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Conceived and designed the experiments: NP. Performed the experiments: NP. Analyzed the data: NP. Wrote the paper: DMW. Conceived and designed the methodology: DMW FB. Performed the computational experiments: DMW. Curated and implemented the IDGenes database: DT. Helped to draft the manuscript: FB WW FJT. Contributed to the manuscript: DT NP. Coordinated the experimental work: WW. Coordinated the theoretical work: FJT.

The authors have declared that no competing interests exist.

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

Understanding brain formation during development is a tantalizing challenge. It is also essential for the fight against neurodegenerative diseases. In vertebrates, the central nervous system arises from a structure called the neural plate. This tissue is divided into four regions, which continue to develop into forebrain, midbrain, hindbrain and spinal cord. Interactions between locally expressed genes and signaling molecules are responsible for this patterning. Two key signaling molecules in this process are Fgf8 and Wnt1 proteins. They are secreted from a signaling center located at the boundary between prospective mid- and hindbrain (mid-hindbrain boundary, MHB) and mediate development of these two brain regions. Here, we logically analyze the spatial gene expression patterns at the MHB and predict interactions involved in the differentiation of mid- and hindbrain. In particular, our analysis indicates that

During vertebrate development, the central nervous system arises from a precursor tissue called neural plate. Shortly after gastrulation this neural plate is patterned along the anterior-posterior axis into four regions, which continue to develop into forebrain, midbrain, hindbrain and spinal cord. This patterning is determined by a well-defined and locally restricted expression of genes, and by the action of short and long range signaling centers, also called secondary organizers

The IsO is characterized by the localized expression of several transcription and secreted factors. In this contribution, we focus on the following eight IsO genes:

(A) Dorsal close-up view of the MHB region in the anterior neural tube of an E10.5 mouse embryo, anterior to the left. The expression pattern of the IsO genes can be subdivided into the six compartments I–VI. (B) Schematic representation of compartments I–VI. The MHB is located between compartments III and IV. (C) Values of the Boolean variables in compartments I–VI. The

The initial expression of

Although a great deal of experimental effort has been made, this regulatory network is not yet understood in full detail. For this reason, we now approach this problem on the systems level. In Systems Biology experimentalists and theoreticians make a concerted effort to unravel the functionality of complex biological systems in a holistic fashion. We closely follow the Systems Biology research cycle as proposed in

In a first step, we describe a methodology for the inference of regulatory interactions by systematic logical analyses of spatial gene expression patterns. As our information about gene expression at the MHB is of purely qualitative nature we base our analysis on

Applied to the wild-type gene expression pattern at the MHB our method predicts several genetic interactions, which well agree with published data. In this context, an unclarified experimental issue is the reported ectopic induction of

In a subsequent step, the results of our previous analysis combined with published data allow construction of spatial models that are able to explain the stable maintenance of the characteristic gene expression patterns at the MHB. These patterns are the result of a refinement and sharpening of initially blurred expression domains. Our models are competent to simulate this process. Moreover, we are able to reproduce the phenotypes of various loss-of-function (LOF) experiments even in a correct spatio-temporal order. We conclude with a robustness analysis of our models.

We subdivided the expression patterns of the IsO genes at E10.5 into six compartments I–VI, cf. the dashed lines in

Only

Only

The crucial point is that this expression pattern is stably maintained by a regulatory network. In the next section we demonstrate that key genetic interactions of this network can be obtained already by analyzing only this spatial information.

Information about the expression patterns of the IsO genes can also be found in our

So far, information about genetic interactions at the MHB has been obtained mainly from the analyses of gene expression patterns (by

These analyses are based on a Boolean model whose species denote, in our case, the (groups of) genes

(A) Necessary interactions. Any valid network includes a mutual inhibition of

In a second approach, we determined as simple a Boolean update function as possible for each species such that the expression pattern is still maintained. This analysis is based on the assumption that the thus obtained minimal network is the core module of the real network. However, we cannot be sure that all the interactions are indeed necessary. The following minimal functions could be derived using a technique called Karnaugh-Veitch maps (see

In the case of

The minimal networks from

To find out if Fgf8 only maintains or de novo induces

(A) E8.0–E8.5 (4–6 somites) mouse anterior neural plate explants (cf. upper

The previous prediction of genetic interactions was based solely on spatial gene expression data from the MHB at E10.5. Here, we complemented this by a careful literature mining for further experimentally validated interactions. Thus, we extended the minimal networks from

The function of this gene regulatory network is threefold. First, it has to maintain the characteristic expression pattern of the MHB. Second, it has to ensure that once expression of a gene has been induced its expression domain is correctly positioned. Third, it has to account for the experimentally observed sharpening of the expression domains and of the whole boundary. In order to check whether these functionalities are ensured, we constructed dynamic models of the interaction network. These are described in the remainder of this section; the

In a first step, we derived Boolean update rules from the regulatory network. The resulting Boolean model was then implemented in six linearly ordered compartments that correspond to the subdivision of the expression patterns shown in

To analyze if the regulatory network is also able to correctly position the expression patterns of the IsO genes after their induction, the Boolean model was transformed into a multi-compartment ordinary differential equation (ODE) model. The variables are now no longer coarse-grained into discrete states but assume continuous values. Continuous variables are necessary as we cannot assume that a gene is induced already to its full expression level. Rather the model should amplify a gene's expression at the right positions while downregulating it elsewhere.

The ODE model is still based on the six compartments shown in

We analyzed if the network constructed in the previous section indeed fulfills its three functions. To this end, we performed several

A simple simulation of the multi-compartment Boolean model (not shown) confirmed that the regulatory network stably maintains the expression patterns from

For different initial conditions,

In lack of quantitative data ad-hoc parameters were used. For different initial conditions

In our network

This result clearly shows the potential as well as the limitations of our model. As soon as there was a predominance of

The blurred expression domains of the IsO genes at E8.5 as well as their sharp expression domains and the clearly demarcated boundary at E10.5 are shown in upper and lower

(A) Expression pattern of the IsO genes at E8.5 (upper figure) and at E10.5 (lower figure). The initially blurred expression domains are refined and a sharp boundary is established. (B, C) Simulations of the PDE model. Initial conditions were chosen to resemble the expression pattern at E8.5, cf. upper Figure A. In lack of quantitative data ad-hoc parameters were used (see

A key feature of the IsO genes studied here is their tight and indispensable interaction for the maintenance of their own expression and of the MHB during midgestational stages, as revealed by the analysis of LOF mouse mutants for these genes

An

Similar simulations of

So far, we used ad-hoc parameter estimates for the simulation of the ODE and PDE model. We now analyse how robust the obtained results are under perturbations of the kinetic parameters.

In the ODE model, each interaction is modeled by a sigmoid Hill kinetic and the effects of different regulators on one species are combined accordingly to the Boolean logic (see

Boolean step functions are replaced by positive and negative sigmoid Hill functions. The continuous homologues of the

The network from

We repeated the computational experiment from

The computational experiment from

In addition to the parameters from the ODE model, the PDE model also contains parameters describing the production and decay as well as the diffusion of the secreted Fgf8 and Wnt1 proteins. The diffusion constants essentially determine the length of the

Using ad-hoc estimates for the parameters, we also investigated the effect of changes in the initial conditions from

Our main objective in this contribution is to elucidate epistatic relationships at the MHB on the transcriptional level. Initially and since ours is the first approach to modeling this biological system, we preferred to include only the eight best-studied and most important IsO genes that are known to constitute the core of the regulatory network maintaining the MHB: the transcription factors

The precise range of diffusion of the Fgf8 and Wnt1 molecules from the murine MHB is still an unresolved issue. With respect to Fgf8, we base our assumptions on the idea that secreted Fgf8, by binding to extracellular HSPGs or by active endocytosis, is a rather sticky protein that does not diffuse very far away from its source (mRNA expression) (see, for example,

In

For our purposes herein, a one dimensional spatial model of the E10.5 neural tube suffices (see

Previous work on the reverse engineering of biological networks has mainly focused on their reconstruction from time-courses of expression levels, see e.g. the reviews

The first strategy is the identification of necessary interactions. In other words, we look for interactions included in all possible networks that are consistent with the data. This is a very strict condition. Consequently, only few predictions will result from this analysis; these, however, can be considered reliable. As in our example of the IsO network, the necessary interactions by themselves will typically not form a valid network explaining the data.

As a second strategy, we propose the determination of all minimal Boolean update functions. Here we assume that these minimal interactions constitute the core module of the unknown network. In the example of the IsO, this assumption is justified, as the minimal networks are fully backed up by experimental data. The condition that an interaction be included in a minimal network is weaker than that of being a necessary interaction. Consequently, our second strategy will typically yield additional predictions, that should be considered less reliable than the necessary interactions. In the IsO example, four necessary interactions could be found, while an additional six interactions were obtained from the minimality analysis. Hence, the analysis of minimal networks yielded valuable insights into the regulatory network between the IsO genes, which go beyond what could be obtained from determining all necessary interactions. A further advantage of the second strategy is that it leads to a functional network which is able to explain the data.

Standard ways of finding minimal Boolean functions are either Karnaugh-Veitch maps

We observe that the literature network from

Remarkable about the structure of the networks from

Summing up our analyses of the literature network, we can describe the role of its single interactions as follows: The mutual inhibition of

In order to evaluate our IsO network we derived spatial dynamical models from it. Theoretical modeling has long been used to explain pattern formation in living organisms. The employed models range from coarse-grained multi-compartment Boolean

In a first approach, we used a parameter-free Boolean model. Boolean models are generally applicable in a purely qualitative context and well able to deal with the exquisite, intricate mechanisms of gene regulation

In the PDE model, a critical point is the modeling of intercellular communication. In

The transformation of a Boolean model into a continuous model entails the introduction of kinetic parameters. Especially in the context of pattern formation it has been shown that a network's function is carried out robustly against perturbations of these kinetic constants

Our modeling pipeline — from Boolean over ODE to PDE models — exemplifies how more and more detailed models of regulatory networks can be built from qualitative information. The transformation methods that we used (see

Apart from the previously mentioned extensions of our models (to include

We demonstrated that similar to temporal expression patterns, also spatial expression patterns can be used to gain information about the structure of regulatory networks. In particular, we showed that the characteristic expression patterns of key IsO genes reveal a maintaining effect of Fgf8 on

Animal treatment was conducted under federal guidelines for the use and care of laboratory animals and was approved by the HMGU Institutional Animal Care and Use Committee.

We think of the neural plate/tube as a chain of cells, which are able to communicate via

Now given update functions

Our goal is now to find update rules, such that model (3) has a steady state pattern

Now, each of our update functions

In a first step, we determine all non-trivial dependencies in the truth tables. Species

In a next step, we look for minimal Boolean expressions describing the partially filled truth tables. In our example, we find the minimal Boolean expressions given in (1) by using the Karnaugh-Veitch maps shown in

Note that in our modeling environment any auto-regulation is excluded. We can now see why this restriction is necessary. Otherwise, the method outlined above would always yield the trivial minimal solution, where each species

Explant cultures of anterior neural plates/tubes (open-book preparations) from wild-type (C57BL/6) mouse embryos were essentially prepared as reported by

Explants were fixed in fresh

From the network shown in

This model was subsequently transformed into an ODE model consisting of the now continuous variables

In order to analyze spatial effects in a continuous manner, the multi-compartment model was transformed into a reaction-diffusion PDE model. The neural plate/tube is now no longer coarse-grained into compartments and we no longer add a subscript

Supplementary text

(3.01 MB PDF)

Simulation of the PDE model under wild-type conditions

(0.60 MB MP4)

Simulation of the PDE model under ^{−/−} mutant conditions

(0.34 MB MP4)

Simulation of the PDE model under ^{−/−} mutant conditions

(0.33 MB MP4)

Simulation of the PDE model under ^{−/−} mutant conditions

(0.26 MB MP4)

Simulation of the PDE model under ^{−/−} mutant conditions

(0.24 MB MPG)

ODE model as MATLAB .m file

(0.01 MB TXT)

Archive containing MATLAB files for simulation of the PDE model

(4 KB ZIP)

We thank Hans-Werner Mewes and Carsten Marr for useful discussions and critical reading of the manuscript as well as Bernd Lentes and Daniela Mayer for technical support on the