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.

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.

An example of a two-node topology. Intracellular regulations (solid lines) act on nodes within the cell; intercellular regulations (dashed lines) act on target nodes in nearby cells.

The histogram of Q values for three-node networks. Each of the 14 348 907 networks is sampled with 100 random parameter sets (black bars). Each of the resulting 156 016 networks with Q>0 is resampled with 1000 random parameter sets (only data with Q>0.1 are shown; red bars). Inset: the same data plotted in log–log scale and in terms of the probability density. The straight line has a slope of −1.37.

The number of networks in a family (A) and the skeleton's estimated Q value (the product of the modules' Q values) (B) versus the Q value of the skeleton.