Rearrangements of 2.5 kilobases of noncoding DNA from the Drosophila even-skipped locus define predictive rules of genomic cis-regulatory logic.

Rearrangements of about 2.5 kilobases of regulatory DNA located 5' of the transcription start site of the Drosophila even-skipped locus generate large-scale changes in the expression of even-skipped stripes 2, 3, and 7. The most radical effects are generated by juxtaposing the minimal stripe enhancers MSE2 and MSE3 for stripes 2 and 3 with and without small "spacer" segments less than 360 bp in length. We placed these fusion constructs in a targeted transformation site and obtained quantitative expression data for these transformants together with their controlling transcription factors at cellular resolution. These data demonstrated that the rearrangements can alter expression levels in stripe 2 and the 2-3 interstripe by a factor of more than 10. We reasoned that this behavior would place tight constraints on possible rules of genomic cis-regulatory logic. To find these constraints, we confronted our new expression data together with previously obtained data on other constructs with a computational model. The model contained representations of thermodynamic protein-DNA interactions including steric interference and cooperative binding, short-range repression, direct repression, activation, and coactivation. The model was highly constrained by the training data, which it described within the limits of experimental error. The model, so constrained, was able to correctly predict expression patterns driven by enhancers for other Drosophila genes; even-skipped enhancers not included in the training set; stripe 2, 3, and 7 enhancers from various Drosophilid and Sepsid species; and long segments of even-skipped regulatory DNA that contain multiple enhancers. The model further demonstrated that elevated expression driven by a fusion of MSE2 and MSE3 was a consequence of the recruitment of a portion of MSE3 to become a functional component of MSE2, demonstrating that cis-regulatory "elements" are not elementary objects.

Euler equation existence, non-uniqueness and mesh converged statistics.

We review existence and non-uniqueness results for the Euler equation of fluid flow. These results are placed in the context of physical models and their solutions. Non-uniqueness is in direct conflict with the purpose of practical simulations, so that a mitigating strategy, outlined here, is important. We illustrate these issues in an examination of mesh converged turbulent statistics, with comparison to laboratory experiments.

Nonideal Rayleigh-Taylor mixing.

Rayleigh-Taylor mixing is a classical hydrodynamic instability that occurs when a light fluid pushes against a heavy fluid. The two main sources of nonideal behavior in Rayleigh-Taylor (RT) mixing are regularizations (physical and numerical), which produce deviations from a pure Euler equation, scale invariant formulation, and nonideal (i.e., experimental) initial conditions. The Kolmogorov theory of turbulence predicts stirring at all length scales for the Euler fluid equations without regularization. We interpret mathematical theories of existence and nonuniqueness in this context, and we provide numerical evidence for dependence of the RT mixing rate on nonideal regularizations; in other words, indeterminacy when modeled by Euler equations. Operationally, indeterminacy shows up as nonunique solutions for RT mixing, parametrized by Schmidt and Prandtl numbers, in the large Reynolds number (Euler equation) limit. Verification and validation evidence is presented for the large eddy simulation algorithm used here. Mesh convergence depends on breaking the nonuniqueness with explicit use of the laminar Schmidt and Prandtl numbers and their turbulent counterparts, defined in terms of subgrid scale models. The dependence of the mixing rate on the Schmidt and Prandtl numbers and other physical parameters will be illustrated. We demonstrate numerically the influence of initial conditions on the mixing rate. Both the dominant short wavelength initial conditions and long wavelength perturbations are observed to play a role. By examination of two classes of experiments, we observe the absence of a single universal explanation, with long and short wavelength initial conditions, and the various physical and numerical regularizations contributing in different proportions in these two different contexts.

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.

Canalization of gene expression and domain shifts in the Drosophila blastoderm by dynamical attractors.

The variation in the expression patterns of the gap genes in the blastoderm of the fruit fly Drosophila melanogaster reduces over time as a result of cross regulation between these genes, a fact that we have demonstrated in an accompanying article in PLoS Biology (see Manu et al., doi:10.1371/journal.pbio.1000049). This biologically essential process is an example of the phenomenon known as canalization. It has been suggested that the developmental trajectory of a wild-type organism is inherently stable, and that canalization is a manifestation of this property. Although the role of gap genes in the canalization process was established by correctly predicting the response of the system to particular perturbations, the stability of the developmental trajectory remains to be investigated. For many years, it has been speculated that stability against perturbations during development can be described by dynamical systems having attracting sets that drive reductions of volume in phase space. In this paper, we show that both the reduction in variability of gap gene expression as well as shifts in the position of posterior gap gene domains are the result of the actions of attractors in the gap gene dynamical system. Two biologically distinct dynamical regions exist in the early embryo, separated by a bifurcation at 53% egg length. In the anterior region, reduction in variation occurs because of stability induced by point attractors, while in the posterior, the stability of the developmental trajectory arises from a one-dimensional attracting manifold. This manifold also controls a previously characterized anterior shift of posterior region gap domains. Our analysis shows that the complex phenomena of canalization and pattern formation in the Drosophila blastoderm can be understood in terms of the qualitative features of the dynamical system. The result confirms the idea that attractors are important for developmental stability and shows a richer variety of dynamical attractors in developmental systems than has been previously recognized.

Dynamic control of positional information in the early Drosophila embryo.

Morphogen gradients contribute to pattern formation by determining positional information in morphogenetic fields. Interpretation of positional information is thought to rely on direct, concentration-threshold-dependent mechanisms for establishing multiple differential domains of target gene expression. In Drosophila, maternal gradients establish the initial position of boundaries for zygotic gap gene expression, which in turn convey positional information to pair-rule and segment-polarity genes, the latter forming a segmental pre-pattern by the onset of gastrulation. Here we report, on the basis of quantitative gene expression data, substantial anterior shifts in the position of gap domains after their initial establishment. Using a data-driven mathematical modelling approach, we show that these shifts are based on a regulatory mechanism that relies on asymmetric gap-gap cross-repression and does not require the diffusion of gap proteins. Our analysis implies that the threshold-dependent interpretation of maternal morphogen concentration is not sufficient to determine shifting gap domain boundary positions, and suggests that establishing and interpreting positional information are not independent processes in the Drosophila blastoderm.

Dynamical analysis of regulatory interactions in the gap gene system of Drosophila melanogaster.

Genetic studies have revealed that segment determination in Drosophila melanogaster is based on hierarchical regulatory interactions among maternal coordinate and zygotic segmentation genes. The gap gene system constitutes the most upstream zygotic layer of this regulatory hierarchy, responsible for the initial interpretation of positional information encoded by maternal gradients. We present a detailed analysis of regulatory interactions involved in gap gene regulation based on gap gene circuits, which are mathematical gene network models used to infer regulatory interactions from quantitative gene expression data. Our models reproduce gap gene expression at high accuracy and temporal resolution. Regulatory interactions found in gap gene circuits provide consistent and sufficient mechanisms for gap gene expression, which largely agree with mechanisms previously inferred from qualitative studies of mutant gene expression patterns. Our models predict activation of Kr by Cad and clarify several other regulatory interactions. Our analysis suggests a central role for repressive feedback loops between complementary gap genes. We observe that repressive interactions among overlapping gap genes show anteroposterior asymmetry with posterior dominance. Finally, our models suggest a correlation between timing of gap domain boundary formation and regulatory contributions from the terminal maternal system.

New directions for Rayleigh-Taylor mixing.

We study the Rayleigh-Taylor (RT) mixing layer, presenting simulations in agreement with experimental data. This problem is an idealized subproblem of important scientific and engineering problems, such as gravitationally induced mixing in oceanography and performance assessment for inertial confinement fusion. Engineering codes commonly achieve correct simulations through the calibration of adjustable parameters. In this sense, they are interpolative and not predictive. As computational science moves from the interpolative to the predictive and reduces the reliance on experiment, the quality of decision making improves. The diagnosis of errors in a multi-parameter, multi-physics setting is daunting, so we address this issue in the proposed idealized setting. The validation tests presented are thus a test for engineering codes, when used for complex problems containing RT features. The RT growth rate, characterized by a dimensionless but non-universal parameter α, describes the outer edge of the mixing zone. Increasingly accurate front tracking/large eddy simulations reveal the non-universality of the growth rate and agreement with experimental data. Increased mesh resolution allows reduction in the role of key subgrid models. We study the effect of long-wavelength perturbations on the mixing growth rate. A self-similar power law for the initial perturbation amplitudes is here inferred from experimental data. We show a maximum ±5% effect on the growth rate. Large (factors of 2) effects, as predicted in some models and many simulations, are inconsistent with the experimental data of Youngs and co-authors. The inconsistency of the model lies in the treatment of the dynamics of bubbles, which are the shortest-wavelength modes for this problem. An alternative theory for this shortest wavelength, based on the bubble merger model, was previously shown to be consistent with experimental data.

Known maternal gradients are not sufficient for the establishment of gap domains in Drosophila melanogaster.

Gap genes are among the first transcriptional targets of maternal morphogen gradients in the early Drosophila embryo. However, it remains unclear whether these gradients are indeed sufficient to establish the boundaries of localized gap gene expression patterns. In this study, we address this question using gap gene circuits, which are data-driven mathematical tools for extracting regulatory information from quantitative wild-type gene expression data. We present new, quantitative data on the mRNA expression patterns for the gap genes Krüppel (Kr), knirps (kni) and giant (gt) during the early blastoderm stage of Drosophila development. This data set shows significant differences in timing of gap gene expression compared to previous observations, and reveals that early gap gene expression is highly variable in both space and time. Gene circuit models fit to this data set were used for a detailed regulatory analysis of early gap gene expression. Our analysis shows that the proper balance of maternal repression and activation is essential for the correct positioning of gap domains, and that such balance can only be achieved for early expression of kni. In contrast, our results suggest that early expression of gt requires local neutralization of repressive input in the anterior region of the embryo, and that known maternal gradients are completely insufficient to position the boundaries of the early central Kr domain, or in fact any Kr-like domain in the central region of the blastoderm embryo. Based on this, we propose that unknown additional regulators must be involved in early gap gene regulation.