Robustness and modular design of the Drosophila segment polarity network

Wenzhe Ma^1,2, Luhua Lai^1,2, Qi Ouyang^1,3 & Chao Tang^1,4

Article highlights

Synopsis

Biological systems have to function robustly and at the same time to be evolvable and adaptive to diverse and changing environments. Functional robustness as a system's property of the underlying network may have consequences on the network's topological structure (Waddington, 1957; Barkai and Leibler, 1997; Li et al, 2004; El-Samad et al, 2005; Wagner, 2005). For a given function, how many distinct network topologies can perform the function and how many can do so robustly (Wagner, 2005)? What are the key features among these robust topologies—what are common and what are variable? How would nature pick among them? Is it possible to construct a 'periodic table' for robust topologies corresponding to a given function?

To address these questions, we carried out an exhaustive computational analysis on the network topologies that perform a specific function. The function was chosen to be a well-defined patterning function in Drosophila embryogenesis, which is highly conserved among all insets, that of setting up a stable periodic pattern of gene expression that defines the sharp boundaries between parasegments (Figure 1F). The biomolecular network responsible for this patterning function in Drosophila is the segmentation polarity gene network (Martizez Arias, 1993; DiNardo et al, 1994; Perrimon, 1994) (Figure 1C). This network has both intra- and intercellular interactions and its core topology consists of three interacting nodes (Figure 1D). We constructed a mathematical model to evaluate the ability of performing the patterning function for each and (almost) all topologies consisting of three nodes with intra- and intercellular interaction.

Figure 1

Segment polarity network and expression pattern of wg and en. (A) The segment polarity gene network model of Ingolia (2004). Ellipses represent mRNAs and rectangles proteins. Lines ending with an arrow and a dot denote activation and repression, respectively. Dashed lines indicate intercellular regulations. The gray line means no direct biological evidence. Nodes are colored into three groups, each of which is represented by one node in (B). (B) The simplified topology of (A). Each node here represents a group of nodes in (A) of the same color. (C) Our model of the segment polarity gene network (see also von Dassow and Odell, 2002). Slp regulates wg positively through the mid gene and its product, which is represented by an arrow from 'S' to 'W' in (D). (D) The simplified topology of (C). (E) The initial condition of the patterning function. In three-node networks, 'S' expresses in the posterior four cells of the parasegment. The pattern is periodic. (F) The final stable pattern. In three-node networks, 'S' is not fixed to be any specific pattern in the final state. (G) zw3 mutant phenotype. (H) ptc mutant phenotype. Note that (E), (F), (G) and (H) are a simple representation of the actual embryo surface, which is extended in both directions and includes 14 segments.

Full figure and legend (309K)Figures & Tables index

In the model, the functional behavior of a topology is modeled by differential equations with parameters characterizing the nodes and the interacting links in the topology. The ability of the topology to perform the function is assessed by randomly sampling the parameters and observing the fraction (Q) of the sampling that enabled the system to achieve the required patterning (von Dassow et al, 2000; Ingolia, 2004). We found that the vast majority of the topologies cannot perform the patterning function (Q0), implying a severe functional constraint on topology (Wagner, 2005). On the other hand, many more than a few distinct topologies are robust in performing this patterning function. Interestingly, a common feature emerges when comparing all the robust topologies—each of them can be understood as a particular combination of three submodules (not to be confused with the three nodes) from three groups (Figure 3B). Each group of submodule corresponds to a subfunction of the patterning function studied.

Figure 3

Skeletons and functional modules. (A) The four skeletons in robust two-node topologies (black lines). The green, orange and red links are neutral, bad and very bad links, respectively. The numbers below the skeletons are its Q value and the size of its family. (B) The three kinds of modules correspond to the three subfunctions in three-node networks. The bold modules are also those of the two-node networks. Many of these modules can be identified as significant network motifs among all networks with Q>0.1 (see details in Supplementary information). The combination of these modules leads to 44 robust core topologies or skeletons. The number under each module is its Q value, the frequency the module is being used in the 44 skeletons.

Full figure and legend (281K)Figures & Tables index

The organization of robust topologies as modular combinations provides nature with multiple choices to solve the patterning problem. Furthermore, each robust core topology can be 'dressed' up with additional neutral and/or nearly neutral links, which enhances the variability and flexibility of robust topologies, making them more evolvable and adaptable (Schuster et al, 1994; Kirschner and Gerhart, 2005). This 'periodic table' of robust topologies also provides us with a framework to suggest and comprehend the topology nature picked. Figure 1A is the segmentation polarity network constructed in previous studies (Ingolia, 2004), which corresponds to the topology of Figure 1B. In this model, the direct autoregulation on wg (W) has no support from the experimental literature but is needed for the model to function robustly. If we disallow this presumed link, the 'periodic table' would suggest an alternative of Figure 1D. The corresponding network of Figure 1C has experimental support on every node and link in it. To further discriminate the two models, Figure 1A and C, we subject both to mutant tests, requiring them not only to generate the wild-type pattern but also to produce the observed mutant patterns when the knockout effect is simulated. Figure 1C and its topology Figure 1D win out in the tests and stand out as a better working model for the Drosophila segmentation polarity network.

The topology Figure 1D is not the most robust among the topologies that can robustly perform the required patterning function. However, it is the top one among all topologies that do not have any direct autoregulation. Direct positive autoregulation may result in a less flexible system, which may impair the other functional abilities of the Hh and Wg pathways. Thus, the topological variety that resulted from the modular combination can be a solution to accommodating biological constraints and pleiotropic effects.

MORE ARTICLES LIKE THIS

These links to content published by NPG are automatically generated.