Thesis
Reference
Mechano-chemical patterning of the skin: a mathematical and statistical perspective
DE OLIVEIRA VILACA, Luis Miguel
Abstract
Ce travail se concentre, dans un premier temps, sur le développement de modèles mathématiques afin de décrire la morphogénèse des poils chez deux rongeurs spécifiques: la souris de laboratoire (Mus musculus) et la souris épineuse (Acomys dimidiatus). Par le biais de nombreuses expériences numériques, il est présenté comment le modèle proposé capture correctement la disposition de ces organes. La comparaison entre notre modèle et le système réel a été permise par le développement d'un pipeline d'analyse statistiques qui décrit quantitativement les caractéristiques spatiales de l'agencement des placodes (précurseurs à l'organe mature) sur la peau de l'embryon. L'évolution spécifique de la formation des poils chez ces deux rongeurs a ainsi été illustrée et l'organisation extraordinairement régulière de ceux-ci chez Acomys dimidiatus quantifiée. En deuxième lieu, le travail présente l'étude basée sur une vaste analyse de mues de serpents, où il a été possible d'entrevoir la diversité des formes générés par ces structures au niveau nanométrique. Les outils de la phylogénie [...]
DE OLIVEIRA VILACA, Luis Miguel. Mechano-chemical patterning of the skin: a
mathematical and statistical perspective . Thèse de doctorat : Univ. Genève, 2019, no. Sc.
Vie 13
DOI : 10.13097/archive-ouverte/unige:117483 URN : urn:nbn:ch:unige-1174838
Available at:
http://archive-ouverte.unige.ch/unige:117483
Disclaimer: layout of this document may differ from the published version.
Département de Génétique et Evolution Professeur Michel C. Milinkovitch
Mechano-Chemical Patterning of the Skin:
A Mathematical and Statistical Perspective
THÈSE
présentée aux Facultés de médecine et des sciences de l’Université de Genève pour obtenir le grade de Docteur ès sciences en sciences de la vie, mention Physique
du vivant par
Luis Miguel de Oliveira Vilaca
de
Lomar, Braga (Portugal)
Thèse N° 13
GENÈVE
Centre d’impression Uni Mail - Université de Genève 2019
process that are relevant to the question being asked.
— John Henry Holland
To my parents and brother. . .
In the following few lines, I wish to acknowledge the people who, by their help or support, allowed me to provide this important work and to defend it today.
I start by thanking the jury members Doctor Denis J. Headon, Doctor Ricardo Ruiz-Baier, Professor Bastien Chopard, Professor Michel C. Milinkovitch for reviewing and evaluating the thesis. Their feedback and questions on the presented material were very helpful and instructive. They showed that there is not just one path for doing science and that many visions can coexist and improve general research.
Special thanks to the Swiss Institute of Bioinformatics (SIB) which, by their fellowship pro- gram of 2014, provided me the financial and administrative support to achieve my PhD thesis without worries. Theconferences, courses and retreat they proposed, even if I did not use them at their full potential, were important tools to improve the quality and the experience gained by their PhD program. Furthermore, the great administrative support from Geneva University and SIB as well as their availability for questions and doubts were a plus to achieve the thesis in the best conditions. I want to thanks also all the secretary staff and to Patricia Palagi (fellowship coordinator) for all their help. Related to the fellowship program, I also want to acknowledge the Foundation Leenaards for proposing the associated grant that enabled me to work during these four years. I hope they will be rewarded by the work submitted.
A special thanks to the Swiss Institute of Bioinformatics (SIB), that by their fellowship program of 2014, provided me the financial and administrative support to perform my PhD thesis without worries. Their proposed conference, courses and retreat, even if I did not use at full potential, were important provided tools to improve the quality and the experience gained by their PhD program. Furthermore, the great administrative support from Geneva University and SIB and their availability for questions and doubts were a plus to perform the thesis in the best conditions. So even if it is not done enough, thanks for all the secretary staff and to Patricia Palagi (fellowship coordinator) for all their help. Related with the fellowship program, I want also to acknowledge the Foundation Leenaards to propose the associated grant that enabled me to work during these four years. Hope that they will be rewarded by the work submitted.
I would also like to mention and thank all the people who took their time to correct my thesis, whether in substance or in form. Thanks to Dr. Athanasia Tzika, Dr José Manuel Nunes, Sandra Reimann, Valérie Hächler, Camille Bor, Guillaume Ferahian, and (future Dr.) Anne-Cécile Ferahian.
I am then grateful to each colleague in the Milinkovitch’s lab. Even I did not have the chance to work with all of them as part of my project or tutoring, their energy, smile, and camaraderie were important to create a great environment and to bring me additional motivation to come to the lab. Even if some people already left, I thank all the people with whom I crossed path.
I will always be grateful to them and I wish them all the best for the rest of their career and life. Although all of them were important, I want to say some special words for Marcelle, my office neighbour. Thanks to all the scientific or less scientific discussions, her support during tough times in my life, her pinch of madness, and her big heart led me to particularly enjoy the years when she was in the lab. She is a great friend and one of few that I considered important in my life. Additionally, her great and clear work on the scales’ nanostructures, in which I participated, eased my personal contribution and enabled me to pick up the project even after she left the lab. So dear Marcelle, thanks a lot!
Talking about friends, I want to present my special thanks to the people that I considered im- portant in my life and who enabled me to endure the difficulties of life with a PhD thesis, when the motivation was not so big. There are so many of them, I probably I will miss some(however I did not forget you), but I want to spend some lines to acknowledge some of them. Firstly and most importantly my friends from Geneva, from the group commonly called ’La Bande’ (even if some of them do not like the name). I have known some of them for 10 years or more, and they have accompanied me in my personal evolution, during good and bad times. Even if I have sometimes more affinities with some of them, I consider all of them as part of my Geneva family and they show me every day how important it is to be surrounded by valuable people.
Thank you ’La Bande’. Secondly, I also want to give thanks to all my EPFL friends. A lot of them also did a PhD, and by looking at their current careers and thesis difficulties, they allowed me to put mine in perspective and brought indirect motivation. Alexandre N., Julien, Aurélien, Paul, Ashwin, Pamela, Alexandre S. and many others, even if we spent less time together, I am grateful for the good energy you brought me. Additionally, I want to thank Juan for his hospitality during my visit in Oxford which made the British weather more shiny and warmer, especially in November...
I also want to take time to say thank you to all the people that I met through my hobbies that are aïkido and dancing. Claude and Jacques, both my aïkido instructors, by their teaching during those many years, taught me the humility and patience necessary to learn such a complex art and those are lessons useful in everyday life. I want to particularly send some love to two beautiful persons that I met there, Muriel and Adriano. You are great role models to me and you were there during my best and worst times. Your smile, happiness and energy are a
motor that I use everyday to be the person that I am today. My travels to Japan and the WWF camps are still some of my best memories, and I wish to have the pleasure of sharing more moments with you in the future. For all the sparing partners, I am also grateful for the good energy that you brought me during so many years and which goes beyond the tatami. Next and not the least important, through dancing, a passion that has been taking a lot of my time these last few years and nowadays, I met a lot of beautiful people who enabled me to evade from life difficulties. A special acknowledgement to the people who were and stillare in the entities Iron Mams Events and Jekizz, who by their different personality, teaching and some carpooling, allowed me to meet people from different domains, with their outspokenness and especially their big heart. Despite heated and long debates (Etienne, Elodie and Camille are the masters of the discipline), this is still a pleasure to be part of such groups who enrich me each time I see them. Thanks also to many other dancers that I encountered and spent some great moments with. I kiss also the "reggeaton duo" Sandra and Laura whom I have know for many years now and with whom I spend so much time for fun, sometimes sad, stories and dances. You are the pinch of madness and good mood that people need to blossom. Many friends still need to be thanked for, but the list will be too long, so for all the people not mentioned, thanks a lot for the great moments that we shared and you are a pivot in my life. And final thanks to the two special women I spent time with in these last 5 years (they will recognize themselves). Even if life eventually separated us, I want to thank you for all the beautiful moments that we spent and your support, and wish you all the happiness and success that you deserve.
Last, but certainly not least, I would like to express my deepest gratitude to my parents and brother who have supported me all my life and pushed me to be a better person. Under any circumstances, they are persons I can always trust and receive support from in the choices I make, and for that, from the bottom of my heart, thank you. A special and ending thanks also to all my family spread out in Europe, you are my sunbeam.
Geneva, May 3, 2019 L. M. O. V.
Nature, by the assembly of forces that governs it, creates systems where the understanding requires collaboration between different actors of science. This interdisciplinary collaboration, although in its infancy only a few years ago, renders nowadays possible to solve complex and once insoluble questions in both natural or human and social Sciences. Among the many examples of cooperation, the field of mathematical and theoretical biology offers us a powerful tool to solve questions that govern both mathematics and biology fields.The blend- ing of both sciences, whose questions seemed at first glance completely different, have now offered important answers in the field of biology, such as the formation of regular and complex structures in certain living beings, the understanding of the dynamics of certain populations or the evolutionary relationship between species. With the increase in computational power and improved algorithms, mathematical and theoretical biology has become increasingly important in scientific research. This thesis is a new example of the encounter between these two sciences.
Since the work of the Scottish Sir D’Arcy Thompson Wentworth "On Growth and Form" in 1917, one of the targeted attention of mathematical biology is the understanding of the processes involved in the genesis of forms and patterns present in Nature. Among the developed models, reaction-diffusion systems have taken an important place in biology. This work, focus on one of these models to describe hair morphogenesis in two specific rodents: the laboratory mouse (Mus musculus) and the spiny mouse (Acomys dimidiatus). Hair formations in many species are formed in successive waves, which create ectodermal thickening, called placodes, which serve as precursors to the future organ. Through many numerical experiments, it has been shown how the proposed model correctly captures the disposition of these placodes during the first wave. The contrast between our model and the actual system was possible by the development of a statistical analysis pipeline that quantitatively describes the spatial characteristics of the placode arrangement on the embryo’s skin. The specific evolution of hair formation was exposed in both rodents and the extraordinarily regular organization of the hair precursors inAcomys dimidiatuswas quantified.
Despite all, the model could not explain the particular arrangement of the observedAcomys’
pelage at a later stage of development. Many studies have shown that these forms often depend on the mechanical forces involved during the development of the living being. In this thesis, it was postulated that the differences in organization between the two rodents probably
were coming from distinct properties and mechanical conditions between these species. To investigate this hypothesis, we have implemented distinct mechanochemical models were implemented to solve the associated mathematical problems, a numerical algorithm that combines both the techniques developed in finite elements and the numerical time integra- tors was created. The algorithm described in this paper had shown its reliability and flexibility for solving the equation of systems inherent to our model. Under certain conditions, it was possible to show that the introduction of these forces into our system does, indeed, offered an unobserved regularity compared to the reaction-diffusion model. In another context, and very recently, these forces were also able to show their influence in the formation of feathers in some birds. Based on experimental findings, a coupled model where skin tissue, described by a poroelastic body, mechanically and chemically interacts with a chemotaxis system of the cells involved in feather morphogenesis were presented.
In another circumstance, and at very different space range, snake scales are another example of organs that present with regular structures. Based on an extensive analysis of snake sheds, it was possible to appreciate the diversity of forms generated by these structures at the nano- metric level. The tools of phylogenetic comparative analysis were used to investigate how this diversity of morphologies as brought about. Although the exact utility of these nanostruc- tures are not fully characterise yet, it was shown statistically that some of the characters used in the description of these forms are correlated. Additionally, it was established that these nanostructures did not had a direct correlation with the environment in which these animals live. Finally, by methods of maximum likelihood, the evolution of these characters and the life habitat of those reptiles were followed.
La nature, par l’ensemble des forces qui la régissent, créée des systèmes dont leur compréhen- sion demande une collaboration entre différents acteurs de la science. Cette collaboration interdisciplinaire, bien que balbutiante il y a peu d’années de cela, permet aujourd’hui de résoudre des questions complexes, autrefois insolubles, aussi bien dans les sciences de la nature que les sciences humaines et sociales. Parmi les nombreux exemples de coopération, le champ des biomathématiques nous offre un outil puissant dans la résolution des questions qui régissent à la fois les mathématiques et la biologie. Contraction de deux sciences dont les questions semblaient à première vue totalement différentes, elles ont offert depuis quelques années des réponses dans le domaine de la biologie, telles que la formation de structures régulières et complexes chez certains êtres vivants, la compréhension de la dynamique de certaines populations ou la relation évolutive entre les espèces. Avec l’augmentation de la puissance de calcul des ordinateurs et l’amélioration des algorithmes, les biomathématiques ont pris de plus en plus d’ampleur dans la recherche scientifique. La présente thèse est un nouvel exemple de la rencontre entre ces deux sciences.
Depuis l’ouvrage de l’écossais Sir D’Arcy Wentworth Thompson «On Growth and Form» en 1917, un des centres d’attention des biomathématiques est la compréhension des processus impliqués dans la genèse des formes et motifs que la nature peut présenter. Parmi les nom- breux modèles développés, les systèmes de réaction-diffusion ont pris une place importante en biologie. Ce travail se concentre sur l’un de ces modèles afin de décrire la morphogénèse des poils chez deux rongeurs spécifiques : la souris de laboratoire (Mus musculus) et la souris épineuse (Acomys dimidiatus). La formation des poils chez de nombreuses espèces se forment par vagues successives, qui créées des épaississements ectodermique, appelés placodes, qui servent de précurseurs au futur organe. Par le biais de nombreuses expériences numériques, il est présenté comment le modèle proposé capture correctement la disposition de ces placodes au cours de la première vague. La comparaison entre notre modèle et le système réel a été permise par le développement d’un pipeline d’analyse statistiques qui décrit quantitative- ment les caractéristiques spatiales de l’agencement des placodes sur la peau de l’embryon.
L’évolution spécifique de la formation des poils chez ces deux rongeurs a ainsi été illustrée et l’organisation extraordinairement régulière de ceux-ci chezAcomys dimidiatusquantifiée.
Malgré tout, le modèle n’a pas pu expliquer l’agencement particulier du pelage d’Acomys dimidiatusobservé à un niveau de développement plus tardif. De nombreuses recherches ont
pu montrer que ces formes dépendent souvent des forces mécaniques en jeu au cours du dé- veloppement de l’être vivant. Dans cette thèse il est postulé que les différences d’organisation entre les deux rongeurs proviennent probablement de propriétés et de conditions mécaniques distinctes entre ces espèces. Pour appuyer ce propos, des modèles mécanochimiques ont été implémentés chez des cobayes afin de tester cette hypothèse. En parallèle, à partir du problème mathématique posé, un algorithme numérique qui combine à la fois les techniques développées pour les éléments finis et pour l’intégration numérique temporelle a été créé. L’al- gorithme décrit dans ce mémoire a su montrer sa fiabilité et sa flexibilité pour la résolution des systèmes d’équations inhérentes à notre modèle. Sous certaines conditions, il a été établi que l’introduction de ces forces dans notre système offre bien une régularité non observée dans le modèle de réaction-diffusion précédemment proposé. Dans un autre contexte, et très ré- cemment, ces forces ont aussi pu témoigner leur influence dans la formation des plumes chez certains oiseaux. Basé sur des conclusions expérimentales, un modèle couplé est présenté, où le tissu de la peau, décrit par un corps poro-élastique, interagit mécaniquement et chi- miquement avec le système centré sur la chemotaxie des cellules impliquées dans le processus.
Dans un autre registre, et à des échelles d’espace très différentes, les écailles des serpents sont un autre exemple d’organes qui présentent des structures régulières. Basé sur une vaste analyse de mues de serpents, il a été possible d’entrevoir la diversité des formes générés par ces structures au niveau nanométrique. Les outils de la phylogénie comparative ont été utilisés afin d’investiguer comment cette diversité des morphologies a été provoqués. Bien que l’utilité exacte de ces nanostructures soit encore sous investigation, il a été possible de montrer statistiquement que certains des caractères utilisés dans la description de ces formes présentent des dépendances. En outre, il a été établi que ces nanostructures ne présentent pas de corrélation directe avec l’environnement dans lequel vivent ces animaux. Finalement, par des méthodes de maximum de vraisemblance, l’évolution de ces caractères a pu être suivie.
Acknowledgements vii
Abstract (English/Français) xi
Contents xv
List of Figures xix
List of Tables xxv
1 Introduction 1
1.1 Mathematics in Biology . . . 1
1.2 Thesis Goals . . . 3
1.3 Main Contribution . . . 4
1.4 Thesis Structure . . . 5
2 Pattern Formation - The Biological Context 7 2.1 Skin appendages morphogenesis . . . 9
3 Pattern Formation - An Exploratory Spatial Analysis 15 3.1 Spatial Point Pattern Analysis . . . 16
3.1.1 Fourier Analysis . . . 17
3.1.2 Voronoi Diagram and Delaunay Triangulation . . . 20
3.1.3 Random Spatial Pattern . . . 24
3.2 Exploratory Spatial Analysis in Rodents’ Hair Follicle Formation . . . 52
3.2.1 Exploratory Analysis . . . 58
3.2.2 Model fitting . . . 74
3.3 Conclusion . . . 80
4 Reaction-Diffusion Models 83 4.1 RD Mechanisms and Biological Applications . . . 86
4.2 Models . . . 92
4.2.1 Rodents Hair Formation Models . . . 92
4.2.2 Coupled Layers Model . . . 95
4.2.3 Growth Model . . . 95
4.3 Numerical Algorithm . . . 96
4.3.1 Introduction to Finite Element Methods . . . 97
4.3.2 Introduction to Runge-Kutta Methods . . . 103
4.3.3 Algorithmic Details . . . 107
4.4 Numerical Experiments . . . 115
4.4.1 Reproducibility . . . 115
4.4.2 Initial Conditions Study . . . 116
4.4.3 Cross-Diffusion . . . 121
4.4.4 Growth . . . 125
4.4.5 Coupled Layers . . . 128
4.4.6 Robustness . . . 134
4.5 Preliminary Comparison betweenin silicoandin vivoPatterning . . . 136
4.6 Conclusion . . . 139
5 Mechanochemical Models 141 5.1 Continuum Mechanics - Theoretical Background . . . 143
5.2 Mechanochemical Interface Model for Skin Patterning . . . 152
5.2.1 Set of Governing Equations . . . 154
5.2.2 A Finite Element Method and its Splitting Scheme . . . 159
5.2.3 Numerical Tests . . . 162
5.3 Elastic Models for Hair Skin Primordia Patterning . . . 177
5.3.1 A Coupled Model of Navier-Lamé and Reaction-Diffusion System . . . . 178
5.3.2 Extension to Finite-Strain Neo-Hookean Solid and Growth . . . 179
5.3.3 Numerical Experiments . . . 182
5.4 Coupled Chemotaxis-Poromechanics for Primordia Patterning . . . 193
5.4.1 A coupled model of linear poroelasticity and chemotaxis . . . 194
5.4.2 Linear stability analysis and dispersion relation . . . 197
5.4.3 Extension to finite-strain poroelasticity and growth . . . 205
5.4.4 Numerical tests . . . 207
5.5 Conclusion . . . 210
6 Scale Nanostructures - A Phylogenetic Comparative Analysis 215 6.1 Nanostructures in Snakes’ Scales . . . 216
6.2 Phylogenetic Comparative Analysis . . . 221
6.2.1 Evolutionary Models . . . 223
6.2.2 Models Fitting . . . 237
6.3 Phylogenetic Mapping of Scale Nanostructure Diversity in Snakes . . . 242
6.3.1 Phylogenetic Mapping . . . 248
6.3.2 Comparative Methods . . . 256
6.4 Conclusion . . . 261
7 Conclusion 267
A Exploratory Spatial Analysis 271 A.1 Model functions . . . 271 A.2 Profile pseudo-likelihood . . . 276 A.3 Exploratory analysis . . . 279
B Reaction-Diffusion 283
C Mechanochemical Models 287
D Scale Nanostructures 291
Bibliography 309
Papers
2.1 Patterns in Nature . . . 8
2.2 Patterns Scales in Biology . . . 9
2.3 Skin Appendages . . . 10
3.1 Fourier Transform . . . 17
3.2 Fourier Transform inR2 . . . 20
3.3 Voronoi Tesselation . . . 21
3.4 Dual Graph . . . 22
3.5 Voronoi Tesselation and Delaunay Triangulation . . . 22
3.6 Local Optimisation Procedure . . . 24
3.7 Examples of Point Pattern Data Sets . . . 25
3.8 Marked Spatial Point Process . . . 25
3.9 Covariate Variable . . . 26
3.10 Complete Spatial Randomness . . . 27
3.11 Homogeneous and Inhomogeneous Poisson Point Process . . . 28
3.12 Pairwise Interaction Point Process . . . 29
3.13 Kernel Intensity Estimation . . . 32
3.14 Edge Effect Correction . . . 34
3.15 Characterisation and Summary Statistics . . . 36
3.16 Inhomogeneity and Summary Statistics . . . 37
3.17 Quadrat-Counting Test . . . 40
3.18 Quadrats-Counting Test Problem . . . 40
3.19 Simulation Envelope . . . 42
3.20 Residual Plots . . . 49
3.21 Q-Q Plots . . . 50
3.22 Pseudo-Compensator and Pseudo-Residuals Analysis . . . 53
3.23 Edge Effect for Voronoi Tessellation . . . 55
3.24 Voronoi Edge Effect Correction . . . 56
3.25 Log-Linear Trend . . . 57
3.26Mus musculusSkin Embryo and its Associated Voronoi Tesselation . . . 60
3.27Acomys dimidiatusSkin Embryo and its Associated Voronoi Tesselation for E21-E22 60 3.28Acomys dimidiatusSkin Embryo and its Associated Voronoi Tesselation for E23-E26 61 3.29 Voronoi Polygons Types Distribution . . . 62
3.30 Voronoi Polygons Types Distribution for Primary and Secondary Follicles in
Acomys dimidiatus . . . 63
3.32 Evolution of Voronoi Area Distribution for Primary Placodes forAcomys dimidiatus 64 3.33 2D Fourier Transform of the Follicles Pattern inMus musculus . . . 65
3.34 2D Fourier Transform of the Follicles Pattern inAcomys dimidiatus . . . 66
3.35 2D Fourier Transform of Primary/Secondary Follicles Pattern inAcomys dimidiatus 66 3.36 Kernel Estimation of Placode Density forMus musculus . . . 67
3.37 Kernel Estimation of Placode Density forAcomys dimidiatus . . . 68
3.38 Simultaneous Envelope Test forMus musculus. . . 69
3.39 Simultaneous Envelope Test forAcomys dimidiatus. . . 71
3.40Ki·Simultaneous Envelope Test for the Lower Back ofAcomys dimidiatus. . . . 73
3.41Ki jSimultaneous Envelope Test for the Lower Back ofAcomys dimidiatus . . . 75
3.42 Test of Independent Components for Lower BackAcomys . . . 76
3.43 Pseudo-Compensator and Pseudo-Residuals Analysis forMus musculusat E14 78 3.44 Pseudo-Compensator and Pseudo-Residuals Analysis forMus musculusat E15 79 4.1 Reaction-Diffusion Model . . . 84
4.2 Positional Information Model . . . 85
4.3 Sketch of Activator-Inhibitor Systems . . . 87
4.4 Nullclines of Gierer-Meinhardt and Schnakenberg Models . . . 88
4.5 Linear Analysis for Gierer-Meinhardt and Schnakenberg Models . . . 91
4.6 Sick Network Model . . . 93
4.7 Painter Network Model . . . 94
4.8 Dhillon Network Model . . . 94
4.9 Schema of Problem (4.18) . . . 98
4.10 Lagrangian Finite Elements forRd, withd=1 . . . 101
4.11 Different Types of Finite Elements . . . 102
4.12 Example of Stiff Equations . . . 105
4.13 Validation with Respect to the Sick and Painter Models . . . 116
4.14 Reproduction Dhillon Model . . . 117
4.15 Reproduction Dhillon Model with Growing Domain . . . 118
4.16 Sets of Regular Initial Conditions . . . 119
4.17 Effects of Initial Conditions for Different Production Rates . . . 120
4.18 Effects of Initial Conditions and Influence of the Basal Inhibitor Production . . 121
4.19 Effect of Initial Conditions for Different Diffusion Coefficients . . . 122
4.20 Effect of Initial Conditions and Comparison of a System with or without Mapping123 4.21 Linear Cross-Diffusion Pattern Space . . . 124
4.22 Linear Cross-Diffusion Pattern Space forρa=0.002 . . . 125
4.23 Nonlinear Cross-Diffusion Pattern Space . . . 125
4.24 Nonlinear Cross-Diffusion Pattern Space forρa=0.002. . . 126
4.25 Evolution of the Activator Pattern in a Square for Different Diffusion Rates . . . 127
4.26 Evolution of the Activator Pattern in a Disk for Different Diffusion Rates . . . . 128
4.27 Comparison through 2D FFT between Fixed and Growing Square Domain for Different Diffusion Rates . . . 129 4.28 Comparison through 2D FFT between Fixed and Growing Disc Domain for
Different Diffusion Rates . . . 130 4.29 Comparison between Uncoupled and Coupled Layer Reaction Diffusion System 131 4.30 Effect of Coupled Layers on the Pattern Structure . . . 132 4.31 Effect of Coupled Layers on the Pattern Structure . . . 133 4.32 Power Spectrum Built on 25 Replicates of Rhombic Layout of Hair Placodes by
Coupling Two Layers of Similar Dhillon Mechanisms . . . 134 4.33 Effect of Coupled Layers Combined with Growth on the Pattern Structure . . . 135 4.34 Effect of Coupled Non-Identical Layers on the Pattern Structure . . . 136 4.35 Replicated Simulations for Classical, Growing and Coupled Dhillon RD Model . 137 4.36 Power Spectrum Built on 25 Replicates for Classical, Growing and Coupled
Dhillon RD Model . . . 138 4.37 Spatial Analysis of Two Waves Induction based on the Dhillon Model . . . 139 5.1 General Motion of a Deformable Body . . . 144 5.2 Traction Vector . . . 147 5.3 Elementary Cauchy Tetrahedron . . . 148 5.4 Schematic Representation of Dermis and Epidermis Layers . . . 154 5.5 Example 2 - Species Concentration . . . 166 5.6 Example 2 - Time Step and Number of Newton Iterations Evolution . . . 167 5.7 Example 2, 2D Case . . . 168 5.8 Exampled 2 - Space Parameter Plots . . . 168 5.9 Example 2, 3D Case . . . 169 5.10 Example 3 . . . 169 5.11 Example 3 - Concentration of Speciesw∗1 and Displacement Magnitude|u∗| . . 171 5.12 Example 3 - Timestep and Newton Iteration Count Evolution for Different Me-
chanical Properties . . . 172 5.13 Example 3 - Timestep and Newton Iteration Coun Evolution for Different Accela-
tion Constants . . . 173 5.14 Example 3 - Species Concentration . . . 173 5.15 Example 4 - Concentration of Speciesw∗1 and Displacement Magnitude|u∗| . . 174 5.16 Example 4 - Concentration of Speciesw∗1 and Displacement Magnitude|u∗| . . 175 5.17 Example 5, 3D Case . . . 176 5.18 Decomposition of the Tensor Gradient of DeformationFinto Pure GrowthFg
and an Elastic Deformation TensorFe . . . 180 5.19 Effect of Boundary Conditions on the Final Pattern Structure for Different Diffu-
sion Rates (Da,Dh) . . . 184 5.20 Effect of BDC0 on the Final Pattern Structure for Different Diffusion Rates (Da,Dh)185 5.21 Effect of BDC3 on the Final Pattern Structure for Different Diffusion Rates (Da,Dh)186
5.22 Effect of BDC3 on the Final Pattern Structure for Different Diffusion Rates (Da,Dh) and loading forcesk . . . 187 5.23 Power Spectrum Comparison Build on 25 Replicates of Mechanochemical Model
(5.36)-(5.42) under BDC3 . . . 188 5.24 Effect of Elastic Parameter on the Final Pattern Structure for Conditions BDC0
and BDC3 . . . 189 5.25 Effect of Elastic Parameter on the Final Pattern Structure for Conditions BDC0
and BDC3 withcf =cg=1 . . . 190 5.26 Finite Strain Neo-Hookean Solid Coupled with Dhillon RD Model for Different
Diffusion Rates (Da,Dh) . . . 191 5.27 Finite Strain Neo-Hookean Solid Coupled with Dhillon RD Model for Different
Diffusion Rates (Da,Dh) in 3D . . . 192 5.28 Painter Avian Model Network . . . 195 5.29 Patterning Space, Parameter Condition and Dispersion Relations for the Uncou-
pled Poro-Mechano-Chemical Model . . . 202 5.30 Patterning Space, Parameter Condition and Dispersion Relations for the Coupled
Poro-Mechano-Chemical Model . . . 203 5.31 Patterning Space for the Coupled Poro-Mechano-Chemical Model . . . 204 5.32 Evolution of the Mesenchymal Cell Concentration, Epithelium, FGF, and BMP
Concentrations under no Deformation . . . 208 5.33 Evolution of the Mesenchymal Cell Concentration, Epithelium, FGF, and BMP
Concentrations under Periodic Traction Applied on the Top Edge of the Domain 209 5.34 Evolution of the Mesenchymal Cell Concentration, Fluid Pressure, Solid Pres-
sure, Epithelium, FGF, and BMP Concentrations under Finite Growth using the Formulation (5.88) . . . 210 5.35 Evolution of the Mesenchymal Cell Concentration, Fluid Pressure, Solid Pres-
sure, Epithelium, FGF, and BMP Concentrations under Finite Growth using the Formulation (5.88) . . . 211 6.1 Snake Epidermis . . . 217 6.2 Microstructure Diversity . . . 218 6.3 Scales’ Microstructure Properties . . . 219 6.4 Diversity Microstructure within Individual Scales . . . 220 6.5 Secondary Microstructure . . . 221 6.6 Phylogenetic Tree of Studied Snakes’ Species . . . 222 6.7 Brownian Motion Introduction . . . 224 6.8 Brownian Motion Rates . . . 225 6.9 Phylogenetic Variance-Covariance Matrix . . . 228 6.10 Hypothetical Pathways of Evolution for Pair of Traits . . . 230 6.11λ-Tree Transformation . . . 231 6.12δ-Tree Transformation . . . 233 6.13κ-Tree Transformation . . . 233
6.14 EB and OU Models-Example . . . 235 6.15 Classification Nanostructures . . . 243 6.16 Cell Surface Structure Identification . . . 245 6.17 Assignment of the Cell Surface State . . . 245 6.18 Mapping Tree of Cell Shape Character . . . 249 6.19 Rate Transition Matrix of Cell Shape Character . . . 249 6.20 Mapping Tree of Cell Border Character . . . 251 6.21 Rate Transition Matrix of Cell Border Character . . . 252 6.22 Sharp Versus Rounded Digitations . . . 252 6.23 Mapping Tree of Cell Surface Character . . . 253 6.24 Rate Transition Matrix of Cell Surface Character . . . 254 6.25 Mapping Tree of Ridge Character . . . 255 6.26 Rate Transition Matrix of Ridge Character . . . 255 6.27 Mapping Tree of Life Habit Character . . . 256 6.28 Rate Transition Matrix of Life Habit Character . . . 257 6.29 Diagnostics Plots of PGLS Model for Cell Border against Cell Surface Character 261 6.30 Diagnostics Plots of PGLM Model for Cell Border agains Cell Surface Character 262 6.31 Geweke-Brooks Plot of PGLM Model for Cell Border against Cell Surface Character262 6.32 Diagnostics Plots of PGLS Model for Cell Border against Ridges Charater . . . . 263 6.33 Diagnostics Plots of PGLM Model for Cell Border against Ridges Character . . . 263 A.1 Area Interaction Process . . . 271 A.2 Connected Component Process . . . 272 A.3 Diggle-Gates-Stibbard Process . . . 273 A.4 Diggle-Gratton Process . . . 273 A.5 Geyer Process . . . 274 A.6 Standardised Cumulative Frequency Histograms for Selected Polygon Parame-
ters forMus musculus . . . 279 A.7 Standardised Cumulative Frequency Histograms for Polygon Area forAcomys
dimidiatus . . . 280 A.8 Standardised Cumulative Frequency Histograms for Polygon Perimeter forAcomys
dimidiatus . . . 281 B.1 Effect of Initial Conditions for Sick Model . . . 283 B.2 Effect of Initial Conditions for Painter Model . . . 283 B.3 Comparison through 2D FFT between Fixed and Growing Square Domain for
Different Production Rates . . . 284 B.4 Comparison through 2D FFT between Fixed and Growing Disc Domain for
Different Production Rates . . . 284 B.5 Replicated Simulations of the Expected Rhombic Layout of Hair Placodes by
Coupling Two Layers of Similar Dhillon Mechanisms . . . 285
C.1 Effect of Boundary Conditions on the Final Pattern Structure for Different Reac- tion Ratesρa . . . 288 C.2 Power Spectrum Build on 25 Replicates of Mechanochemical Model (5.36)-(5.42)
under BDC3 . . . 289 C.3 Finite Strain Neo-Hookean Solid Coupled with Dhillon RD Model for Different
Reaction Ratesρa . . . 289 D.1 Parameter Distribution of Cell Surface Structures . . . 291 D.2 Cell Surface State Distribution . . . 292 D.3 Highly-Organised Nanostructures . . . 292
2.1 Principal Pathways in Induction Skin Appendage Development . . . 12 3.1 Summary Table of Sampled Skin Data . . . 59 3.2 Summary Table of Envelope Tests against an Inhomogeneous Poisson Model . 72 3.3 Summary Table of Independent Component Test forAcomys dimidiatus . . . . 74 3.4 Fitted Models . . . 77 3.5 Estimated Interaction Distances forMus musculus . . . 80 4.1 Butcher’s Tableau . . . 104 4.2 Butcher’s Tableau for the TR-BDF2 Method . . . 109 4.3 Algorthmic Boolean Options . . . 114 4.4 Reference Parameters . . . 115 5.1 Butcher’s Tableau for the TR-BDF2 Method . . . 161 5.2 Example 1 - Parameters for the Mechanochemical Model using (5.21) and (5.23) 163 5.3 Spatial and Temporal Error History Associated with the Discretisation of the
Model Problem . . . 164 5.4 Example 2 - Adaptive Timestep Controller Parameters . . . 165 5.5 Example 5 - Parameters for the Mechanochemical Model using (5.22) and (5.23) 175 5.6 Parameters Sets Used in Numerical Algorithm Implemented in Freefem++ and
FEniCS . . . 182 5.7 Reference Parameters for Navier-Lamé Problem Coupled with RD Dhillon Model 183 5.8 Reference Parameters for Neo-Hookean Problem Coupled with RD Dhillon Model189 5.9 Model Parameters used in the Linear Stability Analysis of the Poro-Mechano-
Chemical System. . . 199 6.1 Summary Table of Characters Regression . . . 260 D.1 Characters and their States for the 353 Studied Snakes’ Species . . . 293 D.2 Macroevolutionary Model Fitting for each Character . . . 307
The complexification of the issues and questions that our society will be asked to answer in the near or distant future, will require increasing knowledge from various fields. Science does not escape this route, and has shown, in the last few years, a border less and less defined between these different branches. The new PhD School of Life Sciences at the University of Geneva is a clear example: biophysics, biochemistry and mathematical biology are all terms that illustrate the ever-growing relationship between classically separate disciplines. This interdisciplinary research, although not new, has benefited from the democratization of science during the twentieth century [Klein, 1990]. Many examples have shown how collaborations between researchers from different disciplines have been able to solve unresolved problems [Gardy and Brinkman, 2003; Pellmar and Eisenberg, 2000; Preziosi et al., 2016]. The present thesis was born in this spirit; to create bridges between sciences. Centered on the application of mathematics in the context of biology, we try to answer questions about the formation of patterns in nature. Mathematical biology, an example of highly multidisciplinary science, includes a wide range of applications and tools that we will try to summarise briefly below.
1.1 Mathematics in Biology
Biomathematics or mathematical biology includes all sub-disciplines that use mathematical tools to study the principles that govern the structure, development, and behaviour of biologi- cal systems. It is based mainly on the description of a model by a system of equations that take into account, as far as possible, all known experimental data of the phenomenon investigated.
Very roughly, these models can be categorized into 2 classes: phenomenological models, empirically constructed, and models that derive from the fundamental theory. Whatever the type, the principle is to give a representation of the biological processes involved in our problem in order to predict behaviour not highlighted by the experimenter, whether it is due to technical limitations, data analysis difficulties, or simply because they have not yet been studied. This allows to guide the experimenter who will confirm, or not, the assumptions, thus strengthening the model. The growth of data sizes obtained in recent years (thanks to genomics, among others), the computing power of computers, or the ethical / administrative
constraints have allowed this discipline to develop in recent years.
Ecology and evolutionary biology are among the first to take advantage of this new develop- ment. The amount of data amassed by new techniques in proteomics, genomics and others has made their analysis increasingly difficult without powerful mathematical tools. Phylogeny, for example, has been possible and has allowed many advances in statistics [Helaers and Milinkovitch, 2010; Holder and Lewis, 2003; Huelsenbeck et al., 2001], in stochastic processes (e.g., Markov process) [Brown and Thomson, 2018; Huelsenbeck et al., 2003; McAuliffe et al., 2004] and others [Gascuel, 2005; Gonzalez and Knight, 2012; Mirkin and Science, 1997; Roch, 2010]. Many other research centres have benefited from the growth of biomathematics and include, for example,
• algebraic biology [Laubenbacher and Sturmfels, 2009; Mishra, 2007; Petoukhov, 2017],
• molecular biology [Gardy and Brinkman, 2003; Ingalls, 2012; Ratushny et al., 2011],
• modelling cell and tissues [Merzouki et al., 2016; Preziosi et al., 2016; Smith, 2016],
• computational neuroscience [Feng, 2003; Sejnowski et al., 1988; Trappenberg, 2009],
• modelling organ [Clayton et al., 2011; Quarteroni et al., 2017b; Siggers et al., 2014],
• spatial modelling [Maini, 2004; Murray, 2011a,b], and so on.
All of these contexts therefore require very different mathematical tools that include
• symbolic computation,
• cellular automata
• algebraic equations,
• ordinary or partial differential equations,
• stochastic process,
• statistics and probabilistic functions, etc.
For example, spatial modelling is expanding rapidly, since the influential D’Arcy Wentworth Thompson’s book "Growth and Form" in 1917. The comprehension of which biological pro- cesses can lead to such robust formation of patterns and form during embryonic development were a mystery that many biologists were focusing on answering during the last century, helped by numerical developments of differential equations solvers. The presented thesis is one of many examples into the large picture of patterning modelling in literature.
1.2 Thesis Goals
The main aim of this thesis is to be able to propose different mathematical models that make it possible to study the formation of regular structures present in certain species. The proposed scope of this thesis includes both statistical analyses of these patterns as well as systems of mathematical equations that model both the chemical signalling and the mechanical forces involved in the associated biological processes. We concentrate here in two types of these patterns. First, we look for the ones generated by the spatial distribution of epidermal and dermal cell condensates during skin appendage morphogenesis. This investigation centres on the hair and feather development in respectively two rodents species,Mus musculusand Acomys dimidiatus, and for the avian group in general. Second, we look for submicron-sized cellular structures formed during the keratinisation of theOberhäutchenspinualea of snakes scales.
Modelisation of skin appendages patterning During development of their coat,Acomys andMuspresent a spatial distribution of their hairs that are quite different. Acomysis an African genus of rodents of the subfamily Deomyinae. These species, commonly called spiny mouse, are able to lose some of their coat and then completely regenerate the tissue damaged by predators. Hair at the dorsal part ofAcomys dimidiatus’ embryos show a regular rhombic disposition, resulting in larger follicles in this area compared to the upper or the lateral part of the animal.Mus musculuspresents meanwhile a random disposition of its hairs. Our task was mainly to understand how such regularity can be reached despite the relative propinquity between the two species.
Comprehension pattern formation mechanisms through numerical models is widespread in the biological field. It is a clear example of interacting science. Biological experiments enable to perfect the mathematical models and these allow to propose new hypotheses to test experimentally. In our case, we concentrate on two types of models: pure reaction-diffusion and mechanochemical systems. Such models lead to two principal mathematical challenges:
numerical integration of nonlinear partial differential equations and linear stability analysis of the related system. We seek to answer to such difficulties by
• proposing a numerical time integrator and spatial discretisation algorithm sufficiently flexible to treat all the different models, and
• computational tools to perform linear stability analysis.
At the same time, the construction of the model needs to take into account new advancements in the field. Recently Ho et al. [2019] shows how signaling pathways and mechanical forces can initiate and create pattern with astonishing regularity in avians. Beyond the challenges that this may engender, the proposed models try as much as they can to maintain a subtle balance between experimental data and the simplicity of the equations that define them.
Finally, in order to compare simulations and real data, we developed a statistical analysis pipeline in order to quantitatively characterise the pattern generated through skin appendage formation.
Phylogenetic comparative analysis for scales nanostructures During cornification of the upper layer of snakes’ scales, cells generate periodic surface deformations whose function is still debated. A sample of sheds from 353 species of snakes coming from different life habits were studied through microscopy. From the obtained data, we establish a characters’ list to describe the observed nanostructures. Our goal was to supply a statistical analysis of the obtained data by
• using a continuous-time reversible Markov model for the phylogenetic mapping of all characters investigated, and
• assessing statistical correlation between characters itself or between characters and life habit.
Additionally, in order to share our data with the majority, we worked on the development of a MySQL database.
1.3 Main Contribution
The results lead to the following main contributions:
I Spatial Analysis of Hair Primodia Patterns
◦ Implementation of a semi-automatic script for Voronoi, Fourier transform and inferential analyses of patterns.
◦ Statistical significant inhibitive interaction between placodes.
◦ Follow of the spatial evolution of hair follicles pattern.
◦ Independence of primary and secondary wave of induction of follicles on the regular pattern inAcomys dimidiatus.
II Pure and Mechanically Coupled Reaction-Diffusion Models
◦ Newton linearisation and high-order adaptive time stepping algorithms built from a Galerkin approximation of nonlinear mechanochemical equations.
◦ Partitioned fixed-point and Schwarz algorithm to decouple a system of two-layers skin.
◦ Influence of cross-diffusion, coupling and growth on reaction-diffusion models.
◦ Coupling between a linear elastic model with advection-reaction-diffusion system and proposition of the ’tension lead to regularity’ hypothesis.
◦ Poro-mechanical model interacting with chemotaxis system of mesenchymal cells in feather morphogenesis and its linear stability analysis.
III Phylogenetic Comparative Analysis of Scales Nanostructures
◦ Characterisation of snakes’ scales nanostructures and construction of a character matrix.
◦ Phylogenetic mapping of the character matrix on a time-calibrated and fully sam- ple phylogeny.
◦ Comparative analysis of the character matrix against itself and snakes’ environ- mental habitat.
◦ MySQL database of the nanostructures, presented through the character matrix and scanning electron microscopy pictures.
1.4 Thesis Structure
The thesis are organised as outlined below:
• Chapter 2will make a brief motivational overview of patterning in biological systems.
• Chapter 3provides an introduction and the results of exploratory spatial analysis of marked skin appendages primordia forMus musculusandAcomys dimidiatus.
• Chapter 4covers the model centred on reaction-diffusion system for hair morphogene- sis.
• Chapter 5presents the mechanochemical models implemented to introduce elasticity properties of the skin.
• Chapter 6will look on phylogenetic comparative methods applied in the context of snakes’ scales nanostructures.
• Chapter 7summarise the different insights and conclusions for each chapter.
Note that, as each chapter is focused on quite different perspectives, we ease the reading by presenting, in each of them, a brief theoretical introduction in order to understand the terminology and the results exposed in a second part. Finally, we conclude the important specific results and bring some associated perspectives.
Context
Nature is undeniably a talented artist. A quick inspection of our surrounding world offers us a wide variety of intricate designs, that includes symmetries, fractals, spirals, waves, bubbles, foams, tessellations, spots, stripes (Fig. 2.1). All these forms and motifs that constitute the shapes of physical, biological or chemical objects are united in the so calledpatterns. Patterns include any kind of regularities forming in different spatio-temporal scale (Fig. 2.1). Since the premises of science, humans attempt to understand how such order can exist and persist in the chaos of nature [Livio, 2008; Thompson, 1992]. Nowadays, some natural patterning can be explained by different mathematical [Attanasi et al., 2014; Hahn et al., 2005; Mandelbrot et al., 1983; Rozenberg and Salomaa, 1976], physical [Beloussov, 2008; Lecuit and Lenne, 2007;
Martins et al., 2018; Montandon et al., 2014; Newman and Comper, 1990; Plateau, 1873; Si et al., 2015], chemical [Feng et al., 2012; Kalinin et al., 2011; Zhang et al., 1993] or even electro- magnetic [Gail¯ıtis, 1977; Müller et al., 2014; Rogacheva et al., 2006] models. For example, Rüther and Olsen [2007] shows that river meanders can be explained by fluid dynamics, while Hernquist et al. [2017] proves how gas composition can lead to create the differences observed in the shapes of galaxies. Nevertheless, plentiful patterning systems are still unexplained and bring together researchers of different fields who aim to resolve such complex problems [Murray, 2000].
In the context of living beings, patterns are the result of the combination of physical, chemical and biological processes [Murray, 2011a,b]. Interactions between these different actors results in a large variety of shapes and colors in organic systems. Fossil records show that these regu- larities are not stuck in time but they are submitted to natural selection that will generate new or remove old patterns, giving sometimes a biological advantage to the associated organism [Daeschler et al., 2006; Nagashima et al., 2009; Prud’homme et al., 2011; Sumrall and Wray, 2007; Yamaguchi et al., 2004]. Even if the utility of such patterns is still debated in the scientific community, in several cases they provide a form of camouflage [Caro et al., 2014; Egri et al., 2012; Stevens et al., 2011], or participate in sexual selection [Leamy and Klingenberg, 2005;
Mäthger et al., 2009], signalling [Koning, 1994; Pérez-Rodríguez et al., 2017], mimicry [Boyden, 1980; Wiens, 1978], and so on. Beyond the usefulness of patterning, scientists attempt to
Figure 2.1 –Patterns in nature. (a) Spiral [Goddard, 2005]. (b) Meanders [ISS056-E-62768, 2018]. (c) Spots [Hamon, 2009]. (d) Fractals [Micheal, 2017]. (e) Tesselation [Munch, 2013]. (f ) Chaos [Zantop, 2005]. (g) Cracks [Pires Nunes Martins, 2018] (h) Stripes [Ramsey, 2007]. (i) Symmetry [Levere, 2014].
explain how order can be maintained and be so persistent among individuals of the same species, despite the stochastic processes involved in the formation of these structures [Hong et al., 2018; Menshykau et al., 2014; Xiong et al., 2014]. Additionally, it is intriguing how very closely-related species (in terms of evolution time) can present very distinguishable patterns (e.g., various color and shapes present in the Anatidae family of water birds [Johnsgard, 1962]) or, in the opposite, how very distantly-related beings present similar structures with iden- tical purposes (e.g., birds’ and bats’ wings [Bell et al., 2011]). The difficulty to understand the complexity of patterning is also increased by the fact that these motifs occur in different spatial ranges going from a very small scale, such as snakes’ scales microstructures, to very large structures composed of more than an one individuals, such as the mesmerizing shapes created by a flock of birds (Fig. 2.2). Key to our comprehension of the processes involved in patterning is to relate how local processes, like cells movements and chemical gradients can generate large scale geometries.
Embryology studies the formation and development of any living organism, from fertilisation to birth. In the same context, morphogenesis is the study of pattern and form development.
How biological systems generate ordering is an important question in embryology but it still remains largly unanswered [Murray, 2000]. Theoretical biologists have proposed two funda- mental patterning models. The first one, namedpre-patterningmechanism suggests that
Figure 2.2 –Patterns’ scales in biology. (a) Snakes’ scales microstructur [Arrigo et al., 2019], scale bar 20µm (b) Embryo ofEchinops telfairi[Werneburg et al., 2013], scale bar 1mm. (c) African leopard (Panthera pardus pardus) [Erkamp, 2007]. (d) Birds flock [COBBS Lab, 2014].
geometrical structures are led by some antecedent pattern [Painter et al., 2012]. Classical ex- amples of such models are theDrosophila[Bieler et al., 2011; Lewis, 2008] or somite [Kerszberg and Wolpert, 2000; Trainor and Krumlauf, 2001] segmentation. Conversely,symmetry-breaking systems hypothesise that patterns are generated intrinsically from the amplification of the inherent noise by chemical or mechanical responses. Thus, it is the initial conditions that distinguish these models. Skin appendages formation is a well-studied domain in morpho- genesis. Indeed, the diverse forms and motifs they present, makes them a rich model for patterning studies [Lin et al., 2009; Widelitz et al., 2006].
2.1 Skin appendages morphogenesis
Skin appendages groups any skin-associated structures that present diverse colors, geometries and biological functions (Fig. 2.3). They cover different systems; hairs, feathers, scales, glands, and nails being the most prevalent [Chuong and Noveen, 1999]. Their large variety is associated to the many functions they serve, like temperature/mechanical protection [Chang et al., 2009;
Figure 2.3 –Skin appendages. (a) Scales inPhysignathus cocincinus[Pixabay, 2017b]. (b) Hair inSaguinus oedipus[Pixabay, 2017a]. (c) Feather inAra macao[Pixabay, 2014]. (d) Schematic drawing of skin appendages types [Chuong and Noveen, 1999].
Holmes, 1981], defensive function [Blackburn et al., 2008; Gonçalves et al., 2018; Seifert et al., 2012], displacement support [Gillette and Lockley, 1991; Heers and Dial, 2012; Lingham-Soliar and Murugan, 2013; Nudds and Dyke, 2010], aesthetic [Andersson et al., 1998; Møller and Höglund, 1991] or sensory perception [Flock, 1971; Saxod, 1996]. Despite this diversification, many of them share similar developmental pathways, [Chang et al., 2009; Widelitz and Chuong, 1999]. Recent literature [Chang et al., 2009; Di-Poï and Milinkovitch, 2016] shows that these similarities can be explained by the fact that they share a common ancestral form, and are thus homologous structures. In [Chuong et al., 1992; Dhouailly et al., 1980], they were able to alter the phenotype determination of feathers in chicken into scale-like structures, or inversely, through protein signalling modulation.
Consequently, whatever the appendage type, the initial developmental events are similar [Chuong and Noveen, 1999; Widelitz and Chuong, 1999]. Starting from an homogeneous system, where dermal (mesenchymal) and epidermal layers are undifferentiated [Hardy, 1992;
Lin et al., 2009; Millar, 2002], skin appendages primordia appear after a ’first signal’ coming from the dermis. The precise nature and properties of this signal is still debated [Millar, 2002], but the main hypothesis suggests that it depends on the initiation of WNT signalling pathway, promoted byβ-catenin, a molecular protein that appears early in skin appendages development [Millar, 2002; Noramly et al., 1999; Zhang et al., 2008]. Jiang et al. [1999] proposes a model where random fluctuations of morphogens and cell density will lead to form small dermal aggregates near the epidermal-dermal border that trigger the formation of cutaneous appendages. This first signal induces then the thickening of the epithelial cells, forming theplacodes. This leads to the initiation of the skin appendages morphogenesis and causes the activation of complex signalling networks that will promote and stabilize the placodes’
formation. All these processes will then constitute the induction step of skin appendages development [Chuong et al., 1996; Widelitz and Chuong, 1999]. Depending of the nature of the integument appendages, placodes and dermal condensates develop further into the mature appendage structure which will first go through a morphogenesis step where their orientation will be fixed with respects to the body of the organism. This step will be followed by a differentiation that will produce differential cell proliferation until the formation of the mature appendages [Millar, 2002; Widelitz and Chuong, 1999]. Despite similarities among appendage formation in terms of developmental stages or involved molecules, many animals will use them in different ways. For example, in many mammalians, hair develop sequentially in a series of ’waves’ leading to different types of hairs [Chase and Eaton, 1959; Sick et al., 2006]. Birds, on the other hand, have feather morphogenesis occurring in precise regional areas, called tracts [Jiang et al., 1999; Lin et al., 2006].
Among the different components involved in cutaneous appendage development, one of the most important and common across species, is the fact that skin appendages are formed through biological mechanisms that imply a multi-layer system that encompasses principally the already cited epidermal and dermal tissues. A large amount of skin reconstitution ex- periments show how the interconnexion between the two layers are crucial to enable their formation [Dhouailly, 1970, 1977; Jiang et al., 1999, 2004]. These experiments conclude that characteristics such as placodes number, size, location or structural identity is set up by the dermis; while epidermis will establish their orientation and their competence state [Chuong et al., 1996; Jiang et al., 2004]. Even if all the details are not completely understood, Shyer et al. [2017] recently proved that the chemical interlayer communication can be triggered by mechanical deformations. These deformations are induced by the internal forces generated by cell movement [Aman and Piotrowski, 2010; Oliver et al., 1994]. [Ahtiainen et al., 2014;
Glover et al., 2017] prove that this intrinsic ability of skin cells to migrate is the main driver of placode morphogenesis. It is hypothesised that the condensation of cells in the dermis can be helped by the mechanical properties of the substrate where cells move. Cells are driven generally by substrate rigidity [Kong et al., 2005; Lo et al., 2000] and by moving they create forms of gradient of extracellular matrix that acts as a positive feedback loop on cell condensation. Mechanical properties of the tissue have thus a non-negligible impact on the possible formation of placodes. Nevertheless, orientation and emphasis of the dermal cells
movement or epidermal deformation is done synchronously thanks to both mechanical and chemical operators [Shyer et al., 2017].
Therefore, beyond the ’first’β-catenin/WNT signalling transduction, skin appendage for- mation is dependent of numerous molecular pathways [Abe and Tanaka, 2017; Mikkola, 2008; Millar, 2002; Veltri et al., 2018; Widelitz and Chuong, 1999; Widelitz, 2008]. During the induction step, signalling acts either by promoting and stabilising dermal and epidermal con- densation or by inhibiting their formation through distinct processes, like cell proliferation, cell migration, and so on. These different signals will then define the spacing, the orientation or the nature of the organ. Thus, the skin appendages primordia formation is centred on a competitive equilibrium of all the involved morphogens [Jiang et al., 1999, 2004; Millar, 2002].
Major part of the concerned signalling chemicals are summarised in Table 2.1. Among the large amount of concerned signalling proteins, the most important ones regroup WNT/β- catenin, FGF, BMP/TGF-β, EDA/EDAR, and Shh, Mikkola [2008]. Even if some pathways can act at different time scales, Hardy [1992] suggests that skin appendage formation proceeds via a chronological event of epidermal-dermal molecular interactions.
Pathways
FGF FGF-1, -2, -4, -10, -20, KGF, FGFR,...
TGFβ TGF-β, BMP-2,-4, noggin, follistatin, BMPR,...
TNF EDA, EDAR, EDARADD, NFκBs, LTβ,...
Protein kinases PKA, PKC
Hedgehog Shh, patched, Gli, Smo,...
WNT WNT-3, -5a, -7a, Lef/TCF,β-catenin, Dkk-1,-4,...
Notch Serrate, Delta, Notch, Fringe, ...
Homeobox Msx, Hox, Dlx,...
Table 2.1 –Principal pathways in induction skin appendage development.
The fibroblast-growth factor (FGF) pathway acts in early stages, principally as a dermal con- densation promoter [Lin et al., 2009]. For example, FGFs 1, 2, and 4 are known activators of feather bud formation, while FGF10 influences the epithelial thickening and size but also the number of placodes [Lin et al., 2006, 2009; Widelitz et al., 1996]. Huh et al. [2013] demonstrates that FGF20 helps the formation of primary and secondary dermal condensations in hair folli- cles primordia development in mice. Cell spatial disposition can be disrupted by mutation of FGF-2 receptors in mammals [Petiot et al., 2003]. During primordia formation FGFs are expressed on the epithelial cells, following the activation through the WNT/β-catenin pathway [Huh et al., 2013]. It acts then as a chemoattractant morphogen of dermal cells, promoting thus their aggregation. However [Petiot et al., 2003; Huh et al., 2013] show that FGF proteins can also have a inhibitory role by deactivating WNT/β-catenin and EDA/EDAR pathways in the mouse’s epithelial layer.
Other activators of placode formation include, for example, the WNT/β-catenin, TGF-β,
EDA, or noggin proteins. Beyond the fact that they might be related to the initiation of the appendages induction, WNT/β-catenin are involved in different processes occurring in distinct spatial locations and developmental time [Millar, 2002; Noramly et al., 1999; Widelitz, 2008]. WNT signalling is involved in proliferation, differentiation, cell migration, cell polarity and cell adhesion during skin organogenesis. In the context of skin appendage development, WNT proteins play a role in the stabilisation of the placode position at early stages, while latter they prompt epithelial cell proliferation and maturation of the dermal condensate to a dermal papilla [Glover et al., 2017; Millar, 2002; Tsai et al., 2014; Sick et al., 2006]. Hair placodes density seems also to be influenced by WNT proteins [Glover et al., 2017]. In feather formation, WNT is suspected to be implicated on tract and placodes physical characteristics (shape, size,...) [Lin et al., 2006; Widelitz, 2008]. For example, WNT-1 signalling can diminish the dorsal tract size, while increasing the size of feather buds developing on it.β-catenin, part of the WNT canonical pathway, is associated with cadherin adhesion protein complexes in the cell membrane in the absence of WNT signalling [Noramly et al., 1999]. Shyer et al. [2017]
demonstrate thatβ-catenin release from cadherin can also be initiated by dermal compression in avian skin, showing how the mechanical and chemical process work in concert. To our knowledge, no similar experiments have been performed in mammals, but Shyer et al. [2017]
suggest that it can be implicated in other situations. [Benham-Pyle et al., 2015; Stewart et al., 2018] demonstrate how mechanical tissue properties have an impact on cells proliferation in skin. WNT is also associated to another proteins, the Dickkopf (Dkk) family. Dkks are antagonists of WNT proteins and their expression depends on the amplitude of WNT signal, Sick et al. [2006]. The most studied proteins are Dkk1, Dkk2 and Dkk4. All of them act as inhibitors of placode formation and are involved in the placode’s fate [Glover et al., 2017; Sick et al., 2006].
The TGFβsuperfamily regroups many subfamilies of proteins such as TGF-βs, activins, inhin- bins and bone morphogenetic proteins (BMPs). TGF-βmolecules are promoters of mesenchy- mal condensation and proliferation [Glover et al., 2017; Li et al., 2003; Widelitz and Chuong, 1999]. In mice, TGF-β2 is probably connected to the initiation of hair follicle induction [Li et al., 2003]. For chick embryos, it is sufficient to generate dermal papillae in the absence of the epithelia [Millar, 2002]. Noggin regulates the influence of BMPs by direct binding and prevent- ing their signal with cell surface receptors [Cheng et al., 2014; Li et al., 2003]. BMPs (BMP-2, BMP-4) act as a negative regulator of the placode formation, Cheng et al. [2014]; Noramly and Morgan [1998], but also in epidermal-dermal interactions [Li et al., 2003]. Recently it has been suggested that BMP can inhibit FGF expression [Painter et al., 2018; Ho et al., 2019] in avians or EDA/EDAR [Mou et al., 2006] in mammals. Nonetheless, like many signalling proteins, BMPs also FGF can operate as appendages induction promoter by dermal cell condensation [Michon et al., 2008].
EDA/EDAR signalling pathway is implicated in skin appendage formation [Cui and Sch- lessinger, 2006; Mikkola, 2008; Mou et al., 2006]. Contrary to precedent molecules, EDA/EDAR is less important for the initiation of the appendages morphogenesis [Cui and Schlessinger, 2006], as experiments show that EDA signalling can be dispensable in order to get appendages
primordia under certain circumstances [Millar, 2002; Mou et al., 2006; Ho et al., 2019]. How- ever, it plays a role in define the type of appendages that will be created. EDA regulation operates by positive activation of the Hedgehog pathway or feedback inhibition of the WNT and BMP pathways by expression of pathway-specific inhibitors [Cui and Schlessinger, 2006;
Mikkola, 2008]. EDA is expressed in the inter-condensation regions in the dermis layer, while for the epidermis, it is expressed inside the placodes [Lin et al., 2006]. Among the signalling pathway promoters,β-catenin and BMP enhanced the expression of respectively EDAR and EDA proteins [Lin et al., 2006].
Montandon et al. have shown that inAcomys dimidiatusthe WNT signalling is involved in the placode formation of the spines of the lower back with increased Lef1 expression in the placodes. They also detected Hoxc6 in the lower back of the animal which we assume provides a positional cue, resulting in larger placodes. They also show increased expression of Dkk1 in Acomysin the larger placodes size of the spines [Montandon et al.].
This non-exhaustive review of the involved signalling pathways in skin appendages formation gives us a small taste of the complexity involved in organ morphogenesis and how patterns can be created. Even if the involved pathways are similar across species, their different expression levels at developmental stages can bring about very diverse results. Additionally, a clear distinction between activators or inhibitors of appendage formation seems difficult, as some of them can act both ways (e.g., FGF, BMP). Therefore, general models will need to combine these molecules/pathways to hypothetical entities. On the other hand, models focusing on specific questions about a particular interaction will need to take into account the properties of each signalling protein. Understanding how a pair of morphogens acts as a system of activator/inhibitor proteins can be modelled as a reaction-diffusion system. Mechanical properties can also be included through pure elastic or chemical-elastic models.
Spatial Analysis
Based on the biological context introduced on Chapter 2, and before any modelling of the biological, chemical and/or mechanical processes that underlie hair follicles patterning for- mation, it is crucial to have a characterisation of the spatial structure and a tool to compare it with future morphological simulations. To have a complete (or at least clear) description of a particular process, informative measures, like nearest-neighbour distances, intensity distribution, or interaction ranges of the placodes that formed the observed pattern, should be computed. Hair follicles primordia can be represented on a domain by a set of points coordinates that gives the locations of their geometrical center. Characterisation and testing of some specific hypotheses that underly the formation of suchspatial datais the center of thespatial analysisorspatial statisticsdomain. The range of applications of spatial analysis is large and spreads in domains such as archaeology [Keron, 2015], astronomy [Jones et al., 2005; Martinez and Saar, 2001], particle physics [Gundersen, 1986; Shepilov et al., 2007; Stoyan and Beneš, 1991] and environmental science [Enright, 1989; Haining, 1993]. Due to advances on the fields (automatisation on data production, computer power, ...) scientific domains as biology [Diggle et al., 2010; Seidler and Plotkin, 2006; Wässle et al., 1981], medecine [Gatrell et al., 1996; Glasbey and Roberts, 1997; Kulldorff and Nagarwalla, 1995] or material science [Ballani et al., 2006; Ballani, 2006] start also to employ largely spatial statistics methods [Illian et al., 2008].
Spatial analysis is commonly classified in three main families of methods [Banerjee et al., 2014; Gelfand et al., 2010; Illian et al., 2008]. First,continuous spatial variationis focused on the analysis of spatially continuous phenomena. In that context, the spatial configuration (location) is known and only the associated stochastic value is studied. This field proves to be useful in different contexts but particularly in geo- or environmental science [Banerjee et al., 2014; Gelfand et al., 2010; Matérn, 1960]. On the other hand,discrete spatial variation, conceptually similar to the continuous variation, focuses on spatially discrete or ’lattice’ data [Besag, 1974]. Discrete means here that the studied domain is partitionned into a finite number of areal units where their boundaries are well defined [Banerjee et al., 2014]. Gelfand et al. [2010] and Matérn [1960] provide some examples of applications. More appropriate
