## Abstract

For us to understand complex regulatory networks and their evolution, they must be modular, at least to some degree. Only modular networks can be partitioned into tractable subcircuits, able to evolve without causing detrimental pleiotropic effects. Traditionally, functional modularity is approximated by detection of modularity in the regulatory structure of a network. However, the correlation between structure and function is loose, and many networks show modular behaviour without evident structural modularity. Here, we partition an experimentally tractable regulatory network—the gap gene system of dipteran insects—using an alternative approach, which transcends these limitations. We show that the gap gene system is composed of *dynamical modules*, each driving a different aspect of the behaviour of the whole network. All these subcircuits share the same regulatory structure, but differ in components and their sensitivity to changes in regulatory interactions, which explains the differential evolvability of expression features of the system.

**This article has not yet been submitted to a journal. Constructive criticism is very welcome!**

## Introduction

Systems biology aims to understand the function and dynamics of complex regulatory networks. This requires some sort of hierarchical decomposition into tractable subsystems, whose properties and behaviour can be analysed and understood in relative isolation (** Simon (1962); Riedl (1975); Lewontin (1978); Bonner (1988); Raff (1996); West-Eberhard (2003); Schlosser and Wagner (2004); Callebaut et al. (2005)**). If each subsystem possesses a clearly delimited and discernible function, the network can be subdivided into

*functional modules*(

**).**

*Raff (1996); von Dassow et al. (2000); Hartwell et al. (1999); Wagner et al. (2007)*The most common strategy to identify functional modules is to partition the graph representing a network into simple motifs (** Shen-Orr et al. (2002); Alon (2007)**) or subcircuits (also called subnet-works or communities;

**). Network motifs are small subgraphs that are identfied through their statistical enrichment (**

*Girvan and Newman (2002); Oliveri and Davidson (2004); Babu et al. (2004); Levine and Davidson (2005); Newman (2006); Davidson and Erwin (2006); Oliveri and Davidson (2007); Erwin and Davidson (2009); Davidson (2010)***, 2006)), while subcircuits are characterised by a high connection density among their component nodes contrasting with sparse connections to the outside (**

*Alon (2007***). In both cases, subsystems are defined in terms of the regulatory structure or graph of the network: they are**

*Girvan and Newman (2002); Radicchi et al. (2004); Newman (2006); Wagner et al. (2007); Fortunato (2010)**structural modules*. This approach presupposes a strong connection between functional and structural modularity (see, for example,

**).**

*Lim et al. (2013)*Strictly interpreted, structural modules are mutually exclusive: they are disjoint subgraphs of a complex regulatory network that do not share nodes among each other (** Girvan and New-man (2002); Radicchi et al. (2004); Palla et al. (2005)**). And yet, such modules can never be fully isolated: their context within the larger network in2uences behaviour and function. The “divide- and-conquer” approach based on structural modularity therefore relies on the assumption that context-dependence is weak and structural modularity is generally pronounced enough to preserve the salient properties of a motif or subcircuit in its native network context.

Structural modularity is widely regarded as a necessary condition for the evolvability of complex networks. “Evolvability,” in the general sense of the term, is defined as the ability to evolve (** Dawkins (1989); Wagner and Altenberg (1996); Hendrikse et al. (2007); Pigliucci (2008)**). More specifically, evolvability refers to the capacity of an evolving system to generate or facilitate adaptive change (

**). Structural modularity can boost this capacity in several ways. Entire modules can be co-opted into new pathways during evolution, generating innovative change (**

*Wagner and Altenberg (1996); Pigliucci (2008)***). Further-more, each module can vary relatively independently, and it has been argued that this accounts for the individuality, origin, and homology of morphological characters as well as their trait-specific variational properties (**

*Raff (1996); von Dassow et al. (2000); True and Carroll (2002); Davidson and Erwin (2006); Erwin and Davidson (2009); Monteiro and Podlaha (2009); Wagner (2011)***). Finally, structural modularity allows for a fine-tuned response to specific selective pressures by minimizing off-target pleiotropic effects (**

*Wagner and Altenberg (1996); Wagner et al. (2007); Wagner (2014)***).**

*Wagner and Altenberg (1996); Pavlicev et al. (2008); Wagner (2011)*The identification and analysis of structural modules has been a great success in many cases. It has been used to understand the regulatory principles of segment determination in *Drosophila* (** von Dassow et al. (2000); Ingolia (2004)**), the origin and evolution of butter2y wings spots (

**and beetle horns (**

*Carroll et al. (1994); Brakefield et al. (1996); Keys et al. (1999); Beldade et al. (2002); Monteiro et al. (2003, 2006))***), and the mechanism and evolution of larval skeleton formation in sea urchins (**

*Moczek (2006)***Davidson (**

*Hinman et al. (2003); Hinman and Davidson (2007); Oliveri et al. (2008); Gao and***). Other examples abound in the literature (see,**

*2008)***, for comprehensive reviews).**

*Raff (1996); Schlosser and Wagner (2004); Callebaut et al. (2005); Peter and Davidson (2015)*In spite of these success stories, structural modularity has a number of serious limitations. Some modelling studies suggest that it is not necessary for evolvability (see, for example, ** Crombach** and Hogeweg (

**). Furthermore, it is notoriously diZcult to identify structural modules and delimit their boundaries with any precision. One reason for this may be that the definition of (sub)system boundaries is fundamentally context- and problem-dependent (see, for example,**

*2008)***et al. (**

*Chu***). More to the point, even the simplest subcircuits tend to exhibit a rich dynamic repertoire comprising a range of different behaviours depending on context (boundary conditions), quantitative strength of parameter values (determining genetic interactions as well as production and decay rates), and the specific form of the regulation-expression functions used to integrate multiple regulatory inputs (**

*2003); Chu (2011)***). Because of this, it is usually not possible to single out subgraphs exhibiting specific behaviours and functions that are robustly independent of their native network context. Maybe then, looking for structural modules is not the best way to subdivide a complex regulatory network.**

*Mangan and Alon (2003); Wall et al. (2005); Ingram et al. (2006); Siegal et al. (2007); Payne and Wagner (2015); Ahnert and Fink (2016)*A recent simulation-based screen of multifunctional gene regulatory networks beautifully illustrates this point (** Jiménez et al. (2017)**). The authors of this study performed a systematic computational search for network structures able to implement two qualitatively different dynamical behaviours given either different interaction strengths or the presence/absence of an external (contextual) signal. They then identified the particular subcircuits that were responsible for either of the two behaviours (functions). What they found is an entire spectrum of structural overlap among functional modules. At one end of the spectrum, “hybrid” networks represent the sum of two completely disjoint structural modules. At the other end of the spectrum, “emergent” networks use exactly the same nodes and connections to implement both dynamical behaviours. Most networks identified by the screen fall somewhere in between these two extremes, that is, they show partial overlap between functional modules. This suggests that most functionally modular networks are not modular in the structural sense.

The limitations of structural modularity can also be illustrated using the real-world example of the gap gene system (see, ** Jaeger (2011)**, for review). This gene regulatory network is involved in pattern formation and segment determination during the blastoderm stage of early embryogenesis in dipteran insects, such as the vinegar 2y

*Drosophila melanogaster*. Its regulatory structure is summarized in Figure 1A. Nodes in this network represent transcription factor encoding genes, which activate or repress each other as indicated by their connections. The gap gene network reads and interprets morphogen gradients formed along the major or antero-posterior (A–P) axis of the embryo by the protein products of the maternal co-ordinate genes

*bicoid (bcd), caudal (cad)*, and

*hunchback (hb)*, resulting in broad, overlapping expression domains for the trunk gap genes

*hb, Krüppel (Kr)*, knirps (kni), and

*giant (gt)*(Figure 1B, C). Extensive gap-gene cross-regulation is essential for the correct dynamic positioning of expression domain boundaries, in particular the dynamic kinematic shifts of posterior gap domains towards the anterior of the embryo (

**,a);**

*Jaeger et al. (2004b***). Maternal co-ordinate and gap genes together then regulate pair-rule and segment-polarity genes, which form a molecular pre-pattern that precedes and determines the formation of morphological segments during later stages of development.**

*Surkova et al. (2008); Manu et al. (2009b); Crombach et al. (2012); Verd et al. (2018)*Due to its high connection density (see Figure 1A), it is impossible to identify structural clusters in the gap gene system. Previous simulation-based analyses identified a number of mechanisms driving gap gene expression, such as the double-negative (positive) feedback loops between *hb/kni* and *Kr/gt* that are responsible for the basic staggered arrangement of gap domains, or the asymmetric repressive interactions between overlapping gap genes (*e. g. kni* on *Kr* and *Kr* on *hb*) that are driving dynamic anterior shifts of gap domains over time (Figure 1A, B) (** Jaeger et al. (2004a**,b);

**). However, these mechanisms are not structurally modular, as they involve different interactions among the same set of genes. Moreover, attempts to explain gap gene expression dynamics with network motifs have failed to correctly account for the integrated behaviour of the whole system (**

*Jaeger (2011); Crombach et al. (2012)***).**

*Zinzen and Papatsenko (2007); Ishihara and Shibata (2008); Papatsenko et al. (2009)*All these diZculties arise because the gap gene network is a strongly emergent network (** Jiménez** et al. (

**). For such networks, analyses based on structural modularity fail to shed insight into the function and the dynamics of the whole intact regulatory system, because the approach is only valid for systems of markedly hybrid character. As we have argued above, these systems only constitute a tiny subset of all possible naturally occurring regulatory networks (**

*2017)***).**

*Jiménez et al. (2017)*Luckily, there is an alternative way to identify network modules that does not rely directly on their structure: we can de1ne them in terms of their activity (** von Dassow et al. (2000); Hartwell et al. (1999)**). Such

*dynamical modules*or dynamical subsystems consist of a group of connected network nodes that implement a particular behaviour (

**). Their (spatio-)temporal pattern of activity specifies a certain type of dynamics—maybe bistable or oscillatory—called a “dynamical regime” (**

*Irons and Monk (2007)***, 2018)). Different dynamical regimes are distinguished by the composition of the underlying phase portraits (**

*Verd et al. (2017***). For instance, they can be generated by different subsets of system attractors (with associated basins) (**

*Strogatz (2014)***). Nodes may be shared between overlapping modules, simultaneously driving different dynamics in the context of different subsystems. No disjoint modular network structure is required or even expected (**

*Irons and Monk (2007)***). Still, each module′s behaviour is relatively independent and clearly distinguishable from that of other dynamical subsystems (**

*Jiménez et al. (2017)***).**

*Irons and Monk (2007)*Note that dynamical modules are not the same as co-expression modules (also called regulatory modules sometimes; see, for example, ** Eisen et al. (1998); Bar-Joseph et al. (2003); Segal et al. (2003); Stuart et al. (2003); Kim et al. (2009); Koch et al. (2017)**). The latter are defined by the correlated or anti-correlated expression of their components. In contrast, the mutual dependence of component expression patterns in dynamical modules is causal, rather than correlative. It can be very complex and obscured by the non-linearity of regulatory interactions (

**). Dynamical modules are not identified by individual expression patterns, but rather by the coherent collective activity of the module as a whole.**

*Irons and Monk (2007)*Just like their structural counterparts, dynamical modules also in2uence evolvability. However, the way they achieve this is fundamentally different. Structural subcircuits can vary independently since they are only loosely interconnected. Dynamical modules, in contrast, drive distinct behaviours that exhibit different levels of sensitivity to changes in system parameters. In particular, some dynamical modules may be in a state of criticality, close to a bifurcation point beyond which their dynamics may change drastically and abruptly; others may be structurally stable, not critical, far from any bifurcation and therefore insensitive or robust to changes in parameters (** Thom (1976); Jaeger et al. (2012); Jaeger and Monk (2014); Jaeger and Sharpe (2014); Strogatz (2014)**). In this way, mutations affecting network components will tend to have a strong effect on certain types of behaviour in a system, leaving others unaltered. This type of dynamic modularity implies that the network is more likely to evolve in certain directions than others. While mutations may be random, their effects on network dynamics are certainly not.

Dynamical modularity transcends the inherent limits of structural approaches and allows us to gain insights into heavily emergent regulatory networks. However, it is far from trivial to apply this theory to the experimental study of specific evolving developmental processes. Irons and Monk (** Irons and Monk (2007)**) have developed an algorithmic method to identify dynamical modules in Boolean network models. Unfortunately, this method is diZcult to generalize and adapt to continuous mathematical frameworks such as differential equation models used to study gradient-driven pattern-forming networks such as the gap gene system.

For this reason, we adopt a pragmatic empirical approach to identify dynamical modules in the gap gene network of *D. melanogaster*. It is based on the observation that only subsets of gap genes are expressed and exert their regulatory in2uence in any one region of the embryo. This defines four localized subsystems, each containing three trunk gap genes that are active in different but overlapping regions of the embryo. Surprisingly, all four subsystems share the same regulatory structure, which identifies them as AC/DC subcircuits (** Panovska-GriZths et al. (2013)**). This structure implements one of the simplest known genetic systems able to produce both switch-like (multistable) and oscillatory behaviour. We show that these AC/DC modules drive distinct dynamical regimes: static domain boundaries in the anterior, versus anteriorly shifting gap domains in the posterior of the embryo (Figure 1C, D) (

**, 2018)). The boundary between these two dynamical regimes is positioned by an AC/DC circuit in a state of criticality. This makes it especially sensitive to changes in the strength of regulatory interactions, and shapes the evolvability of the gap gene system within the Dipteran order (2ies, midges, and mosquitoes).**

*Verd et al. (2017*## Results and discussion

### Modularity

#### Identification of dynamical modules

We take a pragmatic approach to identifying dynamical modules in the gap gene network (Figure 1A, B) (** Jaeger (2011)**). Inspired by

**, we define “dynamical modules” as minimal subcircuits able to recover the main spatio-temporal features of gap gene expression in a given region of the embryo. Dynamical modules may show overlap among each other in terms of component nodes and spatial location where they are active.**

*Irons and Monk (2007)*We used node sensitivity analysis to identify subcircuits relevant for patterning in specific regions of the D. melanogaster blastoderm embryo. Specifically, we focus on cleavage cycle 14A (C14A) during the late blastoderm stage (** Foe and Alberts (1983)**) (see “Materials and methods”). At this stage, zygotic gap gene cross-regulation has become essential for the positioning of domain boundaries, supplanting initial positional cues by gradients of maternal activators (

**). The analysis asks which of the four trunk gap genes (**

*Jaeger et al. (2007); Jaeger (2011)**hb, Kr, kni*, and

*gt*) are required—or, more accurately, which ones are

*not*—to drive correct expression dynamics in nuclei at different positions along the antero-posterior (A–P) axis. It is based on a detailed dynamical model of the gap gene network (the “full model”), which has been fit to quantitative spatio-temporal data of trunk gap gene expression (see Figure 1C) (

**, 2018)). This model implements accurate dynamic mechanisms of gap gene regulation that have been extensively validated against empirical evidence (**

*Verd et al. (2017***,a);**

*Jaeger et al. (2004b***,b);**

*Manu et al. (2009c***, 2018)).**

*Ashyraliyev et al. (2009); Jaeger (2011); Crombach et al. (2012); Verd et al. (2017*The analysis works as follows: we consider the sensitivity of the network to a gap gene negligible, if the node representing that gene in the network can be removed from the model at the onset of C14A without significant consequences to the resulting expression dynamics (see “Materials and methods” for details). If sensitivity is negligible in a given nucleus, we conclude that the gene is not required to drive gap gene expression in that nucleus during C14A.

The results of our analysis are shown in Figure 2A. They reveal three regions that are insensitive to specific gap genes: (1) in the region between 35 and 47% A–P position, developmental trajectories are insensitive to *kni* (red background); (2) between 49 and 59% A–P position, they are insensitive to *gt* (blue), and (3) between 61 and 75% A–P position, they are insensitive to *hb* (yellow). Therefore, the gap gene regulatory network can be reduced from four to three trunk gap genes in each of these regions (Figure 2A and B). Note that regional boundaries are consistent with, but not identical to, observed expression patterns of the gap genes (** Surkova et al. (2008); Jaeger (2011)**).

Next, we identified a minimum set of gap gene interactions that are still able to drive correct expression dynamics in each of these three regions. Surprisingly, the structure of all three resulting subcircuits is qualitatively the same, even though each involves a different set of gap genes (Figure 2B) and therefore distinct strengths of regulatory interactions (Figure 2C). This network structure combines a negative feedback loop between all three genes of a subcircuit with a double-negative (positive) feedback loop between two of them (Figure 2B, C).

Positive feedback loops occur between non-overlapping gap genes Kr and Gt, as well as Hb and Kni; the interactions involved are much stronger than the negative repressive interactions between gap genes with overlapping expression domains (Figure 2C). Previous work has shown that strong positive feedback is required for the basic staggered arrangement of gap domains (“alternating cushions”), while weaker (and hence slower) negative feedback drives anterior shifts in gap domain position over time in the posterior trunk region of the embryo (** Jaeger et al. (2004b**,a);

**). This combination between positive and negative feedback loops is known as the “AC/DC circuit,” which was first described in the context of dorso-ventral patterning in the vertebrate neural tube (**

*Perkins et al. (2006); Ashyraliyev et al. (2009); Jaeger (2011); Crombach et al. (2012); Verd et al. (2018)***).**

*Panovska-GriZths et al. (2013); Perez-Carrasco et al. (2018); Balaskas et al. (2012)*#### The dynamics of AC/DC subcircuits

The next step is to establish whether AC/DC subcircuits are sufficient for patterning in each of the three embryo regions identified in Figure 2A. In order to qualify as a true dynamical module *sensu* Irons (Irons07), each AC/DC subcircuit must recover the expression dynamics as well as the underlying dynamical regime of the full model in the region where it is active. Anterior to the bifurcation boundary at 52% A–P position, this means static gap domain borders governed by multi-stability; posterior of 52% A–P position, this means kinematically shifting domain boundaries governed by a monostable dynamical regime that drives a stereotypical temporal succession of gap gene expression in each nucleus (Figure 1B, C and Figure 3A) (** Verd et al. (2017**, 2018)).

**Figure 3-Figure supplement 1.** Comparison of gene expression dynamic obtained from AC/DC subcircuits and the full model.

**Figure 3-Figure supplement 2.** Comparative dynamical analysis of AC/DC subcircuits and the full model in all nuclei.

All three AC/DC subcircuits reproduce the expression dynamics of the full model with reasonable accuracy in their respective regions of in2uence (Figure 3—figure supplement 1). The only major expression defect involves the anterior boundary of the posterior *gt* domain, which fails to shift in the AC/DC3 model, starting to deviate from the full model around time class T3 (Figure 3 — figure supplement 1). Other defects affect aspects of domain shape or levels of expression (Figure 3 —figure supplement 1), but not the dynamic positioning of boundary interfaces, on which our subsequent analysis will focus.

The range of dynamical regimes that can be implemented by a circuit is constrained by its regulatory structure (see the “Introduction”). Which dynamical regime is realized depends on the strength of regulatory interactions as well as initial and boundary conditions provided by maternal factors. Since all three AC/DC subcircuits exhibit different interaction strengths (Figure 2C) and receive different maternal inputs, they can drive very different expression dynamics (see the “Appendix” for a systematic analysis).

We determine the dynamical regime of each AC/DC subcircuit by calculating and analysing their associated phase portraits. To achieve this, we simulate each AC/DC subcircuit in isolation, using realistic time-variable inputs from maternal gradients (** Verd et al. (2017)**). Such time-dependent boundary conditions render these systems non-autonomous, and cause their phase portraits to change their geometry over time (

**). Previously, we have developed a method for the classification of transient trajectories in non-autonomous dynamical systems (**

*Verd et al. (2017)***), and have used it to study the dynamics of gap gene expression in the full model (**

*Verd et al. (2014)***, 2018)). Here, we redeploy this technique to study each AC/DC subcircuit independently (see “Materials and methods” for details).**

*Verd et al. (2017*AC/DC1 consists of nodes representing *hb, Kr*, and *gt* (Figure 2C, left panel). Its region of in2uence lies between 35 and 47% A–P position (Figure 2A). Phase portraits in this region exhibit multistability with two or more point attractors (shown for the nucleus at 37% A–P position in Figure 3B, left; additional nuclei are shown in Figure 3 —Figure supplement 2A). Comparing these trajectories with those simulated with the full model reveals striking similarities, suggesting that 2ow direction and magnitude is conserved between both models (Figure 3B and Figure 3—figure supplement 2A, A′). In each case, trajectories are shaped by the pursuit of a moving point attractor (** Verd et al. (2014)**), located at equivalent positions in phase space in AC/DC1 and the full model. The only difference between the two models is an additional attracting steady state in the full system, situated at high Kr levels. This attractor is positioned too far from the trajectories to in2uence their shape.

The nodes in AC/DC2 represent *hb, Kr*, and *kni* (Figure 2C, central panel), and it is active between 49 and 59% A–P position (Figure 2A). This region straddles the bifurcation that occurs at 52% A–P position in the full model, designating the division between static anterior and shifting posterior gap domains (** Manu et al. (2009b); Gursky et al. (2011); Verd et al. (2017)**). To be considered a dynamical module throughout this region, the AC/DC2 subcircuit must recover the two distinct dynamical regimes on either side of the bifurcation boundary. The phase portraits of a nucleus anterior and another one posterior of the bifurcation boundary are shown in Figure 3B (centre). Additional nuclei are shown in Figure 3—figure supplement 2B.

The nucleus at 49% is located just anterior to the bifurcation. The phase portrait of AC/DC2 in this nucleus is bistable for all time points except the last one (at T8), when it becomes monostable and only an attractor close to the origin persists (Figure 3B). This is extremely similar to what happens in the full model, where by T8 the steady states at high Kr values have disappeared, and only steady states close to the origin remain. This transition to monostability occurs too late to significantly alter the course of the trajectory. Instead, the geometry of the trajectory in this nucleus—almost identical in AC/DC2 and the full model—first converges towards a saddle during early C14A, and later directly towards a moving attractor (Figure 3B). In the full model, a geometric capture occurs at stages prior to C14A (** Verd et al. (2014**, 2017)), outside the time range of the AC/DC models (

**).**

*Verd et al. (2017)*Steady states in the phase portrait of AC/DC2 at 49% A–P position are restricted to the Hb-Kr plane. They can be mapped onto a subset of steady states present in the full model (Figure 3— figure supplement 2B, B′). Additional steady states in the full model are located on the Hb-Gt plane, at positions which are remarkably similar to those present in AC/DC1 in more anterior nuclei (Figure 3—figure supplement 2A). This suggests that phase portraits of the full model are composites of AC/DC1 and AC/DC2 in the region anterior to the bifurcation boundary at 52% A–P position.

Posterior to the bifurcation at 52% A–P position, the full model exhibits trajectories that curve towards a spiral sink attractor in a monostable phase portrait (** Verd et al. (2018)**). The trajectories of AC/DC2 in this region are very similar (Figure 3B for the nucleus at 59% A–P position, see Figure 3— supplement 2B, B′ for trajectories in other nuclei). Both exhibit a spiralling geometry which is confined to the Kr-Kni plane, with some minor deviations in peak concentration at late stages.

Interestingly, however, the underlying topology of phase space is not equivalent in the AC/DC2 subcircuit and the full model. While both models exhibit monostability in this region, and trajectories in both are shaped by the pursuit of a moving attractor, there are no spiral sinks in AC/DC2 (Figure 3B, and Figure 3—supplement 2B). Spiralling trajectories in this subcircuit are shaped mainly by the movement of a conventional point attractor, with spiral sinks appearing only late, at time points T7 and T8. Attractor movement is much more pronounced than in the full model (Figure 3B, and Figure 3—supplement 2B, B′). Furthermore, the steady states of AC/DC2 are located along the Kni axis with early attractors at high concentration values of Kni, while the full model shows steady states arranged along the Gt axis with early attractors at high concentration values of Gt (Figure 3—supplement 2B′). This is similar to steady states in AC/DC3 in the nucleus at 63% A–P positions (Figure 3B, right panel). At later stages, when the in2uence of attractor position on the geometry of the trajectory becomes more pronounced, the positions of steady states converge between the two models: in each case, they are located at low Kni values, along the Kni axis. This illustrates that identical transient behaviour can be caused by different types of moving attractors in non-autonomous systems.

The nodes in AC/DC3 represent *Kr, kni*, and *gt* (Figure 2C, right panel). It is active between 61 and 75% A–P position (Figure 2). In the nuclei at 61 and 63% A–P position, AC/DC3 faithfully reproduces the dynamics of gap gene expression obtained from the full model (Figure 3B, right, and Figure 3—figure supplement 2C). As in the case of AC/DC2, the system is monostable, and trajectories are shaped by the pursuit of a moving attractor. This attractor is located at high Gt concentrations early on, moving closer to the origin over time with only residual levels of Kni left at late stages (Figure 3B, and Figure 3—figure supplement 2C). Similar to AC/DC2, the type of attractor differs between the full model, where it is a spiral sink at all time points, and AC/DC2, where it is a conventional point attractor during time classes T1–6, only turning into a spiral sink at time points T7 and T8 (Figure 3B, and Figure 3—figure supplement 2C, C′).

Posterior to 63% A–P position, AC/DC3 no longer recapitulates gap gene expression dynamics accurately, because its trajectories fail to switch from the Kr-Kni to the Kni-Gt plane as they do in the full model (Figure 3—figure supplement 1C). It is possible that this switch requires overlap with a fourth AC/DC subcircuit in the posterior sub-terminal region of the embryos, which could not be characterised here since most of its region of in2uence lies outside the spatial range we can analyse in the full model.

In summary, phase space analysis of AC/DC subcircuits establishes that they are true dynamical modules of the gap gene network in the region between 35% and 63% A–P position (** Irons and Monk (2007)**). They faithfully recover the geometry of trajectories in the full model, whose phase portrait can be seen as an overlapping composite of those of the subcircuits in this region. This simple picture is complicated by the fact that spiralling trajectories occur posterior of 52% A–P position despite the absence of spiral sinks in the phase portraits of AC/DC models. The lack of spiral sinks is compensated by more pronounced movements of conventional point attractors, suggesting that distinct non-autonomous phase space topologies can generate equivalent transient dynamics. We discuss this somewhat degenerate relationship between phase space topology and trajectory shapes further in the “Conclusions.”

### Criticality and evolvability

AC/DC1 in the anterior and AC/DC3 in the posterior fall within a consistent dynamical regime across their respective regions of in2uence (Figures 1–3), indicating that these two dynamical modules are structurally stable with respect to inputs from maternal gradients. In contrast, AC/DC2 correctly reproduces the bifurcation observed in the full model at 52% A–P position, which separates static anterior patterning from shifting domains in the posterior (Figure 3) (** Manu et al. (2009b); Gursky et al. (2011); Verd et al. (2017)**). The AC/DC2 subcircuit exhibits two distinct dynamical regimes on either side of the bifurcation boundary. The presence of a bifurcation implies that this dynamical module is in a state of criticality with respect to inputs from maternal gradients. Our analysis therefore suggests that the gap gene network of

*D. melanogaster*is critical to maternal inputs only in the middle of the embryo, where AC/DC2 is active between 49 and 59% A–P position, while it is structurally stable outside of this region. This contrasts with an earlier proposition—based on the quantification of cross-correlations between expression patterns and a set of theoretical models of gap gene regulation—which suggested that the system shows signs of criticality along the entire antero-posterior axis (

**).**

*Krotov et al. (2014)*Understanding criticality in complex regulatory networks is far from trivial, since bifurcations may depend on more than just one of the system′s parameters (see, for example, ** Thom (1976); Scheffer (2009); Kuznetsov (2013)**). This can lead to counter-intuitive effects. For instance, gap gene patterning in the central region of the

*D. melanogaster*blastoderm is robust towards variation in the levels of maternal gradients (Figure 4A) (

**). This is surprising in light of the fact that AC/DC2 is in a critical state. In contrast, the position of the bifurcation boundary between static and shifting gap domains is labile in different dipteran species (see Figure 5 below) (**

*Houchmandzadeh et al. (2002); Gregor et al. (2007a); Manu et al. (2009c); Gursky et al. (2011)***). In the scuttle 2y**

*Jaeger et al. (2004b); Surkova et al. (2008); Manu et al. (2009b); García-Solache et al. (2010); Jaeger (2011); Crombach et al. (2014); Wotton et al. (2015)**Megaselia abdita*(Phoridae), it has been displaced anteriorly compared to

*D. melanogaster*: the region where gap domain shifts occur includes the Hb-Kr interface at around 40% A–P position (

**). The moth midge**

*Wotton et al. (2015)**Clogmia albipunctata*(Psychodidae) shows even more extended and pronounced gap domain shifts (

**).**

*García-Solache et al. (2010); Crombach et al. (2014)*#### Intra-specific robustness

We used the AC/DC2 model to assess the effect of changing the maternal Bcd and Cad gradients on the position of the bifurcation boundary (cf. Figure 4A). In the region between 47 and 61% A–P position, Bcd spans a range of concentrations from 24 to 7 arbitrary units (Figure 4B). For each nucleus in this region, we fixed Cad levels to their values at time point T1, while varying Bcd levels in steps of 0.01 from 0 to 30 arbitrary units. We then calculated the phase portraits of AC/DC2 for each combination of maternal input concentrations (see “Materials and methods”).

The resulting dynamical regimes from AC/DC2 are shown in the phase diagram displayed in Figure 4B. There is a threshold response at a Bcd concentration of approximately 15 arbitrary units: above this threshold, the system is bistable; below, it is monostable. Bcd concentrations in nuclei anterior of 52% fall into the multistable regime, nuclei posterior of this position fall into the monostable regime (see dots in Figure 4B), recovering the same bifurcation position as in the full model (** Manu et al. (2009b); Gursky et al. (2011); Verd et al. (2017)**).

Quantitative measurements of the Bcd gradient find a standard deviation of about 15% relative concentration (** Gregor et al. (2007b); Liu and Ma (2013)**). Plotting this as error bars in Figure 4B, we find that they cross the boundary between dynamical regimes only in nuclei between 51 and 53% A–P position. This very narrow spatial range is in agreement with observed levels of variability in data and previous analyses of gap gene circuits (

**). If we alter the concentration of Bcd by a larger amount, as happens in mutants with varying numbers of**

*Gregor et al. (2007b); Manu et al. (2009c); Gursky et al. (2011)**bcd*copies (

**), we see a more pronounced displacement of the boundary as indicated by the intersection of the dashed gradient profiles with the bifurcation threshold in Figure 4B.**

*Driever and Nüsslein-Volhard (1988); Houchmandzadeh et al. (2002); Liu and Ma (2013)*The situation is very different for the maternal Cad gradient. In the region between 47 and 61% A–P position, Cad spans a range of concentrations from 70 to 110 arbitrary units (Figure 4C). For each nucleus in this region, we fixed Bcd levels to their values at time point T1, while varying Cad levels in steps of 0.1 from 60 to 120 arbitrary units. We then calculated the phase portraits of AC/DC2 for each combination of maternal input concentrations (see “Materials and methods”).

The resulting phase diagram shows that, in contrast to Bcd, the position of the bifurcation boundary is entirely insensitive to Cad concentration. There is an abrupt vertical transition between dynamical regimes around 52% A–P position (Figure 4C). This implies that varying Cad concentration has no effect on the position of the bifurcation, and therefore the extent of the posterior region that exhibits dynamic gap domain shifts. This complements earlier studies which suggested that Cad serves as a general activator in the posterior, but is not directly involved in setting the position of gap gene expression features (** Struhl et al. (1992); Manu et al. (2009b); Verd et al. (2017)**), or in controlling the rate of gap domain shifts in this region of the embryo (

**).**

*Verd et al. (2018)*#### Inter-specific sensitivity

The analysis described in the previous section predicts that increased levels of Bcd should lead to a posterior displacement of the bifurcation boundary between the static anterior and the shifting posterior patterning regime (Figure 4B). Surprisingly, exactly the opposite is observed in *M. abdita*. In this species, the anterior localization domain of *bcd* mRNA is expanded compared to *D. melanogaster*, which presumably leads to an expanded Bcd protein gradient (** Stauber et al. (1999**, 2000);

**). However, despite more Bcd being present, the bifurcation boundary is located further**

*Wotton et al. (2015); Crombach et al. (2015)**anterior*than in D. melanogaster (Figure 5A) (

**). This is impossible to explain if we only take maternal inputs to the gap gene system into account.**

*Wotton et al. (2015); Crombach et al. (2015)*Previous studies have shown that the strength of gap-gap cross-regulatory interactions differs between *M. abdita* and *D. melanogaster* (** Wotton et al. (2015); Crombach et al. (2015)**). Based on this, we sought to identify the specific gap-gap interactions that could account for the altered position of the bifurcation in our models. Our dissection of the gap gene network into dynamical modules narrows this search to interactions within the AC/DC1 and AC/DC2 subcircuits (Figure 2B, C), as their regions of in2uence cover the relevant region of the embryo. First, we focus on AC/DC2, as the bifurcation boundary in

*D. melanogaster*lies within its region of in2uence. We know from previous analyses, that the strong positive feedback between

*hb*and

*kni*is heavily conserved between the two species (

**). We also know that asymmetric interactions between overlapping gap genes (**

*Wotton et al. (2015); Crombach et al. (2015)**kni*on

*Kr*, and

*Kr*on

*hb*) are involved in regulating domain shifts in both species (

**,a);**

*Jaeger et al. (2004b***, 2015)). Therefore, we concentrate our analysis on these two interactions.**

*Ashyraliyev et al. (2009); Crombach et al. (2012*In our D. melanogaster model, repression of *hb* by Kr is negligible (Figure 2C). In contrast, both genetic evidence (** Wotton et al. (2015)**) and models (

**) for**

*Crombach et al. (2016)**M. abdita*indicate that there is a significant net repressive effect of Kr on

*hb*in this species. In Figure 5B, we plot the dynamical regimes of AC/DC2 in nuclei between 47 and 61% A–P position, while varying this regulatory parameter by decremental steps of 0.0005 across a narrow range of values from 0 to −0.0035. If repression remains minimal (below a value of −0.0005), we recover the bifurcation at 52% A–P position, as nuclei anterior to this position are multistable, while more posterior nuclei are monostable (Figure 5B). If the strength of repression is further increased, the bifurcation boundary moves anteriorly and vanishes altogether around a repression strength of −0.0025. In contrast, the position of the bifurcation boundary is largely insensitive to the interaction between

*kni*and

*Kr*(Figure 5C, D). Taken together, the analysis predicts that AC/DC2 in

*M. abdita*should be structurally stable, as net repression of

*hb*by Kr is increased in this species

**, eliminating the bifurcation boundary present in**

*Wotton et al. (2015); Crombach et al. (2016)**D. melanogaster*.

However, the bifurcation boundary in *M. abdita* is located even further anterior, around 40% A–P position, far into the region between 35 and 45% A–P position covered by AC/DC1. As the repression of *hb* by Kr is shared between AC/DC1 and AC/DC2, we asked if increasing its strength would induce a bifurcation in AC/DC1, abolishing its structural stability and rendering it critical. Interestingly, this is not the case, as AC/DC1 remains multistable regardless of repression strength due to the very strong positive feedback between *Kr* and *gt* (Figure 5E). In this subcircuit, further alterations to regulatory interactions are required to render the circuit monostable. This can be achieved, for example, by simultaneously changing both repressive interactions of Kr on *hb* and Hb on *gt* (Figure 5F). In this case, the latter interaction is the critical one. Therefore, both repressive interactions must be stronger in *M. abdita* than in *D. melanogaster* to render AC/DC1 critical in this species. This is consistent with evidence from experiments (** Wotton et al. (2015)**) and modelling (

**).**

*Crombach et al. (2016)*In summary, our analysis suggests that in *D. melanogaster* only AC/DC2 is critical, while in *M. abdita* the bifurcation boundary between static and dynamic patterning regimes falls into the region of in2uence of AC/DC1. Dynamical modules and their criticality have important consequences for the evolvability of the gap gene network. Our work predicts that small changes in the strength of gap-gap cross-regulatory interactions affect the extent of static versus shifting patterning regimes. Other aspects of gap gene patterning, such as the alternating-cushions mechanism of positive feedback between *Kr* and *gt* as well as *hb* and *kni*, are extremely robust towards changes in interaction strengths.

## Conclusions

In the first part of our analysis, we generalize the notion of a dynamical module—first introduced for Boolean network models by ** Irons and Monk (2007)**—and use a pragmatic approach based on sensitivity analysis to identify dynamical modules in the gap gene system of

*D. melanogaster*. The three modules described include distinct sets of regulators, but share a common regulatory network structure. They all correspond to AC/DC subcircuits, able to drive either multistable (switch-like) or oscillatory dynamics depending on parameter values and boundary conditions (

**). We show that each circuit is active in a particular region of in2uence along the antero-posterior axis, where it is able to reproduce the geometry of transient trajectories, and hence the overall expression dynamics, of a full gap gene circuit (Figures 2 and 3).**

*Panovska-GriZths et al. (2013)*Dynamical modules as defined and used here should not be confused with the recent postulation of “static” and “dynamic” modules in the segment determination network of the 2our beetle *Tribolium castaneum* (** Zhu et al. (2017)**). “Static” and “dynamic” in that case refer to slow versus rapid time scales of expression dynamics driven by distinct classes of subcircuits, which are all defined in terms of their disjoint structural components. In contrast, our results indicate that structural modularity is not essential for the evolution of insect segmentation, and that a search for structural modules may be in vain, as it is in the case of the gap gene network of

*D. melanogaster*where only dynamical modules are present in the system. In this sense, dynamical modules provide a powerful complementary alternative to identifying structural modules.

Concerning the dynamical analysis of the AC/DC subcircuits, it is interesting to note that the geometry of transient trajectories generated by different non-autonomous models can be equivalent despite underlying discrepancies in features of phase space (see “Results and discussion”). In particular, posterior subcircuits AC/DC2 and AC/DC3 exhibit spiralling trajectories in the absence of spiral sink attractors (Figure 3, and Figure B, B′, C, C′) (** Verd et al. (2017**, 2018)). In these models, the spiral geometry of the trajectories is generated by a corresponding movement of a point attractor in space, while in the full model it is the consequence of (much less pronounced) attractor movement in addition to the complex eigenvalues of the spiral sink. In other words, there is a certain degeneracy or disconnect between the attractors present in phase space and the resulting transient dynamics driven by the system.

This in turn has important implications for the way in which we generally analyse dynamical systems models in biology. To understand the dynamics of a regulatory network, it is not generally sufficient to perform a steady-state analysis of an autonomous version of the system. Transient dynamics, and the explicit dependence of regulatory structure on time implied by non-autonomy, should be considered the default assumption. Steady-state dynamics and autonomy must first be established before conclusions from classical attractor analysis can be considered valid and applicable. We will discuss this important and general problem in more detail in an upcoming manuscript (in preparation).

In the second part of our analysis, we focus on the consequences of dynamical modularity on the evolvability of the system. Our results shed light on the apparent paradox that gap gene expression dynamics are surprisingly insensitive to maternal gradient concentrations during development within a species, but quite labile when comparing different species on an evolutionary time scale (Figures 4 and 5). We show that this lability depends on changes in downstream regulatory interactions between gap genes.

Bifurcation analysis reveals that AC/DC1 and AC/DC3 are structurally stable in *D. melanogaster*, and therefore insensitive to changes in the parameter values that affect the strength of interactions. In contrast, AC/DC2 is in a state of criticality with regard to the repressive effect of Kr on *hb*. Increasing repression between these two genes leads to a bifurcation event, which changes the dynamical regime of AC/DC2 from multistable, switch-like, behaviour (generating stable gap domain boundaries) to spiralling trajectories (generating shifting transient boundaries). Evolvability of the gap gene system is driven by the presence or absence of criticality in its dynamical modules. This explains why the extent and dynamics of gap domain shifts appear to be highly evolvable, while the basic staggered arrangement of domain boundaries remains remarkably stable, at least during the evolution of cyclorrhaphan (or “higher”) 2ies (** Bonneton et al. (1997); Stauber et al. (2000); Shaw et al. (2002); Lemke et al. (2008); Lemke and Schmidt-Ott (2009); Lemke et al. (2010); Wotton et al. (2015); Crombach et al. (2015)**, see

**for a review).**

*Jaeger (2011)*Our analysis of *D. melanogaster* allows us to infer a number of characteristics of gap gene regulation in other dipteran species, such as *M. abdita*, even though the quality of models we have for that species unfortunately does not allow a direct comparison of phase spaces (** Crombach et al. (2012**, 2015)). Dynamic boundary shifts extend much further anterior in this species than in

*D. melanogaster*(Figure 5A) (

**). The anteriorly displaced position of the bifurcation suggests that AC/DC1, not AC/DC2, must be critical in**

*Wotton et al. (2015)**M. abdita*. Our analysis reveals that changes in several gap-gap interactions are required to render AC/DC1 structurally unstable (Figure 5). This provides a plausible lineage explanation (

**) (or evolutionary trajectory) for changes in dynamical modules that accurately fit what we know about changes in gap gene expression and regulation between**

*Calcott (2008)**M. abdita*and

*D. melanogaster*(

**).**

*Wotton et al. (2015); Crombach et al. (2015)*Outside the cyclorrhaphan lineage, the arrangement of gap domains and the patterning output of the system changes. In the moth midge *Clogmia albipunctata*, for example, there is no posterior domain of Gt protein expression, and the posterior domain of *hb* mRNA only forms after gastrulation (** Rohr et al. (1999); García-Solache et al. (2010)**). Gap domain shifts are much more pronounced in this species compared to

*D. melanogaster*or

*M. abdita*(

**). The beetle**

*García-Solache et al. (2010); Crombach et al. (2014)**T. castaneum*shows an even more dynamic mode of segment determination: only the anterior-most domains form simultaneously, while more posterior domains are generated sequentially by sustained oscillations in segmentation gene expression (

**).**

*Sarrazin et al. (2012); El-Sherif et al. (2012)*An extended analysis of our AC/DC circuits (see the “Appendix”) reveals that they can be induced to drive sustained limit-cycle oscillations with relatively small changes in the values of parameters that determine cross-regulatory interactions (summarized in Figure 6). Although gap genes do not seem to be directly involved in the process of segment determination, they do show repeated waves of kinematically shifting waves of gene expression in the *T. castaneum* blastoderm and germband embryo (** Zhu et al. (2017)**). So do pair-rule genes (

**, 2014)), which are known to be essential for segment determination in this species. Pair-rule genes in**

*Sarrazin et al. (2012); El-Sherif et al. (2012**T. castaneum*may also be regulated by AC/DC-like regulatory subcircuits (

**). These surprising resemblances suggest that ancestral gap and pair-rule expression may have relied on similar regulatory principles, and that the regulatory changes required to turn sequential (short-germband) into simultaneous (long-germband) segmentation may be much more subtle and limited than commonly thought (see**

*Choe et al. (2006)***). Improved empirical and modelling evidence from many additional species will be required to rigorously test this prediction.**

*Tautz (2004); Clark (2017)*Taken together, our results demonstrate the power and versatility of analysing the developmental and evolutionary dynamics of regulatory networks using an approach based on dynamical rather than structural modularity. The main limitation of this approach remains the small number of systems in which this type of analysis is currently possible. Luckily, this is a purely practical problem and the number of potential model systems is increasing. AC/DC circuits involved in neural tube patterning in vertebrates can be considered dynamical modules (** Balaskas et al. (2012); Panovska-GriZths et al. (2013); Perez-Carrasco et al. (2018)**), and so can the growing number of experimentally verified Turing-type pattern generators, for example, those driving digit patterning in the vertebrate limb from sharks to mammals (

**). Based on this, we see an urgent and growing need to identify and characterise dynamical modules in development and evolution. This powerful approach greatly extends the reach of traditional methods by capturing modularity in systems that show no overt clusters in regulatory structure or co-expression patterns. Last but not least, it brings us much closer to the aim of identifying true functional modules in evolving developmental systems (**

*Raspopovic et al. (2014); Onimaru et al. (2016)***), as dynamical behaviour is much more tightly integrated with biological function that the regulatory structure of a network.**

*Wagner (2014)*## Materials and methods

### The full model: a diffusion-less gap gene circuit

What we refer to as the “full model” in this paper corresponds to a diffusion-less gap gene circuit, which is formulated in the connectionist modelling framework first proposed by ** Mjolsness et al. (1991)**, is derived from gap gene circuits with diffusion (

**), and was previously published and described (**

*Ashyraliyev et al. (2009)***, 2018)). Here, we only provide a brief description of the model. See these previous publications for details.**

*Verd et al. (2017*Gap gene circuits consist of a one-dimensional row of nuclei, arranged along the antero-posterior (A–P) axis of the embryo. They are hybrid models that implement continuous dynamics during interphase and mitosis, with discrete instantaneous nuclear divisions occurring at the end of each mitosis. The spatial domain of the model used here extends over a range of 35 to 75% A–P position (where 0% is the anterior pole), covering the trunk region of the embryo. The full model includes cleavage cycles C13 and C14A of the blastoderm stage during early development of *D. melanogaster* (** Foe and Alberts (1983)**). C14A is further subdivided into eight time classes of equal duration (T1–T8) (

**). Division takes place at the end of C13.**

*Surkova et al. (2008)*The state variables of the model represent the concentrations of transcription-factor proteins encoded by trunk gap genes *hb, Kr, kni*, and *gt*. represents the concentration of protein *a* in nucleus at time *t*. The rate of change in gap protein concentration over time is given by the following system of ordinary differential equations:
where *R*^{a} is the rate of protein production, and *λ*^{a} the rate of protein decay. *ϕ* is a sigmoid regulation-expression function which is used to represent the coarse-grained saturating kinetics of transcriptional regulation. It is defined as follows:

Where

*G* = {*hb, Kr, kni, gt*} denotes the set of trunk gap genes, and *M* = {Bcd, Cad} the set of external inputs from protein gradients encoded by maternal coordinate genes (which are not themselves regulated by gap genes). We linearly interpolate quantified spatio-temporal protein expression data (** Surkova et al. (2008); Pisarev et al. (2009); Ashyraliyev et al. (2009)**) to obtain the concentration profiles of the maternal regulators .

Interconnectivity matrices *W* and *E*, with elements *w*^{ba} and *e*^{ma}, represent regulatory weights of interactions between gap genes, and external inputs from maternal gradients, respectively. The effect of regulator *b* or *m* on gap gene target *a* is activating, if the corresponding weight is positive, repressive, if the weight is negative; there is no interaction if the weight is (near) zero. *h*^{a} is a threshold parameter encoding the basal activity of gap gene *a* in the absence of any spatially or temporally specific regulators. The system of equations (1) governs regulatory dynamics during interphase. *R*^{a} is set to zero during mitosis.

Earlier studies have established that diffusion of gap proteins is not essential for pattern formation (** Jaeger et al. (2004b); Manu et al. (2009b); Verd et al. (2017)**). Therefore, it is not included in this version of the model. Omitting diffusion renders each nucleus independent of the others, and reduces the dimensionality of system from 164 (4 gene products in 41 nuclei) to four dimensions, rendering it amenable to phase (or state) space analysis.

#### Fitting of the full model

Values for parameters *R*^{a}, *λ*^{a}, *W, E*, and *h*^{a} were obtained by fitting the model to quantitative spatio-temporal gene expression data as previously described (** Reinitz and Sharp (1995); Jaeger et al. (2004b); Ashyraliyev et al. (2009); Verd et al. (2017)**). Brie2y: we solve model equations (1) numerically, and compare the resulting model output to data by calculating a weighted root-meansquare (RMS) score (

**). This is repeated while changing parameter values until the fit is no longer improving. The difference between model output and data is minimized using a global optimization algorithm called parallel Lam Simulated Annealing (pLSA) (**

*Ashyraliyev et al. (2009)***). Model fitting was carried out on the Mare Nostrum cluster at the Barcelona Supercomputing Centre (http://www.bsc.es). The circuit used here is identical to that published and described in (**

*Chu et al. (1999)***, 2018)). It accurately reproduces the spatio-temporal expression dynamics of gap gene expression (see Figure 1B), and is fully consistent with the available experimental evidence on gap gene regulation (**

*Verd et al. (2017***,a);**

*Jaeger et al. (2004b***,b);**

*Manu et al. (2009c***, 2018)).**

*Ashyraliyev et al. (2009); Jaeger (2011); Crombach et al. (2012); Verd et al. (2017*### Identifying dynamical modules

The full gap gene circuit described above drives two distinct dynamical regimes anterior and posterior of a bifurcation, which occurs at 52% A–P position (** Verd et al. (2017**, 2018)). Anterior of the bifurcation, the system is multistable, positioning static gap domain boundaries through switch-like dynamic behaviour; posterior of the bifurcation, the system is monostable, and the only attractor present is a spiral sink implementing a damped oscillator mechanism driving the observed dynamic anterior shifts of posterior gap domains (

**, 2018)). In the absence of any evident structural modules in the network (see the “Introduction”), we ask whether there are dynamical modules or subcircuits that suZce to reproduce the observed dynamical regimes in different regions of the embryo. We define dynamical modules as subcircuits embedded in the gap gene network that are capable of recovering the dynamics of gap gene expression at a given A–P position (**

*Verd et al. (2017***). Dynamical modules may show overlap in their components, interactions, and regions of influence.**

*Irons and Monk (2007)*#### Node sensitivity analysis

We observe that any specific nucleus along the A–P axis of the embryo only ever expresses two gap genes at the same time, and never more than three different gap genes over the whole duration of cleavage cycle C14A. This implies that only a subset of the four trunk gap genes are required for patterning at any given position. To identify these subsets and their regions of in2uence, we assessed the sensitivity of simulated developmental trajectories to the removal of gap genes in all nuclei between 35 and 75% A–P position. Early gap gene patterning is largely governed by regulatory inputs from maternal gradients, while gap-gap interactions become increasingly predominant during late C13 and C14A. Since our analysis focuses on gap-gap cross-regulation, we limit our analysis to C14A.

Using maternal gradient and simulated gap gene concentrations at the onset of C14A (time class T1) as initial conditions, we numerically solve the full model, and compare it to simulations where one of the gap genes and all its regulatory interactions have been erased from the model. The “node sensitivity” of the system in a given nucleus with respect to a particular deleted gap gene (or node of the network) is then given by the distance between the trajectory simulated using the full model (curve *a*), and the trajectory obtained from a simulation lacking the node representing that gene (curve *b*). Differences are summed over all time classes (T1–T8) in C14A, which results in the following distance metric:

represent simulated concentrations of gap genes at time class *T*_{i}. *j* indicates whether the concentration is derived from the full model (*a*) or simulations with specific nodes removed (*b*). The smaller the distance between trajectories, the smaller the contribution of the removed gap gene to the overall dynamics of gene expression. Sensitivities close to zero indicate that the removed gene is not required for patterning in a given nucleus during C14A. The resulting regions of insensitivity are shown in Figure 2A. The corresponding AC/DC subcircuits associated which each of these regions are shown in Figure 2B and C.

### AC/DC subcircuits

#### Formulation and simulation

We implemented models of the three subcircuits consisting of the components and interactions shown in Figure 2C using the gene circuit modelling formalism described above. Parameter values, including time-variable maternal inputs and autoregulatory weights are taken from the full model. AC/DC1 includes *hb, Kr*, and *gt*, spanning the region of 35 to 47% A–P position; AC/DC2 includes *hb, Kr*, and *kni*, spanning the region of 49 to 59% A–P position; AC/DC3 includes *Kr, kni*, and *gt*, spanning the region of 61 to 75% A–P position. There is a fourth AC/DC subcircuit, consisting of *hb, kni*, and *gt* (AC/DC4), whose region of in2uence lies further posterior, outside the spatial domain of the full model. The AC/DC4 subcircuit was not analysed further in this study. We use maternal gradients and simulated gap protein concentrations from the full model at C141-T1 as initial conditions. Simulations of AC/DC subcircuits include C14A only (no mitosis or division).

#### Phase space analysis

Since they implement time-variable maternal inputs, AC/DC subcircuits are non-autonomous dynamical models (** Strogatz (2014)**). We previously developed a methodology to characterise and classify simulated transient trajectories in such models (

**), which we have used to analyse the full gap gene circuit model (**

*Verd et al. (2014)***, 2018)). Here, we redeploy this method for the comparative analysis of AC/DC subcircuits and the full model. Brie2y, we calculate instantaneous phase portraits for different time points by “freezing” parameter values (maternal inputs) to the value at each given time. Steady states were calculated using a Newton-Raphson algorithm as in**

*Verd et al. (2017***. For each nucleus within an AC/DC subcircuit′s region of in2uence, we compare trajectories and attractor positions between the subcircuit and the full model. The results are shown in Figure 3, and Figure 3—figure supplement 2, and are discussed in further detail the “Results and discussion” section of our paper. A more systematic analysis of dynamical regimes that can be driven by the AC/DC subcircuit is provided in the “Appendix.”**

*Manu et al. (2009b)*## Appendix: Mathematical analysis of the AC/DC circuit

The aim of this analysis is to understand the overall dynamical repertoire of the AC/DC circuit. More specifically, we want to understand the dynamical regimes (as defined in the main text) that this circuit can implement with different sets of parameters determining the strength of its repressive regulatory interactions. We start from the well-studied behaviour of the three-gene repressilator circuit (** Elowitz and Leibler (2000); Müller et al. (2006); Garcia-Ojalvo et al. (2004); Strelkowa and** Barahona (

**), and then extend the analysis to the AC/DC network (**

*2010)***).**

*Panovska-GriZths et al. (2013)*For both repressilator and AC/DC circuit, we characterize the number and type of steady states present for any given set of parameter values. From this, we establish the range of different dynamical regimes that each of these circuits can implement. This comparative approach illustrates how the addition of the backward repressive interaction that distinguishes the AC/DC from the repressilator circuit affects the dynamical repertoire of the network. This analysis reveals that the AC/DC circuit (unlike the repressilator) can implement all the dynamical regimes of gene expression that have been observed during segment determination in different groups of insects.

### Analysis of the repressilator circuit

Let us first consider the three-gene repressilator circuit shown in Figure 7A. Transcription factor X represses gene *y*, whose product, transcription factor Y, represses gene *z*, whose product, transcription factor Z, in turn represses gene *x*. We represent repressive regulatory interactions by bounded, monotonically decreasing, and continuous functions *G*. For example, any sigmoid-shaped regulation-expression function conforms to these general conditions.

We use the following abstract formulation of the repressilator circuit, where *G* stands for a regulation-expression function conforming to the conditions prescribed above, and *d* are gene product degradation rates:

This system has a unique steady state. Let us suppose, for the sake of simplicity, that the degradation rates for all gene products are equal to 1, that is, *d*_{x}=*d*_{y}=*d*_{z} = 1. Then the steady states of the system are solutions to the following system of equations:

By solving for the state variables in the system, and substituting, we obtain:

Due to the fact that the three regulation-expression functions are bounded, monotonically decreasing, and continuous, so is their composition. Therefore, there is a unique solution *X*_{*} to equation (7) (Figure 7B). By substituting backwards, following analogous reasoning, we can establish that there is a unique steady state given by (*X*_{*}, *Y*_{*}, *Z*_{*}).

### Types of steady states: eigenvalue analysis

In order to evaluate the nature of the steady state solution present in the repressilator model (Equation (7)), we linearise the system around its steady state: *X* = *X*_{*} + *x, Y* = *Y*_{*} + *y, Z* = *Z*_{*} + *z*, and evaluate its eigenvalues (** Strogatz (2014); Hirsch et al. (2013)**). This linearisation at steady state (

*X*

_{*},

*Y*

_{*},

*Z*

_{*}) is written in matrix form as: Where because

*G*

_{x},

*G*

_{y}and

*G*

_{z}are monotonically decreasing, and continuous functions.

The eigenvalues associated with the linearised system at steady state (*X*_{*}, *Y*_{*}, *Z*_{*}) are the roots of the characteristic polynomial given by the following determinant:
Let us again consider the special case where all degradation rates are equal, *d*_{x} = *d*_{y} = *d*_{z} = *d*:

Therefore, the complex roots of (*λ* + *d*)^{3} are given by (see also Figure 7C):
It follows that the eigenvalues associated with our system are

*λ*_{1} is always negative. However, this does not need to be the case for *λ*_{2} and *λ*_{3} since

Proof:

This leaves us with two alternative possibilities. From the equations in (18) we know that *Re(A*_{2,3}) *>* 0 if . From this, it follows that *Re(λ*_{2,3}) *>* 0 if Γ *>* 2*d* (this case is illustrated in Figure 7D). A steady state with one negative and two positive eigenvalues is an unstable saddle point. Since the repressilator only has one steady state, this saddle point is unique.

By applying the Poincaré-Bendixson theorem, we can establish that the system must have a limit cycle (** Strogatz (2014); Hirsch et al. (2013)**). The Poincaré-Bendixson theorem in two dimensions predicts the presence of limit cycles. It tells us that if we can find a trapping region—a closed and bounded region of the phase plane that contains a trajectory which remains within this region from a certain time onwards—and this trapping region contains no equilibrium points, then there must be at least one limit cycle within this region. The Poincaré-Bendixson theorem does not normally hold for systems of dimension larger than two. However,

**have shown that, in monotone cyclic feedback systems, the**

*Mallet-Paret and Smith (1990)**ω*-limit set of any bounded orbit can be embedded in ℝ

^{2}. Under these conditions, which are met in our case, and in this two-dimensional subspace, the Poincaré-Bendixson theorem applies, and we can infer that there must be a limit cycle present. In this case, therefore, the dynamical regime of the repressilator is to drive stable limit-cycle oscillations.

There is a second possibility. If Γ < 2*d* ⟹ *Re(λ*_{2,3}) 0 (Figure 7E). The fact that we have three negative eigenvalues, two of which are complex, implies that the unique steady state is a stable spiral sink. Spiral sinks are the hallmark of damped oscillations. In this case, the dynamical regime of the repressilator is to drive damped oscillations.

A third, very unlikely, possibility occurs when eigenvalues have zero real parts. In this case, linear stability analysis breaks down and a full non-linear analysis is required. Due to the small probability of this scenario, we have not pursued this possibility further.

In summary, our analysis shows that the repressilator circuit only has one unique steady state. Depending on the parameter values of the model, this steady state is either a saddle point or a spiral sink. Therefore, the dynamical repertoire of the repressilator consists of two distinct dynamical regimes: stable limit-cycle oscillations, or damped oscillations. No other types of dynamics can be driven by this circuit. In particular, it does not support multistable (switch-like) behaviour.

### Analysis of the AC/DC circuit

We obtain an AC/DC circuit (** Panovska-GriZths et al. (2013)**) by adding an additional backward repressive interaction from Z to Y to the repressilator (compare Figure 7A with Figure 8A). The mathematical formulation of the AC/DC circuit is then as follows:

Regulation-expression functions *G*_{x} and *G*_{z} are assumed to be bounded, monotonically decreasing, and continuous. *G*_{y}(*X, Z*) will typically be taken as the multiplication of the regulation-expression function of X on Y, and that of Z on Y, and its form is explained in more detail below.

Let us begin by finding the steady states of the system:
*Y*_{*} depends on the strength of the backward reaction from Z on Y, as given by *G*_{z}(*Y*_{*}), and on the combination of the reactions of Z on X and X on Y represented by *F (Y*). *F (Y*) (Equation (20)) is monotonically increasing, as it results from the composition of two monotonically decreasing functions, with a general shape as shown in Figure 8B. As the backward reaction on Y is increased, the three scenarios shown in Figure 8C–E become possible. Let us now look at the Jacobian matrix associated with the system:
where
and
*γ*_{yx} refers to the repressive interaction of X on Y, and *γ*_{yz}, to the repressive interaction of Z on Y. Again, for the sake of simplicity, let us assume that *d*_{x} = *d*_{y} = *d*_{z} = *d* and calculate the associated characteristic equation:
From this, it follows that
If we let *d* + *λ* = *η* and , then equation (28) becomes

Now, let *x* = *γ*_{yz}*γ*_{z} > 0, and equation (29) becomes

This implies that the AC/DC circuit has either one or three steady states.

Types of steady states: eigenvalue analysis

The eigenvalues of all steady states are roots of the polynomial in equation (30). In particular, *x* and are evaluated at the steady state. They remain positive since

The simplified characteristic equation shown in (30) defines a depressed cubic (Figure 8F). Its exact shape depends on the values of *x* and when evaluated at each steady state. Given that these parameters always remain positive, we can look at the shape of equation (30), and from it infer the possible roots, as well as the type and sign of every eigenvalue.

From Figure 8F we can see that, when solving for *η*, equation (30) will always have a negative real root and, depending on the magnitude of *x*, either two positive real roots, or two conjugate complex roots, both with positive real parts (Figure 8G, H). The roots of the depressed cubic are given by *η*_{i},*i* ∈ {1, 2, 3}, where *η*_{i} = *λ*_{i} + *d*_{i} and *λ*_{i},*i* ∈ {1, 2, 3}, are the eigenvalues associated with the corresponding steady state of the AC/DC circuit. If we consider *d* = *d*, ∀*i*, we get five possible combinations of eigenvalues, indicating that steady states found in the AC/DC circuit can be unstable spiral sinks (Figure 8I), stable spiral sinks (Figure 8J), point attractors (Figure 8K), saddles_{1,2} (Figure 8L), or saddles_{2,1} Figure 8M).

In summary, the AC/DC circuit can exhibit dynamical regimes that consist of one or three steady states from among the five types described above. This indicates a rich dynamical repertoire, which includes the following regimes that are known to be involved in segment determination in different groups of insects:

Sustained oscillations are observed if the underlying phase portrait has a unique unstable spiral, and therefore a limit cycle (Figure 9, parameter values used for simulation are shown in Table 1).

Damped oscillations are observed if the underlying phase portrait has a unique stable spiral sink (Figure 10, parameter values used for simulation are shown in Table 2).

In addition, the AC/DC circuit can also implement a bi-stable dynamical regime, where the underlying phase portrait will have two point attractors and a saddle

_{1,2}(Figure 11, parameter values used for simulation are shown in Table 3).

We were able to find additional dynamical regimes by sampling the parameter space numerically. These are listed below. Note that a numerical analysis cannot guarantee an exhaustive list of a possible dynamical regimes. The biological relevance of these regimes has not yet been explored.

An unstable spiral can co-exist with a saddle and a point attractor.

A stable spiral sink can co-exist with a saddle

_{2,1}and a point attractor in a bistable regime. The saddle probably needs to have a 2-dimensional unstable manifold, for there to be a basin of attraction with a stable spiral.A bistable regime with two stable spiral sinks.

We conclude that the dynamical repertoire of the AC/DC circuit contains at least six different monostable and multistable regimes, three of which correspond to behaviour observed in insect segmentation.

### Validation of the analysis when using a connectionist model formulation

Our analysis uses an abstract general formulation of the AC/DC circuit to characterize its dynamical repertoire. We validate this analysis for the connectionist formalism described in “Materials and methods.”

Let us consider the AC/DC circuit shown in Figure 12A, and this time formulate it mathematically as follows:

where:
The sigmoidal function (34) is shown in Figure 12B. It is bounded, monotonically increasing, and continuous. *u* represents the sum of all regulatory contributions to a given target gene. For example, the contribution of gene product X on gene *y* is evaluated as the product between the strength of the interaction of X on *y*, the sign of that interaction (negative for repression, positive for activation, denoted by *δ*_{+-}), and the concentration of X. *R* represents the maximum rate of activation for every gene. More generally, we write *u* as:
based on which equation (33) becomes:

In what follows, we will assume that *R*_{x} = *R*_{y} = *R*_{z} = *R, d _{x}* =

*d*

_{y}=

*d*

_{z}=

*d*, and . As in the general case explored above, it is possible to have more than one steady state with the connectionist formulation of an AC/DC circuit: Again,

*F*(

*Y*) is monotonically increasing (

*F*

^{’}(

*Y*)≥ 0, ∀

*Y*), since it is the composition of two monotonically decreasing functions (

*g*is increasing with respect to

*u*, but decreasing with respect to X, Y and Z). Depending on the strength of the back reaction—which is given by the magnitude of parameter

*D*in the connectionist model formulation (Figure 12A)—we will have one or three steady states (Figure 12C). This con1rms our general analysis above.

We will now show that the characteristic equation is the same in both cases. Once this is established, it follows that the same types of steady states occur in both model formulations. Therefore, the dynamical regimes revealed by our general analysis also apply in the connectionist case. Let us look at the associated Jacobian
Where
If we now assume that *d*_{x} = *d*_{y} = *d*_{z} = *d*, we have the same characteristic equation for the connectionist formulation as in the general case (see equation (24)):

This demonstrates that the results of the analysis of a general AC/DC circuit are still valid when the AC/DC circuit is formulated as a connectionist model. Since the characteristic equation is the same in both cases, the dynamical repertoire will be equivalent as well.

## Acknowledgments

We would like to thank Anton Crombach for insightful discussions, and for providing the fitted full model used in this analysis. Other members of the Jaeger Lab in Barcelona, and fellows of the KLI Klosterneuburg provided inspiring and motivating feedback on the project. We thank Ovidiu Radulescu for initial discussions about the dynamical behaviour of the AC/DC circuit. The authors thankfully acknowledge the computer resources, technical expertise and assistance provided by the Barcelona Supercomputing Center—Centro Nacional de Supercomputación.

## Footnotes

↵¶ Centre de Recherches Interdisciplinaires (CRI), Paris, France