multiple levels, molecular, cellular and multicellular.
10Cadherins are transmembrane proteins that depend on calciumC a2+ions to function.
Chapter 1. Introduction
1.1.1 Epithelial cell packing
Epithelium displays a cellular structure composed of polygonal domains separated by well defined boundaries, which ressembles that of many materials including soap froths[11, 12].
Moreover, epithelium has the ability to conserve a topological11steady state despite constant cell division and death. In 1928, Frederic T. Lewis observed for the first time the characteristic cell packing of proliferating cucumber epidermis[13] and showed a correlation between cell division and cell topology and size. Later studies [14] displayed a similarity between the cell packing of a large range of epithelial tissues from metazoa12[15] and plants epidermis[16].
First, the distribution of the cell neighbours number is unimodal with a mode at 6-sided cells13. Nearly 50% of cells have 6 neighbours, approximately 25% of cells have 5 neighbours, 20%
of cells have 7 neighbours and rare cells have 4 or 8 neighbours. Moreover, Lewis showed a correlation between the cell neighbours number and its apical cross-section area. The area of cells increases linearly with the number of their neighbours. Cells with many neighbours are expected to be larger than cells with fewer neighbours.
The mechanisms underlying the common cell organisation in epithelia ranging from certain metazoa to some plants were investigated. The latter can be explained by the way cells divide, rearrange and die (see [14] for a review). Markov chain models representing cell proliferation dynamics [17, 15, 18] managed to predict the equilibrium topology observed experimentally.
In [19], a topological model was developed to compare the cell packing resulting from mul- tiple division orientation strategies. Different division mechanisms were found to generate quantitatively different distributions of polygon types. Using, cell based models of growing epithelium, cell mechanics and cell rearrangement were also demonstrated to influence cell packing [20]. Finally, Aegerter-Wilmsen et al.[21] found that cell mechanics and division strategies that were previously proposed to match the distribution of cell neighbour num- bers actually fail to reproduce the polygon distribution of mitotic cells unless cell growth is regulated by mechanical stress.
1.1.2 Single cell and tissue mechanics
Epithelial tissues are studied at multiple levels: molecular, cellular and multicellular. An increasing number of experiments are performed to characterise the mechanics of single cells and cell monolayers [22, 23, 24, 25], and to understand the way the mechanical behaviour of tissues emerge from cellular and sub-cellular properties [1, 26]. Mechanical properties of single cells are quantified, such as their surface tension and their internal pressure, either in vitro using atomic force microscopy (AFM) [24, 27, 28, 29, 30], or in vivo by measuring (using
11By the topology, we mean the connectivity of a cell. The topology of cell corresponds to its connectivity with other cells, i.e. its neighbours.
12A metazoan is any animal that undergoes development from an embryo stage with three tissue layers, namely the ectoderm, mesoderm, and endoderm.
13A cell can be seen as a polygon with a number of edges equal to its number of neighbours. Ann-sided cell has nneighbours.
1.1. State of the art
fluorescence microscopy) the shape deformation of a droplet introduced between cells in a tissue [23]. The intercellular adhesion is also investigated experimentally using stretching experiments. The work per area required for cell-cell separation is estimated [31].
The influence of cell mechanics, such as cell contractility and intercellular adhesion, on tissue properties are studied. The influence of actin cytoskeleton and myosin contractility on tissue elasticity is investigated [1] by stretching suspended cell monolayers [26] previously perturbed at the molecular level using latrunculin B14 and Y2763215. Studies are performed on the influence of cell contractility and adhesion on the tissue surface tension. They suggest that high surface tensions result from high ratios of adhesion to cortical tension [32]. Moreover, similarly to two immiscible liquids that demix, different tissue surface tensions, which arise from differences in intercellular adhesiveness, are proposed to account for cell sorting [33].
This is the Differential Adhesion Hypothesis (DAH). In 1939, Johannes Hotfreter showed that cells have preferences in their adhesion properties; cells show adhesion affinities [34]. Cells of the same type tend to adhere to one another while differently fated cells tend to separate.
Experimental studies have also shown that cell aggregates surface tensions are a direct linear function of cadherin expression level[35].
Cells adapt to their environment, the extra-cellular matrix (ECM) [36], and to external force applications [37, 38, 39]. Forces applied on cells direct spindle orientation during mitosis [40].
Cells are found to align their mitotic spindle with their longest axis, which limits anisotropic tissue tension [41, 42]. Moreover, studies suggest that mechanical feedback along with mor- phogen signalling [43, 44] regulate tissue growth[45, 21] and can account for specific organ size determination [46]. Eventually, cell overcrowding due to proliferation and migration can induce cell extrusion to control epithelial cell numbers [47, 48]. Cell mechanics are strongly intertwined. Studies suggest that while adhesive molecules mechanically couple the cortices of adhering cells, they can not directly extend the cells surface contact during tissue compaction16. Instead, adhesive molecules can indirectly control interfacial tension by modulating local acto-myosin contractility [49]. Moreover, global cell mechanics are regu- lated by E-cadherin17-mediated force transduction signals [50]. Sensed forces influence the organisation of the cytoskeleton and the stiffness of cells. Due to the importance of forces during morphogenesis (see [51] for a review), experimental techniques for measuring forces in vivo are investigated [52]. Stresses exerted on cells can be estimated from their shapes [53]
and tissue strain along different axes during morphogenesis is measured by severing in vivo adherens junctions around a disc-shaped domain [54].
14Lantrunculin B depolymerises the actin cytoskeleton.
15Y27632 inhibits myosin contractility.
16Tissue compaction is the morphogenetic process by which a tissue adopts a tighter structure.
17E-cadherine stands for epithelial cadherin.
Chapter 1. Introduction
1.1.3 Buckling
Tissue folding is a frequent phenomenon during embryogenesis and organogenesis, from blastula gastrulation [55] to the formation of brain convolutions [56, 57, 58], gut villis [59, 60], cyst [61] and even crocodile head scales [62]. However, the causes of tissue buckling and folding are not well understood. Studies suggest that buckling results from cell mechanical properties. Bielmeier et al. demonstrated that buckling can emerge from the interface contrac- tility between differently fated cells, potentially leading to cyst formation [61]. Tamulonis et al.
showed that the blastula gastrulation can emerge from the endoderm and ectoderm being characterised by different apical cell constriction and intercellular adhesion [55]. ˜Storgel et al. proposed a mechanical model explaining epithelial folds, based on intraepithelial stresses generated by differential tensions of apical, lateral and basal cell sides as well as on the elastic- ity of the basement membrane [63, 64]. Other hypotheses suggest that tissue folding is related to its growth under mechanical constraints. Tallinen et al. suggested that gyrification arises from mechanical instabilities driven by differential growth between the grey and the white matter [56, 57]. Shyer et al. explained that the formation of the gut villis is the consequence of the endoderm expansion under compressive stresses generated by sequential differentiation of distinct smooth muscle layers of the gut [60]. In order to justify such heypotheses, physical mimics using differentially strained composites were developed. Lately, experimental mod- els, where cells are cultured inside hydrogel elastic microcapsules [65, 66], are also used to investigate the effect of mechanics on cell monolayers development.
1.2 Numerical models
Along with the experimental research, computational models of epithelium are used to inves- tigate the contribution of cell mechanics in many biological processes[67, 68], such as cell sorting [69, 70, 71, 72], wound healing [73, 74, 75], embryo development [55] and organ forma- tion [76, 77]. Various cell-based modelling approaches are commonly used for these purposes, including Cellular Potts Models, Boundary Cell Models and Vertex Models. They generally depend on three independent energy terms, related to internal cell pressure, actomyosin cortex contractility and intercellular adhesion[68], while they differ in their cell representation and dynamics.
1.2.1 Cellular Potts Models
The Cellular Potts Models (CPM) [78] are lattice-based models, which are derived from the large-Q Potts model. Space is represented as a regular lattice. Each cell occupies a number of lattice sites. Each site (i,j) is assigned a variable stateσ(i,j) corresponding to the cell ID, called the spin, or to the medium M in which cells are immersed. At each iteration of the simulation, the state of a random site (i,j) can change following the Metropolis rule. The state of the siteσ(i,j) can be assigned one of its neighbours stateσ(i0,j0). The change is accepted with a probabilityp=mi n(1,e−(Enew−Ecur r)/T), whereEcur r is the current configuration energy,Enew
1.2. Numerical models
is the energy of the configuration after changeσ(i,j)←σ(i0,j0) and T is a parameter of the simulation called the temperature. On the one hand, all changes decreasing the energy,Enew<
Ecur r, are accepted with a probabilityp=1. On the other hand, the more energy increase δE=Enew−Ecur r results from a state changes the lower its probabilityp=e−(Enew−Ecur r)/T to be allowed. State changes increasing the energy are accepted to allow mechanisms known as exploration and diversification in the domain of optimisation problem resolution. They prevent the system to get stuck into a local minimum of energy. The temperature parameter T plays an important role in this diversification process. The lower T, the smaller the probability to accept a fluctuation increasing the energy. In the basic form of CPM, the energy function depends cell areas and bonds constraints. Cells have a preferred areaA0, and the deviation of the cell area (number of sites it occupies) from its preferred area increases the system’s energy. A second energy term represents interface tensions. Cells have more or less affinity to be in contact with one another. The original CPM by Graner and Glazier allowed the simulation of cell sorting based on the differential adhesion hypothesis (DAH)[79]. Additions and improvements were made allowing cell growth, division and death [80, 81]. The advantage of CPM is their ease of implementation and their ability to represent arbitrary cell shapes.
However, they may be computationally expensive, especially when cells exert forces with non-local effects, such as apical constriction.
1.2.2 Boundary Cell Models
Boundary cell models are lattice-free models, representing cell boundaries evolving in a con- tinuous space. Each cell is described by a finely resolved elastic polygon. The movement of the polygon vertices drives the dynamics of the model. These vertices are subjected to differ- ent forces associated with the elasticity of the cell membrane and the intracellular pressure.
The integrity of the cell monolayer is provided by adhesion forces exerted between neigh- bour cells vertex-vertex or edge-vertex pairs. These models allow the simulation of complex shapes of cells, as observed during the gastrulation of Nematostella vectensis, the starlet sea anemone [55]. In immersed boundary cell models [82, 83, 84], in addition to the individual representation of deformable cells as finely resolved elastic bodies, the cytoplasm and the interstitial space are represented by a viscous incompressible fluid. Although these models allow a detailed cell representation, with neighbour cells able to adhere and detach from one another, this comes at a computational cost. A permanent detection and management of undesired overlapping of neighbour cells is needed.
1.2.3 Vertex Models
Vertex Models[85, 86] are also lattice-free models. They represent the junctional network defined by intercellular contacts. They were first used in 1978 by Honda to describe cellular patterns [87]. Cells are described by polygons sharing common edges with neighbour cells.
An energy function characterises the cell monolayer. It typically depends on the cell internal pressure, translated to a cell size (area or volume) constraint, on the cell contractility resisting
Chapter 1. Introduction
cell deformations and on intercellular tensions controlling the cell contact affinity. The dynamics of the model are then driven by the movement of the network vertices. The vertex positions evolve at each iteration of the simulation either directly to the closest minimum of the energy function, using high-dimensional minimisation methods such as the conjugate gradient method, or using Newton mechanics by previously deriving the forces exerted on each vertex from the energy function (see [86] for a review). These models usually allow topological changes due to cell division, extrusion and intercalation. They are increasingly used to help understand the mechanics underlying epithelial morphogenesis, such as the emergence of characteristic cell packing [20, 21], the formation of compartment boundaries [70, 71] and the emergence of higher scale tissue mechanics [41, 88]. Vertex models allow the effective modelling of densely packed epithelium tissues with relatively low complexity and reasonable implementation effort. It is the approach we choose to use in this thesis.
1.2.4 3D models
Although most cell-based models are two-dimensional and study planar tissues, they tend to be extended to the 3D space in order to study how cell monolayers deform out of plane.
3D apical vertex models describe the apical side of the cell monolayer as a two-dimensional surface which evolves in a three-dimensional space. These models allowed the simulation of tissue buckling due to contour constriction [89] and the construction of complex three- dimensional shapes from simple sheets of cells due to cell mechanics and cell intercalation [90, 91, 92]. More detailed 3D models allow the representation of the apical, lateral and basal surfaces of cells. These models allowed the study of epithelial shell bending [93] and the simulation of cyst formation due to enhanced interfacial tensions between differently fated cell populations [61]. Coupled with cell signalling, three-dimensional models offer the opportunity to study the development of tubular structures and branching [94, 95].
1.3 Thesis outline
In this thesis, we aim at developing and implementing vertex-models of cell monolayers in order to study the relation between cell biophysical properties and external mechanical constraints on the one hand and tissue mechanics and morphology on the other hand. The remaining of this thesis is structured as follows.
Chapter 2 presents our 2D vertex model of cell monolayers that we implement, use and extend in the remaining of this thesis. The model includes cell mechanical properties, such as cell contractility and intercellular adhesion, cell proliferation and topological rearrangements.
Moreover, it allows us to simulate the existence of different types of cells and their interaction through signalling, as well as the application of external mechanical constraints on the tissue.
Chapter 3 presents our framework for the simulation of confluent cell monolayers, called EpiCells [96]. This chapter details how we implemented our model. It presents the core module
1.3. Thesis outline
managing the cell monolayer description and dynamics, and the different modules allowing more advanced mechanisms such as cell proliferation, signalling and external constraints application.
In Chapter 4, we investigate how cell mechanical properties affect the response of tissues to stretching and inversely how tissue strain may trigger a change in cell mechanical properties.
We compare our simulations of tissue stretching to performed laboratory experiments and propose an algorithm that calibrates our model to match the experimental results.
In Chapter 5, we study the development and morphogenesis of the spine follicles covering the lower back of theAcomys Dimidiatusmouse (also called spiny mouse). We investigate the cell-level mechanisms that drive the Dermal Papilla (DP) cells, located at the centre of the follicle, to flatten and slightly off-centre between the embryonic stages E32 and E36 (32 and 36 days after fertilisation), while the surrounding Matrix cells proliferate.
Chapter 6 investigates the buckling of growing cell monolayers. Cross-sections of spherical cell monolayers are simulated. The study focuses on how cell mechanical properties and proliferation affect the tissue’s geometry, relaxation time and buckling.
Chapter 7 pursues the study presented in Chapter 6 and investigates the buckling of cell mono- layers growing inside spherical elastic environments. Our numerical results are compared to the experimental findings of the Roux Lab, where spherical cell monolayers were cultured inside hydrogel microcapsules.
Chapter 8 presents the extension of our 2D apical model of cell monolayers to the 3D space and its application to study out of plane deformations of simulated tissues and the formation of 3D cell monolayer structures.
Finally, Chapter 9 concludes this thesis by summarising our contributions and presenting our perspectives of future work.
2 2D vertex-model of cell monolayers
In this chapter, we describe our 2D vertex-model of epithelium. This model is used in the next chapters to study the elastic properties of cell monolayers when subjected to stretching, the morphogenesis ofAcomysspine follicles and the buckling of growing cell monolayers. The chapter is organised as follows. In section 2.1, we describe how cells and cell monolayers are represented by the model. In section 2.2, we present the dynamics of the model. In section 2.3, we detail the mechanism of cell proliferation. In section 2.4, we show how cells produce, spread and react to signals. In section 2.5, we describe how external mechanical constraints can be applied on the simulated tissues. In section 2.6, the topological transitions implemented in the model are presented. Finally, section 2.8 summarises the characteristics and the functionalities of our 2D vertex model of confluent cell monolayers.
2.1 Cells and tissue representation
The 2D vertex models can either represent theapicalview of a cell monolayer or thetransver- sal view of a tissue along the cell height (see Fig. 2.1). In both cases, the vertex model is based on a polygonal representation of cells. Each polygon is defined by successive vertices interconnected by straight edges. Two adjacent cells share a common edge. A confluent cell monolayer is represented by a network of vertices.
When modelling the apical side of a tissue, a cell is represented by a polygon with an arbitrary number of edges (see Fig 2.2a). Cell edges represent the surface of contact (i) between two neighbour cells or (ii) between one cell and the external environment of the tissue. If a cell is at the bulk of a tissue, the number of its edges is equal to the number of its neighbours cells.
In contrast, when a cell is at the border of a tissue, the number of its edges is larger than the number of its neighbour cells.
When modelling a transversal section of the tissue along the cell height, cells are represented by quadrilaterals and each cell has a maximum of two neighbour cells. See Fig. 2.2b.
In contrast with the apical models in [20, 41], where periodic boundaries are implemented,
Chapter 2. 2D vertex-model of cell monolayers
(a) (b)
(c)
Figure 2.1 – (a) Apical view of a spherical epithelium monolayer. (b) Cross-section view of the same cell monolayer. These experimental images are from the Roux Lab, Department of Biochemistry, University of Geneva, Switzerland. (c) Schema of an epithelial cell monolayer.
The apical sides of the cells are coloured in yellow. The lateral sides are coloured in magenta.
The cross-section plane along cell height is represented in grey.
we choose to implementopen boundaryconditions, as illustrated in Fig. 2.3. The tissue has borders that separate it from its environment (black line). A cell which is located at the border of the tissue is referred to as aboundary cell(red cells) and has at least oneboundary edge(black edges). Contrary to abulk edge(green edges), which is always shared by two neighbour cells, aboundary edgebelongs to one cell only. Boundary edges represent the interface between the cell and the tissue’s environment. The two vertices linked by aboundary edgeare referred to asboundary vertices(black points). Through these boundary vertices, external mechanical constraints can be applied on the simulated tissue. Section 2.5.2 presents examples of external mechanical contraints application. In the model of transverse sections of cell monolayers along the cell height, each cell has two boundary edges (apical and basal) and two bulk edges (lateral) shared with its neighbour cells (see Fig. 2.4).
2.2 Model dynamics
The tissue is characterised by an energy function H, which depends on the mechanical properties of cells. The energy functionHwill be detailed for each application presented in the remaining of this thesis, namely in Chapters 4, 5, 6 and 7.
2.2. Model dynamics
(a) (b)
Figure 2.2 – Polygonal representation of cells with common edges shared between neighbour cells. (a) Apical model. Cells are represented by polygons with different degrees, i.e. numbers of edges. Edges separate cells from their neighbours and from the external environment. (b) Transversal model. Circular cross-section of a cell monolayer along the cell height. Cells are modelled by quadrilaterals.
One energy term, which is common to all the models presented in this thesis, is related to the cell area elasticityHA(see Eq. 2.1). All cellsαhave a preferred areaA0α. The variation of the actual areaAαof the cell fromA0αis penalised, i.e. it increases the energy of the cell monolayer.
HA= X
al l cel l sα
Kα 2
³
Aα−A0α´2
, (2.1)
whereKαis the area elasticity coefficient of the cellα. Additional energy terms will be related to other cell mechanical properties, such as the cell’s perimeter contractility, the intercellular adhesion and the bending rigidity between neighbour cells.
In many vertex models, as [20], the dynamics are driven by the minimisation of the energy function using mathematical methods for numerical optimisation, such as the Conjugate Gradient Method. The latter allows us to follow the sequence of steady states of the tissue that result from the variation of its mechanical properties, such as the increase of the preferred area of a cell. In contrast, we use Newtonian dynamics in the same spirit as used in Tamulonis et al. [55]. The dynamics we use allows us to follow the physical time evolution of the tissue.
We can study tissues that are not necessarily in equilibrium, such as during the embryogenesis, where the cell proliferation occurs at a fast paste with respect to the time needed for the tissue relaxation.
The forceFiacting on each vertexviat a positionri=
"
xi
yi
#
is derived from the energy function H,
Fi= −d H dri
(2.2)
Chapter 2. 2D vertex-model of cell monolayers
Figure 2.3 – Open boundary conditions. In the apical model, bulk and boundary cells form the tissue. The bulk cells are coloured in yellow. All their edges are bulk edges (coloured in green), which they share with their adjacent cells. Boundary cells are coloured in red. They have at least one boundary edge (coloured in black) that separates the tissue from its external environment.
Figure 2.4 – In the transversal model, all cells have two boundary edges, apical and basal (coloured in black), and two bulk edges (coloured in yellow) that are at the interface with the neighbour cells.
2.3. Cell proliferation
As for the energy functionH, the formulas of the internal forcesFiwill be detailed for each application presented in the remaining of this thesis, in Chapters 4, 5, 6 and 7.
Newton mechanics are used to determine the acceleration dd t2r2i of the vertexvi, d2ri
d t2 = Fi
mi, (2.3)
wheremiis the mass of the vertexvi.
To determine the positionriof a vertexviat the time stept+δtbased on its accelerationdd t2r2i, the differential equation (Eq. (2.3)) is solved using a damped Verlet integration method with a time stepδt,
ri(t+δt)=(2−η·δt)·ri(t)−(1−η·δt)·ri(t−δt)+δt2·d2ri
d t2(t) (2.4)
whereηis the damping parameter controlling the viscosity of the vertex movement during the simulation.
This is equivalent to using the Störmer-Verlet integration method (see Eq. 2.5), and adding a velocity-dependent friction (see Eq. 2.6) to prevent endless oscillations of the system.
ri(t+δt)=2ri(t)−ri(t−δt)+d2ri(t)
d t2 (2.5)
d2ri
d t2 = Fi
mi−ηdri
d t (2.6)
2.3 Cell proliferation
We are interested by the morphogenesis of growing tissues. Therefore, a cell proliferation mechanism is needed in our model. It is the process by which a parent cell yields two daughter cells. Similarly to the model in [20], cells are selected to proliferate and start growing. A cell αgrows as a consequence of an increase of its preferred areaA0αby small increments. When the cell reaches a threshold areaAt hr esh, the cell is divided along a specified axis, creating two daughter cells (see Fig. 2.5 and Fig. 2.6). The division threshold areaAt hr eshis set as the double of the cell size before entering mitosis
Three main strategies are implemented to decide the orientation of the division axis of a cell.
It can be done (i) along a random direction (see Fig. 2.5), (ii) along the shortest axis of the cell, or (iii) along the apical-basal axis when simulating a cell monolayer cross-section (see Fig. 2.6).
The cell proliferation is characterised by two parameters: (i) the probability of a cell to enter
Chapter 2. 2D vertex-model of cell monolayers
Figure 2.5 – Cell proliferation when modelling the apical view of a cell monolayer. (From left to right) A cell grows in size during mitosis, due to an increase of its preferred areaA0. When the cell reaches a threshold sizeAt hr esh(doubled its size preceding mitosis), the cell is divided along a given axis (yellow dashed line). The direction of the division axis is chosen randomly in this case. Cell division creates two daughter cells (blue). The colour of the cells represents the degree of the polygons, i.e. the number of their edges. Grey corresponds to six-sided cells, blue to five-sided cells and red to seven-sided cells.
Figure 2.6 – Cell proliferation when modelling a cell monolayer cross-section. (left) Cells grow in size during mitosis (in yellow). (middle) The cells that reached a threshold size At hr esh (doubled their size preceding mitosis) are divided along their apico-basal axes (dashed line).
(right) Cell division creates two daughter cells (in red).
mitosispmi t osi sand (ii) the cell’s growth rate.pmi t osi scontrols the average number of cells entering mitosis at the same time. The growth rate controls the speed at which the cells on mitosis grow in size.
2.4 Cell signalling
In our model, cells can produce a signal, which is propagated through the tissue and potentially triggers the reaction of other cells, as their proliferation. Each cell is assigned a dimensionless amount of signals. This quantity can be related to the number of a morphogen molecule, as in [44]. This signal is produced, propagated by a diffusion process and degraded. Initially, the cells that are the source of the signal are assigned with a produced signalssour ce(t=0)>0.
The other cells start with a sensed signals(t=0)=0. See Fig.2.7a.
Over time, the signal of the source cells is kept constant,ssour ce(t+δt)=ssour ce(t), while the
