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Conceived and designed the experiments: M Ashyraliyev M Akam J Jaeger. Performed the experiments: M Ashyraliyev K Siggens J Jaeger. Analyzed the data: M Ashyraliyev H Janssens J Blom J Jaeger. Contributed reagents/materials/analysis tools: K Siggens J Blom. Wrote the paper: M Ashyraliyev H Janssens J Jaeger.

The authors have declared that no competing interests exist.

The early embryo of

Currently, there are two very different approaches to the study of pattern formation: Traditional developmental genetics investigates the role of particular factors in great mechanistic detail, while newly developed systems-biology methods study many factors in parallel but usually remain rather general in their conclusions. Here, we attempt to bridge the gap between the two by studying the expression pattern and function of a particular developmental gene—the terminal gap gene

How genes contribute to pattern formation is one of the central questions of modern developmental biology. Traditionally, this question has been addressed using genetic and molecular approaches. Although very powerful, these approaches have several important limitations: First, it is difficult to study expression features which are not specifically affected by a particular mutation (see below). Second, there is always some remaining ambiguity whether an interaction is direct or not

The patterning system we study is the gap gene network of

The expression domains of gap genes in the posterior region of the embryo shift towards the anterior over time

This mechanism suggests that the posterior

The gene circuit method uses mathematical models of gene networks as computational tools to extract regulatory information from quantitative, spatial gene expression data (

(A) Reverse engineering gene regulatory networks using the gene circuit method: A mathematical (dynamical) model of the network is fit to quantitative, spatial gene expression data using combined global and local non-linear optimisation approaches. The resulting gene circuits, consisting of specific estimated sets of parameter values, define regulatory interactions among genes within the network (its regulatory topology). This topology is not defined a priori, but is extracted from the quantitative expression data by the fitting procedure. The resulting dynamical behaviour of the system can be analysed using graphical or numerical methods. (B) Previous gap gene circuit models used concentrations of the protein products of gap genes

To simulate the dynamics of gap gene expression, we use gene circuit models (see

Previous gene circuit models included the gap genes

To avoid such problems, we use a revised model—first introduced in

The modelling framework outlined above does not predetermine any specific regulatory interactions within the gene network. Instead, these interactions—and hence the regulatory topology of the network—are obtained by fitting the model to the data (

The resulting models are analysed in various ways to gain new biological insights. Analysis of the dynamical behaviour of our models allows us to associate specific regulatory interactions and mechanisms with specific features of gene expression (such as the establishment of a new expression domain or the formation, sharpening or shift of an expression domain boundary). This can either be achieved by graphical examination of specific interactions in the model

In the sections that follow, we analyse the protein expression pattern of

Polyclonal antiserum against Hkb protein was raised as follows: A full-length cDNA clone of

Blastoderm stage embryos of

Embryo images were processed to yield integrated expression data as described in the

This figure shows images of representative embryos stained against Hkb protein for each time class (T1–T8) during cleavage cycle 14A (left), with their corresponding quantified Hkb expression profiles (middle). Integrated Hkb expression data for each time class are shown, and compared to integrated profiles of Bcd, Cad, Hb, Kr, Gt, Kni, and Tll from the FlyEx data base

Quantitative integrated expression data for Bcd, Cad, Hb, Kr, Kni, Gt and Tll are taken from the FlyEx database. Concentration measurements were taken at C13, as well as eight regularly spaced time points during C14A (T1–T8)

Gene circuits are hybrid dynamical models with two continuous and one discrete rule: (1) interphase, (2) mitosis and (3) division

Gap gene circuits include cleavage cycles 13 and 14A (ending at the onset of gastrulation;

Kr, Kni, Gt, Tll and Hkb proteins are not present at significant levels before C13 (see

Equation (1) contains

The quality of a fit of the model to the data is measured by the root mean square (

We used a two-step optimisation algorithm to minimise the cost function (3): Global optimisation by the parallel Lam Simulated Annealing (pLSA) algorithm

Based on previous studies using gap gene circuits

Here, we only provide a brief overview of the equations used for calculating confidence intervals and parameter correlations (see

Model optimisation results in a vector

The correlation coefficient between

We quantified expression levels of Hkb protein in blastoderm stage embryos of

Hkb protein can first be detected in both its anterior and posterior domain at C13 (data not shown). Protein levels rapidly increase during early C14A (T1–T3). At this stage, peak levels are very similar in both domains, although the anterior is very slightly weaker than the posterior one. Subsequently, the anterior domain gradually weakens (T5–T8), while protein levels in the posterior domain remain more or less constant (although there may be a slight decrease in concentration at T8). The peaks of both domains remain at a constant position throughout (5% A–P position for the anterior, 95% for the posterior domain). Similarly, the width of both domains remains approximately constant: the anterior domain extends back to about 10–15% A–P position, while the posterior domain reaches as far as 85–90%, both domains covering about 10–15% A–P position in each terminal region. None of the two Hkb domains show any discernible D–V asymmetry at any point in time before gastrulation.

Our quantitative

Local search with the WLS cost function was performed using selected OLS parameter estimates as starting points: the 39 solutions without, and the lowest-scoring 90 solutions with defective

Integrated expression profiles from the FlyEx data base

Gap gene expression patterns produced by circuits from the selected OLS and WLS fits are similar, although variability between different models is somewhat larger for OLS (compare

The distribution of regulatory weights for each regulator (columns) and regulated gene (rows), is shown for OLS fits (above) and WLS fits (below). Number triplets show how many parameter estimates (from independently obtained optimisation solutions) fall into the regulatory categories of ‘repression’ (parameter values

On the other hand, the dynamic expression of

Estimates of regulatory weights obtained by both OLS and WLS fits were classified into the following three categories: ‘activation’ (parameter values

Apart from only two interactions, the predicted regulatory topologies agree between OLS and WLS fits. In the case of OLS, Hkb activates

A similar pattern can be observed when comparing our new 4-gene models with earlier 6-gene circuits (cf.

The regulatory structure of the gap gene system shown in

Results in

(A) Dependent (green) and independent (red) confidence intervals are shown across 39 OLS solutions (horizontal axes) to illustrate a regulatory weight which is well determined (

Parameter determinability analysis based on independent confidence intervals for OLS and WLS fits is summarised in

11 and 12 (out of 32) regulatory parameters cannot be determined for OLS and WLS fits, respectively. Among them are several of the interactions predicted to fall into the ‘no interaction’ category in

Previous quantitative analyses of the gap gene system suggested a set of basic regulatory mechanisms based on broad activation of gap genes by maternal co-ordinate proteins, and spatially specific gap-gap cross-repression

In addition, our current gap gene circuits now accurately reproduce expression in the posterior

Expression profiles from the model (left), regulatory contributions (middle) and change in Hb protein concentration (dashed) vs. Hb protein levels (solid lines; right) are shown in the posterior region of the embryo. Horizontal plot axes represent percent A–P position as in

The posterior

The late initiation of

Our analysis of parameter determinability indicates that those parameters with particularly large confidence intervals could be fixed to specific values—within the non-empty intersections of their dependent intervals—without affecting the quality of the fits. Diffusion rates, for example, show large confidence intervals, despite not being significantly correlated with other parameters (see also below). Therefore, fixing their values during optimisation (to averaged values based on previously found estimates:

We used local search with 60 initial parameter sets arbitrarily chosen from the previously found 117 WLS parameter sets. Additionally, we performed 20 global optimisation runs with these parameters fixed. From the resulting solutions, we selected 66 circuits which have low WLS values (about

The network topology shown for WLS runs in

The occurrence of non-determinable parameters is often caused by correlations between parameters

Finally, the last two interactions which are only weakly determined are the activation of

After regulatory weights of gap gene circuits have been estimated based on wild-type expression data, analysis of mutants can be conducted

The only known alteration of gap gene expression in

Simulated expression profiles of Hb, Kr, Gt and Kni (left to right) in

Embryos mutant for

Our results constitute a comprehensive, integrative analysis of the expression and function of the terminal gap gene

First, we were able to increase the efficiency of optimisation, and the consistency of parameter estimates, by using weighted least squares (WLS) instead of ordinary least squares (OLS) for optimisation. The use of a WLS cost function also reduces the need for human intervention when selecting solutions for analysis, since it prevents the occurrence of minor (but biologically significant) patterning defects such as the ectopic

Second, analysis of parameter determinability

In terms of the biology, we first discuss the expression and regulation of

These results are entirely consistent with what we know about

But how does Hkb affect regulation of other gap genes? The regulatory mechanisms for the expression of the trunk gap genes

Gap domains are shown schematically, with anterior to the left, posterior to the right. Background colours indicate the most prominent activating input to each domain. Auto-activation is indicated by double-arrows. T-bars indicate repressive gap-gap cross-regulation (thickness of the bars indicates repressive strength). See text for details.

The most significant improvement of our models over earlier ones is that they now correctly reproduce the expression and shift of the posterior

While our models reproduce repressive effects on posterior

There is another unresolved question concerning the posterior

For the posterior domain of

Apart from the regulation of the posterior

Little is known about the function and effect of the anterior

Apart from correct posterior

Apart from mutations in

On the other hand, our current models still cannot accurately reproduce null mutants of the trunk gap genes

Why is all this important? After all, our results establish that

This view is supported by the following: First, there are no known mutants that affect any of the gap domain shifts individually. Moreover, evidence from an analysis of the dynamical behaviour of gap gene circuits suggests that all trunk gap genes participate in the shift mechanism in an integrated way

Changes in the regulation of the posterior

Gap gene expression data used for model fitting. Integrated expression patterns (dark lines) with corresponding standard deviations (lightly coloured areas) are shown for Hb (red), Kr (green), Kni (purple) and Gt (blue) at cleavage cycle 13 (C13) and eight time classes (T1–8) during cleavage cycle 14A. Relative protein concentrations are plotted against percent A–P position (where 0% is the anterior pole). All patterns shown are from the FlyEx data base:

(0.41 MB PDF)

Model output compared to quantitative expression data (OLS fits). Integrated expression profiles from the FlyEx data base (

(0.51 MB PDF)

Model output compared to quantitative expression data (WLS fits). Integrated expression profiles from the FlyEx data base (

(0.95 MB PDF)

Parameter determinability analysis: confidence intervals for OLS fits. Columns represent regulators, rows regulated genes. Dependent (green) and independent (red) confidence intervals are shown across all selected 39 OLS solutions (horizontal axes). Vertical axes represent parameter values; note that scales vary between plots.

(0.64 MB PDF)

Parameter determinability analysis: confidence intervals for WLS fits. Columns represent regulators, rows regulated genes. Dependent (green) and independent (red) confidence intervals are shown across all selected 117 WLS solutions (horizontal axes). Vertical axes represent parameter values; note that scales vary between plots.

(1.03 MB PDF)

Parameter determinability analysis: confidence intervals for WLS fits with fixed Hkb weights (WLSfh). Columns represent regulators, rows regulated genes. Dependent (green) and independent (red) confidence intervals are shown across all selected 66 WLSfh solutions (horizontal axes). Vertical axes represent parameter values; note that scales vary between plots.

(0.73 MB PDF)

Mean correlation matrix for WLS fits with fixed Hkb weights (WLSfh). Parameter correlations are arranged in blocks per regulated gene. Abbreviations indicate regulator (for regulatory weights) or parameter (for promoter strength and decay rates). Positive correlations are shown in green, negative correlations in blue. For clarity, only correlation values above 0.5 are shown. Note that most correlations occur between parameters involved in the regulation of the same gene (diagonal blocks of the matrix).

(0.42 MB PDF)

Estimated parameter values are shown for all OLS, WLS and WLSfh optimisation runs.

(0.11 MB ODS)

We thank Zoltan Cseresnyes for help with confocal microscopy, Luke Jostins for contributions to the Matlab code for image segmentation, and Ekaterina Myasnikova for help with background removal for Hkb data. Model optimisation by simulated annealing (pLSA) was performed using the Darwin Supercomputer of the University of Cambridge High Performance Computing Service (