The segmentation gene network in Drosophila embryo solves the fundamental problem of embryonic patterning: how to establish a periodic pattern of gene expression, which determines both the positions and the identities of body segments. The gap gene network constitutes the first zygotic regulatory tier in this process. Here we have applied the systems-level approach to investigate the regulatory effect of gap gene Kruppel (Kr) on segmentation gene expression. We acquired a large dataset on the expression of gap genes in Kr null mutants and demonstrated that the expression levels of these genes are significantly reduced in the second half of cycle 14A. To explain this novel biological result we applied the gene circuit method which extracts regulatory information from spatial gene expression data. Previous attempts to use this formalism to correctly and quantitatively reproduce gap gene expression in mutants for a trunk gap gene failed, therefore here we constructed a revised model and showed that it correctly reproduces the expression patterns of gap genes in Kr null mutants. We found that the remarkable alteration of gap gene expression patterns in Kr mutants can be explained by the dynamic decrease of activating effect of Cad on a target gene and exclusion of Kr gene from the complex network of gap gene interactions, that makes it possible for other interactions, in particular, between hb and gt, to come into effect. The successful modeling of the quantitative aspects of gap gene expression in mutant for the trunk gap gene Kr is a significant achievement of this work. This result also clearly indicates that the oversimplified representation of transcriptional regulation in the previous models is one of the reasons for unsuccessful attempts of mutant simulations.
Systems biology is aimed to develop an understanding of biological function or process as a system of interacting components. Here we apply the systems-level approach to understand how the blueprints for segments in the fruit fly Drosophila embryo arise. We obtain gene expression data and use the gene circuits method which allow us to reconstruct the segment determination process in the computer. To understand the system we need not only to describe it in detail, but also to comprehend what happens when certain stimuli or disruptions occur. Previous attempts to model segmentation gene expression patterns in a mutant for a trunk gap gene were unsuccessful. Here we describe the extension of the model that allows us to solve this problem in the context of Kruppel (Kr) gene. We show that remarkable alteration of gap gene expression patterns in Kr mutants can be explained by dynamic decrease of the activating effect of Cad on a target gene and exclusion of Kr from the complex network of gap gene interactions, that makes it possible for other interactions, in particular between hb and gt, to come into effect.
Citation:Kozlov K, Surkova S, Myasnikova E, Reinitz J, Samsonova M (2012) Modeling of Gap Gene Expression in Drosophila Kruppel Mutants. PLoS Comput Biol 8(8): e1002635. doi:10.1371/journal.pcbi.1002635
Editor: Jennifer L. Reed, University of Wisconsin-Madison, United States of America
Received: January 23, 2012; Accepted: June 25, 2012; Published: August 23, 2012
Copyright: © Kozlov et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding:This work was supported by a grant of the National Institutes of Health [RR07801] (www.nih.gov/ <http://www.nih.gov/>); the Civilian Research & Development Foundation [Grant Assistant Program RUB1-1578] (www.crdf.ru/ <http://www.crdf.ru/>); the State scientific contracts with Russian Ministry of Education and Science N 14.740.11.0166 and 11.519.11.6041 (http://eng.mon.gov.ru/) and the Russian Foundation for Basic Research [09-04-01590-a,10-01-00627-a, 11-04-01162-a, and 11-01-00573] (www.rfbr.ru/rffi/eng <http://www.rfbr.ru/rffi/eng>). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
The segmentation gene network in early Drosophila embryo provides a powerful model system to study the role of genes in pattern formation. This network solves the fundamental problem of embryonic patterning: how to establish a periodic pattern of gene expression, which determines both the positions and the identities of body segments , . The developmental process which performs this task is called segment determination. The fruit fly segments are arranged sequentially along the anterior-posterior axis of the embryo. All segments are determined simultaneously during the blastoderm stage, just before the onset of gastrulation .
The segmentation genes have been subdivided into 4 classes based on their mutant phenotype , . The maternal coordinate genes are expressed from the mother and form broad protein gradients in the anterior, posterior or terminal regions of the embryo –. Other genes, which belong to gap, pair-rule and segment-polarity classes, are zygotic, i.e expressed in the embryo. Most of segmentation genes encode transcription factors, which in turn regulate the expression of many other genes, including segmentation genes themselves. It was demonstrated by genetic analysis that segmentation genes form a hierarchical regulatory cascade, in which genes in higher layers (e.g. maternal coordinate genes) regulate genes in lower layers (e.g. gap genes), but not vice versa. In addition genes in the same hierarchical level interact with each other.
The gap gene system establishes discrete territories of gene expression based on regulatory input from a long-range protein maternal gradients, Bicoid (Bcd) and Hunchback (Hb) in the anterior and Caudal (Cad) in the posterior of the embryo , . Gap genes Kr, kni, hb, gt and tll are expressed in from one to three domains, each about 10–20 nuclei wide . Early gap gene expression of the trunk gap genes Kr, hb, gt and kni is established through feed-forward regulation by maternal gradients, after initial establishment gap domain borders sharpen, moreover both sharpening and maintenance of gap domain boundaries requires gap-gap cross-regulatory interactions . This process is accompanied by the anterior shift of Kr, kni and gt expression domains in the posterior region of the embryo , , .
Kr plays a central role in segmental pattern formation as indicated by strong alteration of expression patterns of almost all zygotic segmentation genes in mutants –. Kr null mutants show deletion of thoracic and anterior abdominal segments as well as frequent mirror duplications in the abdomen , . At the level of gene expression this mutation manifests in the large shift of posterior Gt domain, resulting in overlap of positions of posterior Gt and Kni domains , . During sharpening and maintenance stage of gap gene expression Kr acts a repressor of gt and hb , , . The repression of gt, which expression domains are strictly complementary to those of Kr, is strong, while the effect of Kr on hb is more subtle –. It was observed in assays with cell lines carrying reporter constructs that the regulatory effect of Kr is concentration-dependent: Kr monomer is transcriptional activator, while at high concentrations Kr forms a homodimer and becomes a repressor that function through the same target sequence as the activator. However it is difficult to establish whether such an effect occurs at physiologically relevant regulator's concentrations .
The segmentation gene network is one of the few examples of developmental networks studied using data-driven mathematical modeling , –. These models fall into two categories. The phenomenological models do not require any a priory information about regulatory mechanism ,  and try to reconstruct it by solving the inverse problem of mathematical modelling. A major shortcoming of these models is that their parameters have no explicit connection to the genomic DNA sequence. The second modelling approach seeks to extract information about gene regulation from the sequences of cis-regulatory regions and the measured or inferred binding of sequence-specific transcription factors to these elements –, however it still neglects major features of the transcription process, such as chromatin structure and modifications, binding site orientation and proximity to transcription start site, etc. Current simplifications and unknown features limit the predictive power of these models, but more powerful and complex models may be generated in future using better datasets such as in vivo transcription factors occupancy, relative accessibility of different DNA regions, in vivo data on interplay between different transcription factors, nucleosome and chromatin remodelling enzymes.
In this paper we apply a phenomenological model known as gene circuits to reconstruct the gap gene network in Kr null mutants. This model considers a row of nuclei along the A-P axis of the embryo. Between nuclear divisions the model describes three basic processes, namely protein synthesis, protein decay and diffusion of proteins between neighboring nuclei of syncitial blastoderm. A few basic assumptions about eukaryotic transcriptional regulation were incorporated into the model. First a sigmoid regulation-expression function was used to introduce regulatory inputs into the model. Secondly, each regulatory interaction can be represented by a single parameter which sign indicates the type of regulatory interaction: activation (if it is positive), repression (if negative), no interaction (if it is close to zero). Third it was assumed that regulatory inputs are additive and independent of each other.
The gene circuit models were successfully applied to correctly reproduce the quantitative features of gap gene expression in wild type , . This study revealed five regulatory mechanisms responsible for sharpening and maintenance of gap gene expression domains: broad activation by maternal gradients of Bcd and Cad; gap gene auto-activation; strong mutual repression between gap genes which show complementary expression patterns (hb and kni; Kr and gt); weaker asymmetric repression between overlapping gap genes (Hb on gt, Gt on kni, Kni on Kr, Kr on hb and Hb on Kr) and repression by terminal gene tll at the embryo termini. The asymmetric repression between overlapping gap genes is responsible for shifts of gap gene domains in the posterior region of the embryo. It is important to note that the wild type gap gene circuit model has the predictive power when molecular fluctuations of the input factors are taken into account , .
It is evident that to understand the gap gene network we need not only to describe the mechanism underlying its functioning in intact state, but also to comprehend what happens when certain stimuli or disruptions occur. Recently Papatsenko and Levine (2011) constructed a dynamic model based on a modular design for the gap gene network, which involves two relatively independent network domains with elements of fractional site occupancy. This model requires only 5–7 parameters to fit quantitative spatial expression data for gap gradients in wild type and explained many expression patterns in segmentation gene mutants obtained in studies published mainly in the late 1980s and early 1990s. However these patterns were characterized qualitatively by visual inspection, that may not capture the fine details of gene expression. For example, previous studies based on qualitative visual analysis of gene expression patterns showed that a Kr null mutation results in large shift of posterior Gt domain, overlap of positions of posterior Gt and Kni domains and decrease in the level of gt expression in the second half of cycle 14A . Here we obtained a large dataset on gap gene expression in Kr null mutants and extracted quantitative gene expression data using a data pipeline established previously . The analysis of this data allowed us to characterize the expression of other gap genes at unprecedented level of detail. In particular we showed that the significant decrease in the level of gene expression in the second half of cycle 14A is common to all gap gene expression domains. This novel biological result seems counterintuitive, because genetics studies show that Kr acts as a repressor, and therefore should come under close scrutiny.
The most serious limitation of the gap gene circuit models is their inability to correctly reproduce the expression patterns in trunk gap gene null mutants at quantitative level, although a theoretical study had shown previously that such prediction is possible if gene circuits models were fit to simulated, noise-free data  and simulating null mutants of the terminal gap genes tll and hkb was successful , , . A variety of reasons could be responsible for the failure, of which, from our point of view, the most important is the oversimplified representation of transcriptional regulation in the model. Indeed, as was already mentioned above, the action of regulator on its target gene is represented by a single parameter, whereas it is well known that the cis-regulatory elements (CRE) of segmentation genes often reproduce only one of expression domains of an endogenous gene when placed upstream of a reporter gene –. Moreover different CREs of one gene can have different transcription binding site composition, i.e. different regulatory inputs. For example, computational prediction of transcription factor binding sites showed that regulatory sequences which drive expression of gt in the anterior and posterior domains have different transcription binding site composition: the anterior gt domain has regulatory inputs from Bcd and Kni, while the posterior domain contains inputs from Hb and Cad, which are absent in the sequences responsible for anterior expression . Similar to gt, two CREs essential for hb expression in anterior domain and in central stripe and posterior domain differ in transcription binding site composition –. It is evident that current gene circuits models do not consider the mechanism of gene regulation at such a level of detail. This defect does not interfere with the ability of these models to fit gap gene expression patterns in wild type, however in mutant background with deficient set of regulators the failure of the model to take into account such features may suddenly become essential.
To avoid such problems we use a revised model which builds on separate treatment of domains with different regulatory inputs. This is possible by narrowing down the spatial domain of the model and considering only the posterior half of the blastoderm (region from 47 to 92% embryo length (EL)), in which each of the trunk gap genes is expressed in one domain.
As opposed to previous gap gene circuit models, which have a constant Bcd gradient and did not consider Cad data from late time points just before the onset of gastrulation , , and similar to approach used in , we implement Bcd as a time-variable input and use data on late Cad expression to represent the rapidly changing expression dynamics of these two genes. After cleavage cycle 12 Bcd nuclear gradient starts to decay . Analysis of data from fixed embryos showed that Bcd protein reached its maximal level near the beginning of cycle 14A and thereafter starts to decrease slowly that culminates in an almost twofold decline by gastrulation . From the second quarter of cleavage cycle 14A onward the cad expression in abdominal region start to gradually decrease and by gastrulation cad expression in the posterior region sharpens to a stripe which spans from 75 to 90% EL .
The gene circuit models do not require any assumption about regulatory interactions within a gene network. Instead the regulatory topology of the network is obtained by solving the inverse problem of mathematical modeling, i.e. by fitting the model to the data . To obtain the estimates for regulatory parameters that predict a specific network topology in mutants we fitted the model to gap gene expression patterns in wild type and in embryos with homozygous null mutation in Kr gene simultaneously. The logical justification of such an approach is to use the parameters of the wild type gap gene network as specific constraints on regulatory weights in mutants in order to obtain the consistent parameter estimates for both genotypes on one hand and on the other hand to preserve the characteristic features of gene regulation in mutant. The parameter estimates obtained in such a way were further studied by applying identifiability analysis, that confirmed that fitting to two genotypes simultaneously substantially increases the statistical significance of parameter values.
We use the modeling framework outlined above to explain the characteristic features of gap gene expression in Kr null mutants and in the posterior half of the blastoderm. In what follows we describe the expression patterns of gap genes in Kr null mutants and analyze quantitative gene expression data extracted from these patterns. We then use these data as input to a new gap gene circuit model. We show that in contrast to earlier models, this model correctly reproduces the characteristic features of gap gene expression in Kr mutants. In particular, it reproduces correctly the greater shift of posterior Gt domain than in wild type and significant decrease in the level of gap gene expression in the second half of cycle 14A. We next obtain the parameter estimates for the model (and hence the predicted gap gene network topology in wild type and mutant) and perform identifiability analysis to understand how reliable are these estimates. We study the dynamical behavior of our model and analyze the role of individual regulatory loops in gap gene expression in wild type and mutants. We show that a remarkable transformation of gap gene expression patterns in Kr mutants can be explained by dynamic decrease of activating effect of Cad on a target gene and exclusion of Kr gene from the complex network of gap gene interactions, that makes it possible for other interactions, in particular, between hb and gt, to come into effect. Our model also predicts the derepression of the anterior border of Hb posterior domain in Kr;kni double mutants, that is established in the absence of key repressors. We validate this prediction and show the correctness of network topology inferred in this work.
Gap gene expression in Kr mutants
In wild type Drosophila embryos gap genes are expressed as large intersecting domains along the A-P axis (Figure 1A). In general, all these domains exhibit similar temporal dynamics: after formation they start to grow, reach maximum expression levels around mid-cycle 14A and decline by gastrulation. In the course of cycle 14A gap gene domains change their positions and shift to the anterior . We have shown that the asymmetric gap-gap cross-repression with the posterior dominance is responsible for these shifts .
A,B. Integrated patterns of gap gene expression in wild type embryos (A) and in Kr mutants (B). The red dotted line in (B) shows Kr expression in wild type embryos. C–E. hb, gt and kni expression in Kr mutants and wild type embryos from time class 3. F–H. Expression of the same genes in time class 7.
In Kr null mutants gap gene expression is significantly altered (Figure 1B). It has been previously reported that in these mutants the posterior domain of gt is expanded towards the center of the embryo , . We detect that in the course of cycle 14A the posterior domain of gt shifts dynamically on 15% embryo length (EL) in the anterior direction and overlaps with Kni domain. Thus, by gastrulation, the difference in position of gt domain in mutants and wild type embryos constitutes approximately 10% EL. The anterior shift of the Kni domain maxima in Kr mutants constitutes only 1.8% EL. Hb posterior domain in mutants is formed at the beginning of cycle 14A and shifts on about 3% EL in the anterior direction during this cycle. Thus, the positional dynamics of this domain in mutants and wild type is similar.
The level of hb posterior expression in mutants is nearly the same as in wild type until time class 3, but declines afterwards. By gastrulation it constitutes only a half of the wild type expression level (Figure 1C,F). Gt posterior domain is initially lower than in wild type, it grows up to time class 4 and significantly declines thereafter (Figure 1D,G). The level of kni expression remains constantly low throughout cycle 14A (Figure 1E,H) with a slight decrease at the very end of this cycle (not shown).
The features of the gap gene expression in Kr mutants described above raise many questions. Namely, the regulatory mechanisms underlying a decrease in gene expression levels, as well as a much larger shift of Gt posterior domain should be explained. In the following sections, we describe our modification of the gene circuit model , , ,  that correctly reproduces the gap gene expression in Kr mutants and hence can serve as a tool to answer these questions.
The gene circuit model used in this work differs from previous implementations in several aspects. First we narrowed down the spatial domain of the model by considering only the posterior half of the blastoderm (region from 47 to 92% embryo length (EL)), in which each of the trunk gap genes is expressed in one domain. This allows us to avoid the inherent limitation of the model, in which the action of regulator on its target gene is represented by a single parameter.
Secondly, as opposed to previous gap gene circuit models which have a constant Bcd gradient and did not consider Cad data from late time points just before the onset of gastrulation , , we implement Bcd as a time-variable input and use data on late Cad expression to represent the rapidly changing expression dynamics of these two genes at that stage. We used bcd and cad profiles from FlyEx database for cycle 13 and eight temporal classes of cycle 14A as external inputs to our model equations.
We used the modeling framework outlined above to explain the characteristic features of gap gene expression in Kr null mutants and in the posterior half of the blastoderm. To obtain the estimates for regulatory parameters that predict a specific network topology in mutants the model was fitted to gap gene expression patterns in wild type and in embryos with homozygous null mutation in Kr gene simultaneously. DEEP method was applied to minimize the sum of squared differences between experimental observations and model patterns ,  and find all parameters of model equations, i.e. regulatory weights, synthesis rates, decay and diffusion constants, that allow to reproduce the characteristic features of gap gene expression in Kr null mutants as closely as possible. We performed over 200 runs with different initial parameter approximations and control variables. The search space was sampled uniformly for each parameter in the interval defined by biologically relevant limits. Two step procedure was applied to construct the ensemble of parameter sets. On the first stage, the residual mean square (RMS) was checked and the sets with RMS less than 5% of the maximal gene expression value (equals 255 in our data) were accepted for further analysis. Secondly, we inspected the model expression patterns visually. Consequently, the ensemble of 11 parameter sets was obtained that correctly reproduces the dynamics of gene expression in wild type and mutant embryos, in particular the decrease of gap gene expression levels and the anterior shift of gt domain.
Gene network topology
To infer the topology of regulatory network we classified the estimates of regulatory weights (the elements of T and E matrices) into the following three categories: ‘activation’ (parameter values greater than 0.005), ‘repression’ (parameter values less than −0.005) and ‘no interaction’ (between −0.005 and 0.005). This leads to a predicted regulatory topology of the network based on which category a majority of parameter estimates falls into (summarized in Figure 2).
A. Right panels display simulation results and data for mutant genotype, left panels show simulation results and data for wild type. Top, middle and bottom panels present data and profiles for first, third and seventh temporal classes, respectively. Wild type data was taken from the FlyEx database , . B. Prediction of network topology based on classification of regulatory weights. C. Prediction based on both classification and identifiability analysis of regulatory parameters. Rows correspond to target genes; columns correspond to regulators. Numbers in cell define in how many circuits a given interaction was classified as activation, no action or repression. Red, blue and white cells mark activation, repression and no action correspondingly. Light red and light blue cells define insignificant activation and insignificant repression correspondingly. Cells colored in grey define the situation in which a parameter takes both negative and positive values and therefore the type of regulation is not obvious.
There are several networks in the ensemble called consensus networks, in which the signs of regulatory parameters coincide with the predicted network topology inferred from the fits. Figure 2A shows simulation results together with experimental data and Table S1 presents parameters for one of such networks. It is evident that in spite of some patterning defects especially at early stages the model correctly reproduces the dynamics of gene expression in wild type and mutant embryos.
Some basic features of the gap gene network topology in wild type and mutant become immediately obvious from inspection of Figure 2B and Table S1. First, Cad activates zygotic gap gene expression. Second, hb, Kr, kni, and gt show autoactivation. Third, Bcd activates Kr, gt, kni in all parameter sets, however in case of hb it shows activation in approximately the same number of circuits as it shows repression. Fourth, all reciprocal interactions among trunk gap genes are either zero or repressive. An important exception is activation of hb by Gt. Finally tll represses kni and Kr and weakly activates gt.
Parameter identifiability and correlations
The identifiability analysis was conducted with respect to parameters of the model estimated by fitting to experimental data. The model considers the time evolution of protein concentrations of four gap genes hb, Kr, gt, and kni in two genotypes: wild type and in embryos with homozygous null mutation in Kr gene. The total parameter set that minimizes the cost functional 0 consists of 40 parameters and is denoted as . The set includes four subsets , , , and of 10 parameters each, that describe regulatory action on each target gene. In mutants the model is only fitted to quantitative gene expression data for 3 genes, gt, hb and kni, and hence the parameters from are estimated using data points from wild type embryos only (half of all data points). All the other parameter subsets are estimated from the whole dataset. Due to lack of space we denote the elements of inter-connectivity matrices T and E by single-letter notations of genes, namely, H, K, G, N, B, C, and T stand for gt, Kr, gt, kni, bcd, cad, and tll, respectively. For example, characterizes the regulatory action of Hb on kni.
The sensitivity of the model solution to parameter changes is characterized by the size of confidence intervals. The confidence intervals (2) (see Methods) are constructed under the assumption of normally distributed error in data, that is not satisfied for gene expression data. The error in data almost linearly increases with the mean concentration that is typical rather for the Poisson than for normal distribution. To make the error independent of the mean we applied the variance-stabilizing transform to both data and model solution. The transformed objective functional was minimized using the parameter estimates obtained for non-transformed functional as initial values for the optimization procedure. The new solutions were found in a very close vicinity of initial parameter sets.
The 11 parameter sets, which minimize the transformed model functional, are given in Table S2 and will be referred to as circuit parameter sets. The newly estimated regulatory weights were classified into regulatory categories as described in subsection Gene Network Topology. This classification results in predicted regulatory topology of the network (Figure 2C), which is largely the same as in Figure 2B, however not all the entries in two tables coincide. The estimates of some parameters not-uniquely determine the type of gene regulation in different circuits, i.e. in some circuits the parameter estimates exceed the threshold 0.005 in absolute value, while in the others are below the threshold. It is also true for new parameter sets, however, the number of such circuits is different than those given in Figure 2B.
The confidence intervals for individual parameters are constructed in the vicinity of the model solution. The results for one representative circuit are presented in Figure 3. Most of the values of regulatory parameters are very close to zero, and it is important to make sure whether the value (more precisely, the sign) of a regulatory parameter is significant. The hypothesis that the parameter estimate is non-zero is tested as follows: if a confidence interval includes both positive and negative values, the hypothesis is rejected, otherwise, accepted.
Regulatory weights are labeled by single-letter notations of genes: hb(H), Kr(K), gt(G), kni(N), bcd(B), cad(C), tll(T). The first letter corresponds to the target gene (e.g., HK stands for ).
Our classification method to infer the topology of regulatory network used in this work, was based on comparison of the values of regulatory parameters with the threshold . However, as it has already been mentioned, estimates of some parameters take values, which only exceed the threshold in part of circuits. By exploration of confidence intervals for these parameters we came to the conclusion, that almost all the estimates, that are close in absolute value to the threshold, are insignificant. This result explains the discrepancies between the network topologies presented in Figures 2B and 2C: the conclusions about the type of gene interaction that are based on insignificant parameter estimates are unreliable.
The analysis of confidence intervals conducted for all the circuits (Figures S2 and S3) allowed us to refine the predicted regulatory network topology (Figure 2C). We classify parameters as insignificant activation/repression if the parameter estimates are positive/negative in almost all the circuits but their confidence intervals contain zero, and hence the parameter sign cannot be identified. As a result the non-identifiable regulatory parameters are , , , , , , and and therefore we cannot draw any conclusion about these interactions. Interestingly most of these interactions involve Kr as a target gene or Bcd as a regulator of gap gene domains located in the posterior of the embryo. Other regulatory parameters are well identifiable and, hence, the identifiability analysis corroborates the gene network topology drawn from classifying parameter values only.
It should be stressed that the confidence intervals provide the full information about the parameter estimates only in case of parameter independency, otherwise the intervals are overestimated. Moreover, strong correlation between parameters may lead to their non-identifiability, because a change in one parameter value can be compensated by the appropriate changes of another parameters and, hence, does not significantly influence the solution. In view of this we investigate the dependencies between parameters using the collinearity analysis of the sensitivity matrix. This method allows to reveal correlated and hence non-identifiable subsets of parameters.
The sensitivity matrix defined in Methods was analyzed in the vicinity of 11 points in the parameter space that define the optimal model solutions. The collinearity index (equation (3) in Methods) was computed for all the subsets of dimension k of the parameter set . The threshold value for was chosen equal to 7. This value in case of corresponded to approximately 99% pairwise Pearson correlation between columns of the sensitivity matrix. The method allowed to detect subsets of dimension 2 and 3 with the collinearity index exceeding the threshold value, i.e. subsets of poorly or non-identifiable parameters.
Most of parameter combinations in these subsets were the same for all 11 circuits (see Table 1). Almost all the pairs of parameters in subsets of dimension 2 belonged to , i.e. were related to Kr target gene. To explain this result we additionally compute the collinearity indices between columns of the upper half of the sensitivity matrix, that only include the partial derivatives computed at 1532 wildtype observations. The method detected much more parameter subsets with collinearity indices greater than the threshold, that included the parameters characterizing the input of all four genes. The parameters of the full model fitted to two genotypes thus are better identifiable than those of the model that solely describes the wild type data. However, the parameters from cannot be identified from the full model as are just estimated from the wild type observations.
The subsets of dimension 3 with the highest collinearity indices () are also common for the most of the circuits. Most of these combinations are related to gt and Kr, i.e., include parameters from and .
The two approaches applied to characterize parameter identifiability are closely connected and complement each other. By exploration of confidence intervals we can see to what extent the model solution is sensitive to parameter changes and test the significance of parameter sign, but this method does not give any explanations to the sources of non-identifiabilities. One of such explanations can be provided by collinearity analysis. The correlation between parameters revealed by this approach can clarify insignificance or unreliability of parameter estimates with large confidence intervals. For example, we derive non-identifiability of from the large size of its confidence interval and at the same time the analysis of the sensitivity matrix allows us to detect the subset of parameters and with high mean collinearity index equal to 7.76 (see Table 1). Thus, poor identifiability of can be explained by correlation between two regulatory parameters, that is reflected in their high collinearity index.
The gene network topology inferred from both classifying the parameter values and parameter identifiability analysis is presented in Figure 2C. As Figure S4 shows it is largely in agreement with topologies predicted by earlier models , , , . Strong constraints for mutual repression are present for kni and hb, which show complementary expression patterns. Besides, strong repressive action exert both Kr on hb and Hb on gt. Some previous models had predicted the repressive action of Kr on hb , while most showed no interaction , , . Many repressive interactions between gap genes show weaker constraints toward repression, and interestingly we have found very weak or no dynamical constraints for repression of Gt on Kr, the interaction with strong constraint for repression in all wild type gene circuit models , , , , . In addition our model predicts weak repressive interactions between Kni and gt and Kr and kni. In earlier gap gene circuit models the first interaction was predicted as no interaction , activation  or activation in about half the circuits, and repression in the other half . The repressive action of Kr on kni is only observed in our model, all other models predicted no interaction between the two genes. In addition in current model Bcd shows activation of hb in approximately the same number of circuits as it shows repression, while in all previous models this interaction was predicted as activation. Weak activation of gt by Tll is now present in 10 parameter sets, while previous results predicted this interaction as repression. Finally our model predicts no interaction between Tll and hb. Some previous models had classified this interaction as activation , , while other predicted it as repression or no interaction , , .
Mechanism of alteration of gap gene expression patterns in Kr mutants
Null mutation in Kr gene results in strong alteration of expression patterns of almost all zygotic segmentation genes. In gap gene network this mutation manifests in significant reduction of gap gene expression levels in cycle 14A, as well as in large shift of posterior Gt domain and overlap of positions of posterior Gt and Kni domains.
Previous gap gene circuit models fail to correctly model the gap gene expression patterns in the embryos homozygous for null mutation in a trunk gap gene. A new model introduced here correctly reproduces the characteristic features of gap gene expression in Kr null mutants and in the posterior half of the blastoderm. To investigate the mechanism responsible for strong alteration of the expression patterns of gap genes in Kr null mutants we have performed the detailed graphical analysis of gap gene regulation in the posterior of the embryo. This analysis revealed the following regulatory principles.
In both Kr mutants and wild type the posterior Hb domain is the last gap domain to form; its expression is initiated in cleavage cycle 13 and the domain retracts from the posterior pole at temporal class 2 of cycle 14A. Later the expression level in the posterior Hb domain increases gradually up to temporal class 7 and diminishes at the very end of cycle 14A in wild type embryos, while in mutants the level of expression is nearly the same as in wild type until time class 3, but declines afterwards.
In the model, combined activating inputs by Cad and Gt are responsible for hb expression in early cycle 14A. At later time (from time class 3 onward) hb autoactivation starts to play role and it gradually supplements activation by other factors, which strength decreases in both genotypes. In mutants the positional dynamics of Hb domain resembles that in wild type, while the anterior shift of Gt domain is much larger. Larger anterior shift of Gt domain causes stronger decrease in activation contribution by gt to the Hb domain (Figure S5). This effect together with smaller autoactivation level may cause the fall in accumulation of Hb in Kr mutants. We do not include hkb, the gene responsible for formation of the posterior boundary of the Hb posterior domain, in our model and therefore only the mechanism underlying the formation of the anterior boundary can be analyzed. In wild type this boundary is formed by joint repression by Kr and Kni, while in Kr mutants Kni is the only repressive input, which strength diminishes with time due to decrease in kni expression level.
Gt posterior domain forms in cycle 13. In wild type embryos the expression of gt in this domain reaches maximum at time class 5 and then declines. In Kr null mutants gt expression is lower than in wild type, it grows up to time class 4 and significantly declines thereafter (Figure 1D,G). In mutants the anterior shift of gt posterior domain is much larger than in wild type: by gastrulation, the difference in position of gt domain in mutants and wild type embryos constitutes 10% EL and Gt domain overlaps Kni domain.
Cad and Gt autoactivation contributes activating inputs on posterior Gt domain (Figure 4). Both in Kr mutants and wild type embryos the strength of Cad activating input decreases by gastrulation, however in mutants this reduction is larger, as Gt domain shifts closer to the anterior end of the embryo against gradient of Cad concentration (Figure 4C). It is obvious that weaker activation of gt by Cad will lead to lower level of gt autoactivation within its domain. Indeed, by time class 7 autoactivation contributes strongly to expression of gt only in wild type. This provides a straightforward mechanism for reduction in the level of gt expression in the posterior of the Kr mutant embryos: the decrease of activating contribution by Cad and diminishing gt autoactivation result in downregulation of gt expression.
A. Modeled expression patterns at temporal classes T1, T3 and T7. B. Regulatory contributions in wild type (center) and mutant (right). C. Temporal change in regulatory contributions at position corresponding to Gt domain maximum. Colored areas are given by or in equation (1) and reflect the strength of a given interaction at a specific point in space and time. All plots are based on best scoring solution (circuit C9, see Table S1 for parameters).
It should be noted that in mutant a small level of Hb repression is evident across the middle region of the model spatial domain at all times (Figure 4B). This repression is caused by the spurious expression of hb in the region of 60–77%EL (see Figure 1) and could be responsible for decrease in the gt expression level. However the exclusion of this elevated expression from the model (by setting Gt input into hb expression to zero) does not lead to increase in the gt expression level (Figure S6) in mutants, that makes it unlikely that Hb repression contributes significantly to the low levels of gt.
Kr and Kni repression is involved in the positioning of the anterior boundary of the Gt posterior domain in wild type embryos (Figure 4). In mutants Kni does not significantly contribute to this boundary formation, that can be accounted for its small expression level. Besides by time class 7 Gt shifts to the anterior border of the model spatial domain. All this precludes the conclusions on the possible mechanisms of the anterior boundary formation of Gt domain in Kr mutants.
The posterior boundary of the Gt domain depends almost exclusively on very strong repression by Hb both in wild type embryos and Kr mutants (Figure 4). The accumulation of Hb in the posterior region causes increase in both levels and extent of this repression over time. This in turn leads to an anterior shift of Gt domain. In Kr null mutants the lack of gt repression by Kr and very weak repression of gt by Kni allows Gt posterior domain to move further than in wild type to the territory of kni expression. Thus, the mechanism underlying the shift in posterior Gt domain in Kr mutants is equivalent to those of other gap domains in wild type embryos: shift happens because of the almost absence of repression by the adjacent anterior domain (Kni), while it becomes increasingly repressed posteriorly (by Hb, in this case).
In wild type embryos kni expression is first detected in cycle 13; it reaches maximum by temporal class 5 of cycle 14A. In Kr mutants the level of kni expression remains constantly low throughout cycle 14A (Figure 1E,H) and the anterior shift of the Kni domain maximum constitutes only 1.8% EL.
In the model cad and kni autoactivation provides activating inputs on Kni domain. Both in Kr mutants and wild type embryos the strength of Cad input decreases by gastrulation, however in mutants this decrease is stronger, happens faster and is accompanied by diminishing kni autoactivation. By temporal class 7 autoactivation of kni is present at significant level only in wild type. Similar to Gt domain a small level of hb repression evident across the middle region of the model spatial domain (Figure 5B) is unlikely to contribute to the low levels of kni as the exclusion of this spurious expression from the model does not lead to increase in the kni expression level (Figure S6).
A. Modeled expression patterns at temporal classes T1, T3 and T7. B. Regulatory contributions in wild type (center) and mutant (right). C. Temporal change in regulatory contributions at position corresponding to Kni domain maximum. Colored areas are given by or in equation (1) and reflect the strength of a given interaction at a specific point in space and time. All plots are based on best scoring solution (circuit C9, see Table S1 for parameters).
In wild type embryos Kr repression is responsible for positioning the anterior boundary of Kni domain, while in Kr mutants this boundary forms outside the model spatial domain. The posterior boundary of this domain depends on repression by Gt, Hb and Tll in both genotypes, however Tll repression is only retained in a region posterior of 80% EL. In Kr mutants Gt repression spreads into a territory where kni expression domain forms, preventing the increase of gene expression level in this domain.
The analysis performed above points on the central role of hb in gap gene regulation in embryos. To support the validity of this prediction the in silico experiments were done. In these experiments, instead of setting to zero (as is usually done to model mutant genotype), we multiplied it to the scaling coefficient , which gradually decreases from 1 to 0, and inspected changes in gap gene expression in the posterior of the embryo. It is evident in Figure 6 that decrease in Kr regulatory input is most important for hb dynamics. Besides, this experiment demonstrates the monotonous dependence of change of gap protein concentrations on , that may be an additional argument for validity of numerical results, obtained independently in two genotype modeling. It should be stressed that though such an experiment does not has a special biological sense, it makes it possible to predict, which component of the network is subject to biggest impact by mutation. The last hardly could be revealed in biological experiments.
A. Modeled expression patterns (lines) and experimental data (lines with symbols) on gap gene expression in wild type. B–D. Data on gap gene expression in Kr mutant (lines with symbols) and model output (lines) at the values of equal to 0.6, 0.4 and 0.025. decrease is indicated by arrow.
In this paper we investigate the mechanism of alteration of gap gene expression patterns in null mutants for one of trunk genes, Kr. We applied the integrative approach, which combines on one hand the characterization of expression of gap genes in Kr mutants in a quantitative manner and on the other the gene circuit method, a system-level approach to refine gap gene network topology and reveal mechanisms responsible for the alteration. However before considering these biological questions in detail below, we wish to discuss several improvements in gene circuits method, which for the first time make it possible to correctly reproduce gap gene expression patterns in mutant for a trunk gap gene. We believe that these methodological improvements have important implications for reverse engineering of gene networks in general.
Revised gene circuit model
The failure of gene circuit models to correctly reproduce expression patterns in gap gene null mutants could be due to a variety of reasons: it is possible that our data does not correctly reflect the absolute concentrations of gap gene proteins and some scaling of expression patterns in the data is necessary or production delays may be required in the model to eliminate premature initiation of gap-gap gene interactions. Alternatively, a more detailed consideration of molecular mechanisms responsible for gap gene expression may be required for mutant simulation. At present we do not have a satisfactory understanding of such mechanisms for any of the gap genes, however some important regulatory principles emerged from experiments with reporter constructs, DNAse protection assays, Chip-chip experiments and large-scale computational screens to identify and analyze gap gene enhancers. These experiments demonstrated that cis-regulatory elements (CRE) of segmentation genes often reproduce only one element of an endogenous gene expression pattern when placed upstream of a reporter gene – and that different CRE of one gene can have different transcription binding site composition, i.e. different regulatory inputs. For example, three gt CREs drive reporter gene expression in the posterior (gt_(-3)) and distinct anterior domains (gt_(-6), gt_(-10)), respectively, while another element (gt_(-1)) reproduces endogenous gt expression in both anterior and posterior domains , . Moreover computational screens predict that the anterior and posterior gt domains have different regulatory inputs. It is currently unclear how gt CREs interact in regulation of the endogenous gt gene, however it is obvious that the representation of the gap gene regulatory interactions by a single parameter in the gene circuit model is hardly suitable for theoretical description of such a complex mechanism and should be substituted by more realistic representation.
As a first step in this direction we introduced a revised model which builds on gene circuit method but treats domains with different regulatory inputs separately. A straightforward way to implement such a modification is to narrow down the spatial domain of the model by considering only the posterior half of the blastoderm, in which each of the trunk gap genes is expressed in one domain. Here we demonstrated that the new model correctly reproduces the characteristic features of gap gene expression in Kr mutants, the greater shift of posterior Gt domain than in wild type and significant decrease in the level of gap gene expression in the second half of cycle 14A in particular. The successful modeling of expression patterns in a mutant for a trunk gap gene is a significant achievement of this work. This result also clearly indicates that the oversimplified representation of transcriptional regulation in the previous models is one of the reasons for unsuccessful attempts of mutant simulations.
Most of previous gap gene circuit models represent Bcd as a time-constant gradient and did not consider Cad data from late time points just before the onset of gastrulation , . It was reasonable to implement such an approach when the mechanism for precise positioning of segmentation gene expression domains was investigated, however an intriguing feature of Kr mutants is the reduction in the level of gap gene expression in the second half of cycle 14A, a phenomenon which understanding requires a precise consideration of activators responsible for gap gene expression. As we have shown before , Bcd protein reaches its maximal level near the beginning of cycle 14A and thereafter starts to decline slowly, while Cad expression in abdominal region starts to gradually decrease from time class 3 onward. Accordingly in the model we implement Bcd as a time-variable input and use data on late Cad expression to represent the rapidly changing expression dynamics of these two genes. This allows us to demonstrate that decrease of activating input by Cad and weakening of autoactivation are responsible for reduction in the level of gt and kni expression in the posterior of the Kr mutant embryo.
In gene circuit models the regulatory topology of the network is obtained by solving the inverse problem of mathematical modeling, i.e. by fitting the model to the data . To obtain the estimates for regulatory parameters that predict a specific network topology in mutants we fitted the model to gap gene expression patterns in wild type and in embryos with homozygous null mutation in Kr gene simultaneously. The rationale behind such an approach is that, as it was shown in the Parameter identifiability and Correlations section, using the parameters of the wild type gap gene network as specific constraints on regulatory weights in mutants substantially increases the statistical significance of fitted parameter values.
Our results demonstrate the existence of parameter sets describing gap gene expression in two genotypes simultaneously and thus the applicability of the gap gene circuit formalism to model genotypes of trunk gap gene mutants. One finds hard to say whether the overfitting is the reason why these parameter sets were not discovered during the fit to the wild type data alone. Overfitting is defined by a fine balance between the number of model parameters and the level of details used to describe the system. The qualitative models usually require small amount of parameters. However, when the data under modelling becomes more quantitative, the number of parameters usually increases, and in the general case there are no methods to find the optimal number of parameters, exceeding which will lead to overfitting.
We treat the possible overfitting problem by applying the practical identifiability analysis of the found parameter values. The parameter estimates obtained in such a way were further studied by applying identifiability analysis. Two approaches were used. First the sensitivity of the model to parameter changes and identifiability of parameters in the vicinity of the model solution were analyzed on the basis of confidence intervals of parameter estimates. This analysis showed that the most of regulatory parameters are well identified and their estimates can be used to make conclusions about the type of gene interaction. Secondly, as parameter non-identifiability can be a consequence of their strong correlation we applied the collinearity analysis of the sensitivity matrix to reveal the subsets of correlated parameters. We found that non-identifiability of some parameters detected by the method based on confidence intervals can be explained by the correlations between different parameters. Our analysis also demonstrated that parameters of the model fitted to two genotypes are better identifiable than those of the model fitted to wild type data only.
Mechanism of transformation of gap gene expression domains in Kr null mutants
Our quantitative analysis of gap gene expression in mutants confirms and extends results from earlier studies. It was previously reported that in cycle 14A the shift of posterior Gt domain is much larger than in wild type and that this domain overlaps Kni domain , . It was also demonstrated that the level of Kni expression in Kr mutants is reduced. Here we showed that by gastrulation the difference in position of gt domain in mutants and wild type embryos constitutes approximately 10% EL. Contrary to posterior Gt domain, the anterior shift of the Kni domain maxima in Kr mutants constitutes only 1.8% EL, as a result these domains overlay each other. The level of kni expression remains constantly low throughout cycle 14A. Decrease in the expression levels of both kni and gt in Kr mutants was shown in earlier qualitative studies , , , . High temporal resolution of our dataset enabled us to find out that the decline in gene expression level in the second half of cycle 14A turned out to be an intrinsic property of all gap domains.
The regulatory mechanisms for expression of the trunk gap genes in the posterior of the embryo predicted by our model r summarized in Figure 7A: (1) Cad activates zygotic gap gene expression. (2) Autoactivation is involved in maintenance and sharpening of hb, Kr, kni, and gt domains. (3) Trunk gap genes either repress each other or do not interact. An important exception is activation of hb by Gt. (4) Bcd activates Kr, gt, kni in all parameter sets, however identifiability analysis showed that this activation is insignificant. In case of hb Bcd shows activation in approximately the same number of circuits as it shows repression. (5) In the posterior terminal region of the embryo Tll represses Kr and does not interact with hb.
In A gap domains are shown schematically, with anterior to the left, posterior to the right. Background color indicates the most prominent activating input by Cad. Autoactivation is indicated by double arrows. T-bars indicate repressive gap-gap interactions. Dotted lines show interactions present in wild type only.
In general this regulatory principles are in agreement with the results from previous studies (see Figure S4), however some differences exist. This is not surprising, if to consider that contrary to previous studies the model was fitted to two genotypes simultaneously. The rationale behind such an approach is to use the parameters of the wild type gap gene network as specific constraints on regulatory weights in mutants in order to obtain the consistent parameter estimates for both genotypes on one hand and on the other hand to preserve the characteristic features of gene regulation in mutant.
Previous models predict the mutual repression between non-overlapping gap genes hb and kni, as well as gt and Kr. Indeed, our model showed strong constraints for mutual repression between kni and hb, however identifiability analysis classified the action of Gt on Kr as insignificant repression. This discrepancy can be explained by the fact that in our model the parameters corresponding to Kr gene are estimated using data points from wild type embryos only and therefore their identifiability is inferior to that of parameters estimated from the whole dataset. The non-identifiability of parameters describing the action of Bcd on gap genes in Kr-deficient gap network may be explained by the exclusion from the model of the anterior half of the blastoderm, where Bcd contributes stong activating inputs on the anterior and central gap gene domains , . Bcd activating inputs on posterior domains are smaller and the interactions of gap genes come into play to dynamically position the posterior gap domains.
In the gap gene circuit models the motif which includes Tll is the most variable component of the gap network (see Figure S4). The network reconstructed in this work constitutes no exception to this pattern: our model predicts no interaction between Tll and hb, while some previous models had classified this interaction as activation ,  or predicted it as repression or no interaction , , . In previous models Tll exerts repressive action on gt and kni, however in our model these parameters are non-identifiable. In addition our model predicts repression of Kr by Tll, however in the other model  this parameter was classified as non-identifiable.
Interestingly that in spite of the differences in gene regulation between the models discussed above the asymmetric cascade of cross-repressive interactions between gap genes with overlapping expression domains is preserved in the current two genotype model. As is evident from inspection of Table S1 the regulatory weights and are larger than the reciprocal weights and in all circuits, while the regulatory weight is larger than the reciprocal weight in 9 out of 11 circuits. These asymmetrical interactions lead to anterior shifts in domain positions both in wild type and mutant, as will be discussed below.
The regulatory mechanisms predicted by the model are mainly in agreement with experimental evidences. During cleavage cycle 14A both Cad and Bcd continue to activate gap genes, however the Bcd gradient starts to rapidly decay about 10–15 min before gastrulation , . The evidence for autoactivation of gap genes is not so clear. In Kr and gt mutants expressing non-functional proteins Kr domain is narrowed and weakened , and the intensification of Gt domains during cycle 13 is delayed . Besides computational studies predict that both Kr and Gt bind to some of their own regulatory elements . The autoactivation is not required for expression of the posterior Hb domain . Repressive feedback between hb and kni and gt and Kr is suggested by many experimental results , , . There is experimental evidence for additional repressive interactions between gap genes with overlapping expression domains. Repression of gt by Hb is supported by the fact that the posterior Gt domain fails to retract from the posterior pole of the embryo around mid-cycle 14A , , while no gt expression can be detected in embryos over-expressing hb . The central Kr domain expands posteriorly into regions with reduced or lacking kni activity in mutants , , . There is a Kni binding site in the Kr regulatory region, which overlaps with a Bcd activator site . It has been proposed that Kr is required for kni activation, however, this effect turned out to be indirect , . Kr and hb are the only pair of overlapping gap genes that show mutual repression, however there is some ambiguity in the genetic evidence. Some authors have reported a posterior expansion of the anterior Hb domain in Kr mutants , . Repression of Kr by Hb is suggested by an anterior expansion of the central Kr domain in hb mutants , , , – and multiple Hb binding sites have been identified in the Kr regulatory region . Evidences on repression of kni by Gt and repression of gt by Kni are ambiguous , , , , e.g. a posterior expansion of the abdominal Kni domain was reported in one study , this effect was not seen in another . Terminal gap genes have strong repressive effects on trunk gap gene gt, kni and Kr , , . In contrast the posterior domain of Hb is present and expanded to the anterior in embryos over-expressing tll . This suggests that Tll activates hb expression in its posterior domain, however, this interaction is probably indirect, since posterior Hb is present in tll;kni double mutants.
The important corollary that follows from the inferred topology (Figure 7A) is the prediction that in Kr,kni double mutants the anterior border of Hb posterior domain will not be properly set as both regulators responsible for formation of this border are deficient. We confirmed this prediction in experiment (Figure 7B). This experimental result strongly supports the model.
To explain mechanisms responsible for alteration of the gap gene expression pattern in Kr mutants we implement the analysis of regulatory loops and study the dynamical change of Cad contribution to a target gene regulation in wild type and mutants. Cad is the main activator of gap genes in the posterior of the embryo.
This analysis revealed two mechanisms responsible for alteration of the posterior gt expression pattern in Kr mutants. First, gt expression level decreases because the activating effect of Cad on gt diminishes with time (Figure 4). Secondly, two interactions from Kr to gt and hb responsible for formation of the anterior borders of Gt and Hb posterior domains correspondingly are impaired in embryos (Figure 7A), that makes it possible for Gt posterior domain to move forward to the anterior due to repression by Hb.
The mechanism responsible for reduction of kni expression level in Kr mutants is essentially the same as described for gt posterior domain: Kni level reduces because the activating effect of Cad on kni decreases with time (Figure 5) and because the interaction between Kr and gt is deficient that makes it possible form Gt repression to spread into a territory where kni expression domain forms (Figure 7A).
It should be noted that the difference in Cad regulatory effect on the posterior gt and kni expression between mutant and wild type is not very large (see Figures 4C and 5C). This fact could raise doubt on the role of Cad in the reduction of gap gene expression levels in mutants. However recently it was demonstrated that only a 3% overlap exists between transcription factor occupancy and gene response to TF knockout , . This and other results (discussed in detail in ) point that the relations between TF concentration and function are non-linear and that weak regulatory events and small differences in regulation may play a biologically significant role in the quantitative control of complex biological processes.
Our model can also explain the mechanism, which provides for decrease of the hb posterior domain expression level: such a reduction happens because Gt stops to activate hb due to its shift (Figure S5). However it should be noted that the activation of hb by Gt predicted by all gene circuit models , , ,  is currently not supported by the literature.
In our model hb activation by Gt leads to the elevated and spurious expression of hb in the region of 60–77% EL (see Figure 1). The exclusion of this elevated expression from the model does not cause an increase in both gt and kni expression levels (Figure S6), that makes it unlikely that Hb repression contributes significantly to the low levels of these domains in mutant.
We note finally that the reduction of gap gene expression levels is peculiar not only for Kr mutants. For example hb expression (see Figure 7B) and gt posterior domain levels  are reduced in kni mutants. In wild type embryos the gap gene expression levels stop to grow in the second half of cycle 14A and slightly decrease by gastrulation . Mutations in gap genes aggravate this effect, that underlines the importance of intact network for maintenance of normal gap gene expression levels.
Acquisition of quantitative data on gene expression
We obtained embryos from loss-of-function allele (FlyBase ID FBal0005790) . embryos were collected either from Df(3L)ri-79c or Df(3L)ri-XT1, ru st e ca stocks. Kr;kni double mutant embryos were made by crossing and Df(3L)ri-79c flies.
3–4 hr old embryos from flies carrying mutation  were collected, fixed and stained as described elsewhere , . We used primary antibodies against Kr, Knirps (Kni), Giant (Gt), Hunchback (Hb) and Even-skipped (Eve) ,  and secondary antibodies conjugated to Alexa Fluor 488, 555, 647, and 700 (Invitrogen). Each embryo was additionally stained with either anti-histone H1-4 antibody (Chemicon) or Hoechst 34580 (Invitrogen) to mark the nuclei.
Laterally oriented Kr null embryos, showing zero level of Kr expression and severely transformed Eve pattern , were scanned using Leica TCS SP2 and Leica TCS SP5 confocal microscopes as described . For each experiment, the microscope gain and offset were set on maximum expression level of a given gene in wild type patterns and then these settings were applied for mutants. The 8-bit digital images of gene expression in Kr mutants were acquired for cleavage cycle 14A.
For spatial registration and data integration, embryos from cleavage cycle 14A were distributed into 8 time classes about 6.5 min each on the basis of measurement of degree of membrane invagination, as well as characteristic features of the even-skipped gene expression pattern . The quantitative gene expression data and integrated patterns for each temporal class of cycle 14A were obtained as previously described , , ,  using recently developed packages ProStack and BREReA , . The one-dimensional integrated patterns of gene expression in wild type were taken from FlyEx database (http://urchin.spbcas.ru/flyex/, ).
Gene circuit models
Gene circuit models , , , ,  describe the dynamics of segmentation gene expression in the syncytial blastoderm of Drosophila melanogaster. The circuits used in this paper consider the time evolution of protein concentrations of gap genes hb, Kr, gt, and kni in two genotypes: wild type and in embryos with homozygous null mutation in Kr gene. To make separate treatment of domains with different regulatory inputs possible we narrowed down the spatial domain of the model by considering only the posterior half of the blastoderm (region from 47 to 92% embryo length (EL)), in which each of the these genes is expressed in one domain. We consider a one-dimensional row of nuclei along the anteroposterior axis of the embryo, as anteroposterior (A-P) and dorsoventral (D-V) patterning systems are largely independent of each other in the presumptive germ band of the blastoderm embryo. The modeled region extends over 45% of the A-P axis, from the minimum of gt expression inbetween third and fourth gt stripes to the posterior border of the posterior hb domain (Figure 1).
Gene circuits function according to three rules: interphase, mitosis and division . During mitosis, only protein transport and protein decay govern the dynamics as transcription shuts down and nascent transcripts are destroyed . Mitotic division is modeled as a discrete change in the state of the system. At the end of a mitosis, each nucleus is replaced with its daughter nuclei, the inter-nuclear distance is halved and the daughter nuclei inherit the state of the mother nucleus. During interphase the change in concentration for each gap gene product a in each nucleus i over time t is described by the following system of ordinary differential equations (ODEs)(1)
The three terms on the right-hand side of the equation represent protein synthesis, protein diffusion and protein decay. is the total regulatory input to gene a. is the number of gap genes in the model (hb, Kr, kni and gt), is the number of external regulatory inputs (bcd, cad and tll genes, which are not regulated by gap genes, but regulate these genes). and are genetic inter-connectivity matrices that characterize the action of regulator b or external input e on gene a. The sizes of these matrices are and correspondingly. is a threshold parameter of the sigmoid regulation-expression function . is the maximum synthesis rate, the diffusion coefficient, and the decay rate of the product of gene a.
The gap gene circuits used in this study consider events occurring during cleavage cycles 13 and 14A and ending at the onset of gastrulation . The divisions are carried out according to a division schedule based on experimental data. Time t is measured in minutes from the start of cleavage cycle 13. The interphase of cycle 13 lasts for 16.0 min, and its mitosis from 16.0 to 21.1 min. At min, the thirteenth division is carried out by applying the division rule. The interphase of cycle 14A starts immediately after division, and lasts until gastrulation at min. The cleavage cycle 14A is subdivided into 8 temporary equivalent classes, as a result the model is compared to data at 9 time points, one time point for cycle 13 (C13), and eight points for cycle 14A (time classes T1–T8).
Kr, gt, and kni are exclusively zygotic, and are not present at significant levels before cycle 13 , thus they have initial conditions of zero. For hb, the expression data from cycle 12 is used as the initial condition. hb, which is expressed both maternally and zygotically, shows a large increase in expression in cycle 13 , , indicating commencement of its zygotic expression. Non-zero initial conditions for external inputs Bcd, Cad and Tll are obtained by piecewise linear interpolation of integrated expression data at midpoint of C12 (t = −6.2 min) and midpoint of C13 (t = 10.55 min). Moreover, in order to solve the right hand side of equation (1), the concentrations of external inputs must be supplied for any time in the duration of the model. This is implemented by linear interpolation between data points at C13 and eight time classes of cycle 14A (T1–T8), with data points corresponding to midpoints of C13 and each time class.
As was suggested in previous studies  the zero flux boundary conditions were chosen at both ends of modeling interval because other numerically feasible alternatives, such as periodic boundary conditions, are not biological. The actual flux through the boundaries is nonzero but depends on gene expression in a complicated manner that may add the unneeded overhead for numerical simulations.
The model parameters are estimated by fitting the model output to experimental data. This is performed by minimization of cost function based on the sum of squared differences between gap protein levels in the model and data.
DEEP - Differential Evolution Entirely Parallel method is applied to biological data fitting problem. We introduce a new migration scheme, in which the best member of a branch substitutes the oldest member of the next branch, that provides a high speed of the algorithm convergence .
Differential Evolution Entirely Parallel method.
Differential Evolution (DE) is a stochastic iterative optimization technique introduced by Storn and Price . It is an effective method for minimization of various and complex quality functionals. The power of DE is based on the fact that under appropriate conditions it can attain the global extremum of a functional; the weakness of this method is in high computational demand and dependence on control variables, that provides a motivation for its parallelization. Previous work in this area has produced a number of methods that perform well on particular problems.
DE starts from a set of randomly generated parameter vectors , . The set is called population, and vectors are called individuals. The population on each iteration is referred to as generation. The size of population NP is fixed. The name of the method comes from the fact that the difference between members of the current population is used to generate offsprings (see Figure S1).
Being an evolutionary algorithm, DE can be easily parallelized due to the fact that each member of population is evaluated individually. The whole population is divided into subpopulations that are sometimes called islands or branches, one per each computational node. This eliminates the restriction on the number of individuals. The individual members of branches are then allowed to migrate, i.e. move, from one branch to another according to predefined topology . The number of iterations between migrations is called a communication period.
The Differential Evolution Entirely Parallel (DEEP) method, developed by us , takes into account the age of an individual that is defined as the number of iterations during which this individual survived without changes. The fact that a certain parameter set has not been updated during several iterations indicates that this set corresponds to the local minimum of the quality functional. As we seek the global minimum such a parameter set can be deleted from the population. The set of parameters corresponding to the minimal functional value found so far in a source parallel branch is copied in place of the deleted parameter set in a target branch.
The computational nodes are organized in a ring and individuals migrate from node k to node if it exists and from the last one to the first one. In  we have shown that the migration scheme provides a high speed of the algorithm convergence and the parallel efficiency is about 80% for the 50 nodes and 55% for 100 nodes. The reliability of the method was demonstrated by its ability to recover model parameters with about 1% accuracy.
Other details can be found in Supplementary information (See Text S1).
For the comprehensive analysis of modeling results it is necessary to know how reliable the parameter estimates are. In practice insufficient or noisy data, as well as the strong parameter correlation or even their functional relation may prevent the unambiguous determination of parameter values. Such parameters are related to as non-identifiable.
To reveal non-identifiable parameters the method based on confidence intervals ,  is applied. The confidence intervals are constructed for the parameter estimates in the vicinity of model solution and are given by(2) where is the sensitivity matrix, the matrix of partial derivatives of the model solution with respect to the parameter vector; is the objective functional; is an -quantile of -distribution with m and N-m degrees of freedom. The size of confidence intervals characterize the sensitivity of the solution to parameter changes: the shorter is the confidence interval the more reliable is the parameter estimate. If the most important feature of the parameter estimate is its sign, the identifiable estimate must have the confidence interval bounded away from zero. It is important to mention that the confidence intervals are only estimated precisely in case of independent parameters, if some parameters are strongly correlated the confidence intervals are overestimated. In other words the confidence interval (2) is the whole area of the parameter variation as the other parameters take any possible values from the m-dimensional confidence area (see equation (1) and Figure 1 in Text S2). Besides, correlation of parameters causes calculational errors due to ill-conditionality of the sensitivity matrix. This issue is discussed in more detail in Text S2.
The other method to detect interrelations between parameters is the collinearity analysis presented in . The method is suitable for models with large number of parameters. The aim of the method is to reveal the so-called near collinear columns of the sensitivity matrix, the matrix of partial derivatives of the model solution with respect to the parameter vector, and thus detect subsets of non-identifiable parameters. Identifiability of a parameter subset is characterized by collinearity index defined as(3)
For more detailed description of the methods see Text S2.
Geometric interpretation of Differential Evolution.
95% confidence intervals for estimates of regulatory weights, elements of genetic inter-connectivity matrix T in 11 circuits. Regulators and target genes are gap genes hb (H), Kr(K), gt(G) and kni(N). Graphs are labeled by gene notations, the first letter corresponds to the target gene (e.g., HK stands for ).
95% confidence intervals for estimates of elements of genetic inter-connectivity matrix E in 11 circuits. Target genes are gap genes hb (H), Kr(K), gt(G) and kni(N); external regulators are bcd(B), cad(C) and tll(T). Graphs are labeled by gene notations, the first letter corresponds to the target gene (e.g., HB stands for ).
Comparison of the gap gene network topologies predicted by the current two genotype model (A) and earlier models (B). Dashed lines show interactions with regulatory weights that were either non-identifiable or classified into different categories in different models.
Interactions involved in regulation of Hb posterior domain. A. Modeled expression patterns at temporal classes T1, T3 and T7. B. Regulatory contributions in wild type (center) and mutant (right). C. Temporal change in regulatory contributions at position corresponding to Hb domain maximum. Colored areas are given by or in equation (1) and reflect the strength of a given interaction at a specific point in space and time. All plots are based on best scoring solution (circuit C9, see Table S1 for parameters).
Spurious expression of hb in the region of 60–77% EL is not responsible for decrease in the gt expression level. Patterns for mutant without the spurious domain shown in the right panel were obtained with Gt input to hb expression equal to zero. To compensate the decrease in repression level we set the gt and kni autoactivation to 70% of values found by fitting the model to data. The left panel is shown for comparison; it demonstrates that in wild type embryos the autoactivation level decrease does not lead to reduction of gt and kni expression to the levels observed in mutants.
Parameters of a representative gene network. Rows correspond to target genes, columns to regulators hb (H), Kr(K), gt(G), kni(N), bcd(B), cad(C) and tll(T). R - maximum synthesis rate, D - diffusion coefficient, - decay rate, is a protein half-life measured in minutes. Promoter thresholds h for all genes were fixed at value −3.5. This reduces time needed for DEEP to find the minimum. The values for diffusion coefficient D are given as they appear in the equations so the units are . To relate these values to conventional diffusion coefficient, we should consider the diffusion term in the model equations as a finite difference approximation of the conventional diffusion term containing the second order spatial derivative. The approximation takes place at the 1D mesh of points that coincide with real nuclei. That gives the following approximate formula for the conventional diffusion coefficient , where is the distance between adjacent nuclei. As the real diffusion coefficients of the proteins in question haven't been measured in live embryos or such measurements are not available we can compare our values with results for other proteins. For example, the diffusion coefficient of Bcd was measured in live Drosophila embryos by Gregor et al.  as . By using the above formula, we get the following values for the conventional diffusion coefficient for gap proteins obtained from the fitting: , , , and . These values are realistic.
Circuit parameter sets: 11 sets of parameter values that optimize the transformed functional. Regulatory weights are labeled by single-letter notations of genes: hb(H), Kr(K), gt(G), kni(N), bcd(B), cad(C), tll(T).
Supplementary methods. Parameter estimation.
Supplementary methods. Identifiability analysis.
We thank Johannes Jaeger, Alexander Samsonov, and Vitaliy Gursky for the helpful comments; Manu and Lena Panok for the data on hb expression in Kr;kni and kni mutants (Figure 7B). We thank JUC Chromas, St.Petersburg State University, Russia, and Center for Developmental Genetics, Stony Brook University, USA, for sharing the confocal microscopes.
Conceived and designed the experiments: SS JR. Performed the experiments: SS. Analyzed the data: KK SS EM. Wrote the paper: KK SS EM MS.
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