Whole-Embryo Modeling of Early Segmentation in Drosophila Identifies Robust and Fragile Expression Domains

Abstract

Segmentation of the Drosophila melanogaster embryo results from the dynamic establishment of spatial mRNA and protein patterns. Here, we exploit recent temporal mRNA and protein expression measurements on the full surface of the blastoderm to calibrate a dynamical model of the gap gene network on the entire embryo cortex. We model the early mRNA and protein dynamics of the gap genes hunchback, Kruppel, giant, and knirps, taking as regulatory inputs the maternal Bicoid and Caudal gradients, plus the zygotic Tailless and Huckebein proteins. The model captures the expression patterns faithfully, and its predictions are assessed from gap gene mutants. The inferred network shows an architecture based on reciprocal repression between gap genes that can stably pattern the embryo on a realistic geometry but requires complex regulations such as those involving the Hunchback monomer and dimers. Sensitivity analysis identifies the posterior domain of giant as among the most fragile features of an otherwise robust network, and hints at redundant regulations by Bicoid and Hunchback, possibly reflecting recent evolutionary changes in the gap-gene network in insects.

Introduction

The segmentation of the Drosophila melanogaster embryo is established through the sequential activation of gene regulatory networks, starting with the translation of maternally deposited mRNAs, and subsequent zygotic expression of the gap genes (1). Through the combined actions of localized translation, protein diffusion, degradation, and transport, the Bicoid and Caudal transcription regulators form concentration gradients along the anterior-posterior (A-P) axis (2–8), which provide initial positional information to the gap-gene network. Mutual interaction among the gap genes then leads to relatively broad spatial expression domains along the A-P axis, showing approximate rotational symmetry (9). The initial gap-gene patterning is then gradually refined, as gap genes induce the pair-rule genes and later the segment polarity network, ending with spatial expression domains with a resolution of a single cell (1). Expression of the gap genes starts around cleavage cycle 11 (C11), when the transcription regulators Bicoid (Bcd) and Caudal (Cad) induce the zygotic transcription regulators hunchback (hb), Kruppel (Kr), giant (gt), and knirps (kni) (10). The latter regulators mutually cross-interact, which leads to the formation of spatiotemporal domains in their expression. A few additional genes of the terminal system—notably Huckebein and Tailless, expressed at the poles of the embryo—also contribute to specification of the spatial expression domains (11,12). Patterning of the gap genes is completed at cycle C14A just before cellularization, when the blastoderm is still a syncytium. The gap-gene network has been dissected genetically and functionally (1,3,13–15), providing a detailed and mostly static interaction map for this process. Quantitative measurements of expression profiles (10,16) in space and time have opened the possibility of reconstructing the early segmentation network by fitting generic reaction-diffusion models describing the gap-gene regulatory network (17) with minimal prior constraints on the network connectivity. Although this approach does not model the regulatory interactions explicitly at the level of protein-DNA interactions, inferred networks nonetheless provide insights into design and robustness properties of the system (9,18–21). Because it is thought that head patterns depend on additional genes (Section 2 in the Supporting Material), most current models focus on the trunk of the embryo, i.e., the geometry of the model is restricted to a subinterval of the A-P axis representing about two-thirds of the embryo length (EL) (from EL positions 35–90%) (Fig. 1 C, red line) (18,19,21–25).

Figure 1.

Gap-gene-network model. (A) Experimentally measured maternal protein gradients (Bcd and Cad) and protein expression of the terminal system proteins (Tll and Hkb) are taken as time-varying inputs to the gap-gene network. (B) knirps mRNA at C14A, with the mesh defined by the positions of the nuclei. High and low expression are indicated by red and blue, respectively (the full data set used in this model is shown in Fig. S4). Data are taken from the BDTNP database. (C) Modeling geometry. A-P denotes the anterior-posterior axis and D-V the dorsoventral axis. The thick band along the A-P axis (from EL position 35% to 92%) shows the geometry considered in 1D models (23). Unless stated otherwise, all embryos will be presented in this orientation, and 1D plots will be along the A-P line. (D) The model. mRNA and protein expression levels for the gap genes hb, Kr, gt, and Kni are modeled on the embryo surface; all mRNA and protein species can diffuse (Methods and Fig. S10). Each gap-gene mRNA is transcribed according to a linear model of transcriptional influences (u is a linear combination of the protein expression levels, p_j). A nonlinear transfer function, g(u), models saturation of the polymerase. The proteins are translated from mRNA using a linear model, and all degradations are first-order processes.

Here, we use recent spatiotemporal mRNA and protein expression profiles measured on the entire cortex of the embryo (16) to calibrate a model for the gap-gene network on the entire surface of the syncytial blastoderm. Our model extends previous studies (18,21–23,25) in several ways: 1), we consider the reaction-diffusion model on the curved two-dimensional (2D) surface of the embryo given by the experimentally measured mesh of nuclei (Fig. 1 B); 2), we explicitly model mRNA and protein species, both of which can diffuse on the embryo surface; 3), in addition to simple regulations, we assume that Hunchback monomers and dimers can carry distinct regulations. We obtain what we believe are novel insights into both the mechanisms leading to patterning and the structure of the gap-gene network that make this process robust. Specifically, we identify a network based on reciprocal repression of gap genes that faithfully patterns the embryo, and we evaluate model predictions on gap-gene mutants against experimental data. Finally, sensitivity analysis in this study reveals that the posterior domain of giant belongs to the most fragile features of the patterning process, and that regulatory interactions involving the maternal gradients tend to be highly constrained.

Material and Methods

Integration of several expression data sets

We combine two databases to assemble a spatiotemporal expression data set for mRNA and proteins spanning the cleavage cycles C12–C14A (Fig. S1 in the Supporting Material). Data are taken primarily from the Berkeley Drosophila Transcription Network Project (BDTNP) (16), which contains three-dimensional (3D) measurements of relative mRNA concentration for >80 genes, and the protein expression patterns for Bicoid, Giant, Hunchback, and Kruppel during C14A. These data were registered on the coordinates of 6078 nuclei on the embryo cortex. The second source is the FlyEx database, which has one-dimensional (1D) quantitative data of protein expression for the four gap genes (hunchback, Kruppel, giant, and knirps) and the three input proteins (Bicoid, Caudal, Tailless). These data span cleavage cycles C10–C14A (10), and correspond to a 10% wide A-P stripe located in the middle of the D-V axis (Fig. 1 C). The two databases show highly consistent expression patterns (Fig. S2). To assemble a complete dataset suitable for modeling, extrapolation is necessary, as some of the 3D data needed to model the genetic network from cleavage cycles C12–C14A are missing (Fig. S1). For Bicoid, 3D protein data do not cover the early times of the simulation. At these times, we scaled the initial 3D profile at each A-P position by a scale factor computed from the 1D FlyEx data. For Caudal, Tailless and Huckebein, only 3D mRNA data are available, and they do not cover the whole time interval. We first extrapolate these data in time, assuming that 1), for caudal and tailless genes, the mRNA is constant from C12 to the beginning of C14; and 2), the concentration of huckebein mRNA is zero at C12. We then apply the translation model (Eq. 3) to simulate the protein from the mRNA data, using model parameters (Eq. 3) that are calibrated on the 1D protein data in FlyEx. The initial condition for each protein is taken from the 1D protein data in FlyEx to the whole embryo surface, assuming rotational symmetry. Table S1 summarizes the data assembly (input regulators are shown in Fig. S4 B). Details are given in the Supporting Material.

The reaction-diffusion model for the gap-gene network

The mRNA abundance, $m_i\left(\overrightarrow{x},t\right)$, at position $\overrightarrow{x}$ for each species i follows a production, decay, and diffusion model, as in previous studies (18,21,23):

$$\frac{\partial }{\partial t}{m}_i\left(\overrightarrow{x},t\right)=f_i\left(p\left(\overrightarrow{x},t\right)\right)-{\lambda }_i^m{m}_i\left(\overrightarrow{x},t\right)+D_i^m{\Delta }_S{m}_i\left(\overrightarrow{x},t\right).$$ (1)

The synthesis term f_i depends on a linear combination of the spatial protein abundances $p_j\left(\overrightarrow{x},t\right)$ for eight (or nine for the model with the Hunchback dimer) transcription factors (j = 1…8) Hunchback, Kruppel, Giant, Knirps, Bicoid, Caudal, Tailless, and Huckebein:

$$f_i\left(p\left(\overrightarrow{x},t\right)\right)=R_i^{m}g\left(\sum _{j=1}^8{T}^{ij}{p}_j\left(\overrightarrow{x},t\right)+h_i\right).$$ (2)

Here, the regulatory matrix, T^ij, describes the effect of protein j on (the promoter region of) gene i. If $T^{ij}>0$, then the gene j activates i, and if $T^{ij}<0$, the gene j represses i. The nonlinear function g(u) describes the saturation of the transcription machinery, and the constant h_i is an offset that sets the basal expression level (cf. below). The argument of g(u), $u={\sum }_j{T}^{ij}{p}_j\left(\overrightarrow{x},t\right)+h_i$, is a linear combination of the protein concentrations. Finally, $D_i^m$is the diffusion constant, and Δ_S the Laplace-Beltrami operator on the surface, S, of the embryo, and ${\lambda }_i^m$ is the mRNA degradation rate.

The translation of mRNA into protein follows a linear model:

$$\frac{\partial }{\partial t}{p}_i\left(\overrightarrow{x},t\right)=R_i^p{m}_i\left(\overrightarrow{x},t\right)-{\lambda }_i^p{p}_i\left(\overrightarrow{x},t\right)+D_i^p{\Delta }_S{p}_i\left(\overrightarrow{x},t\right),$$ (3)

where $R_i^p$ is the translation rate, ${\mathit{\lambda }}_i^p$the protein decay rate, and $D_i^p$ the diffusion constant. No posttranslational regulations are taken into account.

The model is integrated numerically (cf. Details in Supporting Material) on the mesh given by the experimentally measured positions of the nuclei and is calibrated from the data using nonlinear optimization as explained below. For simplicity, we take the production rates $R_i^m$ and $R_i^p$ as constants, since most of the simulation time is spent during a single cleavage cycle, C14. Other authors adapt the rates during interphase or mitosis (9,23). In a similar way, the degradation and diffusion rates are taken as constant and should be interpreted as effective rates valid at lengthscales that are large compared to the internucleus distances. The saturation of the transcription machinery is modeled as in Perkins et al. (23) using the nonlinear function $g\left(u\right)=\left(u/\sqrt{u^2+1}+1\right)/2$. We fix h_i = –2 so that the basal production rate equals 5% of $R_i^m$ when the activity of the transcription factors is zero, i.e., when $\sum _j{T}^{ij}{p}_j\left(\overrightarrow{x},t\right)=0$.

Initial conditions

At the beginning of cleavage cycle C12 (t = –30 min), the four gap-gene proteins are essentially not expressed, except for a maternal domain of hunchback (10,26). Thus, we used zero initial conditions for Kruppel, giant, and knirps and considered the hunchback initial mRNA and protein profiles in FlyEx (10). Specifically, we took the 1D protein profile for Hunchback in Fig. S5 and reconstructed the 3D profile by assuming rotational symmetry. In the absence of mRNA data at early C12, the initial mRNA pattern was taken proportional to that of the protein.

Hunchback dimer

Mass conservation implies that the monomeric and dimeric pools add up to the total Hunchback concentration: H_t = H_m + 2H_d. Supposing that the dimerization process is fast compared to the patterning timescale, we calculate the steady state,

$$H_m=1/4\left(-K_d+\sqrt{K_d^2+8K_d\times H_t}\right),H_d=\left(H_t-H_m\right)/2.$$ (4)

Here, K_d is the dissociation constant, which becomes an additional model parameter. For illustration, the different forms of Hunchback are shown in Fig. S6 for K_d = 80, at the middle of cleavage cycle C14A.

Model Calibration and Error Function

We calibrate the model by fitting the predicted patterns to the experimental data (the total number of data points is n = 255,276; see also Table S2) using constrained nonlinear optimization. We applied a two-step optimization to minimize the score, $S\left(\theta \right)$, equal to the squared residuals ${\chi }^2\left(\theta \right)$ plus a bounding term, $B\left(\theta \right)$, to constrain parameters (cf. Table S3). θ represents the model parameters. In the first step, we use a Covariance Matrix Adaptation Evolution Strategy algorithm (27) to globally optimize parameters along a set of 1D A-P stripes. The second step then performs local optimization on the full model using the Downhill Simplex algorithm (28) (see details in the Supporting Material).

Sensitivity analysis

The second-order Taylor expansion of the function $S\left(\theta \right)$ around the minimum $\hat{\theta }$ leads to

$$S\left(\hat{\theta }+\Delta \theta \right)\approx S\left(\hat{\theta }\right)+\frac{1}{2}{\left(\Delta \theta \right)}^{\text{T}}H\left(\Delta \theta \right),$$ (5)

where H is the Hessian matrix related to the covariance $C=\mathrm{Cov}\left[{\theta }_i,{\theta }_j\right]$ through $C=2{\sigma }^2{H}^{-1}$. The eigenvector associated with the lowest eigenvalue of H (all eigenvalues are positive) gives the direction in parameter space that is the least determined by the data (29). Conversely, the eigenvector associated with the highest eigenvalue of H gives the direction in the parameter space along which the output of the model is highly modified, and it thus represents the most sensitive direction (see details in the Supporting Material).

Software availability

Our MATLAB (The MathWorks, Natick, MA) code, data sets, simulations, and parameter files for the models are posted online at http://3d-flies.epfl.ch.

Results and Discussion

A model with dual activity of Hunchback captures the expression patterns on the whole embryo

To calibrate the gene model, we first assembled a complete data set for mRNAs and proteins by combining the BDGP and FlyEX databases (cf. Methods). The model considers the four gap genes, hunchback, Kruppel, giant, and knirps, during a time interval spanning cleavage cycles C12–C14A (Fig. 1). Except for hunchback, the proteins do not show significant levels before C13 (10), and expression of their mRNA starts after cycle C12. The gene-network model uses as regulatory inputs the measured protein expression profiles of the maternal genes bicoid and caudal and the zygotic genes tailless and huckebein. The latter two genes are induced by the terminal maternal system and are thought to be crucial for correct patterning at the poles (11,12,21). The interval from C12 to C14 occurs before cellularization, when most nuclei are localized at the cortex of the embryo. As time delays of 20–30 min are observed between mRNA and protein accumulation (Fig. S4), we explicitly model both mRNA and proteins. The dynamics is restricted to a thin layer at the surface of the ellipsoidal embryo, which we take as the modeling geometry (Fig. 1 C).

The simplest model, in which each transcription factor (TF) contributes independently, as in earlier work (18,21–23,25), yields a best-fit solution in good agreement with the data for hunchback and Kruppel, whereas the agreement is fair for knirps and poor for giant (Fig. S12 A). The giant pattern is the most problematic, with anterior domains (in the head) that are largely absent in the model and a posterior domain that is faint and misplaced. The poor fit of giant could be due to 1), missing genes in the model, for example head gap genes (30–32) (discussed in Fig. S13 legend and in Section 2 of the Supporting Material); 2), the existence of domain-specific enhancers (33); or 3), more complex regulatory interactions, e.g., those that are context-dependent or involve cooperative interactions between transcription factors (9). The whole-embryo data suggest that the poor fit may be a consequence of the complex regulation of giant by Hunchback. Indeed, Hunchback is thought to inhibit giant in the posterior, thereby setting the posterior boundary of the posterior domain in the giant pattern. However, hunchback and giant are both highly expressed in the anterior part of the embryo (Fig. S14), indicating context-dependent regulatory interactions. Complex regulations by Hunchback have been reported; for instance, Hunchback was shown to both activate and repress Kruppel (34). It was further proposed that Hunchback has a concentration-dependent activity (35,36) through dimerization (37). This dimerization may be mediated by dimerization zinc-finger (DZF) domains in the Hunchback protein (38,39). Moreover, the second finger domain was shown to have a specific function for the repression of Kruppel but is not necessary for the repression of knirps (40).

Although more experimental work is needed to clarify the formation and function of Hunchback dimers, we here study a model in which Hunchback exists as a monomer or dimer that carries independent regulatory functions. To implement this assumption, we explicitly model dimerization by splitting the total concentration of Hunchback protein H_t into mono (H_m) and dimer (H_d) pools and treat the dimerization reaction at equilibrium (cf. Methods). As a consequence, the differential abundance of monomers and dimers acquires position dependence and can thus account for the activation in the anterior and inhibition in the posterior part of the embryo (Fig. S6). The best solution for this extended model performs significantly better (mRNA patterns in Fig. 2 A, proteins and cylindrical projection along the 1D A-P band in Fig. S15) compared to the simple model. The overall error with the data decreased from a root mean-squared (RMS) error of 35.8 to 30.8, and agreement is now fair for both knirps and giant. In particular, we observe two clear domains (anterior and posterior) for giant ; the giant pattern is well captured until t = 20 min (error is minimal at t = 20 min), after which the head domain splits into finer subdomains that are not captured (errors for each species are given in Table S7 and Table S8 for the two models).

Figure 2.

Model for the gap gene network calibrated on the whole surface of the embryo. mRNA patterns. (A) Experimental data (red or dark gray) and best fit (green or light gray) are superimposed for three gap genes. Yellow indicates that data and simulations agree perfectly. Notice that the giant domains are much improved compared to Fig. S12; in fact, the agreement is very good up to 20 min and then deteriorates when the head patterns become very fine. (B) Distribution of best-fit parameters across 21 independent solutions (all solutions have RMS error $\sqrt{S\left(\theta \right)/N}<36$; the maximum expression is set globally to 255). Here, only the interaction matrix T^ij is given. Each column shows regulation of one trunk gap gene by the nine regulators. (C) Reconstructed gene network. Gap genes are shown as circles, where green (light gray) indicates self-activation. Input genes are shown above and below. Activating (respectively inhibitory) links are in green or light gray (respectively, red or dark gray). Darker shading indicates smaller errors on the parameters as computed from Hessian matrix (darker shading indicates smaller errors on the parameters) as computed from the Hessian matrix (Section 6 in the Supporting Material). The T matrix and other inferred parameters are given in Table S5 and Table S6.

Thus, the extended 2D model is able to generate better A-P patterns for the gap genes. In particular, the dual regulation by Hunchback is a dynamically plausible scenario to better capture the giant pattern in the head.

Reconstructed gap-gene network

To identify statistically significant network interactions, we ran independent optimizations using random initial seeds. This allowed us to assess the variability in the estimated model parameters (Fig. 2 B). The reconstructed consensus network (Fig. 2, B and C) shows a topology that largely overlaps with experimental studies: the maternal gradients Bicoid and Caudal activate the gap genes (8,13) while the terminal system proteins Tailless and Huckebein act as repressors (11,12). Specifically, 26 of the 32 (81%) regulatory links are in agreement with a recent 1D model (21). An important finding was that the gap genes are mostly mutually inhibitory, which is compatible with results seen in earlier work (18,22,23). Although puzzling mechanistically, gap genes show self-activation, which is supported experimentally in the case of Hunchback (41) and found computationally for Kruppel and giant (18). In fact, the self-activation of gap genes was shown to sharpen the expression patterns, owing to cooperativity in the activation function (41,42). The solutions show that Hunchback monomers and dimers carry opposite regulations (Fig. 2, B and C). Moreover, the model exhibits the typical shifts of posterior domains toward the anterior during cleavage cycle C14 (9,18,43), notably for the Kruppel domain and the knirps (Fig. 2 A). Such shifts were shown to result from sequential repressions (9,18), namely, Hunchback inhibits giant, Giant inhibits knirps, Knirps inhibits Kruppel, as in the consensus model.

Among the differences between our work and previous works, we find that Knirps represses giant. This seems plausible, as giant and knirps show clear anticorrelated patterns on the whole embryo (Fig. S3 B). Moreover, loss-of-function mutants showed that knirps represses giant in the head (15). This interaction can be further tested by ectopically expressing knirps from the promoter of a ventral gene such as twist; we would then expect reduced expression, specifically on the ventral portion of the giant domains. In addition, we find repression of hunchback by Tailless, but we currently have no clear explanation, as Tailless is known to activate hunchback in the posterior (44).

Taken together, the reconstructed network of interactions is largely consistent with previous work and also suggests a few novel interactions. Although there may be alternative ways to model patterns in the head, our approach lends support to the reports that Hunchback may carry independent functions as monomer and dimer. Mutant experiments with two different Hunchback interaction domains (40), one of which was later suggested to mediate dimer formation (37,45), show that the dimerization mutant has a distinct phenotype. To better characterize the possible dual function of Hunchback in patterning gap genes in vivo, a key experiment would now be to perform extensive spatiotemporal measurements of all four gap genes on those putative dimerization mutants.

Alternative and extended models

We considered alternatives to the dimerization of Hunchback. For example, a scenario in which two independent enhancers for giant (33) were modeled did not lead to a comparable outcome. A further possibility would be to assume differential regulation by Hunchback monomers and putative Bicoid-Hunchback dimers (46). Additional interactions, e.g., concentration- or context-dependent regulation, may become necessary for more complex patterns such as those observed for the pair-rule, or segmentation-polarity genes. For instance, the latter are known to use domain-specific cis-regulatory modules (33), in which cooperative interaction and distance constraints play an important role (37,42,47–49). To further improve the fine structure of giant and knirps domains in the head, one may include head gap genes, notably Slp1 (Fig. S13 and Section S2 in the Supporting Material).

The A-P and D-V patterning systems are often described as independent (14), although there are also indications that cross-regulation between the two axes may be important (50). Recent chromatin immunoprecipitation (ChIP) experiments showed that the D-V regulators Twist and Snail bind the promoter of knirps (51,52), which supports the possibility of cross talk. In addition, recent work on the D-V network in the bone morphogenetic protein demonstrated that the pMad signaling output shows A-P-dependent noise characteristics (53). D-V genes can readily be added to the 2D model. In fact, the model showed that the anterior domain of knirps extends too far into the dorsal part of the embryo (Fig. 2 A), which may be due to an overactivation by Bicoid or Hunchback or, more likely, to the absence of regulation by D-V genes. Moreover, the anterior domain of knirps shows a clear ventral asymmetry in the data, which cannot be easily explained by the nearly axially (A-P) symmetric Bicoid profile (Fig. S19 A). Also, the data show a ventral broadening of the posterior domain of knirps (Fig. S19 B). Ventrally expressed D-V regulators such as twist or dorsal may explain these asymmetries. As proof of principle, we included Twist as an additional input regulator in the model. The best-fit solution shows an RMS error (31.4) similar to that in Fig. 2, with the caveat that the posterior hunchback domain is slightly shifted to the anterior (Fig. S9 A). However, the solution shows clear improvements on the ventral domains of the knirps and giant mRNA patterns (Fig. S9 B). As expected from the data, the solution shows that Twist activates knirps and represses giant, consistent with ChIP experiments. In addition, the model predicts that Twist activates Kruppel.

Thus, when Twist is added as a D-V input regulator, the model predicts a strong reduction of the anterior domain of knirps. Partial evidence that this may be correct is given in Zeitlinger et al. (51), i.e., a regulatory region of knirps is shown to be bound by Twist driving expression in the head. The second model prediction is a thinning of the ventral domain of knirps. More generally, it is now realistic to develop 2D models in which the D-V and A-P genes interact reciprocally. Such models should provide novel insights into the regulation of many genes (for example, rho, D, and MESR3) whose patterns suggest the integration and possibly interaction of information from both axes (16).

The model predicts patterns of gap-gene mutants

To assess whether the model correctly predicts patterns in knockout mutants for the gap genes, we simulated the mutants and compared the simulations with experimental data from Kraut and Levine (15) and Hülskamp et al. (40), by quantifying the original in situ hybridizations for giant and knirps from the original images. It is important to note that the mutants were not used for model calibration. Knocking out hunchback in the embryo leads to a small posterior shift of the giant posterior domain in vivo, whereas its anterior domain almost disappears (Fig. 3 A). The model can reproduce these phenotypes. Specifically, the anterior domain totally disappears, probably due to the lack of activation by Hunchback, whereas the posterior domain shifts toward the posterior pole. This mutant also leads to a shift toward the anterior of the posterior domain of knirps, indicating repression of knirps by Hunchback (40), but it also shows a strong reduction of knirps domain in the head, where both genes are coexpressed (Fig. S3 A). Our model, in which Hunchback represses knirps at low concentration and activates it at high concentration is able to reproduce this phenotype (Fig. 3 D). Knocking out Kruppel in the embryo leads to a shift of the posterior giant domain toward the center, whereas the anterior domain is not affected. Our model fails to reproduce this accurately (Fig. 3 B); indeed, before t = 0, the posterior domain of giant shifts toward the anterior, eventually fusing with the anterior domain. Knocking out knirps in the embryo leads to the disappearance of the posterior domain of giant, which seems surprising since Knirps is a repressor of giant. It is interesting to note that the model reproduces this effect; moreover, it suggests a possible mechanism. Namely, it may be that giant is indirectly repressed through the extension of the Kruppel domain into the posterior (Fig. 3 C), due to the absence of the inhibition of Kruppel by Knirps. However, this may not be the only cause, as Kruppel shows only a moderate extension in knirps mutants (54). Among the solutions shown in Fig. 2 B, 18 of 21 showed correct behavior in the knirps mutant, whereas only 5 of 21 showed correct behavior in the hunchback mutant (Fig. S16). No solutions showed correct patterns in the Kruppel mutant. This might be because patterning in this model depends too strongly on the interactions among gap genes instead of the initial activation by the maternal gradients, and that consequently the repressed genes spread too broadly in the mutants. Thus, our best model accurately predicted three of four single-knockout mutants. In comparison, a previously published model (18) correctly predicted one of four (Fig. 3). To make a set of what we consider novel predictions, we compared simulations of all double mutants and found that mutants involving tailless tend to show severe phenotypes (Fig. S20). Data on the kni and Kruppel double mutant indicate that the rescue of the posterior giant domain (55) is too pronounced in the simulation. Validating all pairs would require significantly more experimental work but would provide a highly stringent test for the model.

Figure 3.

3D model predicts patterns of gap-gene mutants. The line-graph data for Giant (A–C) and Knirps (D) patterns in mutants (Gt data) are quantified from Kraut and Levine (15) and Hülskamp et al. The times for the mutant stainings correspond to 30 min. The A-P projections of the modeled proteins at 30 min are also shown (Model), as is a simulation from a published model (Circuit 28008 in Jaeger et al. (18)) (Model 2). In the side views, simulated mRNA patterns for the wild-type (green or light gray) are shown together with the simulated mutant pattern (red or dark gray). Yellow or white indicates no change between the simulated wild-type and the mutant patterns. (A) hb null mutant. The absence of the anterior gt mRNA domain is correctly predicted up to t = 40 min. In a similar way, the posterior extension of the posterior domain is also consistent. (B) Kr null mutant. The gt pattern for this mutant is poorly predicted. (C) kni null mutant. The vanishing of the posterior gt domain is correctly predicted. (D) Kni in hb null mutant. This mutant is correctly predicted.

Sensitivity analysis finds correlated network interactions

To gain further insights into the importance of individual, or groups of, parameters, we performed local sensitivity analysis for the ensemble of optimal models in Fig. 2 B. The analysis of pairwise correlations between model parameters showed that the strongest correlations are shared among the different solutions (Fig. S17, A and B). In particular, we observe expected dependencies between some parameter pairs, for example, higher diffusion in the mRNA can be compensated by lower protein diffusion. Likewise, increased mRNA production can be compensated by increased mRNA degradation, reduced protein production, or inhibition by other genes (Fig. S17). It is of further interest that the anticorrelation between the regulatory influences of Hunchback and Bicoid (Fig. S18 B, arrows) points toward the possible redundancy of these two regulators. Incidentally, comparative studies indicate that Bicoid could have substituted hunchback and orthodenticle (otd) functions during the evolution of Diptera (56), so the observed correlation may reflect an evolutionary origin.

Robust and fragile expression domains in the gap-gene network

The study of the robustness of the gap-gene patterns to different sources of variability, and the notion that the network can buffer noisy inputs (25), for instance, to sharpen the Hunchback pattern (41,57–59), has attracted considerable attention. Here, in a systematic study of how groups of parameters collectively control the gene regulatory network (60), we performed a covariance analysis of the parameters around local minima in the fitting function (60). Eigenvectors of the covariance matrix with small eigenvalues represent constrained, or fragile, directions in parameter space. Specifically, changing parameters along this direction induces large phenotypic changes in the expression domains. Conversely, eigenvectors with large eigenvalues indicate parameter changes that have little impact on the patterning. Analyzing the different solutions, we find that vectors cluster into groups (Fig. 4 A), indicating that the properties of the correlation matrices are shared across solutions, a property that holds for both stiff and soft modes (Fig. 4, B–D). The stiffest mode indicates that regulation of Kruppel and giant by the maternal inputs Tailless and Caudal are highly sensitive interactions in the network and thus need to be tightly controlled (Fig. 4 C). The phenotype resulting from perturbation of parameters along this direction emphasizes that the posterior domain of giant is the most fragile feature of the network (Fig. 4 C, side view). It is intriguing that this coincides with findings on the function of the C-terminal binding protein (CtBP), a broadly acting corepressor that potentiates many repressors in the gap-gene network. Upon loss of CtBP function, which we interpret in our context as increased global noise, the posterior domain of Giant is found to be particularly sensitive (61). On the contrary, the softest mode involves regulation by Huckebein. Perturbations along this direction do not affect patterning (Fig. 4 D, side view). This can be explained by the fact that the Huckebein domain is very small and does not play a dominant role in the model, a conclusion which was also reached by Ashyraliev et al. (21).

Figure 4.

Gap gene network is robust and fragile. (A) Cluster of stiffest eigenvectors. The mean and standard deviation of the cluster are indicated. Arrows indicate the important parameters in that mode (1–6), from left to right (see also C): activation of Kr by Cad (1), repression of Kr by Tll (2), activation of gt by Cad (3), repression of gt by Tll (4), mRNA and protein production of Kr (5), and mRNA and protein production of gt (6). (B) Cluster of softest eigenvectors. The mean and standard deviation of the cluster are indicated. Arrows indicate, from left to right, activation of kni by Hkb (7), and repression of hb (8) and Kr (9) by Hkb. (C and D) Perturbing the network along stiff (C) and soft (D) directions. (C, left) Perturbation associated with the stiffest eigenvector (for the solution in Fig. 2) is shown on the network model. Numbers refer to important parameters defined in A. The perturbation is taken as ${\tilde{p}}_i=p_i+q_i$, where p_i is the optimum and $q_i=\mathit{\epsilon }{v}_i$ is a relative perturbation along the eigenvector $\overrightarrow{v}$, with $\left|\overrightarrow{q}\right|=1$. Components of the eigenvectors determine the color intensity of the links. Green or light gray indicates that the perturbation increases the magnitude of the regulation (positive or negative) and red or dark gray for the opposite. This eigenvector involves mainly the control of Kr and gt by maternal inputs Tll and Cad. (C, right) Simulation of wild-type (green or light gray) and perturbed parameters, ${\tilde{p}}_i$ (red or dark gray), indicates the high stiffness of this mode, as seen by the loss of the posterior gt domain. Yellow or white indicates that the perturbation has no effect. Here, the mRNA at 20 min is shown. (D) Same as in C, but for the softest mode. The perturbation has no effect in this case, even though the perturbation, $\overrightarrow{q}$, has the same magnitude (norm) as in C. Numbers refer to parameters defined in B.

In summary, constrained directions tend to be strongly determined by regulation of the maternal inputs Bicoid, Caudal, and Tailless, whereas robust directions involve compensation mechanisms, e.g., between production and degradation processes or between mRNA and protein diffusion. We would predict that global perturbations such as those induced by general coregulators like CtBP, chromatin modifiers, and also temperature changes, may lead to phenotypes that will resemble those along the computed sensitive directions.

Conclusion

Modeling gene expression networks on the cortex of fly embryos

We followed a data-driven approach to reconstruct the gap-gene network in Drosophila melanogaster on the full geometry of the embryo, which was made possible by the availability of spatiotemporal data sets for the mRNA and proteins involved (16). We calibrated a mathematical model that explicitly includes protein and mRNA species. To keep the model complexity low, we started as in previous 1D models, where transcription regulators contribute independently (18,22,23,25). Due to lack of fit of the simplest model, we assumed that Hunchback monomers and dimers may carry out independent regulatory functions, as suggested experimentally. This led to significant improvements, notably in the anterior giant pattern. The reconstructed network showed a topology largely consistent with experimental studies and earlier modeling work, with a few novel (to our knowledge) predictions. The network solution indicates that patterning of the gap genes follows the principle of activation through the maternal gradients Bicoid and Caudal proteins, repressions by the terminal system proteins Tailless and Huckebein, and mutually inhibitory interactions among the four trunk gap genes. We assessed our model using null mutants for the gap genes. The results showed that we predict a majority of patterns correctly, with some failures, as in the case of the Kruppel mutant. Closer analysis of these failures suggested that the model presented here may overestimate interactions among gap genes compared to initial patterns and maternal inputs. This is also apparent in our analysis of all double mutants, where the knirps and Kruppel double-mutant data still show separate Giant domains (55) that are merged in the simulations (Fig. S20).

Caveats

Like most reaction-diffusion models proposed for the gap-gene network, the type of gene model used here does not attempt to model patterning straight from the regulatory DNA sequences. Although this may become feasible as the regulatory sequences for each of the gap genes become better known (33,52), this model essentially assumes one enhancer per gene, with a simplified linear model of regulatory influences by each factor. These influences are effective and cannot necessarily be interpreted as mechanisms reflecting interactions between transcription factors and cis-regulatory sequences. Approaches more faithful to cis-regulation, which use thermodynamic models for transcription-factor binding as the basis for regulatory interactions, have been proposed (42,47), but they are not typically fully integrated with reaction-diffusion models. Nothing in principle, aside from the need to know the regulatory regions and the specification of a regulatory logic, prevents the development of such fully integrated models.

Though the model presented here identified a consistent regulatory network, a comparison of mutants highlights its current limitations, for example in predicting the Kruppel mutant. There exists a large body of qualitative mutant data in the literature that could be exploited further. As these data tend to be heterogeneous, static, or incomplete, new data on mutants using a similar quantitative acquisition pipeline would clearly help to improve models further. Although our goal was to construct a model of the A-P axis using only a small number of genes, some of the model's shortcomings might be due to missing genes in the network, notably the head gap genes, as discussed in the legend of Fig. S13 and in section 2 of the Supporting Material.

Insights and predictions

The model calibration and sensitivity analysis led to insights into individual interactions predicted to be important for patterning, as, for example, the direct or indirect repression of giant by Knirps, and also identified properties of the full network that would be difficult to anticipate, such as sets of parameters that define either robust or highly sensitive expression domains. Notably, we found that expression domains in this model are most sensitive to regulatory interactions via maternal inputs. We discussed several predictions that call for further experiments. In order of importance, we proposed the following: 1), a test on Hunchback mutants to establish firmly that Hunchback monomers and dimers can carry independent regulatory functions; 2), a test on ectopic Knirps expression to validate the repression of giant by Knirps; 3), phenotyping of all double mutants in the gap genes for model validation; and 4), identification of robust and sensitive domains under global perturbations to validate our sensitivity analysis.

In conclusion, we demonstrated that modeling of segmentation on the entire surface of the early Drosophila can now be performed. Analysis on the network model identified robust and fragile features of the network. For instance, the analysis revealed that the posterior domain of giant is the most fragile feature of the model. One particularly interesting finding is the anticorrelation in the activation of trunk gap genes by Bicoid and Hunchback (Fig. S18). Thus, our model is consistent with the existence of an viable evolutionary path between organisms with and without bicoid. Finally, it should be relatively straightforward to extend such modeling to the pair-rule and other segmentation networks, which show richer patterns than the gap genes.