Master equation simulation analysis of immunostained Bicoid morphogen gradient

Gene regulatory networks in Drosophila early embryonic development as a model for the study of the temporal identity of neuroblasts

Genes belonging to the "gap" and "gap-like" family constitute the best-studied gene regulatory networks (GRNs) in Drosophila embryogenesis. Gap genes are a core of two subnetworks controlling embryonic segmentation: (hunchback, hb; Krüppel, Kr; giant, gt; and knirps, kni) and (hb; Kr; pou-domain, pdm; and, probably, castor, cas). Of particular interest is that (hb, Kr, pdm, cas) also specifies the temporal identity of stem cells, neuroblasts, in Drosophila neurogenesis. This GRN controls the sequential differentiation of neuroblasts during the asymmetric cell division. In the last decades, modeling of the patterning of gene ensemble (hb, Kr, gt, kni) in segmentation was in the center of attention. We show that our previously published and extensively studied model at a certain level of external factors is able to reproduce temporal patterns of (hb, Kr, pdm, cas) in neurogenesis with minor evolutionary explicable modifications. This result testifies in favor of a hypothesis that the similarity of two gene ensembles active in segmentation and neurogenesis is a result of co-option of the network architecture in evolution from the common ancestral form. By means of the model dynamical analysis, it is shown that the establishment of the robust patterns in both systems could be explained in terms of the action of attractors in the gap gene dynamical system. We formulate the common principles underlying the robustness of both GRNs in segmentation and neurogenesis due to the similar functional organization of the gene ensembles as having the same evolutionary origin.

French flag gradients and Turing reaction-diffusion versus differentiation waves as models of morphogenesis

The Turing reaction-diffusion model and the French Flag Model are widely accepted in the field of development as the best models for explaining embryogenesis. Virtually all current attempts to understand cell differentiation in embryos begin and end with the assumption that some combination of these two models works. The result may become a bias in embryogenesis in assuming the problem has been solved by these two-chemical substance-based models. Neither model is applied consistently. We review the differences between the French Flag, Turing reaction-diffusion model, and a mechanochemical model called the differentiation wave/cell state splitter model. The cytoskeletal cell state splitter and the embryonic differentiation waves was first proposed in 1987 as a combined physics and chemistry model for cell differentiation in embryos, based on empirical observations on urodele amphibian embryos. We hope that the development of theory can be advanced and observations relevant to distinguishing the embryonic differentiation wave model from the French Flag model and reaction-diffusion equations will be taken up by experimentalists. Experimentalists rely on mathematical biologists for theory, and therefore depend on them for what parameters they choose to measure and ignore. Therefore, mathematical biologists need to fully understand the distinctions between these three models.

Optimizing bicoid signal extraction

Signal extraction and analysis is of great importance, not only in fields such as economics and meteorology, but also in genetics and even biomedicine. There exists a range of parametric and nonparametric techniques which can perform signal extractions. However, the aim of this paper is to define a new approach for optimising signal extraction from bicoid gene expression profile. Having studied both parametric and nonparametric signal extraction techniques, we identified the lack of specific criteria enabling users to select the optimal signal extraction parameters. Exploiting the expression profile of bicoid gene, which is a maternal segmentation coordinate gene found in Drosophila melanogaster, we introduce a new approach for optimising the signal extraction using a nonparametric technique. The underlying criteria are based on the distribution of the residual, more specifically its skewness.

The organelle of differentiation in embryos: the cell state splitter

The cell state splitter is a membraneless organelle at the apical end of each epithelial cell in a developing embryo. It consists of a microfilament ring and an intermediate filament ring subtending a microtubule mat. The microtubules and microfilament ring are in mechanical opposition as in a tensegrity structure. The cell state splitter is bistable, perturbations causing it to contract or expand radially. The intermediate filament ring provides metastability against small perturbations. Once this snap-through organelle is triggered, it initiates signal transduction to the nucleus, which changes gene expression in one of two readied manners, causing its cell to undergo a step of determination and subsequent differentiation. The cell state splitter also triggers the cell state splitters of adjacent cells to respond, resulting in a differentiation wave. Embryogenesis may be represented then as a bifurcating differentiation tree, each edge representing one cell type. In combination with the differentiation waves they propagate, cell state splitters explain the spatiotemporal course of differentiation in the developing embryo. This review is excerpted from and elaborates on "Embryogenesis Explained" (World Scientific Publishing, Singapore, 2016).

Shaped 3D singular spectrum analysis for quantifying gene expression, with application to the early zebrafish embryo

Recent progress in microscopy technologies, biological markers, and automated processing methods is making possible the development of gene expression atlases at cellular-level resolution over whole embryos. Raw data on gene expression is usually very noisy. This noise comes from both experimental (technical/methodological) and true biological sources (from stochastic biochemical processes). In addition, the cells or nuclei being imaged are irregularly arranged in 3D space. This makes the processing, extraction, and study of expression signals and intrinsic biological noise a serious challenge for 3D data, requiring new computational approaches. Here, we present a new approach for studying gene expression in nuclei located in a thick layer around a spherical surface. The method includes depth equalization on the sphere, flattening, interpolation to a regular grid, pattern extraction by Shaped 3D singular spectrum analysis (SSA), and interpolation back to original nuclear positions. The approach is demonstrated on several examples of gene expression in the zebrafish egg (a model system in vertebrate development). The method is tested on several different data geometries (e.g., nuclear positions) and different forms of gene expression patterns. Fully 3D datasets for developmental gene expression are becoming increasingly available; we discuss the prospects of applying 3D-SSA to data processing and analysis in this growing field.

Making models match measurements: model optimization for morphogen patterning networks

Mathematical modeling of developmental signaling networks has played an increasingly important role in the identification of regulatory mechanisms by providing a sandbox for hypothesis testing and experiment design. Whether these models consist of an equation with a few parameters or dozens of equations with hundreds of parameters, a prerequisite to model-based discovery is to bring simulated behavior into agreement with observed data via parameter estimation. These parameters provide insight into the system (e.g., enzymatic rate constants describe enzyme properties). Depending on the nature of the model fit desired - from qualitative (relative spatial positions of phosphorylation) to quantitative (exact agreement of spatial position and concentration of gene products) - different measures of data-model mismatch are used to estimate different parameter values, which contain different levels of usable information and/or uncertainty. To facilitate the adoption of modeling as a tool for discovery alongside other tools such as genetics, immunostaining, and biochemistry, careful consideration needs to be given to how well a model fits the available data, what the optimized parameter values mean in a biological context, and how the uncertainty in model parameters and predictions plays into experiment design. The core discussion herein pertains to the quantification of model-to-data agreement, which constitutes the first measure of a model's performance and future utility to the problem at hand. Integration of this experimental data and the appropriate choice of objective measures of data-model agreement will continue to drive modeling forward as a tool that contributes to experimental discovery. The Drosophila melanogaster gap gene system, in which model parameters are optimized against in situ immunofluorescence intensities, demonstrates the importance of error quantification, which is applicable to a wide array of developmental modeling studies.

The synthesis-diffusion-degradation model explains Bicoid gradient formation in unfertilized eggs

Precise formation of morphogen gradients is essential to the establishment of reproducible pattern in development. Mechanisms proposed for obtaining the requisite precision range from simple models with few parameters to more complex models involving many regulated quantities. The synthesis-diffusion-degradation (SDD) model is a relatively simple model explaining the formation of the Bicoid gradient in Drosophila melanogaster, in which the steady-state characteristic length of the gradient is determined solely by the rates of diffusion and degradation of the morphogen. In this work, we test the SDD model in unfertilized D. melanogaster eggs, which contain a single female pronucleus and lack the nuclear division cycles and other zygotic regulatory processes seen in fertilized eggs. Using two-photon live imaging as well as a novel method for quantitative imaging based on decorrelation of photoswitching waveforms, we find that the Bicoid gradient is longer and shallower in unfertilized eggs as compared to the gradient at the same time points in fertilized eggs. Using a means of measuring the Bicoid lifetime by conjugation to a photoconvertible fluorophore, we find that the lifetime is correspondingly longer in unfertilized eggs, providing qualitative and quantitative agreement with the predictions of the SDD model.

Genetic variation in the Yolk protein expression network of Drosophila melanogaster: sex-biased negative correlations with longevity

One of the persistent problems in biology is understanding how genetic variation contributes to phenotypic variation. Associations at many levels have been reported, and yet causal inference has remained elusive. We propose to rely on the knowledge of causal relationships established by molecular biology approaches. The existing molecular knowledge forms a firm backbone upon which hypotheses connecting genetic variation, transcriptional variation and phenotypic variation can be built. The sex determination pathway is a well-established molecular network, with the Yolk protein 1-3 (Yp) genes as the most downstream target. Our analyses reveal that genetic variation in expression for genes known to be upstream in the pathway explains variation in downstream targets. Relationships differ between the two sexes, and each Yp has a distinct transcriptional pattern. Yp expression is significantly negatively correlated with longevity, an important life history trait, for both males and females.

Measuring gene expression noise in early Drosophila embryos: nucleus-to-nucleus variability

In recent years the analysis of noise in gene expression has widely attracted the attention of experimentalists and theoreticians. Experimentally, the approaches based on in vivo fluorescent reporters in single cells appear to be straightforward and effective tools for bacteria and yeast. However, transferring these approaches to multicellular organisms presents many methodological problems. Here we describe our approach to measure between-nucleus variability (noise) in the primary morphogenetic gradient of Bicoid (Bcd) in the precellular blastoderm stage of fruit fly (Drosophila) embryos. The approach is based on the comparison of results for fixed immunostained embryos with observations of live embryos carrying fluorescent Bcd (Bcd-GFP). We measure the noise using two-dimensional Singular Spectrum Analysis (2D SSA). We have found that the nucleus-to-nucleus noise in Bcd intensity, both for live (Bcd-GFP) and for fixed immunstained embryos, tends to be signal-independent. In addition, the character of the noise is sensitive to the nuclear masking technique used to extract quantitative intensities. Further, the method of decomposing the raw quantitative expression data into a signal (expression surface) and residual noise affects the character of the residual noise. We find that careful masking of confocal images and use of appropriate computational tools to decompose raw expression data into trend and noise makes it possible to extract and study the biological noise of gene expression.

The role of regulated mRNA stability in establishing bicoid morphogen gradient in Drosophila embryonic development

The Bicoid morphogen is amongst the earliest triggers of differential spatial pattern of gene expression and subsequent cell fate determination in the embryonic development of Drosophila. This maternally deposited morphogen is thought to diffuse in the embryo, establishing a concentration gradient which is sensed by downstream genes. In most model based analyses of this process, the translation of the bicoid mRNA is thought to take place at a fixed rate from the anterior pole of the embryo and a supply of the resulting protein at a constant rate is assumed. Is this process of morphogen generation a passive one as assumed in the modelling literature so far, or would available data support an alternate hypothesis that the stability of the mRNA is regulated by active processes? We introduce a model in which the stability of the maternal mRNA is regulated by being held constant for a length of time, followed by rapid degradation. With this more realistic model of the source, we have analysed three computational models of spatial morphogen propagation along the anterior-posterior axis: (a) passive diffusion modelled as a deterministic differential equation, (b) diffusion enhanced by a cytoplasmic flow term; and (c) diffusion modelled by stochastic simulation of the corresponding chemical reactions. Parameter estimation on these models by matching to publicly available data on spatio-temporal Bicoid profiles suggests strong support for regulated stability over either a constant supply rate or one where the maternal mRNA is permitted to degrade in a passive manner.

A multiscale investigation of bicoid-dependent transcriptional events in Drosophila embryos

Background

Morphogen molecules form concentration gradients to provide spatial information to cells in a developing embryo. Precisely how cells decode such information to form patterns with sharp boundaries remains an open question. For example, it remains controversial whether the Drosophila morphogenetic protein Bicoid (Bcd) plays a transient or sustained role in activating its target genes to establish sharp expression boundaries during development.

Methodology/Principal Findings

In this study, we describe a method to simultaneously detect Bcd and the nascent transcripts of its target genes in developing embryos. This method allows us to investigate the relationship between Bcd and the transcriptional status of individual copies of its target genes on distinct scales. We show that, on three scales analyzed concurrently—embryonic, nuclear and local, the actively-transcribing gene copies are associated with high Bcd concentrations. These results underscore the importance of Bcd as a sustained input for transcriptional decisions of individual copies of its target genes during development. We also show that the Bcd-dependent transcriptional decisions have a significantly higher noise than Bcd-dependent gene products, suggesting that, consistent with theoretical studies, time and/or space averaging reduces the noise of Bcd-activated transcriptional output. Finally, our analysis of an X-linked Bcd target gene reveals that Bcd-dependent transcription bursts at twice the frequency in males as in females, providing a mechanism for dosage compensation in early Drosophila embryos.

Conclusion/Significance

Our study represents a first experimental uncovering of the actions of Bcd in controlling the actual transcriptional events while its positional information is decoded during development. It establishes a sustained role of Bcd in transcriptional decisions of individual copies of its target genes to generate sharp expression boundaries. It also provides an experimental evaluation of the effect of time and/or space averaging on Bcd-dependent transcriptional output, and establishes a dosage compensation mechanism in early Drosophila embryos.

The formation of the Bicoid morphogen gradient requires protein movement from anteriorly localized mRNA

New quantitative data show that the Bicoid morphogen gradient is generated from a dynamic localized source and that protein gradient formation requires protein movement along the anterior-posterior axis.

The Bicoid morphogen gradient directs the patterning of cell fates along the anterior-posterior axis of the syncytial Drosophila embryo and serves as a paradigm of morphogen-mediated patterning. The simplest models of gradient formation rely on constant protein synthesis and diffusion from anteriorly localized source mRNA, coupled with uniform protein degradation. However, currently such models cannot account for all known gradient characteristics. Recent work has proposed that bicoid mRNA spatial distribution is sufficient to produce the observed protein gradient, minimizing the role of protein transport. Here, we adapt a novel method of fluorescent in situ hybridization to quantify the global spatio-temporal dynamics of bicoid mRNA particles. We determine that >90% of all bicoid mRNA is continuously present within the anterior 20% of the embryo. bicoid mRNA distribution along the body axis remains nearly unchanged despite dynamic mRNA translocation from the embryo core to the cortex. To evaluate the impact of mRNA distribution on protein gradient dynamics, we provide detailed quantitative measurements of nuclear Bicoid levels during the formation of the protein gradient. We find that gradient establishment begins 45 minutes after fertilization and that the gradient requires about 50 minutes to reach peak levels. In numerical simulations of gradient formation, we find that incorporating the actual bicoid mRNA distribution yields a closer prediction of the observed protein dynamics compared to modeling protein production from a point source at the anterior pole. We conclude that the spatial distribution of bicoid mRNA contributes to, but cannot account for, protein gradient formation, and therefore that protein movement, either active or passive, is required for gradient formation.

The Bicoid protein gradient plays a crucial role in determining the anterior body pattern of Drosophila embryos. This gradient is the classic example of morphogen-mediated patterning of a developing metazoan and serves as a major topic for mathematical modeling. Accurate modeling of the gradient requires a detailed account of the underlying bicoid mRNA distribution. The classic model holds that mRNA protein gradient arises via protein diffusion from mRNA localized at the anterior of the developing egg. In contrast, recent proposals suggest that an mRNA gradient generates the protein gradient without protein movement. In this study, we introduce a novel mRNA quantification method for Drosophila embryos, which allows us to visualize each individual mRNA particle accurately in whole embryos. We demonstrate that all but a few mRNA particles are confined to the anterior 20% of the egg, and consequently that the protein must move in order to establish a gradient. We further report that the mRNA distribution is highly dynamic during the time of protein synthesis. In numerical simulations, we show that incorporating realistic spatial locations of the individual source mRNA molecules throughout the developmental period is necessary to accurately model the experimentally observed protein gradient dynamics.

Gene expression noise in spatial patterning: hunchback promoter structure affects noise amplitude and distribution in Drosophila segmentation

Positional information in developing embryos is specified by spatial gradients of transcriptional regulators. One of the classic systems for studying this is the activation of the hunchback (hb) gene in early fruit fly (Drosophila) segmentation by the maternally-derived gradient of the Bicoid (Bcd) protein. Gene regulation is subject to intrinsic noise which can produce variable expression. This variability must be constrained in the highly reproducible and coordinated events of development. We identify means by which noise is controlled during gene expression by characterizing the dependence of hb mRNA and protein output noise on hb promoter structure and transcriptional dynamics. We use a stochastic model of the hb promoter in which the number and strength of Bcd and Hb (self-regulatory) binding sites can be varied. Model parameters are fit to data from WT embryos, the self-regulation mutant hb(14F), and lacZ reporter constructs using different portions of the hb promoter. We have corroborated model noise predictions experimentally. The results indicate that WT (self-regulatory) Hb output noise is predominantly dependent on the transcription and translation dynamics of its own expression, rather than on Bcd fluctuations. The constructs and mutant, which lack self-regulation, indicate that the multiple Bcd binding sites in the hb promoter (and their strengths) also play a role in buffering noise. The model is robust to the variation in Bcd binding site number across a number of fly species. This study identifies particular ways in which promoter structure and regulatory dynamics reduce hb output noise. Insofar as many of these are common features of genes (e.g. multiple regulatory sites, cooperativity, self-feedback), the current results contribute to the general understanding of the reproducibility and determinacy of spatial patterning in early development.

Shaping a morphogen gradient for positional precision

Morphogen gradients, which provide positional information to cells in a developing tissue, could in principle adopt any nonuniform profile. To our knowledge, how the profile of a morphogen gradient affects positional precision has not been well studied experimentally. Here, we compare the positional precision provided by the Drosophila morphogenetic protein Bicoid (Bcd) in wild-type (wt) embryos with embryos lacking an interacting cofactor. The Bcd gradient in the latter case exhibits decreased positional precision around mid-embryo compared with its wt counterpart. The domain boundary of Hunchback (Hb), a target activated by Bcd, becomes more variable in mutant embryos. By considering embryo-to-embryo, internal, and measurement fluctuations, we dissect mathematically the relevant sources of fluctuations that contribute to the error in positional information. Using this approach, we show that the defect in Hb boundary positioning in mutant embryos is directly reflective of an altered Bcd gradient profile with increasing flatness toward mid-embryo. Furthermore, we find that noise in the Bcd input signal is dominated by internal fluctuations but, due to time and spatial averaging, the spatial precision of the Hb boundary is primarily affected by embryo-to-embryo variations. Our results demonstrate that the positional information provided by the wt Bcd gradient profile is highly precise and necessary for patterning precision.

A two-dimensional simulation model of the bicoid gradient in Drosophila

Background

Bicoid (Bcd) is a Drosophila morphogenetic protein responsible for patterning the anterior structures in embryos. Recent experimental studies have revealed important insights into the behavior of this morphogen gradient, making it necessary to develop a model that can recapitulate the biological features of the system, including its dynamic and scaling properties.

Methodology/Principal Findings

We present a biologically realistic 2-D model of the dynamics of the Bcd gradient in Drosophila embryos. This model is based on equilibrium binding of Bcd molecules to non-specific, low affinity DNA sites throughout the Drosophila genome. It considers both the diffusion media within which the Bcd gradient is formed and the dynamic and other relevant properties of bcd mRNA from which Bcd protein is produced. Our model recapitulates key features of the Bcd protein gradient observed experimentally, including its scaling properties and the stability of its nuclear concentrations during development. Our simulation model also allows us to evaluate the effects of other biological activities on Bcd gradient formation, including the dynamic redistribution of bcd mRNA in early embryos. Our simulation results suggest that, in our model, Bcd protein diffusion is important for the formation of an exponential gradient in embryos.

Conclusions/Significance

The 2-D model described in this report is a simple and versatile simulation procedure, providing a quantitative evaluation of the Bcd gradient system. Our results suggest an important role of Bcd binding to non-specific, low-affinity DNA sites in proper formation of the Bcd gradient in our model. They demonstrate that highly complex biological systems can be effectively modeled with relatively few parameters.

Parameter estimation and inference for stochastic reaction-diffusion systems: application to morphogenesis in D. melanogaster

BACKGROUND

Reaction-diffusion systems are frequently used in systems biology to model developmental and signalling processes. In many applications, count numbers of the diffusing molecular species are very low, leading to the need to explicitly model the inherent variability using stochastic methods. Despite their importance and frequent use, parameter estimation for both deterministic and stochastic reaction-diffusion systems is still a challenging problem.

RESULTS

We present a Bayesian inference approach to solve both the parameter and state estimation problem for stochastic reaction-diffusion systems. This allows a determination of the full posterior distribution of the parameters (expected values and uncertainty). We benchmark the method by illustrating it on a simple synthetic experiment. We then test the method on real data about the diffusion of the morphogen Bicoid in Drosophila melanogaster. The results show how the precision with which parameters can be inferred varies dramatically, indicating that the ability to infer full posterior distributions on the parameters can have important experimental design consequences.

CONCLUSIONS

The results obtained demonstrate the feasibility and potential advantages of applying a Bayesian approach to parameter estimation in stochastic reaction-diffusion systems. In particular, the ability to estimate credibility intervals associated with parameter estimates can be precious for experimental design. Further work, however, will be needed to ensure the method can scale up to larger problems.

How transcription factors can adjust the gene expression floodgates

The rate of transcription initiation is the main level of quantitative control of gene expression, primarily responsible for the accumulation of mRNAs in the cell. Many, if not all, molecular actors involved in transcription initiation are known but the mechanisms underlying the frequency of initiations, remain elusive. To make the connection between transcription factors and the frequency of transcription initiation, intricated aspects of this complex activity are classified i) depending on whether or not the DNA-bound transcription factors directly activate the commitment to transcription and ii) on the destructive or non-destructive effect of transcription initiation on the stability of promoter complexes. Two possible sources of synergy allowing the combinatorial specificity of transcription factors action are compared, for binding to DNA and for recruiting transcription machineries. Tentative formulations are proposed to discriminate the different micro-reversible modes of DNA binding cooperativity modulating the specificity and dosage of transcription initiation.

When it pays to rush: interpreting morphogen gradients prior to steady-state

During development, morphogen gradients precisely determine the position of gene expression boundaries despite the inevitable presence of fluctuations. Recent experiments suggest that some morphogen gradients may be interpreted prior to reaching steady-state. Theoretical work has predicted that such systems will be more robust to embryo-to-embryo fluctuations. By analyzing two experimentally motivated models of morphogen gradient formation, we investigate the positional precision of gene expression boundaries determined by pre-steady-state morphogen gradients in the presence of embryo-to-embryo fluctuations, internal biochemical noise and variations in the timing of morphogen measurement. Morphogens that are direct transcription factors are found to be particularly sensitive to internal noise when interpreted prior to steady-state, disadvantaging early measurement, even in the presence of large embryo-to-embryo fluctuations. Morphogens interpreted by cell-surface receptors can be measured prior to steady-state without significant decrease in positional precision provided fluctuations in the timing of measurement are small. Applying our results to experiment, we predict that Bicoid, a transcription factor morphogen in Drosophila, is unlikely to be interpreted prior to reaching steady-state. We also predict that Activin in Xenopus and Nodal in zebrafish, morphogens interpreted by cell-surface receptors, can be decoded in pre-steady-state.

Statistical lower bounds on protein copy number from fluorescence expression images

MOTIVATION

Fluorescence imaging has become a commonplace for quantitatively measuring mRNA or protein expression in cells and tissues. However, such expression data are usually relative-absolute concentrations or molecular copy numbers are typically not known. While this is satisfactory for many applications, for certain kinds of quantitative network modeling and analysis of expression noise, absolute measures of expression are necessary.

RESULTS

We propose two methods for estimating molecular copy numbers from single uncalibrated expression images of tissues. These methods rely on expression variability between cells, due either to steady-state fluctuations or unequal distribution of molecules during cell division, to make their estimates. We apply these methods to 152 protein fluorescence expression images of Drosophila melanogaster embryos during early development, generating copy number estimates for 14 genes in the segmentation network. We also analyze the effects of noise on our estimators and compare with empirical findings. Finally, we confirm an observation of Bar-Even et al., made in the much different setting of Saccharomyces cerevisiae, that steady-state expression variance tends to scale with mean expression.

AVAILABILITY

The data are all drawn from FlyEx (explained within), and is available at http://flyex.ams.sunysb.edu/FlyEx/.

Morphogen profiles can be optimized to buffer against noise

Morphogen profiles play a vital role in biology by specifying position in embryonic development. However, the factors that influence the shape of a morphogen profile remain poorly understood. Since morphogens should provide precise positional information, one significant factor is the robustness of the profile to noise. We compare three experimentally relevant classes of morphogen profiles (linear, exponential, and algebraic) to see which is most precise when subject to both external embryo-to-embryo fluctuations and internal fluctuations due to intrinsically random processes such as diffusion. We find that both the kinetic parameters and the overall gradient shape (e.g., exponential versus algebraic) can be optimized to generate maximally precise positional information.

Inferring Drosophila gap gene regulatory network: a parameter sensitivity and perturbation analysis

BACKGROUND

Inverse modelling of gene regulatory networks (GRNs) capable of simulating continuous spatio-temporal biological processes requires accurate data and a good description of the system. If quantitative relations between genes cannot be extracted from direct measurements, an efficient method to estimate the unknown parameters is mandatory. A model that has been proposed to simulate spatio-temporal gene expression patterns is the connectionist model. This method describes the quantitative dynamics of a regulatory network in space. The model parameters are estimated by means of model-fitting algorithms. The gene interactions are identified without making any prior assumptions concerning the network connectivity. As a result, the inverse modelling might lead to multiple circuits showing the same quantitative behaviour and it is not possible to identify one optimal circuit. Consequently, it is important to address the quality of the circuits in terms of model robustness.

RESULTS

Here we investigate the sensitivity and robustness of circuits obtained from reverse engineering a model capable of simulating measured gene expression patterns. As a case study we use the early gap gene segmentation mechanism in Drosophila melanogaster. We consider the limitations of the connectionist model used to describe GRN Inferred from spatio-temporal gene expression. We address the problem of circuit discrimination, where the selection criterion within the optimization technique is based of the least square minimization on the error between data and simulated results.

CONCLUSION

Parameter sensitivity analysis allows one to discriminate between circuits having significant parameter and qualitative differences but exhibiting the same quantitative pattern. Furthermore, we show that using a stochastic model derived from a deterministic solution, one can introduce fluctuations within the model to analyze the circuits' robustness. Ultimately, we show that there is a close relation between circuit sensitivity and robustness to fluctuation, and that circuit robustness is rather modular than global. The current study shows that reverse engineering of GRNs should not only focus on estimating parameters by minimizing the difference between observation and simulation but also on other model properties. Our study suggests that multi-objective optimization based on robustness and sensitivity analysis has to be considered.

Canalization of gene expression in the Drosophila blastoderm by gap gene cross regulation

Developing embryos exhibit a robust capability to reduce phenotypic variations that occur naturally or as a result of experimental manipulation. This reduction in variation occurs by an epigenetic mechanism called canalization, a phenomenon which has resisted understanding because of a lack of necessary molecular data and of appropriate gene regulation models. In recent years, quantitative gene expression data have become available for the segment determination process in the Drosophila blastoderm, revealing a specific instance of canalization. These data show that the variation of the zygotic segmentation gene expression patterns is markedly reduced compared to earlier levels by the time gastrulation begins, and this variation is significantly lower than the variation of the maternal protein gradient Bicoid. We used a predictive dynamical model of gene regulation to study the effect of Bicoid variation on the downstream gap genes. The model correctly predicts the reduced variation of the gap gene expression patterns and allows the characterization of the canalizing mechanism. We show that the canalization is the result of specific regulatory interactions among the zygotic gap genes. We demonstrate the validity of this explanation by showing that variation is increased in embryos mutant for two gap genes, Krüppel and knirps, disproving competing proposals that canalization is due to an undiscovered morphogen, or that it does not take place at all. In an accompanying article in PLoS Computational Biology (doi:10.1371/journal.pcbi.1000303), we show that cross regulation between the gap genes causes their expression to approach dynamical attractors, reducing initial variation and providing a robust output. These results demonstrate that the Bicoid gradient is not sufficient to produce gap gene borders having the low variance observed, and instead this low variance is generated by gap gene cross regulation. More generally, we show that the complex multigenic phenomenon of canalization can be understood at a quantitative and predictive level by the application of a precise dynamical model.

Animals have an astonishing ability to develop reliably in spite of variable conditions during embryogenesis. More than 60 years ago, it was proposed that this property of development, called canalization, results from genetic interactions that adjust biochemical reactions so as to bring about reliable outcomes. Since then, a great deal of progress has been made in understanding the buffering of genotypic and environmental variation, and individual mutations that reveal variation have been identified. However, the mechanisms by which genetic interactions produce canalization are not yet well understood, because this requires molecular data on multiple developmental determinants and models that correctly predict complex interactions. We make use of gene expression data at both high spatial and temporal resolution for the gap genes involved in the segmentation of Drosophila. We also apply a mathematical model to show that cross regulation among the gap genes is responsible for canalization in this system. Furthermore, the model predicted specific interactions that cause canalization, and the prediction was validated experimentally. Our results show that groups of genes can act on one another to reduce variation and highlights the importance of genetic networks in generating robust development.

DuringDrosophila development, the expression patterns of gap genes are much less variable than the Bicoid morphogen gradient. Modeling and experiments show that this specific instance of canalization or developmental robustness occurs by gap gene cross regulation.

Probing intrinsic properties of a robust morphogen gradient in Drosophila

A remarkable feature of development is its reproducibility, the ability to correct embryo-to-embryo variations and instruct precise patterning. In Drosophila, embryonic patterning along the anterior-posterior axis is controlled by the morphogen gradient Bicoid (Bcd). In this article, we describe quantitative studies of the native Bcd gradient and its target Hunchback (Hb). We show that the native Bcd gradient is highly reproducible and is itself scaled with embryo length. While a precise Bcd gradient is necessary for precise Hb expression, it still has positional errors greater than Hb expression. We describe analyses further probing mechanisms for Bcd gradient scaling and correction of its residual positional errors. Our results suggest a simple model of a robust Bcd gradient sufficient to achieve scaled and precise activation of its targets. The robustness of this gradient is conferred by its intrinsic properties of "self-correcting" the inevitable input variations to achieve a precise and reproducible output.