Stabilizing patterning in the Drosophila segment polarity network by selecting models in silico

The segmentation of Drosophila is a prime model to study spatial patterning during embryogenesis. The spatial expression of segment polarity genes results from a complex network of interacting proteins whose expression products are maintained after successful segmentation. This prompted us to investigate the stability and robustness of this process using a dynamical model for the segmentation network based on Boolean states. The model consists of intra-cellular as well as inter-cellular interactions between adjacent cells in one spatial dimension. We quantify the robustness of the dynamical segmentation process by a systematic analysis of mutations. Our starting point consists in a previous Boolean model for Drosophila segmentation. We define mathematically the notion of dynamical robustness and show that the proposed model exhibits limited robustness in gene expression under perturbations. We applied in silico evolution (mutation and selection) and discover two classes of modified gene networks that have a more robust spatial expression pattern. We verified that the enhanced robustness of the two new models is maintained in differential equations models. By comparing the predicted model with experiments on mutated flies, we then discuss the two types of enhanced models. Drosophila patterning can be explained by modelling the underlying network of interacting genes. Here we demonstrate that simple dynamical considerations and in silico evolution can enhance the model to robustly express the expected pattern, helping to elucidate the role of further interactions.     

Robustness and state-space structure of Boolean gene regulatory models

Robustness to perturbation is an important characteristic of genetic regulatory systems, but the relationship between robustness and model dynamics has not been clearly quantified. We propose a method for quantifying both robustness and dynamics in terms of state-space structures, for Boolean models of genetic regulatory systems. By investigating existing models of the Drosophila melanogaster segment polarity network and the Saccharomyces cerevisiae cell-cycle network, we show that the structure of attractor basins can yield insight into the underlying decision making required of the system, and also the way in which the system maximises its robustness. In particular, gene networks implementing decisions based on a few genes have simple state-space structures, and their attractors are robust by virtue of their simplicity. Gene networks with decisions that involve many interacting genes have correspondingly more complicated state-space structures, and robustness cannot be achieved through the structure of the attractor basins, but is achieved by larger attractor basins that dominate the state space. These different types of robustness are demonstrated by the two models: the D. melanogaster segment polarity network is robust due to simple attractor basins that implement decisions based on spatial signals; the S. cerevisiae cell-cycle network has a complicated state-space structure, and is robust only due to a giant attractor basin that dominates the state space.     

Segment polarity genes in neuroblast formation and identity specification during Drosophila neurogenesis

The relatively simple central nervous system (CNS) of the Drosophila embryo provides a useful model system for investigating the mechanisms that generate and pattern complex nervous systems. Central to the generation of different types of neurons by precursor neuroblasts is the initial specification of neuroblast identity and the Drosophila segment polarity genes, genes that specify regions within a segment or repeating unit of the Drosophila embryo, have emerged recently as significant players in this process. During neurogenesis the segment polarity genes are expressed in the neuroectodermal cells from which neuroblasts delaminate and they continue to be expressed in neuroblasts and their progeny. Loss-of-function mutations in these genes lead to a failure in the formation of neuroblasts and/or specification of neuroblast identity. Results from several recent studies suggest that regulatory interactions between segment polarity genes during neurogenesis lead to an increase in the number of neuroblasts and specification of different identities to neuroblasts within a population of cells. BioEssays 21:472–485, 1999. © 1999 John Wiley & Sons, Inc.     

Unique establishment of procephalic head segments is supported by the identification of cis-regulatory elements driving segment-specific segment polarity gene expression in Drosophila

Anterior head segmentation is governed by different regulatory mechanisms than those that control trunk segmentation in Drosophila. For segment polarity genes, both initial mode of activation as well as cross-regulatory interactions among them differ from the typical genetic circuitry in the trunk and are unique for each of the procephalic segments. In order to better understand the segment-specific gene network responsible for the procephalic expression of the earliest active segment polarity genes wingless and hedgehog, we started to identify and analyze cis-regulatory DNA elements of these genes. For hedgehog, we could identify a cis-regulatory element, ic-CRE, that mediates expression specifically in the posterior part of the intercalary segment and requires promoter-specific interaction for its function. The intercalary stripe is the last part of the metameric hedgehog expression pattern that appears during embryonic development, which probably reflects the late and distinct establishment of this segment. The identification of a cis-regulatory element that is specific for one head segment supports the mutant-based observation that the expression of segment polarity genes is governed by a unique gene network in each of the procephalic segments. This provides further indication that the anterior-most head segments represent primary segments, which are set up independently, in contrast to the secondary segments of the trunk, which resemble true repetitive units.     

Ingeneue: a versatile tool for reconstituting genetic networks,with examples from the segment polarity network

Here we describe a software tool for synthesizing molecular genetic data into models of genetic networks. Our software program Ingeneue, written in Java, lets the user quickly turn a map of a genetic network into a dynamical model consisting of a set of ordinary differential equations. We developed Ingeneue as part of an ongoing effort to explore the design and evolvability of genetic networks. Ingeneue has three principal advantages over other available mathematical software: it automates instantiation of the same network model in each cell in a 2-D sheet of cells; it constructs model equations from pre-made building blocks corresponding to common biochemical processes; and it automates searches through parameter space, sensitivity analyses, and other common tasks. Here we discuss the structure of the software and some of the issues we have dealt with. We conclude with some examples of results we have achieved with Ingeneue for the Drosophila segment polarity network.     

Conservation of the segmented germband stage: robustness or pleiotropy?

Gene expression patterns of the segment polarity genes in the extended and segmented germband stage are remarkably conserved among insects. To explain the conservation of these stages, two hypotheses have been proposed. One hypothesis states that the conservation reflects a high interactivity between modules, so that mutations would have several pleiotropic effects in other parts of the body, resulting in stabilizing selection against mutational variation. The other hypothesis states that the conservation is caused by robustness of the segment polarity network against mutational changes. When evaluating the empirical evidence for these hypotheses, we found strong support for pleiotropy and little evidence supporting robustness of the segment polarity network. This points to a key role for stabilizing selection in the conservation of these stages. Finally, we discuss the implications for robustness of organizers and long-term conservation in general.     

Topology and robustness in the Drosophila segment polarity network

A complex hierarchy of genetic interactions converts a single-celled Drosophila melanogaster egg into a multicellular embryo with 14 segments. Previously, von Dassow et al. reported that a mathematical model of the genetic interactions that defined the polarity of segments (the segment polarity network) was robust (von Dassow et al. 2000). As quantitative information about the system was unavailable, parameters were sampled randomly. A surprisingly large fraction of these parameter sets allowed the model to maintain and elaborate on the segment polarity pattern. This robustness is due to the positive feedback of gene products on their own expression, which induces individual cells in a model segment to adopt different stable expression states (bistability) corresponding to different cell types in the segment polarity pattern. A positive feedback loop will only yield multiple stable states when the parameters that describe it satisfy a particular inequality. By testing which random parameter sets satisfy these inequalities, I show that bistability is necessary to form the segment polarity pattern and serves as a strong predictor of which parameter sets will succeed in forming the pattern. Although the original model was robust to parameter variation, it could not reproduce the observed effects of cell division on the pattern of gene expression. I present a modified version that incorporates recent experimental evidence and does successfully mimic the consequences of cell division. The behavior of this modified model can also be understood in terms of bistability in positive feedback of gene expression. I discuss how this topological property of networks provides robust pattern formation and how large changes in parameters can change the specific pattern produced by a network.     

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.     

Construction and analysis of gene-gene dynamics influence networks based on a Boolean model

Background

Identification of novel gene-gene relations is a crucial issue to understand system-level biological phenomena. To this end, many methods based on a correlation analysis of gene expressions or structural analysis of molecular interaction networks have been proposed. They have a limitation in identifying more complicated gene-gene dynamical relations, though.

Results

To overcome this limitation, we proposed a measure to quantify a gene-gene dynamical influence (GDI) using a Boolean network model and constructed a GDI network to indicate existence of a dynamical influence for every ordered pair of genes. It represents how much a state trajectory of a target gene is changed by a knockout mutation subject to a source gene in a gene-gene molecular interaction (GMI) network. Through a topological comparison between GDI and GMI networks, we observed that the former network is denser than the latter network, which implies that there exist many gene pairs of dynamically influencing but molecularly non-interacting relations. In addition, a larger number of hub genes were generated in the GDI network. On the other hand, there was a correlation between these networks such that the degree value of a node was positively correlated to each other. We further investigated the relationships of the GDI value with structural properties and found that there are negative and positive correlations with the length of a shortest path and the number of paths, respectively. In addition, a GDI network could predict a set of genes whose steady-state expression is affected in E. coli gene-knockout experiments. More interestingly, we found that the drug-targets with side-effects have a larger number of outgoing links than the other genes in the GDI network, which implies that they are more likely to influence the dynamics of other genes. Finally, we found biological evidences showing that the gene pairs which are not molecularly interacting but dynamically influential can be considered for novel gene-gene relationships.

Conclusion

Taken together, construction and analysis of the GDI network can be a useful approach to identify novel gene-gene relationships in terms of the dynamical influence.

     

Modeling DNA sequence-based cis-regulatory gene networks

Gene network analysis requires computationally based models which represent the functional architecture of regulatory interactions, and which provide directly testable predictions. The type of model that is useful is constrained by the particular features of developmentally active cis-regulatory systems. These systems function by processing diverse regulatory inputs, generating novel regulatory outputs. A computational model which explicitly accommodates this basic concept was developed earlier for the cis-regulatory system of the endo16 gene of the sea urchin. This model represents the genetically mandated logic functions that the system executes, but also shows how time-varying kinetic inputs are processed in different circumstances into particular kinetic outputs. The same basic design features can be utilized to construct models that connect the large number of cis-regulatory elements constituting developmental gene networks. The ultimate aim of the network models discussed here is to represent the regulatory relationships among the genomic control systems of the genes in the network, and to state their functional meaning. The target site sequences of the cis-regulatory elements of these genes constitute the physical basis of the network architecture. Useful models for developmental regulatory networks must represent the genetic logic by which the system operates, but must also be capable of explaining the real time dynamics of cis-regulatory response as kinetic input and output data become available. Most importantly, however, such models must display in a direct and transparent manner fundamental network design features such as intra- and intercellular feedback circuitry; the sources of parallel inputs into each cis-regulatory element; gene battery organization; and use of repressive spatial inputs in specification and boundary formation. Successful network models lead to direct tests of key architectural features by targeted cis-regulatory analysis.     

A dynamical system for biological development: The case of Caenorhabditis elegans

We show how a simple nonlinear dynamical system (the discrete quadratic iteration on the unit segment) can be the basis for modelling the embryogenesis process. Such an approach, even though being crude, can nevertheless prove to be useful when looking with the two main involved processes: (i) on one hand the cell proliferation under successive divisions; (ii) on the other hand, the differentiation between cell lineages. We illustrate this new approach in the case of Caenrhabditis elegans by looking at the early stages of embryogenesis, up to several hundreds of cells ("lima bean" larval stage). We show how the many results that have been obtained by several groups can be interpreted in terms of values for the parameters controlling the dynamical system. Furthermore, we can extend the model to the cases of genetic mutations. More precisely the teratogenetic and lethal effects are associated with abnormal variation of teh control parameters with time.     

Dimensional reduction of a V1 ring model with simple and complex cells

In this paper, we extend a framework for constructing low-dimensional dynamical systems models of mammalian primary visual cortex to a cortical network model that incorporates the full nonlinear effects of complex cells. The procedure consists of capturing the essential dynamics in a low-dimensional subspace using empirical methods, then recasting the equations in the reduced vector space. Previously, we considered visual cortical network models consisting of only simple cells with nearly linear responses to external stimuli. Here we show that fully nonlinear effects can be incorporated by examining the dimensional reduction of an idealized ring model of V1 with both simple and complex cells. We found it expedient to divide the subspace into four separate neuronal populations: excitatory simple, excitatory complex, inhibitory simple and inhibitory complex. In order to reproduce the fluctuation-driven dynamics in this reduced space, we incorporated (1) white noises with different intensities into individual neuronal populations, and (2) firing rate estimates to capture the probability of firing due to subthreshold fluctuations. With a more accurate, fitted connectivity, our modified dimensional reduced models can reproduce the firing rates, circular variances and modulation ratios observed in the original ring model.     

An efficient algorithm for identifying primary phenotype attractors of a large-scale Boolean network

Background

Boolean network modeling has been widely used to model large-scale biomolecular regulatory networks as it can describe the essential dynamical characteristics of complicated networks in a relatively simple way. When we analyze such Boolean network models, we often need to find out attractor states to investigate the converging state features that represent particular cell phenotypes. This is, however, very difficult (often impossible) for a large network due to computational complexity.

Results

There have been some attempts to resolve this problem by partitioning the original network into smaller subnetworks and reconstructing the attractor states by integrating the local attractors obtained from each subnetwork. But, in many cases, the partitioned subnetworks are still too large and such an approach is no longer useful. So, we have investigated the fundamental reason underlying this problem and proposed a novel efficient way of hierarchically partitioning a given large network into smaller subnetworks by focusing on some attractors corresponding to a particular phenotype of interest instead of considering all attractors at the same time. Using the definition of attractors, we can have a simplified update rule with fixed state values for some nodes. The resulting subnetworks were small enough to find out the corresponding local attractors which can be integrated for reconstruction of the global attractor states of the original large network.

Conclusions

The proposed approach can substantially extend the current limit of Boolean network modeling for converging state analysis of biological networks.