Sharpening of expression domains induced by transcription and microRNA regulation within a spatio-temporal model of mid-hindbrain boundary formation.

BACKGROUND: The establishment of the mid-hindbrain region in vertebrates is mediated by the isthmic organizer, an embryonic secondary organizer characterized by a well-defined pattern of locally restricted gene expression domains with sharply delimited boundaries. While the function of the isthmic organizer at the mid-hindbrain boundary has been subject to extensive experimental studies, it remains unclear how this well-defined spatial gene expression pattern, which is essential for proper isthmic organizer function, is established during vertebrate development. Because the secreted Wnt1 protein plays a prominent role in isthmic organizer function, we focused in particular on the refinement of Wnt1 gene expression in this context. RESULTS: We analyzed the dynamics of the corresponding murine gene regulatory network and the related, diffusive signaling proteins using a macroscopic model for the biological two-scale signaling process. Despite the discontinuity arising from the sharp gene expression domain boundaries, we proved the existence of unique, positive solutions for the partial differential equation system. This enabled the numerically and analytically analysis of the formation and stability of the expression pattern. Notably, the calculated expression domain of Wnt1 has no sharp boundary in contrast to experimental evidence. We subsequently propose a post-transcriptional regulatory mechanism for Wnt1 miRNAs which yields the observed sharp expression domain boundaries. We established a list of candidate miRNAs and confirmed their expression pattern by radioactive in situ hybridization. The miRNA miR-709 was identified as a potential regulator of Wnt1 mRNA, which was validated by luciferase sensor assays. CONCLUSION: In summary, our theoretical analysis of the gene expression pattern induction at the mid-hindbrain boundary revealed the need to extend the model by an additional Wnt1 regulation. The developed macroscopic model of a two-scale process facilitate the stringent analysis of other morphogen-based patterning processes.

Effective parameters determining the information flow in hierarchical biological systems.

Signaling networks are abundant in higher organisms. They play pivotal roles, e.g., during embryonic development or within the immune system. In this contribution, we study the combined effect of the various kinetic parameters on the dynamics of signal transduction. To this end, we consider hierarchical complex systems as prototypes of signaling networks. For given topology, the output of these networks is determined by an interplay of the single parameters. For different kinetics, we describe this by algebraic expressions, the so-called effective parameters.When modeling switch-like interactions by Heaviside step functions, we obtain these effective parameters recursively from the interaction graph. They can be visualized as directed trees, which allows us to easily determine the global effect of single kinetic parameters on the system's behavior. We provide evidence that these results generalize to sigmoidal Hill kinetics.In the case of linear activation functions, we again show that the algebraic expressions can be immediately inferred from the topology of the interaction network. This allows us to transform time-consuming analytic solutions of differential equations into a simple graph-theoretic problem. In this context, we also discuss the impact of our work on parameter estimation problems. An issue is that even the fitting of identifiable effective parameters often turns out to be numerically ill-conditioned. We demonstrate that this fitting problem can be reformulated as the problem of fitting exponential sums, for which robust algorithms exist.

Biologically meaningful update rules increase the critical connectivity of generalized Kauffman networks.

We generalize random Boolean networks by softening the hard binary discretization into multiple discrete states. These multistate networks are generic models of gene regulatory networks, where each gene is known to assume a finite number of functionally different expression levels. We analytically determine the critical connectivity that separates the biologically unfavorable frozen and chaotic regimes. This connectivity is inversely proportional to a parameter which measures the heterogeneity of the update rules. Interestingly, the latter does not necessarily increase with the mean number of discrete states per node. Still, allowing for multiple states decreases the critical connectivity as compared to random Boolean networks, and thus leads to biologically unrealistic situations. Therefore, we study two approaches to increase the critical connectivity. First, we demonstrate that each network can be kept in its frozen regime by sufficiently biasing the update rules. Second, we restrict the randomly chosen update rules to a subclass of biologically more meaningful functions. These functions are characterized based on a thermodynamic model of gene regulation. We analytically show that their usage indeed increases the critical connectivity. From a general point of view, our thermodynamic considerations link discrete and continuous models of gene regulatory networks.

Odefy--from discrete to continuous models.

BACKGROUND: Phenomenological information about regulatory interactions is frequently available and can be readily converted to Boolean models. Fully quantitative models, on the other hand, provide detailed insights into the precise dynamics of the underlying system. In order to connect discrete and continuous modeling approaches, methods for the conversion of Boolean systems into systems of ordinary differential equations have been developed recently. As biological interaction networks have steadily grown in size and complexity, a fully automated framework for the conversion process is desirable. RESULTS: We present Odefy, a MATLAB- and Octave-compatible toolbox for the automated transformation of Boolean models into systems of ordinary differential equations. Models can be created from sets of Boolean equations or graph representations of Boolean networks. Alternatively, the user can import Boolean models from the CellNetAnalyzer toolbox, GINSim and the PBN toolbox. The Boolean models are transformed to systems of ordinary differential equations by multivariate polynomial interpolation and optional application of sigmoidal Hill functions. Our toolbox contains basic simulation and visualization functionalities for both, the Boolean as well as the continuous models. For further analyses, models can be exported to SQUAD, GNA, MATLAB script files, the SB toolbox, SBML and R script files. Odefy contains a user-friendly graphical user interface for convenient access to the simulation and exporting functionalities. We illustrate the validity of our transformation approach as well as the usage and benefit of the Odefy toolbox for two biological systems: a mutual inhibitory switch known from stem cell differentiation and a regulatory network giving rise to a specific spatial expression pattern at the mid-hindbrain boundary. CONCLUSIONS: Odefy provides an easy-to-use toolbox for the automatic conversion of Boolean models to systems of ordinary differential equations. It can be efficiently connected to a variety of input and output formats for further analysis and investigations. The toolbox is open-source and can be downloaded at http://cmb.helmholtz-muenchen.de/odefy.

Spatial analysis of expression patterns predicts genetic interactions at the mid-hindbrain boundary.

The isthmic organizer mediating differentiation of mid- and hindbrain during vertebrate development is characterized by a well-defined pattern of locally restricted gene expression domains around the mid-hindbrain boundary (MHB). This pattern is established and maintained by a regulatory network between several transcription and secreted factors that is not yet understood in full detail. In this contribution we show that a Boolean analysis of the characteristic spatial gene expression patterns at the murine MHB reveals key regulatory interactions in this network. Our analysis employs techniques from computational logic for the minimization of Boolean functions. This approach allows us to predict also the interplay of the various regulatory interactions. In particular, we predict a maintaining, rather than inducing, effect of Fgf8 on Wnt1 expression, an issue that remained unclear from published data. Using mouse anterior neural plate/tube explant cultures, we provide experimental evidence that Fgf8 in fact only maintains but does not induce ectopic Wnt1 expression in these explants. In combination with previously validated interactions, this finding allows for the construction of a regulatory network between key transcription and secreted factors at the MHB. Analyses of Boolean, differential equation and reaction-diffusion models of this network confirm that it is indeed able to explain the stable maintenance of the MHB as well as time-courses of expression patterns both under wild-type and various knock-out conditions. In conclusion, we demonstrate that similar to temporal also spatial expression patterns can be used to gain information about the structure of regulatory networks. We show, in particular, that the spatial gene expression patterns around the MHB help us to understand the maintenance of this boundary on a systems level.

Transforming Boolean models to continuous models: methodology and application to T-cell receptor signaling.

BACKGROUND: The understanding of regulatory and signaling networks has long been a core objective in Systems Biology. Knowledge about these networks is mainly of qualitative nature, which allows the construction of Boolean models, where the state of a component is either 'off' or 'on'. While often able to capture the essential behavior of a network, these models can never reproduce detailed time courses of concentration levels. Nowadays however, experiments yield more and more quantitative data. An obvious question therefore is how qualitative models can be used to explain and predict the outcome of these experiments. RESULTS: In this contribution we present a canonical way of transforming Boolean into continuous models, where the use of multivariate polynomial interpolation allows transformation of logic operations into a system of ordinary differential equations (ODE). The method is standardized and can readily be applied to large networks. Other, more limited approaches to this task are briefly reviewed and compared. Moreover, we discuss and generalize existing theoretical results on the relation between Boolean and continuous models. As a test case a logical model is transformed into an extensive continuous ODE model describing the activation of T-cells. We discuss how parameters for this model can be determined such that quantitative experimental results are explained and predicted, including time-courses for multiple ligand concentrations and binding affinities of different ligands. This shows that from the continuous model we may obtain biological insights not evident from the discrete one. CONCLUSION: The presented approach will facilitate the interaction between modeling and experiments. Moreover, it provides a straightforward way to apply quantitative analysis methods to qualitatively described systems.