EvoDevo in Phase Space:
the Dynamics of Gap Gene Expression
Berta Verd Fern´andez
DOCTORAL THESIS UPF / YEAR 2015
THESIS SUPERVISOR
Dr. Johannes Jaeger
Departament
Comparative Analysis of Developmental Systems Group
Systems Biology Department, CRG
Acknowledgements
I would like to begin by saying that I have had the time of my life. What a wonderful four years! Four years full of new ideas and intellectual development, always in the best of learning environments. I have been incredibly lucky during this time and I am thankful for the people that have surrounded me. Good peo-ple, passionate thinkers with brilliant minds that have always been eager to help and offer guidance. I would like to begin by thanking all the great scientists in the Jaeger Lab. Thank you for welcoming me into the team and for providing the support system I needed to make mistakes, ask stupid questions, make more mistakes and learn so much from you. Thank you Hilde, Astrid, and Barbara for always finding the time and patience to teach me biology. Karl and Eva, thank you for letting me see EvoDevo through your eyes. Damjan, for being a ray of light in times of computational darkness. To Anton, thank you for being my post doc and always helping me find my way. Thank you Manu, Lena Panok and Vitaly Gursky, for your kind help during the early stages of this project.
I want to thank the CRG, and in particular the Systems Biology Department where I always felt communication was easy and growth guaranteed. Thank you to the other PhD students, and in particular to Alba whose drive and enthusiasm I remember from the interviews and has motivated me ever since. Thank you for being my friend. Thank you ’la caixa’ doctoral program, for investing in young scientists like me in a country where no-one will.
Thank you to the Wissenschaftskolleg zu Berlin for the rare chance you gave me to peek into other worlds, meet inspiring people and take a little of it with me. In particular, I’d like to thank Jannie Hofmeyr and Adam Wilkins for the stimulating discussions, and Steven Frank for toughening me up. To the brilliant Simone Reber, for showing me how things get done, for helping me so much and for a wonderful friendship. A big thank you to the members of my committee, for listening and offering advice, for challenging and encouraging me during all these years. Thank you Nick, for all the fun times we had lost in a maze of dynamical regimes.
One thousand times thank you Yogi, for the chance you took on me and your con-tinued investment in me, that has changed my life forever. For turning me into a scientist and a thinker, and teaching me this craft. I am honoured to have been your student, to work with you and to have gotten to know you. I thank you and Hilde for being a home away from home.
Gracias a mis padres, Marisa y Sergio. Yo no ser´ıa nada sin vosotros y a vosotros os lo debo todo. A Fee, por ser como una hermana, y por recordarme que en alg´un momento me tendr´e que poner a trabajar. And last, but certainly not least, thank you Greg for loving me just the way I am: in pyjamas 90% of the time.
Abstract
During insect development segments either form sequentially (short-germband) or simultaneously (long-germband). The gap genes comprise the first zygotic reg-ulatory layer of the segmentation gene hierarchy in flies, which use the long-germband mode. A data-driven mathematical model reveals that two distinct dynamical regimes govern anterior and posterior trunk gap gene patterning in Drosophila melanogaster. Stationary anterior domains rely on multi-stability whilst the observed anterior shifts of posterior domains are an emergent property of a damped oscillator mechanism. Both types of dynamics are recovered by a three-gene sub-network embedded in the gap three-gene regulatory network, which can also sustain oscillations. Oscillations are not found in the gap gene system but are char-acteristic of short-germband segmentation, suggesting that both modes are more similar than previously thought. This insight sheds light on how long-germband segmentation could have repeatedly and independently evolved from the ancestral short-germband mode.
Resum
Durant el desenvolupament dels insectes, els segments es formen seq¨uencial-ment (desenvolupaseq¨uencial-ment de banda curta) o simult`aniaseq¨uencial-ment (desenvolupaseq¨uencial-ment de banda llarga). Els gens gap constitueixen la primera capa de regulaci´o cig`ootica en la jerarquia de la segmentaci´o en mosques, que fan servir un desenvolupament de banda llarga. Amb un model matem`atic posam al descobert que dos r`egims din`amics diferents governen la formaci´o de patrons a les parts anteriors i posteri-ors de laDrosophila melanogaster. Els dominis anteriors s´on est`atics i depenen de la multi-estabilitat del sistema en aquesta part, mentre que el desplac¸ament an-terior dels dominis posan-teriors ´es una propietat emergent d’un osciador amortidor. Una subxarxa de tres gens immersa a la xarxa dels gens gap recupera aquests dos tipus de din`amiques. Aquesta subxarxa tamb´e pot osciar. Al sistema dels gens gap no hi han cap osciacions per`o sabem que s´on caracter´ıstiques de la segmentaci´o de banda curta. Aix`o suggereix que els dos tipus de segmentaci´o s’hi assemblen m´es del que pensavem. Aquest descobriment ens ajudar‘a a entendre com pot haver evolucionat la segmentaci´o de banda llarga repetides vegades de la segmentaci´o de banda curta, que ´es la m´es ancestral.
Table of Contents
List of Figures xvi
List of Tables xvii
List of Boxes xix
I
Introduction
1
II
Classification of Transient Behaviours in a
Time-Dependent Toggle Switch Model
17
1 INTRODUCTION 19
1.1 Dynamical systems . . . 21
1.2 Potential landscapes . . . 24
1.3 The importance of transient dynamics . . . 26
1.4 Non-autonomy: explicit time dependence . . . 28
2 MODEL AND METHODS 31 2.1 Definition of the potential landscape . . . 33
2.2 Calculating quasi-potential landscapes . . . 34
2.3 Approximating non-autonomous trajectories . . . 36
3 TRANSIENT BEHAVIOURS FOUND IN THE TIME-DEPENDENT TOGGLE SWITCH MODEL 39 3.1 Transitions . . . 40
3.2 Pursuit . . . 43
4 CONCLUSIONS 53
III
Full Analysis of a Non-Autonomous Version
of the Gap Gene Model
57
5 INTRODUCTION 59
6 METHODS 67
6.1 The Gene Circuit Model . . . 67
6.2 Model Fitting and Selection . . . 69
6.3 Gap Gene Circuit Analysis . . . 70
7 NON-AUTONOMOUS GAP GENE CIRCUITS WITHOUT
DIFFU-SION 71
8 NON-AUTONOMOUS REGULATORY MECHANISMS FOR GAP
GENE PATTERNING 75
8.1 Gap Domain Borders in the Anterior are Governed by a
Multi-stable Dynamical Regime . . . 75
8.2 Gap Domain Shifts in the Posterior are an Emergent Property of a
Damped Oscillator . . . 80
9 CANALISING PROPERTIES OF THE GAP GENE DAMPED
OS-CILLATOR 85
10 RELAXATION-LIKE OSCILLATORY BEHAVIOUR IMPLEMENTS
FAST-SLOW DYNAMICS 89
11 DYNAMIC MATERNAL GRADIENTS LIMIT LEVEL AND
TIM-ING OF PEAK EXPRESSION 93
12 CONCLUSIONS 97
IV
The AC/DC Circuit
101
14 DYNAMICAL DISSECTION OF THE GAP GENE NETWORK 107
15 MATHEMATICAL ANALYSIS OF THE AC/DC CIRCUIT 115
15.1 Analysis of the Repressilator Motif . . . 115 15.1.1 Types of Steady States: Eigenvalue Study . . . 117 15.2 Analysis of the AC/DC Circuit . . . 122 15.3 Analysis Validation for the Connectionist Model Formulation . . . 132
16 ANALYSIS OF THE AC/DC SUB-NETWORKS ISOLATED FROM
THE GAP GENE NETWORK 137
16.1 Sub-network AC/DC1 drives a bi-stable dynamical regime. . . 138 16.2 Sub-network AC/DC2 produces spiralling trajectories . . . 141 16.3 Sub-network AC/DC3 produces spiralling trajectories via a damped
oscillator . . . 147
17 THE AC/DC SUB-NETWORK IN A REALISTIC SPATIO-TEMPORAL
CONTEXT 151
17.1 Dynamical regimes for sub-network AC/DC1 with maternal gra-dient concentrations. . . 153 17.2 Sub-network AC/DC2 does not drive correct patterning when
ma-ternal gradients are included. . . 156 17.3 Thoughts on the patterning role of the AC/DC circuits . . . 161
18 LANDSCAPE OF DYNAMICAL REGIMES FOR THE AC/DC
CIR-CUIT. 165
19 EVOLVABILITY OF THE AC/DC SUB-NETWORKS. 171
20 APPENDIX 177
List of Figures
0.1 Linking regulatory structure to dynamical regimes and patterning
dynamics . . . 4
0.2 Patterning dynamics and network evolvability . . . 5
0.3 Schematic of the two main modes of segmentation used during insect development . . . 6
0.4 Segment determination inDrosophila melanogaster . . . 8
0.5 Maternal gradients inDrosophila melanogaster . . . 9
0.6 Late gap gene expression . . . 10
0.7 The gap gene regulatory network inDrosophila melanogasterand Megaselia abdita . . . 13
1.1 Waddington’s epigenetic landscape and potential surfaces . . . 20
1.2 The toggle switch model . . . 23
2.1 Dynamical regimes of the toggle switch model . . . 32
2.2 Numerical approximation of non-autonomous trajectories . . . 37
3.1 Transition . . . 41
3.2 Pursuit stabilising the direction of a trajectory . . . 44
3.3 Pursuit altering the direction of a trajectory . . . 45
3.4 Capture due to a change in the topology of the phase portrait . . . 48
3.5 Capture due to a change in the geometry of the phase portrait . . . 49
5.1 Dynamics of gap gene expression . . . 60
5.2 Autonomous versus non-autonomous gap gene patterning mecha-nisms . . . 64
7.1 Regulatory structure of the gap gene network obtained from a non-autonomous gap gene circuit . . . 72
7.2 Values of the parameters in the non-autonomous gap gene circuit
model . . . 73
7.3 Non-autonomous gap gene circuit model fit . . . 74
8.1 Formation of the posterior boundary of the anterior Gt domain . . 77
8.2 Formation of the Hb-Kr interface . . . 79
8.3 A damped oscillator governs posterior gap gene patterning inD.melanogaster 82 9.1 Canalising properties of the gap gene damped oscillator . . . 87
10.1 Fast-slow dynamics in posterior nuclei . . . 90
11.1 Effect of the time-dependence of maternal gradients on gap gene pattern formation . . . 95
12.1 Summary of non-autonomous mechanisms of gap gene pattern formation inD.melanogaster . . . 98
14.1 Gap gene regulatory contributions . . . 109
14.2 Candidate sub-networks . . . 110
14.3 The AC/DC circuit, AC/DC1 and AC/DC2 . . . 111
14.4 Dissecting sub-network AC/DC2 . . . 112
14.5 Sub-network AC/DC3 . . . 114
15.1 The repressilator motif . . . 116
15.2 Uniqueness of Solutions . . . 117
15.3 Complex roots of(λ+d)3 . . . 119
15.4 Eigenvalues of a saddle point . . . 121
15.5 Eigenvalues of a stable spiral sink . . . 121
15.6 The AC/DC circuit . . . 122
15.7 General shape of the function given byF(Y) . . . 123
15.8 No back reaction . . . 123
15.9 Weak back reaction . . . 123
15.10Strong back reaction . . . 123
15.11Roots of a depressed cubic depending on parameter values . . . . 126
15.12Roots of a depressed cubic. Option 1 . . . 127
15.13Roots of a depressed cubic. Option 2 . . . 127
15.15Stable spiral sink . . . 127 15.16Point attractor . . . 128 15.17Saddle1,2 . . . 128 15.18Saddle2,1 . . . 128 15.19Sustained oscillations . . . 129 15.20Damped oscillations . . . 130 15.21Bi-stability . . . 131 15.22AC/DC circuit . . . 132
15.23Sigmoidal regulation-expression function used in the connection-ist model . . . 133
15.24Number of steady states . . . 134
16.1 Dynamics driven by sub-network AC/DC1 . . . 139
16.2 Dynamics driven by sub-network AC/DC2 . . . 142
16.3 Posterior transient dynamics in the full gap gene circuit model . . 144
16.4 AC/DC2 model output versus data . . . 146
16.5 Dynamics driven by sub-network AC/DC3 . . . 148
17.1 Dynamical regimes for AC/DC1 with realistic time-dependent ma-ternal gradients . . . 154
17.2 AC/DC1 with maternal inputs versus data . . . 155
17.3 Dynamical regimes for sub-network AC/DC2 with time-dependent maternal gradients . . . 157
17.4 AC/D2C with maternal inputs vs data . . . 158
17.5 Comparison of phase portraits in the nucleus at 55%A–P position . 159 17.6 Comparison of phase portraits in the nucleus at 69%A–P position . 160 18.1 Landscapes of dynamical regimes associated to sub-networks AC/DC1, AC/DC2 and AC/DC3 obtained using bifurcation parameters A and C . . . 167
19.1 Landscapes of dynamical regimes associated to sub-networks AC/DC1, AC/DC2 and AC/DC3, obtained using A, B, C and D as bifurca-tion parameters . . . 174
20.2 Evolutionary paths of AC/DC circuits illustrate the evolvability of the gap gene system. . . 184
20.3 Oscillatory dynamics occur more anteriorly inM. abditathan in
List of Tables
15.1 Parameter values used to simulate a limit cycle . . . 129
15.2 Parameter values used to simulate damped oscillations . . . 130
List of Boxes
1 Features of phase space in autonomous dynamical systems . . . . 62
Part I
To explain how embryos generate their spatial patterns and temporal programs (...) we need mathematics and detailed measurements, just as we need mathematics and measurements to explain the orbits of the planets or the swing of a pendulum.
Julian Lewis The intricately orchestrated continuation of spatial and temporal gene expres-sion patterns during embryogenesis is one of the main factors responsible for the vast diversity of animal form. Such patterns are dynamically set up as the different developmental processes unfold, and are responsible for the subsequent specifica-tion of distinct regions within the embryo. Not only are these patterns dynamic, but also transient, as are the processes that produce them [1].
Understanding how gene expression forms patterns in space and time is cen-tral to the study of embryonic development. Developmental processes involve complex gene regulatory networks, which drive pattern formation [1]. How they do this, however, is far from obvious, especially when taking into account the full complexity of gene expression dynamics and timing.
The patterning dynamics of a gene network are constrained by the network’s regulatory structure (Figure 0.1A). This means that the nature of the interactions between the genes involved (i.e. repressions, activations...) defines the range of gene expression dynamics possible for such networks, which result in the pattern-ing dynamics observed at the tissue level (Figure 0.1C) [2, 3]. The dynamical regime implemented by a given network determines the dynamics of gene expres-sion that result. While only one dynamical regime is associated to any specific network (defined by its set of parameter values), different dynamical regimes can give rise to the same patterning dynamics (Figure 0.1 B & C). This point will be extensively discussed throughout this thesis. The set of all possible dynamical regimes that can be implemented by networks with the same underlying regula-tory structure is called the dynamical repertoire (Figure 0.1B).
Figure 0.1:Linking regulatory structure to dynamical regimes and patterning dynamics. (A)A regulatory structure is defined by the number of genes and the qualitative nature of the interactions between them. In this example, which will be explained in detail in Part IV of this thesis, there are three nodes corresponding to genes X, Y and Z. The repressive interactions between them are shown by T-bars. All the networks shown in Figure 0.2 have this regulatory structure.
(B)Dynamical repertoire associated to the regulatory structure shown in (A). The dynamical repertoire is the set of all possible dynamical regimes that can be implemented by networks with a given regulatory structure. In this case, the regulatory structure in (A) can give rise to the following dynamical regimes: sustained oscillations, damped oscillations, mono-stability and bi-stability. A specific network (with a concrete set of parameter values) with the regulatory structure shown in (A) will give rise to one, and only one, of the dynamical regimes in the dynamical repertoire. (C)Patterning dynamics result from the gene expression dynamics that unfold in a given dynamical regime. Similar patterning dynamics can be obtained from more than one dynamical regime. The reason for this is that pattern formation is often transient and therefore shaped far from steady state. Dynamical behaviours far from steady state can be indistinguishable in different dynamical regimes.
The dynamical repertoire associated to the regulatory structure of a gene net-work is very limited, and will usually only contain a handful of possible dynamical regimes [4] (Figure 0.1 A & B). Therefore the amount of different gene expres-sion dynamics that can be driven by networks of a certain regulatory structure is constrained. This limits the patterning potential of regulatory structures.
Figure 0.2 shows networks where the relative strengths of the interactions are shown by the thickness of the T-bars between the genes (all the interactions in these examples are repressions). All of these networks have the same underlying regulatory structure shown in Figure 0.1A. Networks with similar ratios between their interaction strengths don’t always produce the same patterning dynamics (as
shown by networks in the intersection of different patterning dynamics in Figure 0.2). This means that only relatively small changes to the strengths of the in-teractions are required to switch between two different patterning dynamics, and therefore we can say that both patterning dynamics are neighbouring each other. For example static and shifting domains are neighbours, as are shifting and oscil-lating domains, but not static and osciloscil-lating domains (Figure 0.2).
It seems that networks can more easily change (via changes to the strengths of their interactions in evolution for example) to patterning dynamics that neighbour their own. Therefore, networks of the same regulatory structure can have differ-ent evolvabilities depending on their currdiffer-ent patterning dynamics. For example, a network driving oscillatory patterning dynamics requires smaller changes to the strengths of its interactions to drive shifting domains, than if it were to drive static ones (Figure 0.2), whilst this is not the case for networks that already drive shifting domains. As a result, networks with different patterning dynamics have different evolvabilities [5]. An understanding of patterning dynamics is therefore essential in order to understand how developmental processes evolve (Figure 0.2).
Figure 0.2: Patterning dynamics and network evolvability. The relative strengths of the interactions are shown by the thickness of the T-bars between the genes. The positions of the networks show which kind of patterning dynamics they can display, given the correct parameter values but within the general ratio between the interactions shown in each network diagram. Networks at intersections represent the potential of networks of that type to produce both kinds of patterning dynamics, and this defines those patterning dynamics as neighbours. Smaller changes to the interaction strengths are required for a network to switch between neighbouring patterning dynamics than otherwise, and this constrains that network’s evolvability.
In this thesis I use the gap gene network in the fruit fly,Drosophila melanogaster, to study how patterns are generated by gene regulatory networks. In particular, I am interested in understanding how gene regulatory networks give rise to
differ-ent dynamics of gene expression and how these in turn, can shape and constrain, the evolution of developmental processes. I will present new methods of analysis and network dissection that make it possible to address these ideas in a real-life, densely connected and complex network such as the gap gene regulatory network. The gap genes are involved in segment determination during early embryogenesis inD.melanogaster[6].
During insect development segments either form sequentially, where new seg-ments are added one at a time to the posterior end of a growing embryo (short-germband), or simultaneously, where the entire length of the embryo is sub-divided into equally sized domains, more or less simultaneously (long-germband) [7, 8, 9] (Figure 0.3). We know that the long-germband mode of segmentation has evolved independently several times from the more ancestral short-germband mode [7, 8, 10]. Dipteran insects (flies, midges and mosquitoes) use the long-germband mode of segmentation to partition their body plan [7].
Figure 0.3: Schematic of the two main modes of segmentation used during insect development. (A)The short-germband mode of segmentation as observed inGryllus bimaculatus. New segments are added one at a time to the posterior end of a growing embryo.(B)The long-germband mode of segmentation as observed inD.melanogaster. Segments form simultaneously by sub-division of the entire embryo length. This picture has been modified with permission from [11]
D.melanogaster uses the long-germband mode of segmentation. Gap genes are expressed before gastrulation when the embryo is still a syncytial blastoderm (a multi-nucleated cell) (cleavage cycles 12–14A, Figure 0.4A). They form the first layer of zygotic regulation in the segmentation gene hierarchy (for a
com-prehensive review of the gap gene network see [6] and references therein). This hierarchy is composed of different regulatory tiers, which were first identified using whole genome mutational screens [12]. The gap genes, together with the maternal coordinate genes, activate the pair-rule genes, which then in turn activate the segment polarity genes (Figure 0.4B).
Figure 0.4: Segment determination inDrosophila melanogaster(A)First 3 hours ofD.melanogasterdevelopment. Cleavage cycle number is shown below each illustration, where cleavage cycle n is defined as the time between mitosis n - 1 and mitosis n. The blastoderm stage lasts from 1 min into cycle 10 to the onset of gastrulation (grey background). The embryo remains syncytial (without membranes between nuclei) until cellularisation occurs during cycle 14A. Cycle 14B denotes the part of cycle 14, which occurs after the onset of gastrulation. Embryos are shown with the anterior pole to the top. (B)TheD.melanogastersegmentation gene hierarchy. Segment determination is based on a molecular pre-pattern established by the segmentation genes, which are active during the blastoderm stage. Maternal co-ordinate genes are expressed in broad gradients (Bcd protein distribution is shown as an example). They regulate the zygotic gap genes, expressed in broad overlapping domains (the central domain of Kr is shown). Gap genes and pair-rule genes together regulate pair-rule genes, which are expressed in 7–8 stripes (shown for Even-skipped (Eve) protein). Pair- rule genes in turn regulate segment-polarity genes whose expression in 14 stripes becomes established just before the onset of gastrulation (shown forenmRNA). These stripes constitute the segmentation pre-pattern and correspond to the positions of para-segmental boundaries later in development. Arrows indicate regulatory interactions between classes of segmentation genes. Circular arrows represent cross-regulation within a class. Embryo images are shown with anterior to the left, and dorsal up. Figure reproduced with permission from [6]
The gap genes are initially activated by the maternal co-ordinate genes, which form long-range gradients of maternal proteins along the A–P axis (Figure 0.4B).
oogen-esis. After fertilisation, the translated protein diffuses towards the posterior of the embryo, setting up an exponential anterior to posterior Bcd gradient [13, 14, 6, 15] (Figure 0.5A). The Bcd gradient represses the translation of the homogeneously expressedcaudal (cad)mRNA to form a posterior to anterior gradient of Cad pro-tein [6, 16, 17, 18, 19, 20]. This gradient is not exponential [20]. Instead it is almost uniformly expressed in the posterior of the embryo, extending roughly two thirds of the whole embryo length (Figure 0.5B).
Figure 0.5:Maternal gradients inDrosophila melanogaster. (A)Bicoid (Bcd) and(B)Caudal (Cad) protein expression profiles at cleavage cycles C12, C13 and C14A (T1–T8). Darker profiles represent earlier time stages (see key). Data for the plots have been taken from the FlyEx database [21, 22].
Similarly, posteriorly locatednanos (nos) mRNA sets up a posterior to ante-rior gradient, which translationally represses the uniformly distributed maternal
hunchback (hb)mRNA to establish an anterior to posterior Hb gradient [6, 23, 24, 25, 26, 27, 28, 29]. Bcd, Cad and maternal Hb all provide initial activation to the gap genes whilst Nos only does so indirectly by setting up the Hb gradient [26]. Further maternal input is provided by geneshuckebein (hkb)andtailless (tll)from the terminal gap gene system, which regulate segmentation gene expression in the posterior pole of the embryo [30].
hunchback (hb), Kr¨uppel (Kr), knirps (kni)andgiant (gt) [6]. Once activated by the maternal gradients, these gap genes cross-regulate each other to form broad overlapping domains of gene expression along the A–P axis of the embryo (Fig-ure 0.6).
Figure 0.6:Late gap gene expression.Protein expression patterns are shown for Hb, Kr, Kni, and Gt at the eight time classes in cycle 14A (T1–T8) [20]. Bottom row plots show integrated one-dimensional expression patterns from the middle 10% along the D–V axis over time, illustrating the anterior shift in boundary position for all expression domains posterior of the central Kr domain. Shifting domains and boundaries have been shaded. Relative protein concentration is plotted against position along the A–P axis (in %, where 0% is the anterior pole). Embryo images and integrated data for plots are from the FlyEx database [21, 22], shown with anterior to the left, dorsal up (see [31] for details on data quantification). Figure reproduced and modified with permission from [6]
Gap-gap interactions introduce feedbacks which result in highly dynamic gap gene expression patterns, especially throughout cycle 14A (Figure 0.6) [6, 32, 33, 34]. During this time, gap domain borders become sharper: in the anterior region
of the embryo thegtdomain splits in two and the posterior border of the anterior
hbdomain refines into a stripe. In the posterior trunk region, gap gene expression boundaries shift anteriorly [35, 36, 33, 34, 37, 20]: covering more than 15% of the embryo length in the case of the posterior border of the posteriorgtdomain [20] (Figure 0.6, shaded regions on bottom row plots). These shifts are not dependent on nuclear movements [6, 20, 38] or any physical mechanisms such as diffusion [36]. Instead, they are driven exclusively by gap gene regulation [36].
Various spatio-temporal data sets of gap gene expression are now available. Quantitative mRNA expression data for Kr, kni, gt and hb are available at all
developmental stages [39, 40, 41, 42]. The FlyEx database (http://urchin.
spbcas.ru/FlyEx) holds a collection of high quality spatio-temporal gap pro-tein expression patterns [21, 31, 22]. Extended gap gene expression data for
D.melanogaster and some non-drosophilid species has also been recently made
available in the SuperFly database (http://superfly.crg.eu) [43, 44].
The relationship between gene regulatory networks and the spatio-temporal patterns they produce is often non-intuitive. Cross-regulation and feedbacks within these networks drive pattern formation in ways that are highly non-linear. This makes it far from trivial to infer the structure of the underlying gene regulatory networks; a task that becomes even more challenging as the number of genes involved increases. Mathematical models are necessary to help us elucidate the principles underlying developmental pattern formation. Models of gene regula-tory networks can be formulated as dynamical systems [45, 46, 47, 48, 49] in order to account for the dynamics of pattern formation.
A complete parts list together with the high quality spatio-temporal data sets available, have made modeling the gap gene network possible. The inference of the regulatory interactions in a gene network from gene expression data is known as reverse-engineering [50, 51, 52, 53, 54, 55, 56]. One particular approach to reverse-engineering – the gene circuit method – has been extensively used to ex-tract information about the dynamical regulatory interactions between the tran-scription factors encoded by the gap genes from the spatio-temporal expression patterns [57, 58, 59]. A model of the gap gene network is formulated using or-dinary differential equations and is fitted to the expression data, from which the
regulatory parameters are obtained. The interactions between the genes predicted by the model are either validated by comparison to the genetic literature [35, 36] or experiments using RNA interference (RNAi) in non-model species [60, 61] and the model is analysed mathematically.
Up until this point, models of the gap genes have been very successful at broadening our understanding of this network’s regulation. The anteriorly shift-ing posterior gap gene domains (Figure 0.6, shaded regions on bottom row plots) were first discovered by analysing quantitative gap gene expression data using gene circuits of the gap genes [36].
We now know that gap gene circuits can also be fitted using data sets based on mRNA combined with a simple semi-quantitative processing protocol [40]. Such data sets are faster to compile and can also be obtained for non-model organisms. This has made comparative studies addressing the evolution of gene regulatory networks possible. The first study of this kind uses gap gene data from the non-drosophilid scuttle flyMegaselia abdita[62] and has revealed that subtle changes in the interaction strengths between the gap genes are responsible for the different
placement of shifting gap gene domains between M.abdita and D.melanogaster
Figure 0.7:The gap gene regulatory network inDrosophila melanogasterandMegaselia abdita. (A)D.melanogaster
andM.abditaphotographed by Karl R. Wotton.B)Early expression of the maternal co-ordinate genes. InM.abdita, thecad
gene is zygotic (and not maternal as inD.melanogaster) and as a result, its expression is delayed.(C)Schematic of gap gene expression. The anterior to posterior order of the gap gene domains is the same in both species. InM.abditathe posterior shifting domains extend further anterior than inD.melanogaster, to include the Hb-Kr interface.(D)D.melanogasterand
M.abditaare both cyclorraphan flies separated by approximately 180 million years of evolution. This figure has been reproduced and adapted with permission from [60]
Gap gene circuits have been analysed using tools and concepts from the theory of dynamical systems. In 2009, work by Manu and colleges [34] used phase space analysis to characterise the different dynamics of gap gene expression along the
A–P axis of theD. melanogasterembryo. Using this approach they were able to
show that different dynamical mechanisms active in the anterior versus the poste-rior of the embryo, underlie static and shifting boundaries respectively.
Gene circuits have also been used to address questions on canalisation. Canal-isation was first introduced by Waddington [63, 64, 65] to describe the resistance of developmental processes to perturbations during embryogenesis, and by exten-sion to explain how discrete and distinct phenotypes are obtained from variable
inputs. Data show that gap gene patterns are consistent and reproducible despite the high variability of the maternal inputs [20, 33], from where it follows that gap gene expression is canalised. The mathematical analysis of gap gene circuits made it possible to prove that canalisation in this network is a direct result of gap gene cross-regulation [33, 66], which corroborated the idea that maternal inputs are not sufficient for the correct placement of gap gene boundaries [32], nor for determining their dynamics [36].
This challenged the long-held belief that morphogen concentration is equiv-alent to positional information. Lewis Wolpert defined positional information as “a mechanism whereby the cells in a developing system may have their position specified with respect to one or more points in the system” [67]. It was assumed that positional information could be encoded by the concentration of morphogen along a gradient [67, 68]. This would imply that the shape and levels of a mor-phogen along a developing tissue are entirely predictive of final pattern [1].
In 1967, Brian Goodwin first presented his phase-shift model for developmen-tal control [69] as an example of how positional information can be transmitted via clocks (or pacemakers as he also referred to them), based on the assumption that cells are already intrinsically temporally organised. In particular, Goodwin proposed [69] that phase differences in oscillators can encode positional infor-mation; a concept that is still at the core of many oscillator-driven mechanisms [70, 71, 72]. However, the discovery of the Bicoid morphogen gradient [12] con-tributed towards extending the view that positional information was transmitted via morphogen concentration read-outs, which prevailed whilst Goodwin’s pro-posal was largely neglected.
It wasn’t until 1997 with the discovery of oscillating gene expression in ver-tebrate somitogenesis [73], that the idea of temporally encoded positional infor-mation first introduced by Goodwin [69] was revived. Oscillatory gene expres-sion placed the focus on how gene networks drive dynamic expresexpres-sion, and by extension, in their potential role encoding positional information. Gene circuits help examine these ideas in the context of gap gene patterning and contribute to broaden our understanding of positional information and the role of morphogen gradients in development.
So far, studies of gap gene expression dynamics have remained limited in one important way: they did not explicitly account for any time-dependence in the system’s inputs. Maternal gradients decay over the same time scales as the gap gene domains are being set up (Figure 0.5). However, in previous gene circuits, maternal gradients were formulated assuming that their profiles were constant and their time-dependence was systematically ignored.
In gap gene circuits, maternal gradient concentrations are introduced as bound-ary conditions to the system. When they are taken as contant over time, their contribution towards gap gene regulation will also be constant. Such circuits can therefore be formulated as autonomous dynamical systems where all the parame-ters are constant over time. This simplification is justified since it facilitates model fitting and analysis greatly.
However, we cannot know whether we fully understand the dynamics of pat-tern formation if the time-dependence of expat-ternal inputs is not accounted for. It seems likely that they could actively shape gap gene expression as well as their dynamics. More generally, it is no longer justified to assumead hocthat the tem-poral temtem-poral dimension of gradients does not shape developmental processes.
In fact, there is some indication that this could be the case from gap gene expression data in D.melanogaster and M.abdita. In M.abdita, the shifting
do-mains extend more anteriorly than in D.melanogaster (Figure 0.7C). We know
that timing of expression of the external inputs differs in both fly species: Cad is maternally expressed inD.melanogasterbut zygotically, and therefore delayed, in
M.abdita (Figure 0.7B). A comparison of the mechanisms underlying gap gene pattern formation between these two dipteran species must therefore take the dy-namics of external inputs explicitly into account, since it is likely they shape the dynamics of gap gene expression in either case [60, 62]. Therefore we need to take into account explicitly the time-dependence of the maternal inputs in order to understand the evolution of the gap gene regulatory network.
For this reason, throughout this study we use a gap gene circuit that incor-porates realistic time-variable maternal gradients of Bcd and Cad (Figure 0.5).
It has been formulated in terms of a non-autonomous dynamical system to ac-commodate parameters that are time-dependent. This has allowed us to examine how these models represent non-autonomous mechanisms of pattern formation. In particular they have allowed us to address explicitly how transient dynamics -dynamics far from steady state - come about, and their role in gap gene pattern formation. I suspect that the gap gene network is not an exception, and that the framework that we have developed will be useful to understand the patterning dy-namics and evolutionary potential of many other regulatory networks.
This thesis is divided into three main parts. In Part II, I present a pragmatic first step towards building a methodology for dealing with transient behaviours in non-autonomous systems. Here I propose a classification scheme for different kinds of such dynamics based on the simulation of a simple genetic toggle-switch model with time-variable parameters. In Part III, I analyse a non-autonomous ver-sion of a gene circuit model for the gap gene network inD.melanogaster. In this model, the time-dependence of the maternal inputs has been taken explicitly into account. I use the classification scheme presented in Part II to characterise the various dynamical mechanisms of gap gene pattern formation along the embryo’s A–P axis. In the last part of my thesis, Part IV, I identify a minimal sub-network that helps explain the different dynamics of gap gene expression and explore its role shaping the evolvability of the gap gene regulatory network.
Part II
Classification of Transient
Behaviours in a Time-Dependent
Toggle Switch Model
Chapter 1
INTRODUCTION
In the study of development we are interested not only in the final state to which the system arrives, but also in the course by which it gets there.
Conrad Hal Waddington “The Strategy of the Genes” 1957 Development in wild-type organisms is robust to genetic and environmental variations. Conrad Hal Waddington introduced the notion of ‘canalisation’ to de-scribe how developmental processes resist perturbations during embryogenesis [63, 64, 65]. The canalised nature of development explains, he argued, why most phenotypes are discrete and distinct. To illustrate these ideas, he developed the epigenetic landscape, one of his most well-known concepts [65, 74].
In Waddington’s epigenetic landscape, the current state of a developing system is indicated by a ball on an undulated surface (Figure 1.1A, top panel) [65, 75, 76]. The topography of this landscape determines the developmental potential or reper-toire of the system. The top-most edge of the surface shown in Figure 1.1A (top panel) represents the initial state of the system given by, for example, a particu-lar set of initial protein concentrations in a cell. Valleys in the landscape sym-bolise the various differentiation pathways that are available. The landscape’s topography—together with the initial state—determine a developmental trajec-tory that follows a particular valley. The structure of the landscape is such that, if the system is slightly perturbed, the sloping valley walls will cause it to correct and re-adjust its trajectory. This behavour is called ‘homeorhesis’—the maintenance of a dynamic trajectory—in analogy to the more static concept of homeostasis—
the maintenance of a (steady) state of the system [65]. The wider and deeper a valley is, the more canalised the developmental trajectory. Waddington named such canalised trajectories ‘chreodes’.
An additional feature of Waddington’s landscape is crucial in our context: it is the fact that its surface is not necessarily fixed, but can change over time due to the influence of genetic or environmental signals. Waddington represented this by pegs connected to the underside of the landscape by ropes (Figure 1.1) [65]. As the genetic or environmental context of the system changes, these ropes pull and stretch the surface, thereby changing its topography and hence its developmental potential.
Figure 1.1:Waddington’s epigenetic landscape and potential surfaces.Two different views of Waddington’s epigenetic landscape taken from “The Strategy of the Genes” published in 1957 [65]. (A)The left panel shows a top view of the landscape. The path that the ball will follow represents the developmental trajectory (chreode) of a given system. Valleys indicate alternative differentiation pathways, branch points imply developmental decisions.(B)This panel shows the view from below the landscape. It illustrates how genes remodel the surface by pulling on it through ropes. Waddington used this sketch to show how the landscape’s topography changes during development and evolution.
Waddington’s epigenetic landscape was conceived as a conceptual tool to il-lustrate the nature of developmental robustness and its effect on evolutionary dy-namics [63, 65, 74, 75, 77]. As such, it remained at a rather metaphorical level of explanation, since complex non-linear biological processes are hard to formu-late and analyse [4, 78, 79, 80, 75, 81, 76]. This is still the predominant way in which it is used in various reviews on contemporary stem cell research (see for example, [82, 83, 84, 85, 86]). However, in order to understand the precise nature of specific developmental trajectories or chreodes in real systems, we have to take Waddington’s landscape a step further: we have to calculate it based on
experi-mental evidence, and use it to characterise the transient behaviours that govern the observed developmental dynamics [87, 88, 89, 90, 91, 92, 93, 94].
The increasing availability of quantitative gene expression data renders this approach feasible. However, before we can successfully apply it, we also re-quire new conceptual and mathematical tools to deal with the analysis of data-driven models that are formulated in terms of non-autonomous (i. e. explicitly time-dependent) dynamical systems. Explicit time-dependence is necessary to reproduce the changing topography of Waddington’s landscape. However, such systems are difficult to study in a rigorous mathematical manner, and few analy-sis tools exist at this point. In this part of the theanaly-sis, I address this challenge by proposing a classification scheme for transient dynamic behaviours observed in a non-autonomous version of a simple gene network model. This scheme is meant to provide the foundation for the analysis of more complex time-dependent mod-els that reproduce the dynamics of specific, experimentally tractable, biological regulatory systems.
1.1
Dynamical systems
I focus on dynamical systems formulated in terms of ordinary differential equations, and illustrate how such models can help explain the function and po-tential of developmental gene regulatory networks in terms of their dynamical repertoires, that is, in terms of the set of behaviours that can be implemented by the system. A system’s behaviour is defined by its trajectories, which represent the change of the state of the system—e. g. consisting of a set of transcription factor concentrations—over time. The shape of these trajectories depends on the structure or organisation of the underlying regulatory network (see, for example, [65, 95, 96, 97, 98, 99, 93, 100, 91, 92]). It is possible to gain a general qualitative understanding of what the system’s trajectories can and cannot achieve without the need to solve the dynamical system analytically [101, 102]. A great deal of this information comes from the geometrical analysis of phase space, i. e. an analysis of the number, nature and relative arrangement of the steady states of the system. The phase space of a dynamical system is an abstract space, in which each
dimension represents the value of a specific state variable. Here we use a well established double-repressive feedback loop model, known as the toggle switch ([92], and references therein), to study transient dynamics in a two-dimensional, time-dependent gene regulatory network (see Figure 1.2, as well as Chapter 2 for the full formulation). In this case, the state variables represent the concentrations of the transcription factors that constitute the network (denoted byX andY; Fig-ure 1.2A). The graphical representation of phase space is called the phase portrait (Figure 1.2C). It shows the rate of change of the system at any given state. This is known as the flow of the system. The flow of the toggle switch model is indicated by arrows of a given length and direction in the phase portrait shown in Figure 1.2C. If we follow the flow from all possible initial states, we obtain the totality of possible dynamic trajectories.
It is evident from the inspection of the flow in Figure 1.2C that trajectories tend to converge to specific points in phase space: the steady states of the sys-tem. There are different kinds of steady states; in general terms, those that are stable, and those that are unstable. The most simple stable steady state is an attractor point [101, 102]. Attractors, as their name implies, draw trajectories to-wards them. Furthermore, they have the special property that once a trajectory has reached an attractor, it will return to it if the system is slightly perturbed. An example of an unstable steady state is a saddle point (Figure 1.2C). Saddle points attract trajectories from some directions, but repel them in others. Usually, the system will move away from a saddle upon perturbation, towards the nearest attractor. The repelling trajectory follows a structure in phase space called an un-stable manifold. This manifold is defined by the the path that links a saddle with an attractor point. Note that unstable manifolds correspond to chreodes at the bot-tom of valleys in Waddington’s landscape.
In our phase portraits, we plot steady states as points coloured according to their stability: attractors in blue, and saddle points in red (Figure 1.2C). The re-gion of phase space around an attractor, from which all trajectories converge to-wards it is called its basin of attraction. Curves known as separatrices set apart the different basins and their attractors (in the case of Figure 1.2C, the separatrix is a straight line indicated in grey). Saddle points are always located on separatrices.
Figure 1.2:The toggle switch model. (A)Diagrammatic representation of the toggle switch network used in the simu-lations. Activating interactions are indicated by arrows, repressing ones by T-bar connectors. See Chapter 2 for detailed parameter descriptions.(B)Mathematical formulation of the toggle switch model.xandyindicate concentrations of the protein products of genesXandY. Ordinary differential equations define the rate of change in protein concentrations (dxdt anddydt). Sigmoid functions with fixed Hill coefficients of4are used to represent auto-activation and mutual repression. Decay and external activation are taken to be linear. Parameters as in (A).(C)Phase portrait for a constant set of parameter values of the toggle switch model in the bistable regime.X- andY-axes represent protein concentrations ofxandy. We use this example to illustrate relevant features of phase space: arrows indicate flow, blue points mark the position of stable steady states (attractors), the red point shows an unstable steady state (saddle) lying on the separatrix that divides the two basins of attraction (grey line). See the main text for detailed descriptions of the highlighted features.(D)Quasi-potential landscape associated to the the phase portrait shown in (C). The steepness of the quasi-potential surface correlates with the flow at each corresponding point on the phase portrait. Attractors, saddle, and separatrix are indicated as in (C). See main text for details.
Attractors and saddles, with their associated basins and separatrices can be created or annihilated through the process of bifurcation [101, 103]. Bifurca-tions represent sudden qualitative changes in the structure of the phase portrait caused by small changes in the values of a given set of control parameters. To date, the best example illustrating the importance and usefulness of geometric ap-proaches for understanding the dynamics and function of specific, experimentally tractable, developmental regulatory systems comes from an analysis of the gap gene regulatory network involved in pattern formation during early embryogen-esis of the vinegar fly Drosophila melanogaster (referred to as D.melanogaster
hereafter). Manu and colleagues [104] identified features of phase space respon-sible for patterning and canalisation of spatio-temporal gene expression in the
D.melanogaster blastoderm embryo. The analysis is based on low-dimensional projections of phase space to study the geometric arrangement of attractors in the four-dimensional system representing the change in protein concentration for four transcription factors encoded by the gap geneshunchback (hb),Kr¨uppel (Kr),
knirps (kni), and giant (gt). Gene expression domain boundaries that remain at a constant position over time could be attributed to movements of attractors in phase space, or nuclei switching between attractor basins depending on their po-sition in the embryo. In contrast, more posterior expression boundaries, which keep shifting position over time, were associated with transient behaviours along a canalising unstable manifold, forming the equivalent of a valley in Waddington’s landscape [104].
1.2
Potential landscapes
Phase portraits of systems with two state variables can be visualised in terms of their associated potential landscape [101] (Figure 1.2D). In this representation, the steepness of the potential landscape corresponds to the flow of the system (compare Figure 1.2C and D). Trajectories travel downhill towards the attractor points. The topography of the potential landscape is therefore a direct result of the topology (number and nature of steady states) and the geometry (relative positions of the steady states and size of the flow) of the underlying phase portrait.
Potential surfaces can be thought of as mathematical representations of Wadding-ton’s epigenetic landscape [105, 106, 87, 89, 107, 108, 93, 94, 109]. Attractors represent differentiated states (in agreement with earlier postulates, [110, 63, 111, 112, 113, 114, 115, 82, 83]), separatrices form the ridges between the valleys, which are formed by unstable manifolds representing their associated chreodes (see Figure 1.2C and D), and the flow is represented by the steepness of the slopes on the landscape. Canalisation is explained by the depth and width of each valley in the potential landscape. This provides a way of probing and un-derstanding the features that confer robustness to the system (see, for example, [114, 116, 117, 87, 88, 90, 93]). Branching valleys can arise through particular types of bifurcation events. In particular, new attractor states can branch off from existing ones during development or stem cell differentiation through
(supracriti-cal) pitchfork bifurcations[101, 118, 116, 119, 88, 89, 120, 94].
Due to this analogy to Waddington’s epigenetic landscape, potential land-scapes are becoming increasingly popular as explanatory tools in fields such as evo-devo, developmental biology, and especially in stem cell research. In the case of stem cells, the positioning of the valleys in the landscape relative to each other explains which differentiation pathways can be reached by a given stem cell, and which cell fates can (or cannot) be trans-differentiated into each other [105, 106, 121, 122, 82, 83, 84, 85, 123, 109]. This illustrates how specific biolog-ical meaning can be gained from studying the topography of potential landscapes, which ultimately, is nothing more than a visually accessible way of studying the underlying phase portrait.
Waddington meant his epigenetic landscape as a “diagrammatic representa-tion” of development and warned explicitly against interpretations that were too rigorous or literal [64, 65]. Detailed topographical interpretations of Wadding-ton’s landscape may be quite inaccurate and even misleading. Ferrell [94] points out that Waddington’s landscape, where valleys progressively split into an increas-ing number of branches, does not help explain many realistic cell differentiation processes. In particular, many inductive processes in development (e. g. vulval
induction in the roundworm Caenorhabditis elegans, or mesoderm induction in
vertebrates) involve saddle-node rather than pitchfork bifurcations, which corre-spond to the disappearance of valleys rather than to their creation by branching. The system shifts to a new attractor only once the old one has vanished. This is why it is important to move from metaphorical uses of Waddington’s epigenetic landscape to accurately calculated potential surfaces whenever possible, or more importantly, to the detailed analysis of the underlying phase portraits, which is where the dynamics of specific regulatory networks are determined.
Potential landscapes can only be calculated and visualised explicitly for a re-stricted range of dynamical systems, belonging to the class of gradient systems [101]. Note that the notion of a gradient system is defined mathematically by the absence of limit cycles or any other complex attractor structures in their phase por-traits, and has nothing to do with biochemical or other biological gradients (see Chapter 2 for a detailed explanation). In cases where we do not know whether
the system under study is a gradient system or not, we can approximate the actual potential using various numerical methods [117, 124, 88, 89, 107, 108]. The re-sulting approximations are called quasi-potential landscapes.
However, even such quasi-potential landscapes can only be visualised directly when the number of state variables of the system does not exceed two. This is not true for most biologically realistic systems. Nevertheless, (quasi-)potential land-scapes are still useful as conceptual tools for the analysis of higher-dimensional regulatory networks. In some cases, it is possible to reduce a high-dimensional system to a lower-dimensional one (see, for example, [125, 126, 127]). But even if this is not the case, the concept of the potential landscape provides two ad-vantages. First, as discussed above, quasi-potential surfaces link Waddington’s intuitively accessible concept of the epigenetic landscape to the biological in-terpretation of (high-dimensional) phase space analysis. And second, potential surfaces are useful as visual guides to diagnose features of the underlying phase portrait of the system that are characteristic of specific dynamic behaviours of the regulatory network under study.
1.3
The importance of transient dynamics
Many models of biological systems are formulated with the assumption that the relevant dynamics occur near or at a steady state. For instance, Thom’s pi-oneering systematic and rigorous analysis of morphogenesis in terms of catas-trophe theory explicitly and strongly relies on this assumption [4]. Similarly, work on robustness and evolvability, using ensembles of simulated networks, stan-dardly assumes that the steady state pattern produced by a model can be taken as a satisfactory and realistic representation of the phenotype of the system (e. g. [128, 129, 130, 131]). Furthermore, stem cell models that make use of potential landscapes are analysed with a strong focus on how their attractors govern system behaviour [116, 117, 88, 89, 125, 127].
In some cases, the steady state assumption makes obvious sense based on bi-ological reasoning. One example is the analysis of the segment polarity network inD.melanogaster, which amplifies and maintains a periodic input, and thus
per-forms an intrinsically stabilising patterning function [132, 133, 134]. In most cases, however, biological pattern formation is highly dynamic and far from equi-librium, and the steady state assumption is justified based on methodological, rather than biological considerations. Focussing on asymptotic behaviour at or near steady state greatly simplifies the analysis of the system. First, it discretises and reduces phase (and hence phenotype) space into a small number of possible states—represented by the system’s attractors (see Figure 1.2C). Second, it en-ables the powerful toolkit of linear stability analysis to be employed to examine the characteristic properties of system states [101, 102].
However, there are both theoretical and practical reasons indicating that steady state analysis misses essential and biologically relevant systems behaviours. One line of theoretical reasoning is provided by Waddington himself, who reminds us that “[i]n the study of development we are interested not only in the final state to which the system arrives, but also in the course by which it gets there.” [65]. His concept of homeorhesis and his representation of developmental trajectories or chreodes as descending valleys in the epigenetic landscape places the focus ex-plicitly on the transient dynamics of cells on their way to their final, differentiated state. In the same spirit, other authors have suggested that phenotypes should be defined over developmental trajectories, rather than representing some sort of ‘fi-nal’ outcome or ‘end state’ of the system [135, 91]. To decide what a final pattern is, and when exactly it occurs, is always arbitrary to some degree, while tran-sient features (such as intermediate stages and the timing of their transitions) are clearly important when considering the function and dynamics of developmental processes.
There are also practical reasons to consider transient dynamics explicitly. For developmental processes that consist of continuous transitions between patterns rather than the production of a final output, it is impossible to decide a priori whether the system is representing a non-autonomous succession of steady states (see Section 1.4 below), or whether its behaviour is truly transient (i. e. far from steady state). In the case of gap domain shifts, we have evidence for the latter [104], although there is no reason to assume that the two situations need to be mutually exclusive. The gap gene model analysed by Manu and colleagues [104] exhibits boundary shifts that are caused by trajectories following a canalising
un-stable manifold. Assuming steady state dynamics would collapse the trajectories representing these shifts into a single attractor point at the final configuration of gene expression. Any such analysis would miss the relevant underlying features of phase space (the transient manifold), and therefore fail to provide a proper char-acterisation and explanation for the observed gene expression dynamics (the shift in domain position over time).
Other examples of developmental processes, where transient dynamics are clearly important are dorso-ventral patterning of the vertebrate neural tube, which involves boundary shifts strikingly similar to those of the gap domains [136, 137], and vertebrate somitogenesis or short-germband segmentation in arthropods where transient travelling waves of gene expression are an essential component of the underlying clock-and-wavefront patterning mechanism [70, 138, 139, 140, 141, 142, 72]. Incidentally, similar considerations can be made for models deal-ing with ecological networks, and several examples exist in the literature that con-sider transient dynamics explicitly (see models of coupled oscillating population dynamics between species [143, 144], or [145, 146, 147]).
1.4
Non-autonomy: explicit time dependence
Another important aspect of biological regulatory processes, which receives surprisingly little attention, is the explicit time dependence of these systems. As soon as we consider cellular dynamics, development, or evolution over a large-enough time span, the organisation of the underlying regulatory system starts to change. This affects the parameters—not just the state variables—of the system. Such explicitly time-dependent dynamical systems are called non-autonomous [101, 148, 149]. Time-dependent signalling cues and environmental conditions have long been known to shape many processes in the fields of evolutionary and developmental biology. Obvious examples of such phenomena are induc-tive processes or external (e. g. seasonal) cues that are essential to trigger many developmental pathways (as described in standard textbooks such as [150, 151]), or evolutionary dynamics driven by changing environmental conditions (exam-ples, based on the simulation of gene regulatory network models, can be found in [152, 153, 154, 155]).
Still, it is rare to find studies based on explicitly non-autonomous models in the literature, and most authors avoid the challenge of dealing with dynamical systems where the parameters representing external cues are time-dependent. This is the case in the study of gap domain shifts by Manu and colleagues [156, 104], where maternal morphogen gradients providing regulatory input to the system were as-sumed to reach steady state before gap gene boundary positioning was analysed. Such simplifications can be risky, especially when describing biological phenom-ena where the time scales of change in parameters and state variables are of similar order. In such cases, time scales should not be separated, nor quasi-steady states considered since it is easy for dynamical properties and behaviours of the system to be missed or misinterpreted under these conditions.
Recently there have been some attempts at including non-autonomy in bio-logical models. Corson and Siggia [157], for example, offer an explanation for
vulval development in C.eleganswhich explicitly considers temporal parameter
changes due to inductive signalling cues in their model. Their model considers differentiation of cells into three differentiated states depending on inputs from two signalling pathways. Signalling inputs are encoded in the model by altering values of system parameters, which change and distort the geometry of the phase portrait by displacing separatrices from their original position (see Figures three– nine in [157]). When a cell receives a signal, its developmental trajectory comes to lie within another basin of attraction, inducing an alternative cell fate to that which would have been reached in the absence of the signal. This study illustrates the importance of non-autonomous dynamics in development. However, it remains somewhat limited in its implementation of explicit time-dependence. Although parameter change is included in the model, signalling occurs before cells em-bark on their developmental paths, and trajectories develop purely autonomously thereafter. Therefore, this approach does not fully capture the transient dynam-ics of cell differentiation. In other words, although their model is able to offer an explanation for cell differentiation in vulval development, it cannot capture the full non-autonomous nature of the developmental process, since it does not reproduce developmental trajectories, chreodes, in an accurate and fully time-dependent manner.
Similarly, most of the few other examples of non-autonomous models in the biological literature do not explicitly consider the effect of parameter changes on transient behaviour (e. g. [158, 159, 160]). This simplification may be justified in many cases and is necessary for any kind of rigorous analytical treatment of a model. In many situations, however, it fails to capture essential features of the system. For instance, a truly accurate analysis of gap gene regulatory dynamics requires the inclusion of both non-autonomy from the rapidly changing maternal morphogen gradients, and transient dynamics, which are known to underlie the temporal shifts in domain position. Before we can undertake such an analysis we must first build a conceptual toolkit for phase space analysis of transient, non-autonomous dynamics. Due to the limited amount of analysis possible in such systems, this toolkit will need to be developed in a pragmatic and empirical man-ner, using numerical simulation and exploration of a simple conceptual model as its basis.
In the following chapters, I present such a simulation-based attempt at devel-oping concepts to classify transient, non-autonomous behaviours. For this pur-pose, we use a simple two-component model of a genetic toggle switch, whose potential landscape can be explicitly visualised. We use time series of graphs and animations of systems dynamics on this potential to identify mechanisms leading to state transitions, and other forms of pattern formation. From this, we are able to identify four basic types of dynamical mechanism and behaviour— transitions, pursuits and two types of captures of trajectories—that can be used to classify and understand dynamical behaviour in more complicated and realistic models, such as the full non-autonomous version of the gap gene model presented in Part III.
Chapter 2
MODEL AND METHODS
To develop our methodology for analysing transient behaviours in non-autonomous dynamical systems, we use a simple toggle switch model (see [91], and references
therein) with time-dependent parameters. We consider two interacting genes X
and Y (Figure 1.2A). Concentrations of the corresponding protein products are
labelledxandy. X andY mutually repress each other, are linearly activated by external signals and can auto-activate themselves (Figure 1.2A). Protein products decay linearly dependent on their concentration. The mathematical formulation of our toggle switch model is thus given by
dx dt = " αx+ x4 a4+x4 # " b4 b4+y4 # −λxx dy dt = " αy+ y4 c4+y4 # " d4 d4+x4 # −λyy (2.1)
where parametersαx andαy represent the external activation on genesX andY
respectively. Sigmoid functions with Hill coefficients of 4are used to represent auto-activation and mutual repression, where parametersaandcdetermine auto-activation thresholds, whileb andd determine thresholds for mutual repression. Protein decay rates are represented by parametersλx andλy.
The toggle switch model (2.1) exhibits different dynamical regimes depending on the values of its parameters (Figure 2.1A–C). Its name derives from the fact that
it can exhibit bi-stability over a wide range of parameters. When in this bistable region of parameter space, the underlying phase portrait has two attracting states and one saddle point (Figure 2.1B). All phase portraits associated with parameters in the bistable range are topologically equivalent to each other, meaning that they can be mapped onto each other by a continuous deformation of phase space called a homeomorphism [103].
Figure 2.1:Dynamical regimes of the toggle switch model.The toggle switch model can exhibit three different dynamical regimes depending on parameter values. (A)In the monostable regime, the phase portrait has one attractor point only (represented by the blue dot on the quasi-potential landscape). At this attractor, both products ofXandY are present at low concentrations.(B)In the bistable regime, which gives the toggle switch its name, there are two attractor points (shown in different shades of blue) and one saddle (red) on a separatrix (black line), which separates the two basins of attraction. The attractors correspond to highx, lowy(dark blue), or lowx, highy(light blue). The two factors never coexist when equilibrium is reached in this regime. (C)In the tri-stable regime, both bistable switch attractors and the steady state at low co-existing concentrations are present (shown in different shades of blue). In addition, there are two separatrices with associated saddle points (red). These regimes convert into each other as follows (double-headed black arrows indicate reversibility of bifurcations): the monostable attractor is converted into two bistable attractors and a saddle point through a supercritical pitchfork bifurcation; the saddle in the bistable regime is converted into an attractor and two additional saddles in the tri-stable regime through a subcritical pitchfork bifurcation; the bistable attractors and their saddles collide and annihilate in two simultaneous saddle-node (or fold) bifurcations to turn the tri-stable regime into a monostable one. Graph axes as in Figure 1.2D
The toggle switch model has two other dynamical regimes: monostable and tri-stable. Phase portraits associated with parameters in the monostable range have only one attractor point (Figure 2.1A), while those in the tri-stable range
have three attractor states and two saddle points (Figure 2.1C). Again, phase por-traits within each regime are topologically equivalent to each other. While phase space can be geometrically deformed within each regime (through movements of attractors or separatrices), its topology only changes when one regime transitions into another through different types of bifurcations [101, 103] (Figure 2.1). The transition from monostable to bistable is known to be governed by a supracritical pitchfork, the transition from bistable to tri-stable involves a subcritical pitchfork bifurcation, and the transition from tri-stable to monostable takes place through two simultaneous saddle-node bifurcations involving the two attractors labelled in darker blue in Figure 2.1C.
2.1
Definition of the potential landscape
Potential landscapes can only be calculated explicitly for the class of dynam-ical systems called gradient systems [101]. A two-variable gradient system is a dynamical system
dx
dt = f(x, y) dy
dt = g(x, y) (2.2)
which satisfies the following relationship between partial derivatives
fy(x, y) = gx(x, y). (2.3)
For gradient systems, it is possible to calculate a closed-form (explicit) poten-tial function,V(x, y)such that
Vx = −dx dt Vy = −dy dt . (2.4)
The local minima on the two-dimensional potential surface given by V(x, y)
correspond mathematically to the steady states of the system in (2.2) since, if (x∗, y∗) is such that Vx(x∗, y∗) = Vy(x∗, y∗) = 0, (2.5) then dx dt(x ∗ , y∗) = 0 dy dt(x ∗ , y∗) = 0, (2.6)
and, therefore, (x∗,y∗) is a steady state of (2.2).
2.2
Calculating quasi-potential landscapes
Condition (2.3) will not always be met. In particular, dynamical systems rep-resenting gene interaction networks are not in general gradient systems, and there-fore an associated potential function and landscape may not exist. In such cases, we can still take advantage of the visualisation power of potential landscapes by approximating the true potential using a numerical method. The numerical ap-proximation method we adopt for this part of the study was developed by Bhat-tacharya and colleagues [107] using a toggle switch model very similar to the one used here. This allows us to calculate a quasi-potential landscape for any specific set of fixed parameter values.
