PhD Thesis

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C. Grierson. In addition, in collaboration with A. Champneys, the bifurcation criticality analysis ...... [18] N. A. CAMPBELL, Biology, Benjamin Cummings, 2002.
Modelling Initiation of Plant Root Hairs A Reaction-Diffusion System in a Non-Homogenous Environment

By V ÍCTOR F RANCISCO B REÑA –M EDINA

Department of Engineering Mathematics U NIVERSITY OF B RISTOL

A dissertation submitted to the University of Bristol in accordance with the requirements of the degree of D OCTOR OF P HILOSOPHY in the Faculty of Engineering.

J UNE 2013

Word count: thirty-seven thousand four hundred forty-nine

L EY

DE LA VERDAD ABSOLUTA

Lo que es, es y lo que no es, no es. L OS

TRES POSTULADOS

1. De la unicidad: Una cosa es una cosa y otra cosa es otra cosa. 2. De la temporalidad: Las cosas duran hasta que se acaban. 3. De la irrebatibilidad: Hay cosas que ni qué.

VA

A BSTRACT

reaction-diffusion system, which can be considered as a generalised Schnakenberg-like model, is studied mathematically in 1D and 2D. This system models an initiation process within a root-hair cell which involves biochemical interactions of the G-proteins, known as Rho of Plants, or ROPs. These proteins attach to the cell membrane prompting a localised patch which, in consequence, induces cell wall softening and subsequently hair growth. This model assumes that the auxin provided is the key catalyst. Auxin is a plant hormone which is known to be cause of many different features in plant morphogenesis. Also, this hormone is experimentally known to enter the cell inducing a spatially dependent gradient. Numerical bifurcation analysis is carried on in order to explore solutions which resemble all features that the G-proteins and auxin are known to cause. The main bifurcation parameters are taken to be the overall auxin rate, and the cell length. The analysis is backed up by full numerical simulations and asymptotic analysis using semi-strong interaction theory. The asymptotics not only provides existence of solutions and explains numerical proprieties, but also sheds light on transition mechanisms via the theory of competition instability and transverse instability of homoclinic stripes. The analytical results are found to agree favourably with numerical simulations, and to give further explanation of the agreement between the model and biological data for different scenarios. From a mathematical point of view, pattern formation of non-homogeneous reaction-diffusion systems is a subject that is not yet well understood. However, upon using the theory of semi-strong interactions, light is shed on the dynamics and instabilities that spatially dependent coefficients bring about. As a consequence, transitions between different spot-like patterns and the dynamics of their location can be explored and theoretically explained.

A

iii

A CKNOWLEDGEMENTS

his work is dedicated to those who have made my P H D studies a quite enjoyable learning experience. First of all, for his guidance, patience and expertise, my special thanks to my supervisor Alan Champneys, who always knew how to put me on the right way. I am honoured for learning from you. Thank you so much. Also, I would like to thank my second supervisor Claire Grierson for her support and willingness to discuss biology and mathematics. Through this journey I have met and collaborated with great people, without them this thesis would be undoubtedly still in progress: Michael Ward for his brilliant ideas, which made this thesis possible; Daniele Avitabile for showing me how to tackle the, sometimes, intricate paths of numerical analysis; and, Jens Rademacher for his insightful suggestions. Thank you all for giving me this opportunity and for your warm hospitality when I visited you in the UBC, the UoN, and the CWI, respectively. I would like to thank my reviewers, Arjen Doelman and Martin Homer, for a quite splendid Viva experience. The core of this thesis, which consists of the non-homogeneous 1D system analysis, has been submitted for publication to SIADS; this is a joint work with A. Champneys, M. Ward, and C. Grierson. In addition, in collaboration with A. Champneys, the bifurcation criticality analysis of the homogeneous system will be included in a work to be submitted for publication soon. Also, the results presented in Chapter 4 will be shortly submitted for publication to SIAP, as a joint work with A. Champneys, M. Ward, and D. Avitabile. To the A PPLIED N ONLINEAR M ATHEMATICS group, my gratitude for providing a great working environment, the nonlinear seminars and those first-Wednesday-of-[every]-month. To the people of the BUNCAER: Pablo, Piotr, Petros, Jakub, Steffi, Tom I, Oscar, Steve, Roz, Tom G, Helge, Linford, John L–M, Tom T, Yani, Antoni, Alexandro, Neeraj, Vassil, and Dan, thanks for sharing frustrations, laughs and helping me out when I most needed it, and making quite a wonderful experience all these almost-four years. Ana, there are no words that can express all I feel, thank you so much for sharing your life with me. To my parents and sister, thank you for giving me your confidence and complete support, this achievement is also yours. I would also like to thank to those who, through their friendship, have made my stay in Bristol a pleasure: Henrietta, Irene, Anaid, Priscilla, Karen, Emylee, Karin, Don Miguel, Marisa, Ulises, Mauro, José, JC, Ángel, Desis, Miguel, Mona, Ester, Michael, Son, and the recently meet-up banda chilena. I appreciate the financial support of the CONAC Y T fellowship for P H D studies, and a EPSRC complementary financial support.

T

v

A UTHOR ’ S DECLARATION

declare that the work in this dissertation was carried out in accordance with the requirements of the University’s Regulations and Code of Practice for Research Degree Programmes and that it has not been submitted for any other academic award. Except where indicated by specific reference in the text, the work is the candidate’s own work. Work done in collaboration with, or with the assistance of, others, is indicated as such. Any views expressed in the dissertation are those of the author.

I

SIGNED: .................................................... DATE: ..........................................

vii

TABLE OF C ONTENTS

Page Abstract

iv

Acknowledgements

vi

Author’s Declaration

viii

List of Tables

xiii

List of Figures 1

Introduction

1

1.1

Pattern formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.1.1

Reaction and diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.1.2

Localised patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

Plant root hairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.2.1

Cell specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.2.2

Root hair initiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

1.2.3

Root hair growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

1.2.4

Mutant studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

The ROP model system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

1.3.1

ROP kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

1.3.2

The gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

1.3.3

Fundamental model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

1.2

1.3

1.4 2

xv

Bifurcation Analysis and Initial Simulations

21

2.1

Turing instability conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

2.2

The homogeneous system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

2.2.1

Diffusion driven instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

Pitchfork bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

2.3.1

30

2.3

Criticality condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

TABLE OF CONTENTS

3

2.4

Conservation principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

2.5

Initial parameter sweep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

2.6

Numerical Bifurcation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

2.7

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

Analysis of Localised Patches in 1D

45

3.1

Asymptotic Rescaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

3.2

Asymptotic analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

3.2.1

Single patch solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

3.2.2

Boundary patch solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

3.2.3

Multiple patch solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

Competition Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61

3.3.1

A nonlocal eigenvalue problem . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

3.3.2

The two-patch case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

3.3.3

Boundary-interior patch case . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

3.A Appendix: Local operator proprieties . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

3.3

3.4

4

3.A.1

A spike . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73

3.A.2

Key identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

Breakup Instability of Stripes in 2D

77

4.1

Numerical investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

4.1.1

Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

4.1.2

Bifurcation diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

Breakup instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

4.2.1

Interior stripe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

4.2.2

Boundary stripe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

94

Stripes into spots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

4.3.1

98

4.2

4.3 4.4

A richer zoo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.A Appendix: A NLEP numerical computation . . . . . . . . . . . . . . . . . . . . . . . . 102 5

Dynamics and Transitions in 2D 5.1

5.2

Asymptotic regime for spot formation . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.1.1

Multiple-spot profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.1.2

Spots and baby droplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

Dynamics of spots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5.2.1

5.3

107

Solvability condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

Fast O (1) time-scale instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 x

TABLE OF CONTENTS

5.4

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.A Appendix: Neumann G-function for Ω2 . . . . . . . . . . . . . . . . . . . . . . . . . . 126 6

More Robust 2D Patches 6.1

Auxin transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

6.2

A Particular gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

6.3

Further results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 6.3.1

6.4 7

129

Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

Concluding Remarks

137

A O (1) Time-scale Criticality Condition in 1D

143

Bibliography

145

xi

L IST OF TABLES

TABLE

Page

3.1

First and second parameter sets in the original and re-scaled variables. . . . . . . . .

47

4.1

Third parameter set in the original and re-scaled variables. . . . . . . . . . . . . . . . .

91

xiii

L IST OF F IGURES

F IGURE

Page

1.1

Image of Arabidopsis RHs and a RH cell with a hair that is at to its final length. . . .

5

1.2

Developmental zones of an Arabidopsis root. . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.3

Transverse section showing the position-dependent pattern of RH cells and non-RH cells, and a sketch of tissue organisation and auxin pathway. . . . . . . . . . . . . . . .

8

1.4

Switching mechanism related to active GTP-bound and Inactive GDP-bound ROPs. .

9

1.5

Sketch of the switching fluctuation between active-GTP and inactive-GDP ROPs. . .

9

1.6

Sketch of the binding process nearby the cell membrane. . . . . . . . . . . . . . . . . .

10

1.7

Hair-forming mutant cells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

1.8

Arabidopsis wild-type. A patch of surface bound ROP, and time lapse of patch drift. .

13

1.9

Sketch of the model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

1.10 Sketch of the auxin flux in the RH cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

2.1

Turing bifurcation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

2.2

Numerically computed bifurcation diagram and pitchfork criticality condition. . . . .

30

2.3

Localised Turing activator-substrate patterns. . . . . . . . . . . . . . . . . . . . . . . . .

32

2.4

Initial simulations under a spatially dependent gradient. . . . . . . . . . . . . . . . . .

34

2.5

Hysteresis for two different transition speeds varying growth patch region length and auxin acceleration rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

2.6

Bifurcation diagram varying k 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

2.7

Direct numerical stability simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

2.8

Hopf-like bifurcation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

2.9

Patch location diagram, and comparison of the bifurcation diagram for the boundarypatch and single-interior-patch branches. . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

2.10 Bifurcation and location diagrams of the single interior patch solution under variation of other physical parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

2.11 Length and auxin two-parameter continuation. . . . . . . . . . . . . . . . . . . . . . . .

42

2.12 Two-parameter bifurcation diagrams as k 20 varies along with another parameter. . .

43

3.1

48

Schematic plot of the inner and outer solution for a steady-state solution. . . . . . . . xv

L IST

OF

F IGURES

3.2

Comparison between the asymptotic and numerics of the single interior spike quasisteady-state solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3

Comparison between asymptotic and numerics of the interior-spike location bifurcation diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.4

51 54

Comparison between the asymptotic and numerics of the boundary spike quasi-steadystate solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

3.5

Solution multiplicity of DAE system (3.48) and (3.49). . . . . . . . . . . . . . . . . . . .

58

3.6

Comparison between the asymptotic and numerical of profile and time-dependent spike locations for the quasi-steady-state solution. . . . . . . . . . . . . . . . . . . . . .

59

3.7

A single-patch and two-patch outcome for several lengths and two k 20 values. . . . . .

62

3.8

O (1) time-scale competition criticality condition as varying D 0 . . . . . . . . . . . . . . .

68

3.9

O (1) time-scale competition instability of a two-interior-patch quasi-steady-state solu-

tion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

3.10 Numerical results for an O (1) time-scale competition instability of a quasi-steady-state solution consisting of a boundary patch and a single interior patch. . . . . . . . . . . .

71

3.11 The complex ν-eigenplane, and first two eigenpairs of local operator L 0 . . . . . . . . .

75

4.1

Snapshots of a travelling front breaking up into a travelling pinned spot. . . . . . . .

79

4.2

Snapshots of a pair of stripes breaking up into an asymmetrical array of spots. . . . .

80

4.3

Comparison of bifurcation diagrams between homoclinic stripes and 1D-spike scenarios. 82

4.4

Example of a breakup instability of a homoclinic stripe into one spot and two peanutforms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

4.5

Sketch of a dispersion relation and unstable band. . . . . . . . . . . . . . . . . . . . . .

89

4.6

Criticality condition of an interior stripe. . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

4.7

Dispersion relation of an interior homoclinic stripe for several ratio lengths and location points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.8

Breakup instability and secondary O (1) time-scale instabilities of an interior homoclinic stripe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.9

92 93

Criticality condition of a boundary stripe, and critical mode m c as γ increases for several values of parameter s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

4.10 Breakup instability of a boundary stripe for two different values of k 20 . . . . . . . . .

96

4.11 Dispersion relations numerically computed for particular steady solutions marked in bifurcation diagram in Figure 4.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

98

4.12 Breakup instability and final stable state solutions of each extended-solution kind. .

99

4.13 Bifurcation diagram: spots and a boundary stripe. . . . . . . . . . . . . . . . . . . . . . 100 5.1

Constant χ (S c ) as varying the source parameter, and radially symmetric canonical core solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.2

Bifurcation diagram of spots and baby droplets. . . . . . . . . . . . . . . . . . . . . . . . 115 xvi

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OF

F IGURES

5.3

Numerical solutions of the adjoint BVP (5.30) for several values of S c . . . . . . . . . . 118

5.4

Location dynamics of two spots for a y-independent gradient. . . . . . . . . . . . . . . . 121

5.5

Two snapshots of a simulation where a peanut form merges into a spot. . . . . . . . . 122

5.6

Competition instability and spot self-replication. . . . . . . . . . . . . . . . . . . . . . . 123

6.1

Sketch of a idealised 3D RH cell, and its projection on a 2D rectangular domain. . . . 130

6.2

Comparison of bifurcation diagrams as varying k 20 in 2D and 1D. . . . . . . . . . . . . 131

6.3

Bifurcation diagram with α = α (x, y). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

6.4

Time dependent transitions, controlled by a y-dependent gradient. . . . . . . . . . . . 134

xvii

HAPTER

C

1

I NTRODUCTION

T

his thesis shall consider a particular biological model of morphogenesis at the sub-cellular level. This is a cell polarity and patterning process in the particular cells on the epidermis of a root that grow hairs. The central question is what triggers hair formation and

what determines where within the cell it occurs. We shall provide theory to suggest this is a self-organised process. Morphogenesis is the biological process that organises growth, organisation and differentiation of the structure of an organism or parts of an organism. This process consequently gives rise to new life forms. That being so, a fundamental understanding of morphogenesis is vital for making advances across life sciences from agriculture up to human health. In agriculture for instance taking cognisance of flowering, self-pollination and nutrient uptake from the soil by plants, or stem cells specialisation and apoptosis can eventually help with crop yield or reforestation. Morphogenesis lies at the heart of many diseases such as cancer, for example. Many rich and intricate biochemical and physical interactions take part in the development of organisms. Cell differentiation changes the shape and size of cells, and also alters their ability to react to external factors and to synthesise complexes; in consequence, these modifications lead to more specialised cells. However, differentiation is only a piece of the whole picture, multicellular arrangements and connections between tissues play an important role as well. As a result of this, complex structures need to be synthesised at a specific time and place in order for different tissues and anatomy to arise. In other words, a group of initially similar cells are susceptible to many different signals from cells outside the group. So, through cell memory, some signals can have a persisting effect on the internal state of a cell. Indeed, sequences of simple signals having an impact on cells inside the group at different times and places give rise to tissues, which are formed from a population of cells, in consequence organs will be constructed from populations of tissues [3]. 1

CHAPTER 1. INTRODUCTION

All these assembles and mechanisms are the consequence of an extremely precisely controlled cell self-organisation phenomena.

1.1

Pattern formation

From a purely physical point of view the spontaneous formation of spatial, temporal and spatiotemporal structures, also known as patterns, occurs in open systems outside of thermodynamical equilibrium. As a result of this, pattern formation has been observed on several physical scales and explained theoretically. Examples include, coherent Raman and Brillouin scattering (e.g. [2, 24]), Taylor–Couette flow (e.g. [52, 142]), Rayleigh–Bénard convection (e.g. [11]), and a quite diverse range of meteorological and astrophysical phenomena are examples brought out from physics; Briggs–Rauscher reaction (e.g. [107]) and Belousov–Zhabotinsky reaction (e.g. [57, 132]) from chemistry; traffic and material flow networks, ordered defects in crystals, neural networks, and interactions on the internet, from technological applications; growth, competition and extinction of companies, and formation of public opinion are instances of concern in economics and politics. Moreover, from a linguistics point of view, vowel systems (e.g. [27]) and language evolution (e.g. [127]) are also been considered as examples of pattern formation. Nonetheless, biology brings up a yet larger variety of interesting pattern formation examples that certainly shine by themselves, for example: skins of animals (fishes, snakes, birds), butterfly wings, bones of vertebrates, cognition, growth of brains, and neuronal dendrites are examples of morphogenesis that have been at least superficially explained using pattern formation theories (see, for instance, the textbooks [48, 99, 100] for a detailed discussion). In addition, matters of population dynamics such as competition, extinction, and collective behaviour (e.g. [58, 122]), and mechanism of disease spreading (e.g. [114]), or visual hallucinations (e.g. [129]), biological clocks (e.g. [36]) and synchronisation (e.g. [53, 94]) also deserve to be mentioned as magnificent examples of pattern formation phenomena. All these cases are outstanding illustrations of self-organising mechanisms (see, for instance, [4] and references therein contained). That is, steady states under natural local perturbations (such as from genetic variation, normal environmental fluctuations, and the randomness of biochemical processes), biological systems appear to undergo a transition to an ordered structure thermodynamically far from equilibrium. To maintain these structures biological systems constantly consume sources of energy.

1.1.1

Reaction and diffusion

The Scottish zoologist Sir D’Arcy Wentworth Thompson became a pioneer of mathematical biology. He emphasised the fundamental role of physical laws in growth and form of living organisms. This point of view is elegantly stated in his book On Growth and Form [130] published in 1917: 2

1.1. PATTERN FORMATION

Cell and tissue, shell and bone, leaf and flower, are so many portions of matter, and it is in obedience to the laws of physics that their particles have been moved, moulded and conformed. [...] Their problems of form are in the first instance mathematical problems, their problems of growth are essentially physical problems, and the morphologist is, ipso facto, a student of physical science. Thirty years later an English mathematician, Alan Mathison Turing, after having established the foundations of computers, and consequently given a new direction to modern mathematics, turned his interests radically into physiology. Probably influenced by Thompson’s ideas, Turing published in 1952 a ground-breaking paper: The chemical basis of morphogenesis [131], where a remarkably simple model is proposed. This study led to the onset of mathematical theory of morphogenesis. Diffusion and reaction are two processes that separately show no symmetry breaking by themselves. However, another story occurs when both processes act together. Indeed, Turing proposed an interaction of two chemical agent concentrations, also called morphogens, whose spatiotemporal concentrations are characterised by means of a reaction-diffusion system. This process of interaction is assumed to occur via an active agent that is inhibited by another one diffusing slower. The resulting system can be derived, in terms of modern theory, by taking into account the law of mass conservation, yet can also be deduced, from a mere stochastic dynamics perspective, from the Fokker–Planck equation in its simplest case (e.g. [108]). Either approach assumes the thermodynamical limit, an important issue for multi-scale systems. Turing’s mechanism is based upon two main assumptions: asymptotic stability under lack of diffusion, and instability otherwise. Following a linear analysis, a threshold is provided for a spatially homogenous stable steady state to be destabilised. In simple terms, diffusion breaks symmetry, which therefore induces a heterogeneous spatial array. In this way, the transition from stability towards instability is known as Turing bifurcation. Even though the validity of this mechanism for describing observed morphogenesis is still under discussion in the scientific community, it can give very interesting theoretical questions and can shed light upon other biological theories. Indeed, the understanding of a wide range of biological pattern formation problems have been approached through Turing bifurcation analysis (see, for instance, [4, 100]). Another idea underlying morphogenesis is that of spatial localisation. Theory shows that autocatalysis (short-range positive feedback) interactions merged with long-range inhibition stimulates aggregation (e.g. [47, 93, 100]). In fact, these two competing processes bring a large variety of patterns into play. Notwithstanding, when autocatalytic interactions take part on concentrations playing a role as substrate (depleting process), localisation may occur as an alternative mechanism. This process leads to a huge diversity of patterns as well, see [81, 90, 100] for instance. However, there are other different mechanisms that seem to favour localisation. By way of illustration, far from the theoretical Turing viewpoint, spots, targets, stripes, and extremely 3

CHAPTER 1. INTRODUCTION

complicated, and not yet completely understood, spatiotemporal dynamics (e.g. [116]) can be found in simple reaction-diffusion models such as the Gray–Scott system (e.g. [54]) or the Gierer– Meinhardt system (e.g. [72]). These systems of equations are broadly similar to the Schnakenberg model studied in this thesis. In addition, from a biological context, cells appear to be organised in such fashion that long-range gradients teach morphogens where to be placed [4, 75]. The resulting patches seem to be predetermined by positional information coming from boundaries, see [68] for example. Dynamics and stability of these localised structures cannot be understood theoretically by Turing bifurcation analysis, yet gives an essential framework to begin the understanding of localised pattern formation.

1.1.2

Localised patterns

The benefit of a mathematical analysis rather than a purely numerical approach is that in most cases the precise details of the biology are unknown and those models mentioned above are at best a gross simplification of many different biochemical pathways. So, a mathematical analysis can uncover the generic features that models of a wider class can display, without being bound by particular parameter sets or functional forms. Also, as we shall see, within a given model, asymptotic methods can lead to full parametrisation of qualitative features, such as pattern width and instability thresholds, that should then enable new experimental hypotheses to be formed and tested. As shall be seen further, in an example of this, in multiple root hair formation, a correlation of location proximity to apical end of the cell and progress to tip growth has been observed by Jones & Smirnoff [70]. As we shall see, asymptotic analysis can actually shed light on these observations. Even though a wide variety of models present localised structures as solutions, we shall focus our attention on a generalisation of the Schnakenberg system, which will be derived in §1.3. As we shall see further, this system not only captures the biological features we will study, but also it is amenable to be analysed. Localised pattern formation using this model and related reaction-diffusion systems has been widely investigated in a variety of contexts. For example, Chen [22, §2] reduces the Gray–Scott model to an ODE-PDE Stefan-type problem, and Ehrt et al. [37, 118] analyse first- and secondorder interface interactions providing methods to describe the location and dynamics of spikes in Schnakenberg-like systems. In addition, Doelman et al. [31, 32] and Iron [66, §2] analyse the 1D Gierer–Meinhardt system where, specifically, Iron derives a non-local eigenvalue problem (NLEP) to estimate spike stability. In a similar vein, Ward & Wei [135, 136] analyse existence and stability of asymmetric patterns for the Schnakenberg system. Results exist also in 2D domains, see e.g. [22, 72, 73, 141]. Note that all these studies consider the spatially homogeneous problem, nonetheless. In general, pattern formation in the presence of spatial inhomogeneity is less well understood, yet some interesting approaches have been made. For instance Glimm et al. [49, 50] consider a 4

1.2. PLANT ROOT HAIRS

(a) (b)

F IGURE 1.1. (a) Image of RHs on an Arabidopsis root. Figure reproduced from [115]. (b) A RH cell with a hair that is at to its final length. Figure reproduced from [56].

reaction-diffusion system where a chemical gradient in the independent kinetic term is assumed to be spatially linear. In the same way, Page et al. [110, 111] explore self-organised spatial pattern formation outside the classical Turing regime by varying the kinetic coefficients via a discrete jump [111] or a periodic modulation [110] for the Gierer–Meinhardt system. On the other hand, far from a Turing perspective, Matties & Wayne [89] prove existence of stationary and pinned waves in a more general context, using averaging and homogenisation. Similarly, Ward et al. [134] analyse a reduced Gierer–Meinhardt system with inhomogeneous linear kinetic terms, in addition using matched asymptotic method.

1.2

Plant root hairs

The majority of the surface area of the root system of a typical flowering plant is provided by its hairy outgrowths, see for example Figure 1.1(a). Root hairs (RH)s are protuberances that outgrow from root epidermal surfaces. In Arabidopsis, these protuberances can grow up to more than 1mm in length and approximately 10–20µ m in diameter, see Figure 1.1(b). Thus the study of RHs is agriculturally important for understanding and optimisation of both nutrient uptake and anchorage. For instance, once humid soil is set, mineral salts and water can enter through plant roots. Higher densities of longer RHs are produced in regions where relatively low amounts of less mobile nutrients, such as phosphorus, are available. Roots grow through soil inducing a larger surface area between roots and soil. Consequently, a more advantageous absorption of nutrients is provided. 5

CHAPTER 1. INTRODUCTION

F IGURE 1.2. Developmental zones of an Arabidopsis root. Figure reproduced from [56].

More widely, RH formation and growth in Arabidopsis has established itself as a key biological model problem within development and cell biology. This is mainly because RHs are found far from the plant body and become visible early in seedlings, so they are particularly amenable to scientific study due to the ease with which they can be imaged. In addition, the formation and growth of a RH represents an important problem in single-cell morphogenesis. Also, Arabidopsis is a model genetic organism for which there is an internationally coordinate project, providing large collections of mutants available for analysis [86]. RHs grow quickly, at a rate of nearly 1µ m per minute, allowing us to analyse efficiently the various phenotypes that arise under many different genetic mutations. Besides, a detailed analysis of the cellular changes can be experimentally tracked during the RH formation process, because their development is such that cells are organised in a single-file along the developing root epidermis. So young RH cells are found near the root tip and cells of increasing age and stage of growth are found further away in the same longitudinal stripe, see [56, 119] and references therein. For the model plant Arabidopsis there is a vibrant experimental literature surrounding each step of RH formation: differentiation of the root epidermis into hair and non-RH cells (e.g. [123, 124]); the initiation and growth of a single RH from wild-type hair cells (e.g. [96]); and the behaviour of a wide variety of mutants that form deformed or multiple hairs per cell, or no hairs at all (e.g. [68, 70]). Let us now deal with each of the processes in detail.

1.2.1

Cell specification

Root hairs develop in predictable and organised files that allow detailed analysis during the whole process. As can be seen in Figure 1.2, three different zones can be identified: the meristematic zone located in the root tip where new cells are formed in the meristem, which is covered by root 6

1.2. PLANT ROOT HAIRS

cap cells; the elongation zone, where cellular expansion occurs, and the differentiation zone where RHs are produced. Root hair cells are intercalated with cells, non-RH cells, that do not develop RHs. Specification from a recently-formed epidermal cell into a RH cell or non-RH cell is a topic which has attracted particular attention. That is, the assimilation of this process leads to the better understanding of cell-type patterning in plants. The root epidermis is covered by a specific arrangement of RH and non-RH cells, both kinds are distributed over the root surface in such way that RH cells are located between cortical cells, whereas non-RH ones lie on top of a single cortical cell, see Figure 1.3. In this way, cell type differentiation and cell position strongly suggest a presence of cell-to-cell communications which are crucial for formation in the root epidermis. Patterning information is provided at an early stage in epidermis development, immature epidermal cells destined to become root-hair cells are distinguished from their counter part, non-RH cells, prior to hair outgrowth. The differentiating RH cells present a greater cell division rate, smaller length, greater cytoplasmic density, lower vacuolation rate and distinguishable cell wall epitopes; see [12, 34, 44, 45, 56, 70]. In addition, plant hormones known as auxins are thought to play a crucial role in almost all aspects of a plant’s life. Auxins stimulate growth, and regulate fruit setting, budding, side branching, and the formation of delicate flower parts among many other morphogenic tissue-level responses. In a sense, RH outgrowth is one of the simplest and most easily studied effects that is stimulated by auxin [68, 96]. Auxin flows along the root from the core to the surface, see a sketch in Figure 1.3(b).

1.2.2

Root hair initiation

After being formed in the meristem, cells that will develop a RH become highly specialised. As the root continues to grow and the meristem moves away, the RH cells get wider, longer and deeper; this is done by diffuse growth, then the RHs forms. This process can be divided in two main stages: initiation, when a small disc-shaped area of the cell is softened given rise to a swelling, and tip growth, when a protuberance grows by targeted secretion. The first visible sign that this process has begun is the formation of a patch of small G-proteins known collectively as Rho of plants (ROP)s at the location on the cell membrane at which the hair will form, as shown further in Figure 1.8(a). Just prior to cell wall bulging, to form a RH stub, this patch is observed to be 10–20% of the cell length away from the apical (root tip) end of the cell. ROPs are examples of Rho-family small GTPases, a group of proteins whose role seems to be that of transmitting chemical signals in- and outside a cell in order to effect a number of changes inside the cell. Working as molecular “connectors”, these proteins shift between active and inactive states. ROPs control a wide variety of cellular processes, they contribute strongly to crucial cellular level tasks such as morphogenesis, movement, wound healing, division and, of particular interest here, cell polarity generation [40, 68, 96, 102, 112]. These proteins are 7

CHAPTER 1. INTRODUCTION

(a)

(b)

F IGURE 1.3. Arabidopsis root. (a) Transverse section showing the position-dependent pattern of RH cells and non-RH cells. Figure reproduced from [56]. (b) Sketch of a plant root showing tissue organisation and auxin pathway from the root core to root surface. Figure reproduced from [68].

unique to plants, although are related to Rac, Cdc42 and Rho-family small GTPases that control morphogenesis of animal and yeast cells, see [17] and references within. In particular we are interested in the role of ROPs in forming patches that produce local cell outgrowths. That is, in the initiation stage, small GTP-binding proteins appear at the growth region. Exchange factor regulators ARF GTPases known as ARF-GEFs not only are inhibitors preventing ROP localisation, but also are required for the proper polar localisation of PIN1, a candidate transporter of the plant hormone auxin [46]. ROP localisation starts to produce a protuberance, at the same time the pH of the cell wall 8

1.2. PLANT ROOT HAIRS

(a)

(b)

F IGURE 1.4. (a) Mutants when active GTP-bound cannot be turned off; the bar represents 200µ m. Figure reproduced from [70]. (b) Mutants where the ROPGDI1 is a protein that can bind to the GDP-bound ROP and keep it in that inactive form and in the cytoplasm; the bar represents 50µ m. Figure reproduced from [20].

GEF

GAP inactive-GDP

active-GTP

F IGURE 1.5. Sketch of the switching fluctuation between active-GTP and inactive-GDP ROPs.

falls, which leads to softening of the cell wall where a hair is about to be formed. Moreover, plants overexpressing a mutant ROP that is permanently in the active form have balloon-shaped root hairs, see Figure 1.4(a). On the other hand, protein ROPGDI1 that can bind to the GDP-bound ROP and keep it in that form and in the cytoplasm, has been inactivated, see Figure 1.4(b). This suggests that ROPs are able to cycle from the GTP-bound form, which are attached to the cell membrane, to the GDP-bound form, which lie in the cytoplasm. In other words, ROPs switch between two states: active when are bound to GTP, and inactive when are bound to GDP, see [96, 97]. The switching fluctuation between both active- and inactive-ROP forms is depicted in Figure 1.5. The biochemical process by which the process of RH formation is initiated within an Arabidopsis RH cell is depicted in Figure 1.6; blue and red balloons represent active-GTP ROP and inactive-GDP ROP respectively, ROPs which do not appear in the growth patch region (orange box) are represented by yellow balloons. Active-GTP ROP and Free ROP classes are able to transition to become either inactive-GDP ROP class or get unbound. This is illustrated by a double arrow connecting red and right-hand yellow balloons. Similarly the process shown in Figure 1.5 is shown as a link between blue and red balloons. A hypothesised autocatalytic process 9

CHAPTER 1. INTRODUCTION

Cell wall

Prenyl and S–acyl groups related to binding and activation biochemical processes

Cell membrane GTP-bound GEF

Free ROPs

GDP-bound GAP

Cell wall softening and root hair initiation Growth patch region Free ROPs

GEF: Guanine nucleotide exchange GAP: Rho GTPase– accelerating proteins

Cytoplasm

Auxin interior pathway

F IGURE 1.6. Sketch of the binding process nearby the cell membrane. Autocatalytic activation and catalysis effect by auxin are coloured in purple.

is symbolised by a purple loop-arrow attached to the active-GTP balloon. In addition, the green box encloses active-GTP class members that go to cell wall softening. Cell membrane binding activation processes are generally represented by snaking curves in blue and red accordingly. The interior auxin pathway is indicated by a purple arrow so the auxin gradient is represented by a decreasing purple shade in the same direction. As shall be seen further, we are interested in the binding process in the growth region.

1.2.3

Root hair growth

As mentioned above, once a swelling is set in the RH cell, a tip growth occurs on the bulge site. ROPs are not only strongly involved in the initiation process, but also participate in the direction of the tip growth. That is, as mentioned above, ROPs are active when bound to GTP and inactive when bound to GDP. When ROPs cannot be switched off or their inactive processes are modified their tip growth direction is affected [96], see Figure 1.4. Arabidopsis RHs normally develop at 1µ m per minute, and stop growing when a critical length is reached. At this point, the cytoplasm at the tip spreads out and the vacuole expands. Also, it has been experimentally observed that ROPs are involved in the branching and deformation of tips that is observed in some mutants (e.g. [70]). Thus, genetic analysis of this processes has identified specific proteins that are strongly engaged in mechanisms of RH initiation and tip growth. See [42, 56] and references therein.

1.2.4

Mutant studies

The Arabidopsis Information Resources is a database of genetic and molecular biology data for 10

1.2. PLANT ROOT HAIRS

(a)

(b)

(c)

(d)

(e)

F IGURE 1.7. (a) A mutant RH cell. Asterisks show multiple sites of RH initiation in a single root hair cell (indicated by the arrows). Figure reproduced from [120]. (b) Hair-forming cell with three RH initiation locations. The bar represents 50µ m. Figure reproduced from [87]. (c) Large bump in mutant rhd1. Figure reproduced from [56]. (d) Mutant overexpressing gene ROP2; from right-hand to left-hand, numbers indicate progressive snapshots at different times. RH initiation sites are indicated by the arrows. The bar represents 75µ m. Figure reproduced from [69]. (e) Mutants affected by auxin. On the left-hand side, RH site is farther away from the apical end (left arrow cap); on the right-hand side, multiple RH locations (arrows). Figure reproduced from [115].

11

CHAPTER 1. INTRODUCTION

the model plant Arabidopsis thaliana [128]. This contains data on a large number of genes that particularly affect RHs can be seen. Here we will briefly review only a few of those that are relevant to the purposes of this thesis. The normal role of the WER, TTG, and GL2 genes is either to stimulate non-RH cell differentiation or to repress RH cell differentiation. These genes are named after being present on the mutants: werewolf (wer), transparent testa glabra (ttg) and glabra2 (gl2). WER and TTG genes might be earlier acting components required for position-dependent non-RH and RH cell differentiation [29, 45, 80, 88]. Nevertheless, root epidermal cell specification is also influenced by genes related to the non-RH roothairless mutants: rhl1, rhl2, and rhl3, the ectopic RH mutants: erh1, erh2/pom1, and erh3, and the tornado mutants: trn1 and trn2, see [25]. Moreover, genes such as AXR2, that are linked to auxin resistant mutants such as dwarf (dwf) and auxin resistant2 (axr2), imply that auxin participates in RH formation [95]. This hormone is also known to be involved in early stages of epidermal cell type definition, which would imply it participates in the RH location [19, 87]. A list of features we are interested in that are affected by some genes that take part in the RH formation process is summarised as follows: i. Number of swellings. Wild-type produces a single swelling leading to a single hair. Even though mutants RH defective6 (rhd6) are mainly hairless, a small proportion of cells on these plants have extra hairs. This suggests that RHD6 gene is involved in controlling the establishment and the number of swellings. Examples of mutants with multiple RH sites are depicted in Figure 1.7(a), 1.7(b) and 1.7(d). ii. RH location. The Arabidopsis gene RHD6 determines the number of cells that will form hairs as well as the position of the hair on the hair cell [87]. In addition, auxin signalling induces hairs to form at the apical end of the cell, it also affects the location of hair initiation; the site of hair emergence on rhd6 mutant roots is restored from its more basal position to normal by treatment with auxin [88], see Figures 1.7(d) and 1.7(e). iii. Swelling size. The root hair defective1 (rhd1) mutant have larger bumps than the wild-type. This suggests that the RHD1 gene restricts swelling size (e.g. [113]), see Figure 1.7(c). For further detailed discussion on topics discussed in this section, see for instance [42, 56, 119] and references therein.

1.3

The ROP model system

A model of the initiation process discussed in §1.2.2 was proposed by Payne & Grierson [115] in the form of a Schnakenberg-type reaction-diffusion system for inactive (free) and active (membrane bound) ROPs. The key feature of the model is that the activation step is postulated to be dependent on the concentration of the plant hormone auxin. Auxin diffuses much faster 12

1.3. THE ROP MODEL SYSTEM

(a)

(b)

F IGURE 1.8. (a) A patch of surface bound ROP imaged using GFP in wild-type just prior to cell wall bulging. Figure reproduced from [115]. (b) Time lapse of patch drift within a single cell from end-wall to wild-type location. Figure reproduced from [39].

than ROPs [79] and is too small to be imaged on its own and also very similar to the amino acid tryptophan, an essential component in many proteins which makes it impossible to make a sensor that is specific for auxin. However, Jones et al. [68] found a surprising difference between the location of auxin in- and out- pumps in RH and non-RH cells on the root epidermis. From this data, using the auxin flow model introduced by Kramer [76], Jones et al. were able to surmise that there is a decreasing gradient of auxin from the apical end of each RH cell. Assuming that such a gradient mediates the autocatalytic activation of ROPs, simulations by Payne & Grierson showed that the active-ROP variable tends to form patch-like states towards the apical end, as in Figure 1.8(a). Moreover, various patch states can be found that show a close qualitative match with observations on the location, width and distribution of multiple hair cells in a variety of mutants. Grierson et al. [39] presented the new experimental data reproduced in Figure 1.8(b) which sheds light on the dynamic process by which ROP patches form. The process by which a patch of active ROP migrates from the apical cell boundary towards its wild-type position, from where a RH forms, takes place within a timescale of minutes. It appears to be triggered within the growing root as the RH cell reaches a combination of a critical length and a critical overall auxin concentration. In this thesis we shall show that the model derived by Payne & Grierson 13

CHAPTER 1. INTRODUCTION

is particularly effective in capturing this dynamical process. Specifically, a boundary patch of active ROP becomes unstable in favour of an interior patch as either the cell length or the auxinmediated activation reaction rate are increased. This instability is hysteretic, which suggests a robust biological mechanism. Grierson et al. [39] also showed that the effect of too much auxin is to produce mutants that spontaneously form two or more hairs, in keeping with a trend seen in the experimental literature. Moreover, if auxin concentrations are increased too rapidly, then multiple-hair mutant states were found to become stable in the simulations, via a similar process. These and other related features are seen in following chapters.

1.3.1

ROP kinetics

Let us now give a fresh justification of the Schnackenberg model used by Payne & Grierson [115] and in particular give the physical meaning of each of the variables and terms. ROPs can become attached to the cell membrane through prenylation and S-acylation, these chemical modifications determine the steady-state distribution between the cell membrane and cytosol, the membrane interaction dynamics, as well as function [126]. In other words, each ROP protein comes in two states: activate and inactivate. Inactivate ROP may be either bound to the cell membrane or in the cytoplasm. Once bound though, there is a good chance of phosphorylation, leading to the active form of ROP. In our model we shall not model the binding mechanism per se but shall differentiate only between the active form which can only diffuse within the confines of the cell membrane, and the inactive form, the majority of which is free to diffuse in the cytoplasm. We will also approximate the long-thin RH cell by a domain (either 1D or by a rectangular domain in 2D with high aspect ratio), and shall distinguish between membrane and cytoplasm only through separate diffusion constants, D 1 and D 2 for the active-ROP and inactive-ROP populations respectively. That is, we model a mathematical domain (z, T) ∈ Ωd × R+ ∪ {0}, which

Ωd is either the interval Ω ≡ [0, L] or the rectangle Ω2 ≡ [0, L x ] × [0, L y ] where L x > L y , and let ˜ T) represents the concentration of bound/active and V˜ (z, T) the unbound/inactive ROP. In U(z, the model we shall also reflect the fact that the kinetic processes occur considerably faster than the changes to cell length and auxin concentration levels. Therefore we shall assume that the latter are effective bifurcation parameters that vary quasi-statically. Figure 1.9(a) is a replica of Figure 1.6 with variable names added and assumes schematically the entire process, the auxin gradient is represented by a shade in the cytoplasm, where the purple arrow indicates decreasing direction. This is also the symplastic pathway, which is the path that auxin, along with other light molecules and water, diffuses through the cytoplasm. We shall assume that the active Rho state is deactivated at rate κG AP , and equivalently the inactive Rho state is activated at rate κGEF depending upon the presence of GEFs or GAPs respectively; see sketch in Figure 1.9(b). To derive a specific form for the activation and deactivation steps we shall take a model that is inspired by similar processes for Rho proteins in mammals and yeast. There, it is widely held that 14

1.3. THE ROP MODEL SYSTEM

(a)

κGEF

κG AP





(b) Prenyl and S–acyl groups related to binding and activation biochemical processes

Cell wall Cell membrane

U˜ a1 Free ROPs

D1  D2 GEF

r Cell wall softening and root hair initiation

c GAP



b

K U˜ ; κ1, κ2, κ3

Growth patch region



a2 Free ROPs

GEF: Guanine nucleotide exchange GAP: Rho GTPase– accelerating proteins

Cytoplasm

Auxin interior pathway

F IGURE 1.9. (a) Sketch of the switching fluctuation between active- and inactive-ROP densities. (b) Sketch of the model where autocatalytic activation and catalysis effect by auxin are coloured in purple.

the activation GEF activation step involves positive feedback (autocatalysis) whereas the GAP step is thought to be passive; see for example [97]. Both Jilinke et al. [67] and Otsuji et al. [109] have derived models for active process of cross-talk between the three different mammalian/yeast Rhos — cdc42, RhoA and Rac — which process is thought to underlie the polarity determination process. Like ROPs, each protein is supposed to come in two states, an activated state (which is bound to the cell membrane) and a deactivated state (either bound or in the cytosol). Similarly the transition between these two states is via GEFs and GAPs. In [67, 109], the GEF-induced activation step of each Rho species is modelled by a Hill function: (1.1)

˜ ≡ κ1 + κGEF (U)

κ2U˜ q , 1 + κ3U˜ q

where κ1 is the activation rate, κ2 represents autocalatytic acceleration, κ3 is a saturation coefficient, and the power q is typically taken to be an integer but need not be. In [67, 109] the 15

CHAPTER 1. INTRODUCTION

typical value q = 2 is chosen which preserves the odd power of the nonlinearity in their models. Payne & Grierson took a simplified version of (1.1) by taking κ3 = 0 and allowing only one species of ROP to be modelled. This simplification can be justified as follows. In plants, although there are several different kinds of ROP, their activation is not thought to involve cross talk. Also, by including a constant production rate of the inactive ROP and a constant probability of recycling or further processing of active ROP, there is no need for an explicit saturation term. In fact, as we shall later show in Proposition 2.4, the total amount of active ROP is a conserved quantity in any steady solution. As we referred at the beginning of this section, Jones et al. [68] found experimental evidence that suggests a longitudinal spatially decaying gradient of auxin which Payne & Grierson postulate modulates the autocatalytic step. That is, we suppose then the parameter κ2 is spatially ¡ ¯ ¯¢ ¡ ¯ ¯¢ dependent: κ2 = k 20 α z; ¯Ωd ¯ . Here k 20 measures an overall concentration of auxin and α z; ¯Ωd ¯ is a smooth function that represents the spatial distribution of auxin, normalised so that (1.2)

α(0, y) = 1,

∂α ∂x

< 0,

for all (x, y) ∈ (0, 1) × (0, 1).

In contrast, we suppose that κ1 = k 1 a constant. All other processes are supposed to follow the simple law of mass action, with the GEF-induced deactivation/unbinding rate given by a constant κG AP = c. Furthermore, b is assumed to be the constant rate of production of inactive ROP and r

the rate at which active ROP is recycled or used up to produce other complexes, including those that go on to produce cell wall softening; see Figure 1.9.

1.3.2

The gradient

Auxins are a class of plant hormones, strongly involved in growth plant processes, that travel through the cytoplasm and across the cell walls that separate plant cells. Auxin transport is a process that occurs inside and outside the plasma, these pathways are also known as symplastic and apoplastic pathways respectively (see e.g. [18]). There are three families of genes in charge of this polar transport: PIN in efflux, AUX/LAX in influx and MDR/PG in both. As a consequence, a complete description of auxin transport should contain interactions where these three families participate. Such a description also needs to contain both, symplastic and apoplastic auxin diffusion and localisation of auxin in- and out-flux carriers inhomogeneously distributed along the cell membranes. These carriers act like permeability gates permitting auxins leave the cells, see [55, 71, 76, 77] and references therein contained, and [78] for a review of experimental findings and models of auxin polarisation. For instance, Band et al. in [7] derive a one-dimensional continuous PDE system from a two-dimensional spatially discrete approach. To do so, they asymptotically analyse a model which consists of a set of epidermal, cortical and endodermal cells where auxin travels through each cell; a nondimensionalised measure of the length along the whole tissue is considered as the spatial variable. That is, epidermal cells express AUX1 influx carriers in every cell membrane, and also 16

1.3. THE ROP MODEL SYSTEM

PIN1 and PIN2 efflux carriers in cell membranes allowing a transport of auxin from the tip to the elongation zone. Cortical and endodermal cells only express efflux carriers in its membranes, PIN1 and PIN2 carriers for the former and only PIN1 carriers for the latter, in such way that polarisation occurs towards the root tip (see also [76]). Their results show that the tissue-scale auxin flux is determined by the carriers, where physical proprieties as cell length is particularly relevant. This approach however is an extension from the model proposed by Grieneisen et al. in [55], where a rather discrete model is analysed. Their results support the existence of a steep auxin gradient. Such a model considers to regulate auxin transport in RH cells as a leaky auxin transport flux. From these studies condition (1.2) can be derived. That is, the auxin flux getting in to root cells that are located in the differentiation zone occurs via auxin permeability, this is governed by ˆ J p = (P e − P i ) α˜ n,

(1.3)

ˆ is a vector normal to the where α˜ is a generic variable to represent auxin concentration, n boundary, and P i and P e are influx and efflux permeability rates, respectively. As auxin can leave the cell whenever there is a PIN (auxin carrier) at the membrane, these compounds are not uniformly distributed along the boundary, but are mainly situated at the apical end [55]; thus, the efflux permeability is greater than influx permeability, P e > P i , at places where PINs are located, otherwise P e < P i . From (1.3), auxin RH-cell dynamics is controlled by an advection equation of the form ∂α˜ ∂t

ˆ · ∇) [(P e − P i ) α˜ ] = 0. + (n

In addition, auxins are small molecules that are assumed to diffuse within the cell at a rate ¡ ¢ D α˜ ∼ O 102 µ m2 /s , see [79], which is much faster than the ROP diffusion rates assumed in [97, 115, 117]. ROPs and auxin speeds belong to different time-scales. This therefore implies that auxin distribution in a RH cell can be determined by condition (1.2). A sketch of the auxin flux inducing a decreasing gradient in the RH cell along longitudinal direction is depicted in Figure 1.10. In all computations that follow, we typically take values of α(x) as in [115], α(x) = e−ν x ,

ν = 1.5,

for the 1D case. Simulation results in Chapter 2 show that the qualitative results are insensitive to different piecewise-smooth non-increasing functional forms of α, for the same difference α(0) − α(1).

On the other hand, for the 2D case, we take into account two different scenarios: (1.4)

α(x, y) = e−ν x sin (π y)

17

and α(x, y) = e−ν x .

CHAPTER 1. INTRODUCTION

Apical end

Basal end

Pi ˆ n

∂α˜ ∂t

Pi ˆ n

ˆ · ∇) [(P e − P i ) α˜ ] = 0 + (n PIN

Cytoplasm

Pe

Pe

F IGURE 1.10. Sketch of the auxin flux in the RH cell; azimuthal view. The gradient is coloured as a purple shade. Cell wall and cell membrane are coloured as in Figures 1.6 and 1.9.

The first one, as shall be seen in Chapter 6, favours aggregation towards the middle of the transversal direction (this situation is not sketched in Figure 1.10). The second form in (1.4) does not give any transversal preferential aggregation sites. However, as shall analytically and numerically be seen, equivalent results to the y-dependent gradient scenario can still be reproduced. This form also sheds light on experimental results of mutants discussed in §1.2.4 and provides a natural mathematical transition from the 1D case to the 2D case. All this shall be thoroughly analysed in Chapter 4.

1.3.3

Fundamental model

Under the above assumptions, using the law of mass action allied to Fick’s law of diffusion in a standard way, we obtain the reaction-diffusion system of equations (1.5a)

Active-ROP:

∂U˜ ∂T

³ ¯ ¯´ ¯ ¯ = D 1 ∆U˜ + k 20 α z; ¯Ωd ¯ U˜ 2 V˜ − (c + r)U˜ + k 1 V˜ ,

in Ωd (1.5b)

Inactive-ROP:

∂V˜ ∂T

³ ¯ ¯´ ¯ ¯ = D 2 ∆V˜ − k 20 α z; ¯Ωd ¯ U˜ 2 V˜ + cU˜ − k 1 V˜ + b.

We also impose the homogeneous Neumann boundary conditions ∂U˜ = 0, ˆ ∂n

∂V˜ =0 ˆ ∂n

on ∂Ωd ,

which supposes that the large ROPs do not diffuse through the cell wall. This system is a generalisation of the Schnakenberg model [125], which is doubtless one of the most widely studied models undergoing Turing-like pattern formation schemes. In contrast to other models such as the Brusselator [104], which takes into account four chemical reactions, 18

1.4. THESIS OUTLINE

the Schnakenberg one is a simplification where three hypothetical chemical reactions are represented, amongst which one is autocatalytic. As a consequence, it is not only simpler, but also it is already known that the Schnakenberg system can display features that are considered biologically relevant [125]. The usual Schnakenberg system can be obtained from (1.5) by setting ˜ k 1 and c to zero, adding a constant production term to the U-equation, and taking α ≡ 1; this transformation can be seen as an homotopy between both systems. Furthermore, both systems are derived from simple reactions where an autocatalytic process is present. In addition, as can be straightforwardly explored, both describe activator-substrate processes. In other words, near activation regions the activator U˜ aggregates so that the substrate V˜ is consumed quickly, as a result substrate valleys occur where activator peaks do. This will be reflected in numerical and analytical analysis in the chapters to follow. Additionally, it can present a very rich variety of self-organising dissipative structures, and inherent dynamics, that shed light on sub-cellular or ecological processes, for example; see [84] and references therein for a review. From a theoretical point of view there is an extensive range of research on the Schnackenberg model. For instance, Ward & Wei have analysed existence and asymmetry of spikes [135]; stability of symmetric N-spiked steady-states is deeply examined in [66]; self-replication spots and their dynamics in 2D are studied in [73]; and for findings on chemical reactors see e.g. [35, 133], where this model plays a central role. Non-homogeneous approach, apart from studies previously mentioned in §1.1.2, there have been mathematical analysis in superconductivity (e.g. [13, 14, 21, 82]). Reaction-diffusion equations with spatially dependent coefficients of nonlinear terms are not easy to study analytically, see e.g. [134]. Nevertheless there have been some previous work both analytical and numerical: for the Schnakenberg system [9, 10, 83], the Gierer–Meinhardt and Brusselator systems (e.g. [60, 61]), and in models of mitosis in cytokinesis [62, 63]. In this thesis we analyse the Schnakenberg-like system (1.5) with spatially dependent coefficients which modulate the nonlinear terms. As shall be seen in the further chapters, we shall thorough undertake a study of this system, exploring the formation and dynamics of patches of activator both numerically and analytically.

1.4

Thesis outline

This thesis is outlined as follows: in Chapter 2 we present some preliminary results focusing mainly on the homogeneous case. As is common in systems with widely different diffusion rates, we find the occurrence of Turing bifurcations. It is shown however that in the parameter region of interest, all bifurcation are subcritical. Then, also in Chapter 2, we proceed to explore the non-homogeneous case where numerical simulation data point to the existence of localised spike solutions that represent patches of high concentrations of active ROPs in 1D. This is followed by a numerical bifurcation analysis to identify parameter regions where different forms of 19

CHAPTER 1. INTRODUCTION

solution profile exist, and the forms of instability that cause transitions between them. Chapter 3 then goes on to present matched asymptotic analysis, to provide predictions of the location and width of single- and multiple-patch states. We also explore thoroughly the competition instability of multiple spikes in an O (1) time-scale regime, in order to reach an explanation of the criticality threshold beyond which a single spike solution becomes unstable. In Chapter 4 we analyse numerically and analytically the so-called transverse instability of trivial extensions of 1D solutions onto a 2D domain, this instability also belongs to an O (1) time-scale scheme. An extensive numerical bifurcation analysis for the 2D case is carried out in Chapter 5 showing the existence of stable spot localised solutions. In addition, a differential algebraic system is derived in other to characterise the location dynamics of a multi-spot array, and O (1) time-scale instabilities are numerically shown. In Chapter 6, a transversally and longitudinally varying auxin gradient is proposed to model the RH initiation process in a 3D RH cell. As we only concern about the ROP interactions in the cell membrane and the cell wall, the growth patch region, a projection onto a 2D rectangular domain is taken. A numerical bifurcation analysis and direct simulations are implemented, which shed light on a more biologically realistic scenario. Chapter 7 contains discussion, concluding remarks and suggestions for future work. Finally, in Appendix A we reproduce a criticality result from [136], which is used in the analysis of the competition and transverse instabilities in Chapters 3 and 4.

20

HAPTER

C

2

B IFURCATION A NALYSIS AND I NITIAL S IMULATIONS

A

first approach to the analysis of system (1.5) under an environmental gradient will be performed in this chapter. System (1.5) derived in Chapter 1, which is a generalised Schnackenberg system, like the Gray–Scott system, naturally inherits features that come

from the Turing bifurcation. As can be straightforwardly shown, the Schnakenberg system can be ˜ inferred from (1.5) by setting k 1 and c to zero, adding a constant production term to U-equation, and taking α ≡ 1. The Gray–Scott system can be similarly derived. That is, there is homotopy between systems, which come from the fact they are derived from modelling simple reactions

where an autocatalytic process is present. In this chapter an analytical bifurcation analysis of the homogeneous case is performed, from which a criticality condition is then derived. This condition will determine whether a bifurcation is subcritical or supercritical. Light is therefore shed on the properties that initially trigger the formation of the localised structures which will be studied further. Then, we shall perform a numerical bifurcation analysis of the kind of attributes that play the role of an underlying backbone to the system when a spatially dependent gradient is taken into consideration. This will provide a strong numerical framework which will lead to insights of relevant solutions and instabilities that the model presents. These features shall be thoroughly explored in further chapters by means of direct simulations and asymptotic analysis.

2.1

Turing instability conditions

To begin the analysis of the Turing bifurcations that gives rise to localised structures, let us briefly summarise a linearised analysis (see, for instance, [100]) that will be relevant later. Consider a generic reaction-diffusion system in a homogeneous domain D for U and V . These variables represent chemical concentrations at position x and time t > 0, which take the role of an activator 21

CHAPTER 2. BIFURCATION ANALYSIS AND INITIAL SIMULATIONS

and an inhibitor/substrate respectively, (2.1a)

ρU U t = DU ∆U + F(U, V ) ,

(2.1b)

ρV Vt = D V ∆V + G(U, V ) ,

where parameters DU and D V are positive diffusion coefficients, and evolution parameters ρU , ρV are non negative. Homogenous Neumann boundary conditions are taken into account. To proceed with the linear analysis, we assume existence of an isolated spatially homogeneous solution: (U0 , V0 ). This solution, also named an equilibrium solution or steady state, is obtained from intersection of nullclines F(U, V ) = 0 and G(U, V ) = 0. Now, upon substituting the incremental variables U = Uˆ + U0 and V = Vˆ + V0 into system (2.1), and dropping hats, we get at first order (2.2a)

ρU U t = DU ∆U + a 11U + a 12 V ,

(2.2b)

ρV Vt = D V ∆V + a 21U + a 22 V ,

where coefficients a i j are the entries of J, the Jacobian matrix, a 11 =

∂F ∂U

,

a 12 =

∂F ∂V

a 21 =

,

∂G ∂U

a 22 =

,

∂G ∂V

,

at the steady state. ¡ ¢ Next, consider the space of vector-valued functions, C n,ν D , R2+ , whose derivatives of order n

belong to the space of functions on the closure of the domain D that are ν-Hölder continuous, for some 0 < ν ≤ 1. Hence, the system (2.1) can be re-written in terms of the functional (2.3)

t

φ (U, V ) ≡ diag (ρU , ρV )

Ã

∂ ∂t

U

!

V

Ã

+L

U

!

V

+ N (U, V ) ,

¡ ¢ ¡ ¢ ¡ ¢ where φ t : C 2,ν D , R2+ −→ C ν D , R2+ . The linear operator, L : X −→ C ν D , R2+ , is given by Ã

(2.4)

L

U V

!

Ã

= −diag (DU , D V ) ∆

U V

!

Ã

−J

U

!

V

,

where, from (2.2), the Jacobian entries are given as (J) i j ≡ a i j , and the nonlinear terms are gathered in N (U, V ). A suitable domain of the operator L gives a bounded Fredholm operator of index zero (see [51] and references within for a detailed discussion). As shall be seen further, this choice is essential for the application of the Lyapunov–Schmidt reduction method, which will be used to calculate the normal form of the steady state functional. Upon considering boundary conditions we therefore choose (2.5)

¯ ½ ¾ ¡ ¢¯ ∂V 2, ν 2 ¯ ∂U X = (U, V ) ∈ C D , R+ ¯ = =0 . ∂n ∂n

22

2.1. TURING INSTABILITY CONDITIONS

The steady state functional comes from (2.3) by setting the evolution parameter matrix to zero, i.e. Ã φ (U, V ) ≡ L

(2.6)

U V

!

+ N (U, V ) .

From (2.2), when no diffusion is involved, linear stability is obtained if conditions (2.7)

tra J < 0,

det J > 0 ,

are satisfied at the steady state. However, once diffusion is present, the steady state stability is disrupted. In order to derive its consequences, we define the vector w ≡ (U, V )T . Thus, solutions to the boundary problem −∆w = m2 w,

¯ ∂w ¯¯ ∂n ¯

∂D

=0

provide the subspace Σ ≡ {wm (x) : m ∈ N}, which spans the eigenspace of L. Indeed, for each m ∈ Z, the orthogonal set Σ is complete (see, for instance, [41]). As a consequence, the operator L on this subspace is given by the matrix Ã

(2.8)

L| Σ =

m2 DU − a 11

−a 12

−a 21

m2 D V − a 22

!

.

This implies that instability conditions driven by diffusion arise from the eigenvalues of (2.8), which are found by computing its trace and determinant, ¡ ¢ (2.9a) E m2 ≡ tra L|Σ = (DU + D V ) m2 − tra J , ¡ ¢ (2.9b) H m2 ≡ det L|Σ = DU D V m4 − (D V a 11 + DU a 22 ) m2 + det J . Notice that, from (2.9a), the homogenisation nature of diffusion is straightforwardly reflected ¡ ¢ by the inequality E m2 > |tra J|, which holds for all m ∈ N. In addition, as a function of m2 , the determinant (2.9b) has a negative minimum wherever (2.10a)

D V a 11 + DU a 22 > 0 ,

and (2.10b)

D V a 11 + DU a 22 > 2

p

DU D V det J .

Inequalities (2.7) and (2.10) are the so-called Turing instability conditions or diffusion driven instability conditions. As a first consequence, for instability we note from (2.7) and (2.10a) that DU 6= D V . Henceforth, we should assume without loss of generality that DU < D V . In other words, to get a spatially organised structure from small perturbations upon a homogeneous state, concentration U must diffuse slower than concentration V ; for a more detailed discussion on this subject, see [100]. The trace and the determinant in (2.9) are not only essential for Turing bifurcation analysis, but also for Hopf–Turing bifurcation analysis. These two formulae contain the dispersion relation associated to system (2.1) in finite domains, which consequently plays a central role on the linear analysis of the reaction-diffusion systems. 23

CHAPTER 2. BIFURCATION ANALYSIS AND INITIAL SIMULATIONS

2.2

The homogeneous system

In order to begin the study of system (1.5) when there is no spatial gradient, we are first interested in its steady state solutions. As is assumed in this chapter, we consider an one-dimensional domain

Ω = [0, L] with L > 0. As the reader shall note, these results can be easily extended to higher spatial dimensions. At first sight, it would seem that the model with D 1 < D 2 is well set up for pattern formation through Turing bifurcations as shall be analysed here. To begin such an analysis, consider the homogeneous problem α(x) ≡ 1. Thus, the system (1.5) takes the form (2.11a)

U t = D 1U xx + k 20U 2 V − (c + r)U + k 1 V ,

(2.11b)

Vt = D 2 Vxx − k 20U 2 V + cU − k 1 V + b .

in Ω.

2.2.1

Diffusion driven instability

First of all, we start from the steady solution U0 ≡

(2.12)

b , r

V0 ≡

br(c + r) , k 20 b2 + k 1 r 2

upon substituting the incremental variables U = Uˇ + U0 and V = Vˇ + V0 into system (2.11), and dropping hats we get the linearised system ¡ ¢ U t = D 1U xx + a 11U + a 12 V + k 20 U 2 V + V0U 2 + 2U0UV , ¡ ¢ Vt = D 2 Vxx + a 21U + a 22 V − k 20 U 2 V + V0U 2 + 2U0UV ,

(2.13a) (2.13b)

where the coefficients a i j depend on k 20 as follows: ¡ ¢ (c + r) k 20 b2 − k 1 r 2 k 20 b2 + k 1 r 2 (2.14a) a 11 = , a = , 12 k 20 b2 + k 1 r 2 r2 ck 1 r 2 − k 20 b2 (c + 2r) k 20 b2 + k 1 r 2 (2.14b) a 21 = , a = − . 22 k 20 b2 + k 1 r 2 r2 Next, we examine stability when no diffusion is present. From (2.14), the trace and determinant of J are k 1 r 2 + b2 k 20 det J = , r

¡ ¢ ¡ ¢2 r 2 (c + r) k 20 b2 − k 1 r 2 − k 20 b2 + k 1 r 2 ¡ ¢ tra J = , r 2 k 20 b2 + k 1 r 2

which, from conditions (2.7), show that the steady state (U0 , V0 )T is asymptotically stable in absence of diffusion if c + r < 8k 1 . Otherwise, it also is asymptotically stable wherever k 20 < k− 20 or k 20 > k+ 20 , where (2.15)

k± 20 ≡

´ p r2 ³ c + r − 2k ± + r) + r − 8k , (c (c ) 1 1 2b2

24

c + r > 2k 1 .

2.2. THE HOMOGENEOUS SYSTEM

System (2.11) presents no Hopf bifurcation points unless the bifurcation parameter crosses k± 20 as is increased. As can be seen further from the linear analysis, diffusion does not contribute to oscillatory behaviour unless conditions above are satidfied to get k 20 = k± 20 . Only pitchfork bifurcations are expected, otherwise. Nevertheless, according to the Turing conditions revised in §2.1, the steady state is destabilised by diffusion. To quantify this, we consider the eigenspace of the operator L which is spanned by the subspace (

Σ := cos

(2.16)

³ π mx ´

Ã

L

A B

!

) 2

2

: m ∈ N, A + B > 0 .

Thus, for each m ∈ Z, it is sufficient to take into account the orthogonal set {cos (π mx/L) : m ∈ N}. This subspace is complete for scalar functions. Hence, it provides also a complete set for vector functions in Σ. The operator L acting on subspace (2.16) gives the matrix L|Σ as in (2.8). Hence, from (2.9), the trace and determinant take the form µ 2 2¶ π2 m 2 π m (2.17a) E = + D − tra J, (D ) 1 2 L2 L2 Ã ! ¡ ¢ µ 2 2¶ (c + r) k 20 b2 − k 1 r 2 π m π4 m 4 k 20 b2 + k 1 r 2 π2 m2 = D1 D2 (2.17b) H − D2 + det J . − D1 L2 L4 k 20 b2 + k 1 r 2 r2 L2 Next, from (2.10b), in order to find Turing instability the inequality (2.18)

¡ ¢ ¡ ¢2 p D 2 r 2 (c + r) k 20 b2 − k 1 r 2 − D 1 k 20 b2 + k 1 r 2 ¡ ¢ > 2 D 1 D 2 det J r 2 k 20 b2 + k 1 r 2

∗+ must also hold for D 2 (c + r) > 8D 1 k 1 and k∗− 20 < k 20 < k 20 , where

´ p r2 ³ D + r) − 2D k ± . D + r) + r) − 8D k (c (c (D (c ) 2 1 1 2 2 1 1 2D 1 b2 ¯ ¯ ¯ ∗± ¯ ¯ ¯ ¯ Condition (2.18) implies D 1 < D 2 , and ¯ k± 20 < k 20 , as can easily be verified. In addition, the

(2.19)

k∗± 20 =

determinant (2.17b) has a global minimum at the critical value ³ ¡ ¢ ¡ ¢2 ´ D 2 r 2 (c + r) k 20 b2 − k 1 r 2 − D 1 k 20 b2 + k 1 r 2 L2 ¡ ¢ (2.20) m2min = . 2π2 D 1 D 2 r 2 k 20 b2 + k 1 r 2 Using the parameter set as given in Table 3.1(a) (see §3.1), the determinant of L|Σ as function of π2 m2 /L2 for several values of k 20 is depicted in Figure 2.1(a) as well as the dispersion relation, which is shown in Figure 2.1(b). There bold solid black curves correspond to a certain value of k 20 at which (2.17b) has a single root. In addition, Figure 2.1(c) shows a bifurcation diagram where positive critical modes, m± c , at which the determinant has real roots. Notice that, below a critical value, k∗20 , there is no m c mode at which the determinant has a root. Also, two different U-steady state solutions are plot in Figure 2.1(d). These states not only bifurcate, but also correspond to wave numbers as depicted in Figure 2.1(c). 25

CHAPTER 2. BIFURCATION ANALYSIS AND INITIAL SIMULATIONS

(a)

(b)

1

2

8D 1 k 1 , ∗+ and k∗− 20 < k 20 < k 20 . For reaction parameters satisfying either c + r < 8k 1 and inequality (2.10b), + or c + r > 8k 1 and condition (2.10b) whenever k 20 < k− 20 or k 20 < k 20 , then system (2.1) presents

26

2.3. PITCHFORK BIFURCATION

Turing bifurcations upon varying k 20 . Hopf bifurcations occur if D 2 (c + r) = 8D 1 k 1 and the inequality (2.10b) is reversed. As Proposition 2.1 indicates, system (2.1) presents Turing bifurcations for the parameter set as given in Table 3.1(a), however no Hopf bifurcations were found. Despite this, later in Figure 2.9, we shall see breathing-like patterns when a gradient is spatially dependent for very small values of k 20 . This might come from some Hopf bifurcation triggered by the gradient.

2.3

Pitchfork bifurcation

We shall now compute a criticality condition for the Turing instability that system (2.11) undergoes. To do so, we apply the Lyapunov–Schmidt reduction method to calculate a normal form of the functional (2.6); see [51] for a detailed discussion and [5, 16] for examples where this method has been also applied. Then, we derive a criticality condition from which this system presents either a super- or a sub-critical pitchfork bifurcation. Recall, the main bifurcation parameter k 20 represents the auxin overall rate in the cell. We construct a function g = g (z, λ) such that z parametrizes the kernel of L, and the translation λ = k 20 − k∗20 , which is defined as the bifurcation parameter by simplicity. Next, as defined in §2.1, we choose the vector w = (U, V )T to hence get the reduced function ­ ® g (z, λ) ≡ w∗ , φ (zw, λ) , RL where the usual inner product, 〈 f 1 , f 2 〉 = 0 f 1 (x) f 2 (x) dx, is used. Notice that, due to the fact the Fredholm alternative guarantees that (range L)⊥ = ker L∗ holds for compact linear operators (see e.g. [51]), we consider the kernel of L∗ , which is spanned by w∗ , rather than the range of L. Here we denote L∗ as the adjoint operator of L, and φ : R2 × R −→ R2 as the smooth mapping (2.6) of the system (2.13). Recall we have assumed a trivial translation to the steady state (U0 , V0 ), thus w = 0 is the steady solution of functional (2.6), from which we have g(0, λ) ≡ 0. Then, taking into account subspace Σ, defined in (2.16), and assuming a bifurcation point for a certain value of the parameter such that the determinant det L|Σ is zero, the range of the parameter is such as m± c is real, see Figure 2.1(c). In so doing, we have g(0, 0) = 0 and g z (0, 0) = 0. Next, without loss of generality, we choose a mode m = m± c , which comes from the fact the spatial structure of the solutions to bifurcate has the form cos (π mx/L), see Figure 2.1(d). Thus, from matrix (2.8) and entries of L|Σ in (2.14) a basis of ker L|Σ is (2.21a)

³ πm x ´ c w = cos L

Ã

Aˆ Bˆ

!

,

we analogically get one for ker L|∗Σ , Ã ! ³ π m x ´ Aˆ ∗ c ∗ , (2.21b) w = cos L Bˆ ∗

Aˆ = a 12 ,

Bˆ =

π2 m2c

L2

π2 m2c Aˆ ∗ = D 2 − a 22 , L2

27

D 1 − a 11 ;

Bˆ ∗ = a 12 ,

CHAPTER 2. BIFURCATION ANALYSIS AND INITIAL SIMULATIONS

where k 20 -dependent coefficients a i j are evaluated at the bifurcation, λ = 0. As w ∉ range L the inner product 〈w∗ , w〉 6= 0, hence the correct vector orientation is given for all k 20 as ­

µ 2 2¶ 2 2 ® π mc π mc w , w = a 12 E > 0. 2L L2 ∗

Notice, as can be seen from (2.14a), the trace (2.17a) as well as a 12 , is positive for all values of the bifurcation parameter. To calculate the coefficients of the reduced function expansion at the bifurcation point, we note that the steady solution at the singularity is uniquely mapped onto the origin, which implies coefficients g λ (0, 0) and g λλ (0, 0) are zero. Now, the problem defined by the system (2.11) together with homogeneous Neumann boundary conditions commutes with the reflection x → L − x. This is a symmetry which plays an important role on the expansion of the reduced function. As a matter of fact, from bases (2.21), the eigenfunction cos (π m c x/L) is invariant when m c is even, and despite the nonlinear quadratic terms in (2.13), the bifurcation remains a pitchfork even when m c is odd, see [51, §VI] for a detailed discussion. Indeed, this implies that as symmetry is broken, a pitchfork bifurcation must occur. We hence seek for the normal form of a pitchfork bifurcation. Now, from (2.21a) once again, we obtain (2.22a)

(2.22b)

³ πm x ´ c d φ(w, w) = F0 cos L 2

2

Ã

1

!

,

−1 Ã ! ³ ´ 1 3 3 πmc x , d φ(w, w, w) = G0 cos L −1

¡ ¢ F0 ≡ −2k∗20 V0 Aˆ + 2U0 Bˆ Aˆ ,

G0 ≡ −6k∗20 Aˆ 2 Bˆ ,

where Aˆ > 0 for any value of k∗20 . In addition, even though F0 in (2.22a) is not necessarily zero, the ­ ® coefficient of the quadratic term in the reduced function expansion, g zz (0, 0) = w∗ , d 2 φ(w, w) , vanishes. We consequently calculate the coefficients of terms λ z and z3 of the reduced function expansion. These are given by formulae ­ ¡ £ ¤¢® (2.23a) g zzz (0, 0) = w∗ , d 3 φ(w, w, w) − 3d 2 φ w, L−1 P d 2 φ(w, w) , ­ ¡ ¢ ¡ ¢® (2.23b) g λ z (0, 0) = w∗ , d φλ · w − d 2 φ w, L−1 Pφλ , where P is a projection operator from Y = (ker L) ⊕ (range L) onto range L. To proceed, notice that, from trigonometric identity for double angles, coefficient (2.22a) is a member of the set defined in (2.5), where here D ≡ Ω, and orthogonal to the basis (2.21a), so that d 2 φ(w, w) ∈ range L. Thus, d 2 φ(w, w) ∈ X is trivially decomposed as d 2 φ(w, w) = d 2 φ(w, w) + 0 , where d 2 φ(w, w) ∈ range L and 0 ∈ ker L in the right-hand side. We therefore have that the projection P maps (2.22a) onto itself, £ ¤ P d 2 φ(w, w) = d 2 φ(w, w) . 28

2.3. PITCHFORK BIFURCATION

As a consequence, in order to compute the second term of the right-hand side in (2.23a), from (2.22a), we set the problem Ã

L

U¯ V¯

!

³ πm x ´ c = F0 cos L

Ã

1

2

!

U¯ x = V¯ x = 0 ,

,

−1

at

x = 0, L ,

which, upon using a 12 = −a 22 , has solution ¶ 2π m c x K u cos F0   µL ¶ = 2  − 1 + K cos 2π m c x v a 22 L 

Ã

(2.24)

!

U¯ V¯

µ

  , 

where ¶ µ 4π2 m2c 1 a + a − D 11 21 1 , M M L2 L2 µ ¶µ ¶ π2 m2c π2 m2c M = a 11 − 4 D 1 a 22 − 4 2 D 2 − a 12 a 21 . L2 L

Ku =

4π2 m2c

Kv =

D2 ,

Therefore, from (2.24), we get ¡ £ ¤¢ d 2 φ w, L−1 P d 2 φ(w, w) = à ! ¶ ¸ · µ ³ πm x ´ 1 ¡ ¡ ¢ ¢ U0 ˆ 2π m c x c ∗ ˆ ˆ ˆ + A cos , = − k 20 F0 A K u V0 + K u B + K v A U0 cos L a 22 L −1

to then obtain, from (2.21b) and (2.22b), " ¡

ˆ∗

ˆ∗

g zzz (0, 0) = A − B

¢

Ã

k∗20 F0

Ã

! ! # ˆ 3π2 m2c G 2 A 0 Aˆ K u V0 + K u Bˆ + K v Aˆ + U0 + . a 22 2 4L

Next, to compute (2.23b), first notice that à φλ = −

a011U + a012 V a021U + a022 V

Ã

! ¡ ¢ ¡ ¢ − U 2 V + V0 + λV00 U 2 + 2U0UV

1 −1

!

,

a0i j ≡

da i j dλ

,

¡ ¢ which maps w = 0 onto zero, so that d 2 φ w, L−1 Pφλ = 0. From (2.14), the equations a021 = −a011

and a012 = −a022 are satisfied, hence we have ¡ ¢¡ ¢ π2 m2c g λ z (0, 0) = a021 Aˆ − a012 Bˆ Aˆ ∗ − Bˆ ∗ . 2L

Therefore, the reduced function g(z, λ) nearby the bifurcation point k∗20 takes the pitchfork normal form (2.25)

g(z, λ) = g λ z (0, 0)λ z + g zzz (0, 0)z3 . 29

CHAPTER 2. BIFURCATION ANALYSIS AND INITIAL SIMULATIONS

(a) 3.0

(b) 0.3

σ >0, k20∗ =0.0129972 σ =0, k20∗ =0.0140607 σ 0 σ, ×10−6

0.1 ||U||2

2.0

0.0

1.5 0.1 1.0 0.005

0.010

0.015

k20

0.020

0.2 0.010

0.025

Supercritical: σ 0 and wξ (0) = 0 .

A spike

Problem (3.79) has an unique homoclinic solution. This comes from the fact that the system can be rewritten as a first order ODE system whose solutions lie on the phase space defined by p and q. That is, from (3.79), let p = q ξ and q = w to get p ξ = q − q2 .

q ξ = p,

(3.80)

Locally, this system has a saddle at (p 0 , q 0 ) = (0, 0) and a centre at (p 1 , q 1 ) = (0, 1). Nonetheless, from a global analysis, (3.80) also has an unique homoclinic that satisfies: Proposition 3.14. Problem (3.79) has the unique homoclinic solution µ ¶ 3 2 ξ w(ξ) = sech , 2 2

(3.81) with the following proprieties: (i)

R∞

2

dξ =

R∞

(ii)

R∞

3

dξ =

36 5 .

(iii)

R∞

−∞ w

−∞ w

−∞ w d ξ = 6.

6 2 −∞ wξ d ξ = 5 .

Proof. From (3.80), we obtain 1 2 1 2 1 3 p − q + q = 0, 2 2 3

(3.82)

because of q −→ 0 as |ξ| −→ ∞ and p(0) = 0. Now, without loss of generality, for q, p > 0 this equation can be rewritten as p r

q

2 1− q 3



=

r

w

2 1− w 3

= 1,

which by explicit calculation, after the change of variables w = 23 z2 , yields to (3.81). 73

CHAPTER 3. ANALYSIS OF LOCALISED PATCHES IN 1D

Notice that poperty (i) is satisfied by integrating (3.79), then the second integral follows from direct calculation. Next integrate (3.82) to get Z∞

(3.83) −∞

2

p dξ =

Z∞

−∞

2 q dξ − 3 2

Z∞

q 3 d ξ;

−∞

now, multiply by wξ equation (3.79) and after integrating by parts, we obtain Z∞

Z∞ Z∞ 2 2 wξ d ξ = − w dξ + w 3 d ξ. −∞ −∞ −∞

(3.84)

So replacing p and q by wξ w in (3.83) then subtracting to (3.84) and because of (i), we find (ii)



and therefore (iii).

3.A.2

Key identities

From (3.79), a linearisation around a spike w, we obtain the local operator L 0 φ ≡ φξξ + 2wφ − φ, and its eigenvalue problem is defined as follows: (

(3.85)

L 0 φ = νφ ,

−∞ < ξ < ∞ ,

φ −→ 0 , as |ξ| −→ ∞ .

Local operator L 0 is Hermitian, which implies that its eigenvalues are real. Let |ξ| −→ ∞, ¡ p ¢ thus (3.85) in the limit takes the form φξξ −φ ≈ νφ so φ ∼ exp − 1 + ν ξ , which only exponentially decays if ν > −1. In Figure 3.11, the first two eigenfunctions of (3.85) and a sketch of its spectrum are depicted. Proposition 3.15. The first two eigenpairs to (3.85) satisfy: (i) ν0 = 5/4 and φ0 = sech3 (ξ/2). (ii) ν1 = 0 and φ1 = wξ . ¡ ¢ Proof. From (3.79), we have that wξξ ξ − wξ + 2wwξ = 0 which directly implies that L 0 wξ = 0,

hence ν = 0 and φ = wξ is eigenpair. Now, as can be seen in Figure 3.11(c), φ = wξ has one nodal line, this implies that ν = 0 is the second eigenvalue. In consequence, there must be an eigenpair whose eigenfunction has no nodal line. Hence, since the first eigenvalue is always simple, thus

ν0 > 0. Explicit calculation gives φ0 = sech3 (ξ/2), see Figure 3.11(b). Next, we investigate the following L 0 -identities. Proposition 3.16. The local operator L 0 satisfies: (i) L 0 w = w2 . ¡ ¢ (ii) L 0 w + 1/2ξwξ = w 74



3.A. APPENDIX: LOCAL OPERATOR PROPRIETIES

(a)

(b)

(c)

-1

ν1 =0

φ1

0

φ0

=(ν)

Continuous spectrum 0

ν0 =5/4 0 0

0

ξ

L y . We will follow the methodology developed in [72, 121]. There homoclinic stripes and mesa-stripes are analised for the Gierer–Meindhart system, where a linearised problem is studied in order to investigate stability of such structures irrespective of whether a saturation effect is present or not. A homoclinic stripe is a localised structure that lies on a planar curve in the domain. Its name comes from the fact that it can be constructed by trivial extension onto a 2D domain from a solution that is a homoclinic orbit of some ODE system of an appropiate spatial phase space. For the Gierer–Meindhart and Gray–Scott systems, homoclinic stripes are typically unstable. The main instabilities related to homoclinic stripes can be grouped as: breakup instabilities and zigzag instabilities (see [72]). We shall examine whether system (1.5) shows breakup instabilities and the role the gradient plays in their development. Notice that although we follow auxin flow assumption presented in §1.3.2, in the x-direction, in order to provide a theoretical approach, the gradient here will be assumed to be constant in y. The outline of this chapter is as follows. In §4.1 we explore the gradient effects by running full numerical simulations and perform a numerical bifurcation analysis using the parameter set in Table 3.1(a). Then, in §4.2, we analyse breakup instabilities by constructing trivial quasisteady-solution extensions from the 1D case for both interior stripe and boundary stripe solutions of (3.2). A nonlocal eigenvalue problem (NLEP) is derived, from which we study and compute dispersion relations for several scenarios. We then perform numerical experiments by using a third parameter set in order to predict the number of spots which arise after breakup instabilities. Finally, in §4.3, we compute dispersion relations for each case and further simulations revealing a new collection of solutions. In Appendix 4.A a P YTHON script that numerically computes dispersion relations from the NLEP of an interior stripe is presented.

4.1

Numerical investigations

In order to provide a more accurate biological description, we shall run full simulations of the system (1.5) with the parameter set given in Table 3.1(a). In addition, to uncover the dynamics and instability we shall seek, we consider the rectangular domain Ω2 defined previously in Chapter 1, and a 2D auxin-gradient, which is decreasing along the x-direction and constant along the y-direction. That is, from §1.3.2, we take α(x, y) = e−ν x with ν = 1.5. Also, a small random perturbation from the homogeneous steady-state (2.12) is considered as initial condition.

4.1.1

Simulations

In Figure 4.1 snapshots of a full simulation for k 20 = 0.1 are depicted. The initial condition for the active-ROP concentration is shown in Figure 4.1(a); then, as time increases, a front is formed at the boundary; this front resembles a boundary homoclinic stripe, see Figure 4.1(b). After this, the 78

4.1. NUMERICAL INVESTIGATIONS

(a)

(b)

(c)

(d)

(e)

(f)

F IGURE 4.1. Snapshots of a travelling front breaking up into a travelling pinned spot. (a) Initial condition. (b) Front formed at the boundary. (c) Travelling front. (d) Breakup into a peanut form. (e) Travelling spot. (f) Final pinned spot. Parameter set as given in Table 3.1(a) with L y = 20.

79

CHAPTER 4. BREAKUP INSTABILITY OF STRIPES IN 2D

(a)

(b)

(c)

(d)

(e)

(f)

F IGURE 4.2. Snapshots of two stripes breaking up into an asymmetrical array of spots. (a) Early localised stripes. (b) Stripes moving apart. (c) Stripes breaking apart. (d)-(e) Counterclockwise rotation and travelling peanut-form. (f) A pinned spot-like pattern. Parameter set as given in Table 3.1(a) with L y = 20.

80

4.1. NUMERICAL INVESTIGATIONS

front travels slowly into the interior of Ω2 , see Figure 4.1(c), to be then broken up into a transitory peanut form as shown in Figure 4.1(d). This structure collapses onto a travelling spot, which finally gets frozen at some location longitudinally closer to left-hand boundary, see Figure 4.1(f). In this simulation, as similarly observed in the 1D case (see Chapter 2), different spatiotemporal scales occur: two spatial scales define the localised regions of the U-concentration, one time-scale describes how the steady-state is quickly destabilised, and another one provides a slower frame for the location dynamics. In addition, on this slower time-scale, a breakup instability seems to take place. It would seem that this diversity of spatiotemporal scales are inherited from the 1D case described in Chapter 3; moreover the auxin gradient still seems to control location, an expected dynamical property, nonetheless. In other words, once a localised structure is rapidly formed as a boundary stripe, it starts to move towards domain interior. Then, in a process that is not captured by the 1D theory this structure breaks-up into a spot once it is far from the boundary, before slowly setting into its equilibrium location. To investigate whether the bifurcation parameter k 20 affects the dynamics of the system in a manner similar to the 1D case, we shall now increase this parameter to k 20 = 0.4. Snapshots at different times can be seen in Figure 4.2; initial conditions are as before. In Figure 4.2(a) a stripe-like state is formed at the boundary and a second one quickly emerges further towards the interior. These structures then move away from each other, see Figure 4.2(b). An instability occurs in such a fashion that both stripes break up, see Figure 4.2(c), into two half-spots at the boundary and a counter-clockwise rotating peanut form, see Figures 4.2(d)–4.2(e). When the peanut-form gets aligned longitudinally, it travels slowly far away from the half-spots at the left-hand side boundary, see Figure 4.2(f). In analogy with the case above, a process seems to rapidly stimulate aggregation into a quasi-stable pattern. In this case, the k 20 value is larger, which induces a two-stripe transitory pattern. Which, when it breaks up induces a multiple spot pattern. This behaviour is not surprising because localised structures are expected in regions where the nonlinear terms dominate, see e.g. [141]. Nonetheless, because a 2D domain provides many more degrees of freedom than a 1D domain, once these stripes are far enough from each other, they break apart into a seemingly natural spatial arrangement. In other words, system (1.5) tends to form spots, this is a property that comes from the Schnackenberg system (see e.g. [73]). In addition, rotation and location dynamics suggest that O (1) time-scale instabilities can occur, in addition to the competition instability, which we already analysed in 1D. Analysis of the location dynamics and instabilities which explains transitions will be numerically and analytically explored further in §4.2 below.

4.1.2

Bifurcation diagram

We now shall numerically perform a bifurcation analysis of homoclinic stripes as k 20 is slowly varied. To do so, we numerically construct the set of steady-state homoclinic stripes solutions 81

CHAPTER 4. BREAKUP INSTABILITY OF STRIPES IN 2D

0.50

B

Mutant with basal end patch Wild type with single interior patch Mutant with basal end and interior patch Mutant with two interior patch (1D)

A

0.45 0.40

Stable-stripe branch

C

||U(x;y0 )||2

0.35

D

0.30

E

0.25 0.20 0.15

0.0

0.2

0.4

0.6

k20

0.8

1.0

F IGURE 4.3. Comparison of bifurcation diagrams between homoclinic stripes and 1Dspike scenarios coloured dashed portions of the diagram indicated where stable 1D solutions are unstable to transverse instabilities (labelled B - E ). A narrow stable window is found, given by the solid black curve labelled A . Parameter set as given in Table 3.1(a).

from steady-state spike solutions by a trivial extension along the y-direction. That is, let (U0 , V0 ) a steady solution of the BVP in 1D, see §2.6, in such fashion that U0,y (x, y) = U0 (x) ,

V0,y (x, y) = V0 (x) .

Which implies that U0,y and V0,y are also steady solutions of the BVP associated to (1.5) when

∆ = ∂ xx + ∂ yy. From now onwards, these solutions will be called extended solutions. Therefore, the bifurcation diagram of such solutions is entirely equivalent to Figure 1.5. However, the stability proprieties become dependent on perturbations in the y-direction. This can be seen as follows. We take into account a perturbation of the form (4.1)

U˜ = U0 + eλ t+ im y/L y U ,

V˜ = V0 + eλ t+ im y/L y V ,

where the variable m is determined by the homogeneous Neumann boundary conditions at y = 0 ¡ ¢ ¡ ¢ and y = L y . That is, because the perturbation is e im y/L y , hence ℜ e im y/L y = cos m y/L y . Then, equivalently to (2.30), we use M ATLAB to numerically analyse the spectrum of the discretised operator (4.2)

µ µ ¶ µ ¶ ¶ my 2 my LW ≡ D x − s cos I Idiff + F0 (W0 ) ; Ly Ly

82

4.2. BREAKUP INSTABILITY

¡ ¢2 ¡ ¢ here s is the square length ratio defined as s ≡ L x /L y , Idiff ≡ diag ε2 , D is the diffusivity

matrix, D x ≡ diag (∂ xx , ∂ xx ) is the x-direction discretised diffusion operator, and the Jacobian matrix evaluated at W0 = (U0 , V0 ) is given as F0 (W0 ). The bifurcation diagram for homoclinic stripes is depicted in Figure 4.3. We use the L 2 -norm of the active component as solution measure. To compute stability, we fix y = y0 and then compute eigenvalues of matrix (4.2) for each extended solution as m is varied. As can be seen there, stable branches related to the cases studied in previous chapters are plotted as dashed coloured curves accordingly to their classification: boundary stripe (purple), interior stripe (dark green), boundary and one interior stripe (light brown), and two interior stripes (light green). All these branches are unstable apart from a small window plotted in black. This indicates that, other than the stripes in the black window, all 1D localised patterns tend to destabilise under transverse perturbations. The stability boundaries of the stable extended pattern (the black spots in Figure 4.3) are pitchfork bifurcation points. Pitchfork bifurcation is a type of local bifurcation that comes out from some symmetry breaking. This is because a basic property, as the bifurcation parameter crosses a critical value, the number of solutions jumps from one to three (see e.g. [51]); Turing bifurcation belongs to this class. In particular, this window is a subinterval of the boundary stripe branch, which implies that the gradient does not locally vary. Thus, as it can be checked from results in Chapter 2, the following results yields: Proposition 4.1. System (1.5) is stripe-unstable under transverse perturbations. In addition, there exists two pitchfork bifurcation points that define a small stable boundary-stripe branch. To get a flavour of the kind of O (1) time-scale instability we shall look for, we take an interior homoclinic stripe as initial condition and run a direct simulation, see Figure 4.4. We see that first the stripe breaks up into a spot and a “semi-stripe” set at the initial location (Figure 4.4(a)). Then the recently formed spot splits (Figure 4.4(b)) giving way to two baby droplets. These two spots move away from other each while the semi-stripe collapses into a peanut form (Figure 4.4(c)). Finally, (Figure 4.4(d)) a spot is formed from the semi-stripe besides two peanut-forms. Here two different instabilities are present: a breakup instability, which destabilises the homoclinic stripe to form spots, and another instability that creates peanut forms from spots. We will investigate breakup instability from a numerical point of view in the next section.

4.2

Breakup instability

The theory of stripe patterns has usually been addressed from a near-Turing bifurcation parameter regime (see e.g. [33]), but this approach does not apply to the ROP problem because system (3.2) has a spatially dependent parameter that modulates the nonlinear term. However, ideas like those presented in [72] can be used. In this section we derive a NLEP to theoretically explain the stability features numerically observed for the stripe solutions as shown in Figure 4.3. 83

CHAPTER 4. BREAKUP INSTABILITY OF STRIPES IN 2D

(a)

(b)

(c)

(d)

F IGURE 4.4. Example of a breakup instability of a homoclinic stripe into one spot and two peanut-forms. (a) The stripe breaks up into a semi-homoclinic-stripe and a spot. (b) Spot splits up. (c) Breakup of the semi-homoclinic-stripe into a peanut-form, and spots move away from each other. (d) A spot and two peanut-forms are finally formed. Parameter set as given in Table 3.1(a) with D 2 = 0.025 and k 20 = 0.1, which corresponds to a stripe location in x0 = 18.5, and L y = 20.

We begin with rescaling system (1.5). Thus from (3.1), (3.4)–(3.5) and upon defining nondimensional time and space variables (4.3)

t = (c + r)T,

¢ 1 ¡ p (x, y) = z1 , s z2 , Lx

Lx s≡ Ly µ

¶2

,

so that (x, y) ∈ [0, 1] × [0, 1], where s > 1, and rescaling variables U, V and D as in §3.2, we obtain (4.4a) (4.4b)

¡ ¢ ε2 u t = ε2 u xx + su yy + α(x)u2 v − u + v , τγ ¡ ¢ ¡ ¡ ¢ ¢ −1 2 τγ α(x)u v − u + βγ u , ετv t = D 0 v xx + sv yy + 1 − εv − ε

84

4.2. BREAKUP INSTABILITY

with homogeneous Neumann boundary conditions; also, we write the gradient as before, α = α(x), because it is assumed to be y-independent. The relation between the dimensionless parameters τ, γ, β, and D 0 and the original parameters is as further given in Table 4.1.

4.2.1

Interior stripe

We shall proceed by performing a thorough asymptotic analysis of an interior homoclinic stripe. In the limit ε −→ 0 we construct a quasi-steady y-independent equilibrium solution comprising the 1D interior patch solution in x, centred at some 0 < x0 < 1 for a given γ value. That is, from Proposition 3.1, we obtain u∼

(4.5a)

ε−1

4βγ

sech2

ε−1 (x − x0 )

2



,

    − x0 (x − x0 ) ,



(4.5b)

µ

 6βγ 1 1 2 v ∼ ε  α (x ) − 2D (x − x0 ) + D  0 0 0 

(1 − x0 ) (x − x0 ) ,

0 ≤ x ≤ x0 , x0 < x ≤ 1 .

   

Now, we extend this solution trivially in the y-direction to make a stripe, and look for an O (1) time-scale instability. Hence, to determine the stability of the stripe solution, we introduce a

transverse perturbation u = u s + eλ t+ im y ϕ(x),

(4.6)

v = vs + eλ t+ im y ψ(x) ,

where ϕ, ψ ¿ 1, and the wavenumber m is determined by the homogeneous Neumann boundary conditions. We thus require m = kπ for k ∈ Z, so the perturbation takes the form of cos (kπ y), which implies that k = m/π. Substituting (4.6) into (4.4) we get the eigenvalue problem (4.7a) (4.7b)

ε2 ϕ xx − ϕ + 2α(x)u s vs ϕ + α(x)u2s ψ +

ε2

¡ ¢ ψ = λ + sε2 m2 ϕ ,

τγ ¡ ¢ ¡ ¡ ¢ ¢ D 0 ψ xx − sm2 − ε−1 τγα(x)u2s ψ − εψ = ε−1 τγ 2α(x)u s vs ϕ − ϕ + βγϕ + ετλψ ,

where ϕ x = ψ x = 0 at x = 0, 1. According to [72], there are two distinct classes of eigenvalues each giving rise to a different kind of instability: small ones and large ones. Small eigenvalues correspond to zigzag instabilities and the large ones to breakup instabilities. Simulations and numerical analysis in §4.1 suggest that breakup instabilities dominate, we hence seek large eigenvalues λ = O (1) as ε −→ 0. This implies we seek not necessarily an orthogonal to the spike w = 3/2sech2 (ξ/2), which is a localised eigenfunction of the form (4.8)

¡ ¢ ϕ(x) ∼ ϕ0 ε−1 (x − x0 ) ,

Here, non orthogonality guarantees that

ϕ0 −→ 0

R∞

−∞ ϕ0 wd ξ 6= 0,

as

|ξ| −→ ∞ .

which will allow us to capture the role

that the spike and consequently the gradient play on this kind of instability. 85

CHAPTER 4. BREAKUP INSTABILITY OF STRIPES IN 2D

Performing the same calculations as in (3.60) form §3.3 to (4.7a) we get L 0 ϕ0 +

n20

¡ ¢ α (x0 ) w2 ψ (x0 ) = λ + sε2 m2 ϕ0 ,

62 β2 γ2

−∞ < ξ < ∞ ,

with L 0 φ ≡ φξξ + 2wφ − φ being the local operator defined in (3.27) and n0 =

6βγ . α (x0 ) v0

Thus (4.9)

L 0 ϕ0 +

¡ ¢ w2 2 2 £ ¤2 ψ (x0 ) = λ + sε m ϕ0 , 0 α (x0 ) v

−∞ < ξ < ∞ .

In this equation we just need to find ψ (x0 ), hence as in (3.61) from (4.7b) we obtain an equation from which ψ (x0 ) will be found, that is (4.10)

¡ ¢ τ n2 α (x0 ) D 0 ψ xx − sm2 ψ − 0 2 δ (x − x0 ) ψ = 2τγδ (x − x0 ) 6β γ

Z∞ µ

−∞

µ ¶¶ 1 β w− 1− ϕ0 d ξ . 2 τ

Therefore, solving equation (4.10) with homogeneous Neumann boundary conditions is equivalent to solving

(4.11)

  ψ xx − sm2 ψ = 0, 0 < x < 1,    ψ x (0) = ψ x (1) = 0 ,  ¡ ¡ ¢ ¡ ¢¢ a 0   ψ0 + γ b 0 ,  D 0 ψ x x0+ − ψ x x0− = γ

where ψ0 ≡ ψ (x0 ), and substituting n 0 in terms of α (x0 ), we have 6τγ2 a0 ≡ £ ¤2 , α (x0 ) v0

b 0 ≡ 2τ

Z∞ µ

−∞

µ ¶¶ β 1 ϕ0 d ξ . w− 1− 2 τ

We now solve problem (4.11) using a Green’s function. Hence let G m,s (x; x0 ) be defined by ( ¡ ¢ G m,s xx − sm2 G m,s = −δ (x − x0 ) , (4.12) ¡ ¢ G m,s x = 0 at x = 0, 1 , where m > 0 for existence; this not only implies that k ∈ Z+ , but also reflects the fact that m = 0 is equivalent to a trivial, rather than a stripe, solution. Integrating (4.12) in a neighbourhood of x0 , we obtain the jump condition ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ G m,s x x0+ − G m,s x x0− = −1 .

Let be ψ(x) = AG m,s (x; x0 ) for some constant A. Then, upon substitution into (4.11), from the previous jump condition we find £¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢¤ a 0 D 0 A G m,s x x0+ − G m,s x x0− = AG 0m,s + γ b 0 , γ

86

4.2. BREAKUP INSTABILITY

where G 0m,s ≡ G m,s (x0 ; x0 ). This implies A=− a

γb0 0

γ

G 0m,s + D 0

,

therefore ψ0 = −

(4.13)

γ2 b 0

a 0 G 0m,s + γD 0

G 0m,s .

Substituting (4.13) into (4.9) and from the definitions of a 0 and b 0 , we obtain Ã

(4.14)

L 0 ϕ0 − 2χw

2

! Z∞ µ

G 0m,s D 0 + 6χG 0m,s

−∞

µ ¶¶ ¡ ¢ 1 β w− 1− ϕ0 d ξ = λ + sε2 m2 ϕ0 , 2 τ

where χ≡

τγ £ ¤2 , α (x0 ) v0

v0 =

α (x0 )

6β2 γ2

.

Now it is then convenient to define a parameter that will contain terms which depend upon location, µ≡

(4.15)

12χG 0m,s D 0 + 6χG 0m,s

.

As a matter of fact, the spike moves slowly in time, and because we aim to look for instabilities on a fast O (1) time-scale, we can consider x0 to be frozen. Indeed, the parameter defined by (4.15) implicitly depends on the gradient via G 0m,s , which is determined once x0 reaches equilibrium. This is described in Proposition 3.3. Then, (4.14) can be written as follows L 0 ϕ0 −

(4.16)

µ

6

¡ ¢ w2 (I 1 − κ I 2 ) = λ + sε2 m2 ϕ0 ,

where κ, I 1 , and I 2 , are equivalently defined as in (3.71) µ ¶ β 1 1− , κ= 2 τ

Z∞

I1 =

wϕ0 d ξ ,

Z∞

I2 =

−∞

ϕ0 d ξ .

−∞

In addition, similarly to (3.70), equation (4.16) has two separate non-local terms. Hence, we proceed as before and integrate across the entire domain, this gives I2 =

(4.17) where we recall

R∞

−∞ w

2

2−µ I1 , λ + 1 + sε2 m2 − µκ

d ξ = 6. In consequence, eliminating I 2 from (4.16), the NLEP charac-

terising breakup instabilities for an interior homoclinic stripe is determined by the following proposition: 87

CHAPTER 4. BREAKUP INSTABILITY OF STRIPES IN 2D

Proposition 4.2. The stability on an O (1) time-scale of a quasi-steady-state interior stripe solution of (4.4) is determined by the spectrum of the NLEP Z∞

(4.18)

wϕ0 d ξ

−∞ L 0 ϕ0 − θh (λ; m) w2 ∞ Z

¡ ¢ = λ + sε2 m2 ϕ0 ,

−∞ < ξ < ∞ ;

w2 d ξ

−∞

ϕ0 −→ 0 ,

as

|ξ| −→ ∞ ,

where θh (λ; m) is given by θh (λ; m) ≡ µ

(4.19) µ≡

12χG 0m,s D 0 + 6χG 0m,s

,

λ + 1 + sε2 m2 − 2κ λ + 1 + sε2 m2 − µκ

χ≡

τα (x0 )

62 β2 γ

,

,

G 0m,s ≡ G m,s (x0 ; x0 ) ,

and G m,s (x; x0 ) is defined by (4.12). Here the number of resonant modes, k ∈ Z+ , in the y-direction is determined by k = m/π. The NLEP (4.18) is non-standard since it is not self-adjoint, it has a nonlocal term and it is a nonlinear eigenvalue problem. However, our goal is to show that, from Proposition 4.2, there ¡ ¢ exists a band of unstable modes 0 < m low < m < m up with m low = O (1) and m up = O ε−1 for which the spectrum of NLEP is similar to the sketch shown in Figure 4.5. We remark that, as shall be proven further, m up depends on the length ratio s, and m low is the lowest unique mode that defines the left-hand border of the unstable band. In addition, the expected number of spots can be estimated from the maximum growth rate m∗ , which we conjecture to be unique; further computations shall confirm this assumption. We will first find the upper bound m up of the instability band. To do this, G m,s (x; x0 ) must be calculated. From (4.12), a simple calculation gives ( ¡p ¢ ¡p ¢ cosh s mx cosh s m (1 − x0 ) ; 0 ≤ x < x0 1 ¡p ¢ G m,x (x; x0 ) = p , ¢ ¡p ¢ ¡p s m sinh s m cosh s mx0 cosh s m (1 − x) ; x0 < x ≤ 1 which yields ¢ ¡p ¢ s mx0 cosh s m (1 − x0 ) ¡p ¢ (4.20) = . p s m sinh s m ¡ ¢ On the one hand, from this formula we get G 0m,s = O 1/m2 for m ∼ 0+ , which implies G 0m,s −→ ∞ ¡ ¢ ˜ = 1/m into (4.20), we also obtain G 0m,s = O 1/m2 as m −→ 0+ . On the other hand, substituting m ¡ ¢ for m À 1. Hence, θh (λ; m) = O 1/m2 for m À 1. ¡p ¢−1 Now, let m = s ε m 0 with m 0 = O (1) and then substitute into (4.18) to get

G 0m,s

(4.21)

cosh

¡p

¡ ¢ L 0 ϕ0 − O (ε) = λ + m20 ϕ0 .

88

0 such that ¡p ¢−1 p p λ < 0 if m 0 > ν0 . Therefore the upper threshold is given by m up = s ε ν0 , which in terms of the original parameters depends on the length ratio as follows: s ν0 (c + r) L y . m up = D1 Lx Next, to compute the lower threshold m low , we suppose that m = O (1), so we ignore sε2 m2 terms to leading order in (4.19). Then, from Proposition A.1, we obtain that instability occurs whenever (4.22)

θh (0; m) = µ

1 − 2κ < 1. 1 − µκ

Thus, m low is the root of θh (0; m) = 1, however this criticality condition is equivalent to finding the roots of (4.23)

G 0m,s =

6βD 0 γ . α (x0 )

From (4.20), G 0m,s decreases with m increasing for any fixed x0 , which is straightforwardly ˜ = 1/m to then apply the L’Hˆopital’s rule for m ˜ −→ 0+ . verified via a transformation of the form m Which therefore implies that there is an unique root of (4.23). Notice that m low increases as the gradient α is increased. Figures 4.6(a)–4.6(b) show monotony of the lower threshold as a function ¡ ¢2 of x0 and γ for several values of s = L x /L y and parameter sets as given in Table 3.1. 89

CHAPTER 4. BREAKUP INSTABILITY OF STRIPES IN 2D

(a) 101

(b)

101

s =6.00 s =5.50 s =5.00 s =4.50

s =6.00 s =5.50 s =5.00 s =4.50

0.1

0.2

0.3

0.4

x0

0

1

γ

2

100

3

0.1

0.2

x0

0.3

(c)

O(1)

x0 =0.3903, γ =0.5600 x0 =0.3712, γ =0.7094

0

1

x0 =0.1853, γ =2.3172 x0 =0.4302, γ =0.3435

2.0

time-scale stability

2

3

γ

4

x0 =0.4682, γ =0.1240 x0 =0.4824, γ =0.0515

O(1)

time-scale stability

O(1)

time-scale instability

1.5

θh (0;m)

1.5

1.0

0.5

1.0

0.5 O(1)

0.0

100

s =5.50

x0 =0.4387, γ =0.2591 x0 =0.4106, γ =0.4054

2.0

0.4

(d)

s =5.50

θh (0;m)

s =6.00 s =5.50 s =5.00 s =4.50

101

mlow

101

mlow

s =6.00 s =5.50 s =5.00 s =4.50

0

5

time-scale instability 10

15

m

20

25

30

0.0

35

0

10

20

30

m

40

50

60

70

F IGURE 4.6. Lower threshold m low as function of x0 and γ for several values of the length ratio parameters: parameter set as given in (a) Table 3.1(a) and (b) Ta¡ ¢ ble 3.1(b). Comparison of the criticality condition when O sε2 m2 terms are ignored (solid curves) and otherwise (dotted curves) for different locations and a s = 5.5: parameter set given as in (c) Table 3.1(a) and (d) Table 3.1(b). Both scenarios favourably agree when m is small.

In addition, from Proposition 4.2 the wavenumber of the unstable mode m is given by k = m/π. Hence the number of spots expected is given by the number of maxima of cos (k max y) when m low < k max < π

s

This result is summarised as follows:

90

ν0 (c + r) L y π2 D 1

Lx

.

4.2. BREAKUP INSTABILITY

Original

Re-scaled

D 1 = 0.075 D 2 = 20 L x = 70 L y = 29.848 k 1 = 0.008 b = 0.008 c = 0.1 r = 0.05 k 20 = 0.001 . . . 2.934

2

ε = 1.02 × 10−4 D = 0.51

s = 5.5 τ = 18.75 β = 6.25

γ = 150 . . . 0.051

T ABLE 4.1. A parameter set in the original and re-scaled variables.

¡ ¢2 Proposition 4.3. Let ε ¿ 1. For a fixed O (1) s = L x /L y > 1 let m low be the unique root of the

equation G (m) ≡

and m up ≡

¡p

6β D 0 γ − G 0m,s = 0 , α (x0 )

¢−1 p sε ν0 . Then, an interior homoclinic stripe is always unstable, in the sense of a

breakup instability to modes with transverse wave number m satisfying m low < m < m up . The criticality parameter (4.19) is depicted in Figure 4.6(c) for the parameters given as in Table 3.1(a), and equivalently in Figure 4.6(d) for Table 3.1(b). Solid curves represent θ (0; m) as m varies when sε2 m2 terms are ignored, otherwise it is plotted as dotted lines. As can be seen, in both parameter sets, it shows that θh (λ; m) − 1 = 0 is satisfied at some mode m. In addition, there is better agreement for small modes in Figure 4.6(d) rather than in Figure 4.6(c). This comes from the fact that parameter set in Table 3.1(b) is closer to the asymptotic limit. Notwithstanding, these figures reinforce the assumption of uniquely finding m low as indicated in Proposition 4.3. In addition, it also provides an estimation of the number of spots that an interior homoclinic stripe breaks into. In order to test this proposition, we choose the parameter set given in Table 4.1, which is a further modification of the set in Table 3.1(a). From Proposition 4.2, we compute the spectrum of the NLEP for an interior homoclinic stripe under transverse perturbations. To do so, we discretise equation (4.18) to obtain a nonlinear eigenvalue problem, and then apply a backwards iterative process on m. A P YTHON script (see [43] for documentation) that computes the NLEP is given in Appendix 4.A. In order to perform this computation, m is treated as a continuous variable. The outcome is depicted in Figure 4.7. Dispersion relations for a fixed homoclinic stripe location and different length square ratio values are depicted in Figure 4.7(a). Note that, a rightwards horizontal translation of ℜ (λ) curves as s decreases seems to occur, in consequence the most unstable mode increases

with s. Since parameter s determines domain width for each fixed value of L x , the larger in y 91

CHAPTER 4. BREAKUP INSTABILITY OF STRIPES IN 2D

(a)

(b)

x0 =0.40

0.35

s =5.50 s =4.50 s =5.00 s =5.50 s =6.00

0.30

x0 =0.350 x0 =0.375 x0 =0.400 x0 =0.425

0.5 0.4

0.25

0.3



1+ς , 2κ

ς≡

p 1 + 2κ (1 − κ) .

This proposition determines whether mode m c , at which the threshold is reached, exists. Notice, critical mode grows smaller as γ increases as Figure 4.9(a) shows so regardless any width. As an illustration, we set a boundary stripe as initial condition for a very small value of parameter k 20 and another one considerably larger. In Figure 4.10(a) a stripe begins to collapse into a spot, which is formed at the middle of transversal length, see Figure 4.10(b). Analogously, in Figure 4.10(c) is depicted a scenario where the value of k 20 is larger, the boundary stripe begins to snap in two. Then, these split once more to give place to four spots at the boundary, see Figure 4.10(d). All this suggests that boundary stripes collapsing through a breakup instability form a larger number of spots as k 20 is greater. In other words, because the main bifurcation parameter controls nonlinearity influence in the system, higher values induce more intensity of the gradient at the boundary, which favours symmetry breaking.

4.3

Stripes into spots

In the previous section a theoretical analysis of breakup instabilities was performed, indeed favourable agreement between the theory and computations was also found for the parameter set in Table 4.1. However, numerical bifurcation analysis initially depicted in Figure 4.3 shows that solution branches, which comes from the 1D analysis (see §2.6), are no longer stable under transverse perturbations as has been analysed here. Indeed, computations in Figures 4.8 and 4.10 indicate stripes are susceptible to fall over into spots. In order to verify that unstable solutions in Figure 4.3 similarly break forming spots-like, we choose a solution from each branch, and respectively compute its dispersion relation. In Figure 4.11 each curve is labelled accordingly to solution kind (see also labels in Figure 4.3): A stable boundary stripe, B unstable boundary stripe, C an interior stripe, D boundary and interior stripe, and E two interior stripes. Upon using each of these steady-states as initial conditions and performing a direct time-step computation, we confirm that those labelled from B up to E are indeed unstable, and solution labelled by A is stable. Figure 4.12 shows initial snaps induced by breakup instabilities and final stable states, each of these can respectively be seen in Figures 4.12(a), 4.12(c), 4.12(e) and 4.12(g), so are final states in Figures 4.12(b), 4.12(d), 4.12(f) and 4.12(h) as well. Even though, according to dispersion relations in Figure 4.11, the most unstable mode would lead to number of spots the stripe splits, it does not occur to this particular parameter set, which makes “fat” patterns. 97

CHAPTER 4. BREAKUP INSTABILITY OF STRIPES IN 2D

E

25

D

20

0 centred at the origin, B²0 (0), and use the Green’s second identity to obtain ·³

Z

(5.32)

lim

²0 −→∞ B²0 (0)

´T ¡ ³ ´¸ ¢ T ∗ T ∗ ˆ ˆ ˆ ∆Q1 + M c Q1 − Q1 ∆Pcos,sin + M c Pcos,sin d ξ =

P∗cos,sin

Z

= lim

²0 −→∞ ∂B²0 (0)

à ³

P∗cos,sin

´T ∂Q ∂P∗cos,sin ˆ1 T ˆ − Q1 ∂ρ ∂ρ

!¯ ¯ ¯ ¯ ¯

dξ , ρ =²0

which, from (5.29)–(5.31), becomes lim

Z2πZ²0 ³

²0 −→∞

P∗cos,sin

´T

à !# ¡ ¢ ! ¡ ¢ 2 −1 − u0c x˙ j cos θ + y˙ j sin θ ρ dρ dθ = + a 1 ρ cos θ + a 2 ρ sin θ u c v c τ /β 0



0 0

(5.33)

= lim

Z2π³

²0 −→∞

P∗cos,sin

´T

Ã

Z2π

!¯ ¯ ¯ ¯ ˆ ˆ ζ j1 cos θ + ζ j2 sin θ ¯

0

d θ − lim

ρ =²0

0

²0 −→∞

ˆT Q 1

¯ ∂P∗cos,sin ¯¯ ¯ ∂ρ ¯

dθ ;

ρ =²0

0

here we have used the notation z˙ = dz/d η for the slow-time derivative and the shorthand (a 1 , a 2 ) =

∇α j αj

,

for the gradient-dependent coefficient. ³ ´T ¡ ¢ ¡ ¢ Now, from (5.30), we see that P∗cos,sin tends either to 0, 1/ρ cos θ or 0, 1/ρ sin θ as ρ −→ ∞ depending which polar direction is considered. As a consequence, the first right-hand term in (5.33) becomes lim

Z2π³

²0 −→∞

0

P∗cos,sin

´T

Ã

!¯ ¯ ¯ ¯ ζˆ j1 cos θ + ζˆ j2 sin θ ¯

0

ρ =²0

119

(

dθ = π

ζˆ j1 ζˆ j2

.

CHAPTER 5. DYNAMICS AND TRANSITIONS IN 2D

Upon considering that the far-field condition (5.29b), equation (5.31b) and the average of cos2 θ and sin2 θ in [0, 2π] imply that the second term is Z2π

lim

²0 −→∞

0

¯ ∂P∗cos,sin ¯¯ ˆT Q ¯ 1 ∂ρ ¯

(

d θ = −π

ρ =²0

ζˆ j1 ζˆ j2

.

Analogously, the left-hand side terms yield

lim

Z2πZ²0 ³

²0 −→∞

P∗cos,sin

´T

Ã

¡ ¢ ! − u0c x˙ j cos θ + y˙ j sin θ

0

0 0

ρ d ρ d θ = −π

Z∞

(

ˆ ∗ u0 ρ d ρ φ c

0

x˙ j y˙ j

,

and,

lim

Z2πZ²0 ³

²0 −→∞

P∗cos,sin

´T ¡

Ã

a 1 ρ cos θ + a 2 ρ sin θ

¢

u2c v c

0 0

−1 τ/β

! ρ dρ dθ =

( µ ¶ Z∞ a1 τ ˆ∗ ˆ∗ 2 2 . = π u c vc ψ − φ ρ dρ β a2 0

Hence, we obtain that (5.32) takes the form −π

Z∞

ˆ ∗ u0 ρ d ρ φ

µ ¶ Z∞ ∇α j τ ˆ∗ ˆ∗ 2 2 + π u c vc ψ − φ ρ dρ = 2πζˆ j , dη β αj

dx j

c

0

0

then upon defining the integrals

(5.34)

n1 ≡ −

¶ µ Z∞ τ ˆ∗ ˆ∗ 2 2 u c vc ψ − φ ρ dρ β

2 Z∞

,

n2 ≡

0

Z∞

ˆ ∗ u0 ρ d ρ φ c

,

ˆ ∗ u0 ρ d ρ φ c

0

0

we get an ODE system from which the location dynamics of N spots is characterised. Note n 1 and n 2 depend on the source parameter and τ/β, this implies that the activation process takes part in the spot location dynamics via modulating the terms related to spatial features: spot-to-spot interaction, boundary interaction and the gradient. We summarise this result in the following proposition: Proposition 5.2. Under the same assumptions as Proposition 5.1, the slow dynamics on a timescale η = ε2 t of a collection of N radially symmetric spots satisfy the ordinary differential system consisting of the ODE’s (5.35)

dx j dη

= n 1 ζˆ j + n 2

∇α j αj

120

,

j = 1, . . . , N ,

5.2. DYNAMICS OF SPOTS

(a)

(b) 55 50 45 40 35 30 25 20 15 10

x1 -asymptotics x2 -asymptotics

13.5 13.0 12.5 12.0 11.5 11.0 10.5 10.0 9.5

y1 -asymptotics y2 -asymptotics

50 30

x1 -numerics x2 -numerics

20

x1 ,x2

x1 ,x2

40

10

25 20 15 10 5 0

y1 -numerics y2 -numerics

y1 ,y2

y1 ,y2

0

0

5

10

15

20

t, ×103

25

30

35

40

0

2

4

t, ×103

6

8

10

F IGURE 5.4. Location dynamics of two spots for a y-independent gradient; x1,2 -coordinates (top panels) and y1,2 -co-ordinates (bottom panels): (a) direct simulation, and (b) asymptotic solutions. Parameter set as given in Table 4.1.

where constants n 1 and n 2 are defined by (5.34). In addition, system (5.35) is coupled to the nonlinear algebraic system (5.23), and from (5.26b), we also have defined the source-interacting vector Ã

ζˆ j ≡ −2π S c j ∇x R j +

N X

!

S ci ∇x G ji .

i 6= j

Proposition 5.2 describes the asymptotical dynamics of a set of N radially symmetric spots under a spatially dependent gradient. That is, system (5.35) gathers all the key elements identified in previous chapters. Indeed, as is thoroughly analysed in Chapter 3, the gradient controls location, a feature that is replicated for the 2D case. As expected, the domain geometry strongly contributes on the location. Several key terms deserve closer attention. First of all, notice that the vector ζˆ j bring together geometrical features and interactions between spots. In other words, note that the gradient of the regular part of the Green’s function corresponds to the characterisation of self-interactions; equivalently, the second term describes the interactions between spots, and the source parameters modulate these two terms. The source-interacting vector delineates the influence of the shape and form of the domain on the aggregation points. This, in fact, provides a full description of the role that the v-component plays on the location dynamics. On the other hand, notice the second term in the right-hand side of (5.35) is only associated to the auxin gradient, which does not depend on gradient intensity but rather its slope. This reinforces the conclusion that the spatially dependent gradient plays a more relevant role on the location rather than its size. 121

CHAPTER 5. DYNAMICS AND TRANSITIONS IN 2D

(a)

(b)

F IGURE 5.5. Two snapshots of a simulation where a peanut form merges into a spot. (a) Peanut form and a spot. (b) Two spots. Parameter set as given in Table 4.1.

On the top of all, the gradient and interaction terms are regulated by the constants given in (5.34), which depend on parameter τ/β, which is a quotient of parameters which correspond to the linear terms of the original system (1.5), and S c j parametrizes the solutions coming from the nonlinear CCP. Thus, the integrals n 1 and n 2 balance the geometrical and nonlinear roles over the location dynamics. As an illustration, we take two spots in two different locations as initial condition. Figure 5.4 shows the dynamics of a two-spots configuration where x j - and y j -co-ordinates are depicted in the top and bottom panels, respectively. In Figure 5.4(a) a direct simulation is depicted. As expected, the gradient induces a drift setting up a final non-symmetric location along the longitudinal direction (top panel), this is also seen for the 1D case, see Figure 3.6. However, the opposite occurs along the transversal direction, a certain quasi-symmetric location position is obtained; this seems to be qualitatively described by system (5.35). On the other hand, Figure 5.4(b) shows the solution of the asymptotical approach derived in Proposition 5.2. Even though there is not full agreement between asymptotics and the direct simulation, notice that solution depicted in Figure 5.4(b) seems to also replicate location characteristics seen in Figure 5.4(a) for a long time-scale. This disagreement comes from the fact that there is translational invariance along the transversal direction. Despite this issue, the asymptotical approach partially reproduces the location dynamics at some extent. However, as already seen in §4.3.1, a gradient of this kind produces a diversity of patterns which are not biologically relevant. We shall come back to this issue in Chapter 6. 122

5.2. DYNAMICS OF SPOTS

(a)

(b)

(c)

(d)

(e)

(f)

F IGURE 5.6. Competition instability: (a) baby droplets and (b) a spot gets suppressed. Spot self-replication: (c) one spot, (d) early stage of peanut form, (e) well-formed peanut form, and (f) two spots moving away from each other. Parameter set as given in Table 4.1.

123

CHAPTER 5. DYNAMICS AND TRANSITIONS IN 2D

5.3

Fast O (1) time-scale instabilities

In Figure 5.2(d) a steady state, consisting of a spot and a peanut form is shown, which is obtained as a result of performing continuation over the bifurcation parameter k 20 . This solution belongs to the unstable branch labelled 1 in Figure 5.2(a), which is confined between two fold bifurcations (not explicitly shown). Upon taking this solution as initial condition, we ran a simulation using central finite differences in M ATLAB. In this simulation, we noticed that the peanut-form merged into a single spot, see see Figure 5.5(a). Together with the other spot (the right-hand one), it transforms into a stable two-spot solution, see Figure 5.5(b). As expected, this is a result of the overlapping of the stable branch A with an unstable one. Nevertheless, other instabilities occur. So, in order to explore them, and particularly whether a peanut-form instability occurs, we perform a direct simulation of an unstable steady-state solution from branch 9 in the bifurcation diagram depicted in Figure 5.2(a). We shall not pursue further such a formal analysis here, however. The evolution of a solution consisting of two spots followed by four baby droplets is depicted at a different times in Figure 5.6. As mentioned earlier, two O (1) time-scale instabilities simultaneously act on the initial steady-state. There, due to a competition instability, baby droplets (Figure 5.6(a)) get suppressed. Then, the right-hand spot disappears by means that seems to come from the fact that it is initially located far away from the left-hand boundary, where the gradient is stronger (Figure 5.6(b)). Later, a single spot is temporarily formed (Figure 5.6(c)) to then give rise to a peanut form (Figure 5.6(d)). Notice that splitting direction seems to align along the decreasing gradient direction, remind the gradient has been taken y-constant. In Figures 5.6(e) and 5.6(f), another spot emerges from the single-spot as a result of a two-spot stationary solution as described by splitting instability. Both spots move away from each other, which later sets up Proposition 5.2. Competition and splitting instabilities are two examples of fast time-scale instabilities and occur in such way that the gradient shows signs of being strongly relevant to these transitions. For instance, Kolokolnikov et al. [73] analysed existence and the dynamics of spots for the Schnakenberg system in a circular and a rectangular domain. There, besides thoroughly performing an asymptotic analysis showing existence of radially symmetric solutions, the splitting direction is analysed by means of a NLEP, which is obtained from a small radial perturbation to a spot. Due to the complexity of the resulting equation, no analytical results of this are given in [73], but the authors did succeed in numerically solving the splitting NLEP and determining the splitting direction, the latter from an analytical approach. Here the gradient not only adds robustness to solutions, which results in the overlapping of branches, but also controls the location of spots (see §5.1.2 and §5.2). That is, once the spot is self-replicated the splitting direction seems to be completely determined by the direction along which the gradient varies, see Figure 5.6. This is in contrast to that found in [73], where the splitting direction is perpendicular to the direction of motion. In other words, as system (5.35) stipulates, spots get pinned according to the gradient 124

5.4. SUMMARY

and domain shape, and because competition instability also plays a role, the direction of motion is such that two surviving spots will arrange themselves in such fashion that there is enough room between them and such that the gradient is not too weak. This phenomenon deserves a more detailed study, which will not be pursued here.

5.4

Summary

In Chapter 3, a matched asymptotic method, thoroughly developed in e.g. [72, 135, 141], was used to undercover the role that the gradient plays on the location dynamics of ROP patches. We used a similar approach to obtain existence of solutions and stability of solutions for the 2D case assuming that auxin concentration varies only in one spatial direction, as was done in the previous chapter. In this fashion, we have derived a DAE system which theoretically describes the location dynamics of multi-patches. The contrast with Chapter 3 is that here we look for localised solutions rather than ones that are localised only in the direction of the concentration gradient. The analysis earlier begun in Chapter 4, where steady stripe-states were found to be unstable to transverse perturbations, results presented here are also a continuation from that one in Chapter 4. That is, due to interior stripes break up into spots, we have also performed a numerical bifurcation analysis that provides steady states in the form of spots as expected. The bifurcation diagram shown in Figure 5.2 confirms overlapping of stable branches, which comes from bistability and bifurcations close to subcritical pitchfork points (see §2.3.1 for the analysis of the homogeneous system). In addition, reminisces of a homoclinic snaking structure are also indicated. Moreover, O (1) time-scale instabilities come to light as well as competition instabilities similar to the 1D scenario in Chapter 3. Also, as an example of attributes not observed in the 1D case, early self-replication also occurs as is depicted in Figure 5.6. Even though we do not pursue a further analysis in this direction, the gradient seems to become relevant due to its control of the dynamics of location.

125

CHAPTER 5. DYNAMICS AND TRANSITIONS IN 2D

5.A

Appendix: Neumann G -function for Ω2

The analysis performed in §5.1.1 and then in §5.2 depend on the Neumann G-function, which is defined in (5.14a), and its expansion in (5.14b). A thorough description of some features for the unit disk and a rectangular domain has already derived in [73], we shall briefly reproduce the highlights of the G-function calculation for the rectangular domain Ω2 ≡ [0, 1] × [0, d y ]. We begin with

∆G =

(5.36)

1 − δ (x − y) dy

in Ω2 ,

∂G

=0

∂n

on

∂Ω2 ,

where x = (x1 , x2 ). Now, upon applying the separation of variables method, the solution to

∆Ψm,n + λm,n Ψm,n = 0 in Ω2 ,

∂Ψm,n ∂n

on

=0

∂Ω2 ,

which, as can be straightforwardly verified, is satisfied by ¶ µ m 2 π2 mπ x2 Ψm,n (x) = cos (nπ x1 ) cos , λm,n = n2 π2 + . dy d 2y As a consequence, this solution provides a representation of the G-function as follows: G (x; y) =

(5.37)

∞ X

A 0,n Ψ0,n +

n=1

∞ X

B m,0 Ψm,0 +

n=1

∞ X ∞ X

C m,n Ψm,n ,

n=1 m=1

where A 0,n =

2 dy

Z

G Ψ0,n dx ,

B m,0 =

Ω2

2 dy

Z

G Ψm,0 dx ,

C m,n =

Ω2

4 dy

Z

G Ψm,n dx . Ω2

Upon substituting (5.37) into (5.36), we get the Fourier representation of the G-function ¡ ¢ ¡ ¢ ∞ cos π x cos π y ∞ cos mπ x /d cos mπ y /d 2 X 2 X (n 1 ) (n 1 ) 2 y 2 y (5.38) G (x; y) = + + d y n=1 d y m=1 n 2 π2 m2 π2 /d 2y ¡ ¢ ¡ ¢ ∞ X ∞ cos (nπ x ) cos (nπ y ) cos mπ x /d cos mπ y /d 4 X 1 1 2 y 2 y + . d y m=1 n=1 n2 + m2 π2 /d 2y Now, from the identity ∞ cos π z) X π2 (k = h (z) , 12 k2 k=1

h (z) ≡ 2 − 6| z| + 3z2 ,

| z| ≤ 2 ,

we get ∞ cos π x cos π y X (n 1 ) (n 1 ) h (x1 − y1 ) + h (x1 + y1 ) = , 2 2 2 n π n=1

and, upon using the identity µ ¶ ∞ cos π z) X π cosh (aπ (1 − | z|)) 1 (k = − 2, 2 + a2 2a sinh π k 2a (a ) k=1

126

| z| ≤ 2 ,

5.A. APPENDIX: NEUMANN G -FUNCTION FOR Ω2

the last two terms together in (5.38) can be written as µ ¶ µ ¶¶ ∞ µ 1 X π m (x2 − y2 ) π m (x2 + y2 ) F m (5.39) Σ (x, y) ≡ cos + cos , 2π m=1 dy dy m where ¡ ¢ ¡ ¢ cosh mπ (1 − | x1 − y1 |) /d y + cosh mπ (1 − | x1 + y1 |) /d y ¡ ¢ . Fm ≡ sinh mπ/d y

Therefore, from (5.39), the Fourier representation of the G-function is rewritten as G (x; y) =

(5.40)

h (x1 − y1 ) + h (x1 + y1 ) + Σ (x, y) . 12d y

The term Σ (x, y), will now be rewritten in such way that the singular and regular parts of the G-function are easy to be written. To do so, we recall the definitions of cosh and sinh, and the identity that relates trigonometric and hyperbolic functions. Next, we define µ ¶ µ ¶ µ ¶ π r ±,± π p ±,± 2π q ≡ exp − (5.41a) , z±,± ≡ exp , s ±,± ≡ exp , dy dy dy where (5.41b) (5.41c)

r +,± ≡ − (x1 + y1 ) + i (x2 ± y2 ) ,

r −± ≡ − | x1 − y1 | + i (x2 ± y2 ) ,

p +,± ≡ x1 + y1 − 2 + i (x2 ± y2 ) ,

p −,± ≡ | x1 − y1 | − 2 + i (x2 ± y2 ) ,

as we are taking into account that x1,2 , y1,2 ∈ R+ . Then, note that q < 1, which implies ( ) ∞ X ¯ ¯ (q n )m = − log ¯1 − q n ¯ ℜ m=1 m holds. Thus, we obtain (5.42)

∞ © ¯¯ ¯¯ ¯¯ ¯¢ª ¡¯ 1 X log ¯1 − q n z+,+ ¯ ¯1 − q n z+,− ¯ ¯1 − q n z−,+ ¯ ¯1 − q n z−,− ¯ − 2π n=0 ∞ © ¯¯ ¯¯ ¯¯ ¯¢ª ¡¯ 1 X log ¯1 − q n s +,+ ¯ ¯1 − q n s +,− ¯ ¯1 − q n s −,+ ¯ ¯1 − q n s −,− ¯ . − 2π n=0

Σ (x, y) = −

Here, as (5.42) diverges at the singularity, the only singular part term is n = 0 for z−,− . That is, ¯ ¯ this comes from the fact that log ¯1 − z−,− ¯ does not converge as x −→ y. However, also note that the second term in the right-hand equation ¯ ¯ ¯ ¯ ¯ ¯ ¯ 1 − z−,− ¯ ¯ log ¯1 − z−,− ¯ = log ¯ r −,− ¯ + log ¯¯ r −,− ¯

is regular as x −→ y, as can be straightforwardly verified as notice that ¯ ¯ ¯ ¯ 1 − z−,− ¯ ¡ ¢¯ ¯ ¯ = log ¯¯ π + O r −,− ¯¯ , as r −,− −→ 0 . log ¯ r −,− ¯ d 127

CHAPTER 5. DYNAMICS AND TRANSITIONS IN 2D

Therefore, from (5.40) and (5.42), we conclude G (x; y) = −

1 log |x − y| + R (x, y) , 2π

where r −,− = |x − y| and the closed form for the regular part of the G-function is ¯ ¯ ∞ ¯ 1 − z−,− ¯ 1 X ¯ ¯ 1 h (x1 − y1 ) + h (x1 + y1 ) ¯ ¯− − log ¯ log ¯1 − q n z−,− ¯ − R (x, y) = ¯ 12d y 2π x−y 2π n=1 ∞ X ¯ ¯ ¯ ¯ ¯¢ ¡¯ 1 log ¯1 − q n z+,+ ¯ ¯1 − q n z+,− ¯ ¯1 − q n z−,+ ¯ − − 2π n=0 ∞ ¯¯ ¯¯ ¯¯ ¯¢ ¡¯ 1 X log ¯1 − q n s +,+ ¯ ¯1 − q n s +,− ¯ ¯1 − q n s −,+ ¯ ¯1 − q n s −,− ¯ . − 2π n=0

Now, from above, we only need to calculate the self-interaction term, which is given by ¯ µ µ ¶ µ ¶ ¶¯ ∞ ¯ 1 1 2nπ ¯¯ 1 π 1 X 2 ¯ log ¯1 − exp − R (y, y) = − y1 + y1 − log − − dy 3 2π dy 2π n=1 dy ¯ ∞ ¯¯ ¯¯ ¯¢ ¡¯ 1 X 0 ¯¯ n 0 ¯¯ n 0 ¯ log ¯1 − q n z+ − ,+ 1 − q z+,− 1 − q z−,+ − 2π n=0 ∞ ¯¯ ¯¯ ¯¯ ¯¢ ¡¯ 1 X − log ¯1 − q n s0+,+ ¯ ¯1 − q n s0+,− ¯ ¯1 − q n s0−,+ ¯ ¯1 − q n s0−,− ¯ , 2π n=0 0 0 where z± ,± and s ±,± are obtained from (5.41) as x −→ y, where

r 0+,+ = −2y1 + 2i y2 , p0+,+ = 2y1 − 2 + 2i y2 ,

r 0+,− = −2y1 ,

p0−,− = −2 ,

r 0−,+ = 2i y2 ,

p0+,− = 2y1 − 2 ,

128

p0−,+ = −2 + 2i y2 .

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M ORE R OBUST 2D PATCHES

W

e are now ready to consider a slightly more realistic features of auxin transport that might lead to the RH-initiation process. The ROP-model attributes, derived in Chapter 1 and thoroughly studied in subsequent chapters, seem to depend crucially on the spatial

gradient of auxin. The key assumption we made was a decreasing auxin distribution along the x-direction. Indeed, according to our results in previous chapters, such an auxin gradient controls the x-co-ordinate patch location such that the larger the overall auxin concentration is, the more patches are likely to occur. However, as was discussed in Chapter 4 and, briefly, in §5.3, there are instabilities that take place when an extra spatial dimension is present. In other words, when a RH cell is modelled as a two-dimensional flat and thin cell, certain pattern formation attributes become relevant that are not present in a one-dimensional setting. In particular, stripes tend to break into spots, but there is no preferred y-co-ordinate to the spot location. Due to the symmetry of Neumann boundary conditions, they equally will be along the centreline of the cell or at the edge. Nevertheless, a different story may occur when the gradient is spatially dependent of the transverse direction. In this final chapter, we shall briefly explore such a possibility.

6.1

Auxin transport

Auxin transport models are studied as a polarisation event between cells, however little is known within a cell. In §1.3.2 a description of the role of the auxin flux in the RH cell can be seen, and a sketch of the auxin flux is depicted in Figure 1.10. There a simple transport equation is assumed to govern the auxin flux through a RH cell where the diffusion term is neglected because auxin diffuses much faster than ROPs in the cell. This assumption comes from the fact that a in- and out-pump mechanisms inject auxins into RH cells, which occurs as a consequence of 129

CHAPTER 6. MORE ROBUST 2D PATCHES

Apical end Projection onto a 2D rectangular domain

Auxin symplastic pathway

ˆ n

Basal end

Pi ˆ n

PIN

ˆ n

ˆ n

Pe

F IGURE 6.1. Sketch of a idealised 3D RH cell, which auxin gradient is shown as a consequence of in- (solid arrows) and out pump (dashed arrows) mechanisms. Influx and efflux permeabilities are depicted by orange and blue arrows respectively; the auxin symplastic pathway and auxin gradient level curves are plotted in purple.

intercommunication between RH and non-RH cells, see e.g. [7, 55, 59]. As we are interested in the biochemical interactions that prompt RHs, rather than model an auxin transport in the roof we shall hypothesise a specific form of the auxin in a 2D azimuthal projection of a 3D RH cell. As a RH cell is a epidermal cell, and AUX/LAX influx and PIN efflux carriers are distributed along the cell membrane, the auxin flux travels from the tip of the cell towards the elongation zone. As already discussed in §1.3.2, the auxin follows apoplastic and symplastic pathways, we are interested in the latter which travels within the cell and consequently induces the switching process. The auxin gradient, as is experimentally known, is longitudinally decreasing; this is a consequence of the inhomogeneous distribution of carriers in the cell membrane. In Figure 6.1, a sketch of a idealised 3D RH cell is depicted. The auxin flux is represented by purple arrows which schematically indicate a longitudinal decreasing symplastic pathway. A projection onto a 2D rectangular domain of this cell is also shown; because there is no transversal auxin efflux, we shall assume a decreasing gradient away from the longitudinal central line, this is represented by purple level curves. 130

6.2. A PARTICULAR GRADIENT

(a)

(b)

300 280

Boundary patch One interior patch Boundary and one interior patch Two interior patch Boundary and two interior patch

1

260

||U||2

220 200

2

3.5 3.0

3

2.5

2.0

180 160 140 0.00

Mutant with basal end and interior patch Mutant with two interior patch

||U||2

240

Mutant with basal end patch Wild type with single interior patch

4.0

4 0.05

0.10

0.15

0.20

k20

0.25

1.5

5 0.30

0.35

1.0

0.40

0.0

0.5

1.0

k20

1.5

2.0

F IGURE 6.2. Comparison of bifurcation diagrams of system (1.5) as varying k 20 with (a) an auxin gradient (6.1) in a 2D-rectangular domain, and (b) 1D domain reproduced from Chapter 2. Parameter set as given in Table 3.1(a).

6.2

A Particular gradient

In order to mathematically capture these features, we perform a numerical bifurcation analysis of the system (1.5) for Ω2 ≡ [0, L x ] × [0, L y ] and homogeneous Neumann boundary conditions but with the gradient will however be assumed of the form µ ¶ µ ¶ −ν x πy α (x) = exp (6.1) sin . Lx Ly Such a distribution represents a decreasing concentration of auxin in the x-direction, as is biologically expected, and a greater longitudinally concentration of auxin along the middle of a flat rectangular cell. Note the gradient (6.1) is positive for all x ∈ Ω2 as models a non uniform decreasing distribution of auxins. To perform this computation, we use once again a 2D continuation code written in M ATLAB as in Avitabile’s code [6]. In particular, we compute steady-states using pseudo arclength continuation and finite differences in space. Stability is computed a posteriori using M ATLAB eigenvalue solutions. The resulting bifurcation diagram is depicted in Figure 6.2(a). Notice there are similarities between the bifurcation diagram for the 1D case, from Chapter 2, which is reproduced in Figure 6.2(b). That is, no other than fold bifurcations were found for the branches depicted. Because of similitudes with the homoclinic snaking theory on a finite domain (see e.g. [26]), all branches seem to lie on a single curve, which consequently gives rise to an overlapping of stable branches. Hence, in quasi-static parameter-variations, transitions between solution kinds would be hysteretic, as a consequence of bistability. To see more detail, the bifurcation diagram in Figure 6.3(a) has been split in two parts: the top panel for lower values of k 20 , and bottom panel for larger values. In fact, a trend similar to the 1D case occurs here, for very low values 131

CHAPTER 6. MORE ROBUST 2D PATCHES

(a) (b) 400 350 300 250 200 150 0.00

Boundary patch One interior patch Boundary and one interior patch

2

3

0.02

170 165 160 155 150 145 140 135 0.10

15

y

||U||2

1

0.04

10 5

0.06

0.08

0

0.10

0

5 0.15

0.20

10

20

0.25

0.30

k20

0

0.35

10

20x 30

10

10

y

y

15

5

5

0

0

0

10

20

20x 30

x

40

30

50 0

40

5

10y

50

15

0

30 25 20 15 U 10 5 0 20

10

0

10

20

20x 30

(e)

15

15 10

y

y

50 0

5

10y

15

x

40

30

50 0

40

5

10y

50

15

25 20 15 U 10 5 0 20

k20 = 0.3174

20

10 5 0

40

40 30 20 U 10 200

(f)

k20 = 0.2408

20

50

k20 = 0.0583

20

15

10

40

(d)

k20 = 0.0322

0

30

0.40

(c) 20

x

Two interior patch Boundary and two interior patch

4

||U||2

k20 = 0.0122

20

5 0

10

0

10

20

20x 30

x

40

30

50 0

40

5

10y

0

50

15

14 12 10 U 648 02 20

0

10

0

10

20

20x 30

x

40

30

50 0

40

5

10y

50

15

14 12 10 8U 46 02 20

F IGURE 6.3. Bifurcation diagram with α = α (x) as given in (6.1). (a) Stable branches are plotted as solid lines, and filled circles represent fold points; top and bottom panels show overlapping of branches of each steady solution: (b) boundary patch, (c) one interior patch, (d) boundary and one interior patch, (e) two interior patch, and (f) boundary and two interior patch. Parameter set as given in Table 3.1(a).

132

6.3. FURTHER RESULTS

of the overall auxin rate, a boundary patch as shown in Figure 6.3(b) emerges; steady states as such are gathered in branch 1 . Unlike the homoclinic stripe solutions though, this state is fully localised in 2D. Such a solution on a slowly increasing parameter k 20 , the interior patch branch 2 comes into play, see Figure 6.3(c) for an example of a single interior patch. As the bifurcation parameter increases, this patch family persists until stability is lost. Branch 3 gathers stable steady states consisting of one interior spot and a boundary spot, see Figure 6.3(d). As k 20 continues increasing, this behaviour goes on further, hence another spot emerges in the interior, Figure 6.3(e) and branch 4 , and then an additional one is formed at the boundary, see Figure 6.3(f) and branch 5 . Indeed, this trend follows attributes already explored for the 1D case, namely a creation-anihilation cascade, see §2.6 for details.

6.3

Further results

In order to analyse the initiation process in a 1D cell, a direct simulation sweep in length and auxin overall rate was performed in Chapter 2. Also, to initially analyse transitions between the different scenarios exposed, a two-parameter continuation was performed. For a 2D domain, the later is numerically more complex and we will not pursue such an analysis here. However, as learnt in previous chapters, some of the features arising as a consequence of increasing parameter k 20 , as can be seen in §6.2. As shall be seen, similar instabilities analysed in Chapter 4 are also found here. Indeed, as we have already seen, the auxin gradient controls RH location and determines the number of RHs to occur. In other words, the more auxin is in the cell, the more RHs give rise, which are located accordingly to an inhomogeneous distribution of auxin in the cell. Moreover, due to similarities related to the theory of homoclinic snaking, the overlapping of stable branches is favoured, which gather all different physically observable states.

6.3.1

Instabilities

In addition, transitions from unstable to stable branches seem to be determined by means of fold bifurcations, and controlled by O (1) time-scale instabilities. In order to illustrate the above, we choose a steady-state which belongs to an unstable branch as initial condition, and run a simulation using a finite differences method in M ATLAB; snapshots at key times of the outcome are depicted in Figure 6.4. An unstable steady-state as initial condition is shown in Figure 6.4(a). As we have analysed in Chapter 3 for the 1D case, such a steady-state is expected to be unstable because of closeness of two spots. That is, these two spots start to merge at the boundary, see Figure 6.4(b). As this merging process continues, a new-born homoclinic stripe shape emerges near the right-hand boundary end, as can be seen in Figure 6.4(c). In Figure 6.4(d), the homoclinic stripe shape give rises to a transversally aligned peanut form, whilst another one longitudinally occurs at the left-hand boundary. The later seems to be a consequence of the merging process 133

CHAPTER 6. MORE ROBUST 2D PATCHES

(a)

(b)

(c)

(d)

(e)

(f)

F IGURE 6.4. Time dependent transitions, controlled by a y-dependent gradient, from an unstable steady-state to a two interior and a boundary patch: (a) a boundary and one interior patch as initial condition, (b) spots merging, (c) a new-born homoclinic stripe shape, (d) peanut forms, (e) an interior spot coming from a collapsed peanut form, and (f) two interior and a boundary patch. Parameter set as given in Table 3.1(a) and k 20 = 0.0209.

134

6.3. FURTHER RESULTS

and the drift that the gradient imposes in regions where is stronger. The right-hand peanut form shows signs of being persistent, however, it does not break up in the end, the left-hand one does, see Figure 6.4(e). The dynamics is finally completed, as can be seen in Figure 6.4(f), where no peanut forms survive and spots get frozen in a location governed by the gradient. In the instabilities and dynamics described above, the gradient apparently play a very important role. Indeed, competition instability occurs in such way that the initial state is diluted, this suggests that distance between spots still remains relevant to the dynamics. In fact, both patches merge together rather than to have a spot suppressed, which indicates that the strong gradient favours aggregation rather than annihilation. On the other hand, a homoclinic stripe is formed to quickly collapse into a peanut form in regions where the auxin gradient is weaker. The gradient provides more robustness to the pattern formation, which induces almost radially symmetric spots but in locations controlled by the gradient. The study of O (1) time-scale instabilities sheds light on the understanding of the phenomena observed here. For instance, peanut forms and location dynamics can be studied by following a similar formulation derived in Chapter 5. However, an analytical approach similar to that one performed in Chapter 4 seems to become rather more complex for this case. The main difficulty lies in the fact that the gradient depends on the y-direction, which makes it impossible to derive a quasi-steady-stripe-state and, consequently, an appropriate NLEP from which a dispersion relation can be obtained. Despite this, considering the variable y as a parameter of the location point x0 along the x-direction, we might be able to apply a matched asymptotic method. In so doing, the u-component can be rewritten as ¡ ¢ u = U ε−1 (x − x0 ) ; y ,

and, consequently, the equation for the u-component would take the form 2

U xx + ε U yy + α0 U

2

v0 − U = 0 ,

α0 ≡ α (x0 ; y) ,

which has an asymptotic solution of the form U=

¡ ¢ 1 w ε−1 (x − x0 ) , α0 v0

where w is the usual spike-solution in 1D. The ε2 -term to leading order implies a much lower diffusion in the y-direction. The difficulty araises when v0 is calculated in the outer scale as this method would end up dealing with an integral equation. On the other hand, upon assuming that α (x0 ; y) can be written as α (x0 ; y) = α (x0 ) (1 + ²U (y)) ,

for ² ¿ 1 ,

and similarly for v0 , the outer scale equation would be transformed to a distinguished ansatz for

U

is dependent on y, satisfying appropiate boundary conditions. In other words, by taking 135

CHAPTER 6. MORE ROBUST 2D PATCHES

the assumption above, the x-direction is suggested to be the main and most relevant direction of the auxin gradient, and variable y would play as a parameter characterising the transversal direction of the gradient.

6.4

Summary

In this chapter we have considered a gradient which contains a more relevant geometrical feature. That is, the 2D domain models a projection of a 3D cell membrane and cell wall, where the auxin gradient is affected by this consideration. In order to propose a more realistic model that leads to the understanding of the role that the auxin plays on the biochemical interactions giving rise to RHs, we have numerically performed a bifurcation analysis where the auxin gradient is transversally and longitudinally dependent. Indeed, bifurcation features already noticed in previous chapters are also observed. As a consequence, robustness is favoured by means of stable branches overlapping; hysteresis is therefore obtained as noted in previous chapters. In addition, we have observed similarities in the instabilities that are brought about over a long time-scale. Simulations shown here suggest that, as in Chapter 4, homoclinic stripes are also unstable, which hence favours fully localised solutions. These simulations also indicate that location is strongly controlled by the gradient.

136

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thorough mathematical analysis of an inhomogeneous reaction-diffusion system has been performed in this thesis. The model shows results that match the behaviour of wild type and mutants of the original biological problem of root hair (RH) initiating for plausible

parameter regions. A wide diversity of different scenarios are analysed where ROP initiation occurs, either in a 1D domain in a homogeneous domain and in a 2D domain inspired by a 2D projection of a long thin root hair cell. The biology involved in the biochemical interactions that prompt root hair initiation is briefly reviewed in Chapter 1. The process is proposed to be catalysed by auxins, whose transport is experimentally known to cause an inhomogeneous auxin distribution in the cell. As we are interested in the biochemical processes and geometrical features that participate in RH initiation, the ROP model is derived via a reaction-diffusion approach in a non-homogeneous environment, putting some detail behind the brief derivation by Payne and Grierson (see [115]). The analysis of the homogeneous case throws light on the process that initially triggers the pattern formation. Turing bifurcation conditions are computed for the parameter sets used, and a criticality condition derived in Chapter 2 shows that subcritical patterning instabilities undergo fold bifurcations favouring localised solutions. This permits us to numerically analyse the non-homogeneous 1D system via bifurcation analysis and direct simulation. As the two key parameters auxin and cell length increase in the developing root hair cell, the process of forming an active-ROP patch that will go on to bulge into a root hair bud proceeds via a number of abrupt transitions, interspersed with periods of slow drift. The first transition is to form a patch of active ROP at the apical end of the cell. Then, as this patch becomes unstable, a new patch is formed in the interior of the cell, strongly biassed towards the apical end. These transitions are hysteretic, which means that once a state is reached, a reversal process would not easily occur. 137

CHAPTER 7. CONCLUDING REMARKS

This is biologically relevant, such a result indicates that the all biochemical processes would be able to survive stochastic fluctuations and influences from outside the cell or the plant. In other words, through the asymptotic analysis and the two-parameter bifurcation diagrams, we have shown that the process described is robust to relaxation of some of the underlying assumptions that have gone into the model. The asymptotic analysis suggests that multiple RH formation is a matter of timing, amount of auxin in the cell and cell length. That is, the more auxin arrives to the cell, the more multipatches can occur. As the patch drifts towards the interior, before its locking in place occurs, a new transition occurs prompting new active ROP patches. There is an inverse relationship between auxin and cell length, this means that if there is greater cell lengthening, multi-RHs can be caused. However, the asymptotic stability analysis of multi-patches shows that if there is insufficient lengthening or excess of auxin to cause multiple root hairs, then we should expect that there will be further drift of the wild-type patch location from the apical end. This seems to resemble experimental findings, see e.g. [69]. The auxin dependency of the activation process is only hypothesised. There are many other parameters that could play a similar role to that of k 2 in controlling this hysteretic transition process. Also, the precise shape of the auxin gradient is not important, providing it is monotically decreasing. However, similar behaviour occurs for the same model posed on a flat, thin mathematical domain. The growth patch region gathers all the biochemical interactions involved in the RH initiation process, which occurs in such a fashion that ROP gets attached to the cell membrane to then induce cell wall softening. This strongly implies that geometrical considerations must be taken into account. As a consequence, a 2D rectangular domain is considered as a projection of the cell membrane and cell wall of a 3D cell. In such a domain, pattern formation can consists of lateral stripes and spot-like states depending on the auxin gradient dependency. Here, we have proven that, for a transversally independent gradient, all lateral stripes are unstable states in the interior of the cell. Nonetheless, aggregation of active ROP can be obtained as a combination of spots and a lateral stripe at the apical end. This is a consequence of having a strong presence of auxin at the boundary and a transversally constant auxin distribution. In addition, the role of the gradient seems just to be that of controlling the location of the localised pattern, through a slow-time-scale patch-drift equation. The method we have applied shows that the gradient indirectly plays a role on transverse instability via the location points. That is, the gradient controls location and is also strongly involved in specifying then number of transverse unstable modes. This suggests that active ROP aggregation occurs in the form of spots, which can be reformulated as a maximising-entropy thermodynamical process. As these mechanisms indicate that the geometry of the cell strongly influences pattern formation, an asymptotic analysis is further performed in order to capture these features in the dynamics of spot location. Not only does the gradient control the location, but also the inactive ROP distribution plays an important role. Effectively, the inactive ROP acts as a substrate, 138

which gathers all shape and form interactions within the domain. In other words, source points, which depend on the switching process, characterise the pattern formation as inactive ROP gets activated and therefore bound to the cell membrane. As expected, a numerical bifurcation analysis shows that hysteresis still remains when a 2D domain is considered. Stable branches overlap, which implies that abrupt transitions between pattern types occurs through fold bifurcations. In addition, due to reminiscences of the homoclinic snaking theory, a creation-annihilation cascade provides configurations where spots and baby droplets can occur. However, O (1) time-scale instabilities determine whether multi-spot arrays survive. That is, competition and peanut-form instabilities occur. In Chapter 6, a more realistic projection of the cell membrane and cell wall onto a 2D domain is considered. In so doing, we claim the transversal curvature does not significantly affect to the switching process, but it does affect auxin distribution. In addition, we propose that, because of active diffusion of auxin from non-RH to RH cells, auxin concentrations in the latter are distributed in such way that a longitudinal gradient direction is more relevant than a transversal one, and switching process occurs in the cell where the auxin distribution is higher, rather than at the transversal boundaries. We briefly analyse this scenario; our bifurcation analysis reproduces all the features previously found — analysed in previous chapters — when a non transversally dependent gradient is taken into account. Moreover, more robust patches are obtained as a consequence of hysteresis and the fact that the auxin concentration decays in the transversal directions. For the same reason, no active ROP stripes seem to take place. Any attempt to analyse asymptotically would need the location dynamics to depend on the gradient in both directions. Such an analysis is beyond the scope of this thesis.

Theoretical approach From a mathematical point of view, several novel results have been found. That is, analyses related to pattern formation via reaction-diffusion systems in non homogeneous domains are not well understood yet. However, it seems that there is a strong backbone to our results that are related to what is knew in homogeneous systems. We have performed a linear analysis that provided the necessarily settings to seek for a particular kind of solutions. Indeed, we found subcritical Turing bifurcations which strongly favour localised solutions rather than periodic, and is a key ingredient of the homoclinic snaking theory (see e.g. [15, 26, 143]). Here, nonlinear reaction terms, which come from bistability inherent to the system, are central because of the balance between the cubic and quadratic terms of the normal form, see §2.3. Further results for the homogeneous system will appear elsewhere. The above however does not provide of the main features that solutions, coming from direct simulations, present. That is, even though subcritical bifurcations seem to be essential for localised solutions, another key feature of our system is the asymptotic scaling that takes it into the realm of semi-strong interaction theory (e.g. Ward et al. [65, 72–74, 101, 103, 134–136]). 139

CHAPTER 7. CONCLUDING REMARKS

Moreover, solutions as presented in this thesis also have certain similarities with theory of wave pinning (see [97, 98]). As can be seen in Proposition 2.4, the ROP system (1.5) satisfies a certain conservation principle as well, where the source term b and recycling r play a crucial role, regardless of the inhomogeneity. However, there is a main feature that should be marked, solutions described here get pinned, due to a non homogeneous coefficient rather than due to a wave-propagation mechanism. This is evident from the fact that the v-component of the saturation of the solution shown in Propositions 3.2 and 3.5 relies on the gradient at interactions with location points on the inner scale. On the outer scale, the v-component is determined by boundary conditions and form and shape of the domain as equation (5.16) shows, where geometrical proprieties are evident. The matched asymptotic method applied here has shed light on each feature the system plays on the solutions. As can be seen in the analysis in both 1D and 2D, the gradient is not only crucial for the location dynamics, but also prohibits uniform amplitudes in a multi-spike and multi-spot set. As matter of fact, the further from regions where the gradient is strong localised solutions are, the higher the amplitude. Furthermore, because gradient controls location, a symmetrical multi-array is unlikely to occur. By means of bifurcation analysis, transitions between states are properly described. Also, upon performing a stability analysis we have used the theory of NLEP in several situations, which is usually applied to homogeneous systems. O (1) time-scale instabilities are therefore analysed, from which competition, transverse and peanut-form instabilities were found.

Further issues There are many possible avenues for future work, both mathematical and biological. Each of our predictions is ripe for further experimental investigation. For example, preliminary unpublished data from Grierson’s lab suggests ROP patches do indeed drift, and they do undergo sudden abrupt changes, e.g. from boundary-patch to interior-patch positions. More realistically, this model is a mere approximation of complex biochemical processes, and we have only addressed a thermodynamically macroscopic viewpoint; that ignores single ROP interactions. Again preliminary results with an agent based model are promising. More importantly, even though the hormone auxin is proposed only as a catalyst, no deep analysis was performed in order to investigate whether there are other terms where its gradient might become relevant as well. These could be related to other mechanisms yet unknown and not in consideration here. In addition, the model only considers a single RH cell, considering the auxin in a single cell rather than a set of cells. In such a scenario, the auxin transport should not be modelled simply by a spatially dependent coefficient. This however may represent a challenge because of the gap between ROP and auxin diffusion rates. Upon pursuing a reaction-diffusion approach, instead of studying a two-component system, an extra equation for the auxin would be needed, where cell curvature and 3D effects may become more important and boundary conditions would 140

presumably be of Robin type. To analyse a three-component system with homogeneous Neumann conditions for active and inactive ROP components and Robin conditions for the auxin component would become more complex. See [85] for an asymptotic analysis in this direction. In a slightly different trend, the Lyapunov–Schmidt method was applied to compute a criticality condition for the homogeneous system, which implies the calculation of a normal form. To do so, an expansion at third order was achieved here, and further terms discarded. Nevertheless, computing at fifth term has not been yet performed. This would be an interesting investigation direction, as would a detailed investigation of the codimension two point where a Turing bifurcation occurs precisely where the third-order coefficient vanishes. Such a point on a long domain satisfies the ingredient required for homoclinic snaking providing the fifth-order coefficient is of the right sign. On the other hand, the O (1) time-scale instabilities numerically explored in Chapter 5 deserve a deeper analysis; competition, transverse and peanut-form breakup are relevant not only because shed light on transitions between states, but also from a mathematical point of view. As seen there, the gradient changes crucial characteristics seen in homogeneous systems such as the splitting direction, see e.g. [73]. Finally, there are many other problems involving morphogenesis inside the cell, involving Rho-like G-proteins (e.g. [17, 20, 40, 67, 87, 102]). It may well be that the mechanisms uncovered in this thesis, namely subcritical Turing bifurcation, localisation and patch location through inhomogeneity may well play a wide role across biology.

141

APPENDIX

A

O (1) T IME - SCALE C RITICALITY C ONDITION IN 1D

T

he criticality result for NLEP which is implicit in [136], used in Chapters 3 and 4, is reproduced in this appendix. The result characterises O (1) time-scale stabilities, it provides a sufficient condition to test for an instability, with the criticality condition on

the parameters obtained by setting ζ(0) = 1. When ζ depends on λ, it is considerably more difficult to provide necessary and sufficient conditions for stability, and we shall not pursue this direction here. The technical difficulty with such an analysis is that whether a non-self adjoint NLEP as (A.1) has any complex eigenvalues in the right-hand half-plane, ℜ(λ) > 0, must be examined.

Proposition A.1. Let 1/ζ(λ) be holomorphically extendable in the right-hand half-plane ℜ(λ) > 0. (i) If ζ(0) < 1, then the NLEP Z∞

(A.1)

−∞ L 0 φ − ζ (λ) w2 ∞ Z

wφ d ξ = λφ ,

−∞ < ξ < ∞ ,

w2 d ξ

−∞

has an unstable real eigenvalue in ℜ(λ) > 0. (ii) If ζ(0) = 1, then φ = w an eigenfunction of (A.1) corresponding to a zero eigenvalue. Proof. Let φ = w, and from the identity L 0 w = w2 , we get Z∞

L 0 w − ζ(0)w2

−∞ Z∞

w2 d ξ = w2 − ζ(0)w2 = 0 .

w2 d ξ

−∞

143

APPENDIX A. O (1) TIME-SCALE CRITICALITY CONDITION IN 1D

The last equiality is satisfied if ζ(0) = 1. This proves the second part of Proposition A.1. R∞ Now, for the first part, recall −∞ w2 d ξ = 6, then in (A.1) we write φ as φ=

ζ(λ)

6

Z∞

wφ d ξ (L 0 − λ)−1 w2 .

−∞

We then multiply both sides of this equation by w and integrate over the real line to obtain that the eigenvalues λ of the NLEP (A.1) are the roots of the transcendental equation g(λ) = 0, where 1 E (λ) ≡ 6

1 − E (λ) , g(λ) ≡ ζ(λ)

Z∞

w (L 0 − λ)−1 w2 d ξ .

−∞

Notice that ψ = (L 0 − λ)−1 w2 solves (L 0 − λ) ψ = w2 with ψ −→ 0 as |ξ| −→ ∞. In addition, as shown in Proposition 3.15 in the appendix of Chapter 3, when λ = ν0 , the operator L 0 − λ is not invertible. This therefore implies that E −→ ∞ as λ −→ ν− 0. Now, we write w2 = L 0 w, hence 1 E (λ) = 6

Z∞

£

−1

w (L 0 − λ)

((L 0 − λ) w + λw) d ξ = 1 + ¤

−∞

λ

6

h(λ) ,

h(λ) =

Z∞

w (L 0 − λ)−1 wd ξ ,

−∞

and E (0) = 1. In order to conclude, we just need to prove that E 0 (λ) > 0 on 0 < λ < ν0 . So ¡ ¢ differentiating, we obtain E 0 (λ) = h(λ) + λ h0 (λ) /6 with 0

h (λ) =

Z∞

−2

w (L 0 − λ)

wd ξ =

−∞

Z∞

¡

w (L 0 − λ)−1 w

¢2

dξ ,

−∞

where we have integrated this formula by parts to obtain the second equality. Since h0 (λ) is positive for 0 < λ < ν0 , we only have to prove that so is h(λ) on the same interval. Then, from R∞ Proposition 3.16 and identity −∞ w3 d ξ = 36/5, Z∞

Z∞ µ

h(0) = wL 0−1 w d ξ = −∞ −∞

2

w +

ξ¡

4

w

2

¢

ξ



dξ =

9 , 2

where we have integrated by parts to get the last two equalities. Thus, E 0 (λ) > 0 on 0 ≤ λ < ν0 . Since 1/ζ(λ) is holomorphically extendable in the right-hand half-plane ℜ(λ) > 0, there must exist a real positive root λ∗ of g(λ) = 0 on 0 < λ < ν0 whenever ζ(0) < 1. This completes the proof of



Proposition A.1.

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