High bicoid levels render the terminal system dispensable for Drosophila head development

In Drosophila, the gradient of the Bicoid (Bcd) morphogen organizes the anteroposterior axis while the ends of the embryo are patterned by the maternal terminal system. At the posterior pole, expression of terminal gap genes is mediated by the local activation of the Torso receptor tyrosine kinase (Tor). At the anterior, terminal gap genes are also activated by the Tor pathway but Bcd contributes to their activation. Here we present evidence that Tor and Bcd act independently on common target genes in an additive manner. Furthermore, we show that the terminal maternal system is not required for proper head development, since high levels of Bcd activity can functionally rescue the lack of terminal system activity at the anterior pole. This observation is consistent with a recent evolution of an anterior morphogenetic center consisting of Bcd and anterior Tor function.     

Dynamic modeling of gene expression in prokaryotes: application to glucose-lactose diauxie in Escherichia coli

Coexpression of genes or, more generally, similarity in the expression profiles poses an unsurmountable obstacle to inferring the gene regulatory network (GRN) based solely on data from DNA microarray time series. Clustering of genes with similar expression profiles allows for a course-grained view of the GRN and a probabilistic determination of the connectivity among the clusters. We present a model for the temporal evolution of a gene cluster network which takes into account interactions of gene products with genes and, through a non-constant degradation rate, with other gene products. The number of model parameters is reduced by using polynomial functions to interpolate temporal data points. In this manner, the task of parameter estimation is reduced to a system of linear algebraic equations, thus making the computation time shorter by orders of magnitude. To eliminate irrelevant networks, we test each GRN for stability with respect to parameter variations, and impose restrictions on its behavior near the steady state. We apply our model and methods to DNA microarray time series' data collected on Escherichia coli during glucose-lactose diauxie and infer the most probable cluster network for different phases of the experiment. ELECTRONIC SUPPLEMENTARY MATERIAL: The online version of this article (doi:10.1007/s11693-011-9079-2) contains supplementary material, which is available to authorized users.     

Modeling the precision and robustness of Hunchback border during Drosophila embryonic development

During anterior-posterior axis specification in the Drosophila embryo, the Hunchback (Hb) protein forms a sharp boundary at the mid-point of the embryo with great positional precision. While Bicoid (Bcd) is a known upstream regulator for hb expression, there is evidence to suggest that Hb effectively filters out “noisy” data received from varied Bcd gradients. We use mathematical models to explore simple regulatory networks which filter out such noise to produce a precise Hb boundary. We find that in addition to Bcd and Hb, at least one freely evolving protein is necessary. An automated search yields a number of examples of three-protein networks exhibiting the desired precision. In all such networks, Hb diffuses much slower than the third protein. In addition, the action of Hb on the third protein is the opposite of the action of the third protein on hb (i.e. if Hb activates the third protein, then the third protein inhibits hb expression, and vice versa). Most of the discovered systems satisfy the known biological properties, that Bcd activates hb, and that Hb activates its own expression. We find that all network topologies satisfying these constraints arise among the networks exhibiting the desired precision. Investigating the dynamics of these networks, we find that under a general class of non-uniform initial conditions, Bcd can be eliminated from the system and the spatiotemporal evolution of these two proteins alone is sufficient to recapture the dynamics. We hypothesize that Bcd is needed only to spatially disturb the gradient of the third protein, and then becomes unnecessary in the further evolution of the Hb border. This provides a possible explanation as to why the Hb dynamics are robust under perturbations of the Bcd gradient. Under this hypothesis, other proteins would be able to assume the role of Bcd in our simulations (possibly in the case of evolutionary divergences or a redundancy in the process), with the only constraint that they act to positively regulate hb.     

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.     

Bicoid - morphogen function revisited

Bicoid (Bcd) functions as a morphogen during Drosophila development. Accordingly, bcd mRNA is maternally localized to the anterior pole of the embryo, and Bcd forms an anterior/posterior gradient, which functions in a concentration dependent fashion. Thus, nuclei receiving identical amounts of Bcd should express the same target genes. However, we found that ectopic, uniform expression of Bcd causes anterior gene expression in the posterior with mirror image polarity, indicating that one or several additional factors must provide positional information. Recently, we have shown that one of these factors is Capicua (Cic), a ubiquitous maternal repressor that is down-regulated at the embryonic termini by maternal Torso, a key component of the maternal terminal system. Cic acts on Bcd dependent enhancer elements by repression and thereby controls the posterior limit of Bcd target gene expression. Based on these new findings, we propose that spatial control of gene expression in the anterior region of the embryo is not solely the result of Bcd morphogen action. Rather, it relies on a "morphogenic network" that integrates the terminal system and Bcd activities, providing both polarity and spatial information to the prospective head region of the developing embryo.     

Morphogen gradient formation and action: insights from studying Bicoid protein degradation

In a recent publication, we identified a novel F-box protein, encoded by fates-shifted (fsd), that plays a role in targeting Bcd for ubiquitination and degradation. Our analysis of mutant Drosophila embryos suggests that Bcd protein degradation is important for proper gradient formation and developmental fate specification. Here we describe further experiments that lead to an estimate of Bcd half-life, < 15 min, in embryos during the time of gradient formation. We use our findings to evaluate different models of Bcd gradient formation. With this new estimate, we simulate the Bcd gradient formation process in our own biologically realistic 2-D model. Finally, we discuss the role of Bcd-encoded positional information in controlling the positioning and precision of developmental decisions.     

Morphogen gradient formation and action

In a recent publication, we identified a novel F-box protein, encoded by fates-shifted (fsd), that plays a role in targeting Bcd for ubiquitination and degradation. Our analysis of mutant Drosophila embryos suggests that Bcd protein degradation is important for proper gradient formation and developmental fate specification. Here we describe further experiments that lead to an estimate of Bcd half-life, &lt;15 min, in embryos during the time of gradient formation. We use our findings to evaluate different models of Bcd gradient formation. With this new estimate, we simulate the Bcd gradient formation process in our own biologically realistic 2-D model. Finally, we discuss the role of Bcd-encoded positional information in controlling the positioning and precision of developmental decisions.     

PCA based population generation for genetic network optimization

A gene regulatory network (GRN) represents a set of genes and its regulatory interactions. The inference of the regulatory interactions between genes is usually carried out using an appropriate mathematical model and the available gene expression profile. Among the various models proposed for GRN inference, our recently proposed Michaelis–Menten based ODE model provides a good trade-off between the computational complexity and biological relevance. This model, like other known GRN models, also uses an evolutionary algorithm for parameter estimation. Considering various issues associated with such population based stochastic optimization approaches (e.g. diversity, premature convergence due to local optima, accuracy, etc.), it becomes important to seed the initial population with good individuals which are closer to the optimal solution. In this paper, we exploit the inherent strength of principal component analysis (PCA) in a novel manner to initialize the population for GRN optimization. The benefit of the proposed method is validated by reconstructing in silico and in vivo networks of various sizes. For the same level of accuracy, the approach with PCA based initialization shows improved convergence speed.     

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.