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. 2007 Aug 21;104(34):13591-6.
doi: 10.1073/pnas.0705396104. Epub 2007 Aug 9.

Innovation and robustness in complex regulatory gene networks

S Ciliberti  1 , O C Martin, A Wagner
Affiliations

Innovation and robustness in complex regulatory gene networks

S Ciliberti et al. Proc Natl Acad Sci U S A. .

Abstract

The history of life involves countless evolutionary innovations, a steady stream of ingenuity that has been flowing for more than 3 billion years. Very little is known about the principles of biological organization that allow such innovation. Here, we examine these principles for evolutionary innovation in gene expression patterns. To this end, we study a model for the transcriptional regulation networks that are at the heart of embryonic development. A genotype corresponds to a regulatory network of a given topology, and a phenotype corresponds to a steady-state gene expression pattern. Networks with the same phenotype form a connected graph in genotype space, where two networks are immediate neighbors if they differ by one regulatory interaction. We show that an evolutionary search on this graph can reach genotypes that are as different from each other as if they were chosen at random in genotype space, allowing evolutionary access to different kinds of innovation while staying close to a viable phenotype. Thus, although robustness to mutations may hinder innovation in the short term, we conclude that long-term innovation in gene expression patterns can only emerge in the presence of the robustness caused by connected genotype graphs.

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Conflict of interest statement

The authors declare no conflict of interest.

Figures

Fig. 1.
Fig. 1.
Neutral networks in transcriptional regulation. (a) A transcriptional regulation network. Solid black bars indicate genes that encode transcriptional regulators in a hypothetical network of five genes. Each gene is expressed at a level that is influenced by the transcriptional regulators in the network. This influence is usually exerted through the binding of a transcriptional regulator to a gene's regulatory region (horizontal line). The model represents the regulatory interactions between transcription factors j and genes i through a matrix w = (wij). A regulator's effect can be activating (wij > 0, red rectangles) or repressing (wij < 0, blue rectangles). Any given gene's expression may be unaffected by most regulators in the network (wij = 0, white rectangles). The different hues of red and blue correspond to different magnitudes of wij. The highly regular correspondence of matrix entries to binding sites serves the purpose of illustration and is not normally found, because transcription factor binding sites usually function, regardless of their position in a regulatory region. (b) The topology on the space of genotypes induced by single mutations. The center network shows a hypothetical network of five genes (Upper) and its matrix of regulatory interactions w (Lower), if genes are numbered clockwise from the uppermost gene. Red arrows indicate activating interactions, and blue lines terminating in a circle indicate repressive interactions. The leftmost network and the center network differ in one repressive interaction from gene 4 to gene 3 (dashed gray line, black cross, and large open rectangle). The rightmost network and the middle network differ in one activating interaction from gene 1 to gene 5 (dashed line, black cross, and large white rectangle). Each of the three networks corresponds to one node in a graph as indicated by the large circle around the networks. These circles are connected because the respective networks are neighbors; i.e., they differ by one regulatory interaction. [a and b were reproduced with permission from Ciliberti et al. (31) (Copyright 2007, Public Library of Science)]. (c) The neutral network for a given phenotype. Each node corresponds to a network of a given topology, and two nodes are connected by an edge if they differ at one regulatory interaction (n = 3 genes, 4 ≤ M ≤ 5 regulatory interactions, and Hamming distance of S(0) and S∞ of d = 2/3). This neutral network is connected and the number of edges incident on a node is highly variable. Note that neutral networks for greater numbers of genes typically have a huge number of nodes. The number of nodes in a neutral network can be counted, because different nodes differ only in the signs of their regulatory interactions.
Fig. 2.
Fig. 2.
The ability to innovate depends on a genotype's position in the neutral network. (a) A tradeoff between robustness and innovation. The horizontal axis shows mutational robustness Rμ, the fraction of a network's w topological neighbors that share the same equilibrium expression state, S∞, with w. For each network w whose robustness is displayed on the horizontal axis, the vertical axis shows the fraction of networks of genotype distance D < 0.1 around w, whose equilibrium state is different from S∞. This fraction declines with increasing robustness. n = 8, c = 0.25, and d = 0.5. (b) The horizontal axis shows genotype distance D12 of two networks (w1 and w2) with the same phenotype. The vertical axis shows the mean fraction f of unique new phenotypes, as defined in the main text, found in a k-neighborhood (see legend for k) around these networks. If f is close to zero, then all or most of the phenotypes of networks in the two neighborhoods are identical. If f is close to one, then almost all phenotypes in the two neighborhoods are different. Standard deviations around each data point are no greater than 8 × 10−3. (c) Like b, except for a sample of 2,210 network pairs (w1 and w2) chosen at random from the neutral network and with mutational robustness Rμ in the interval (0.45, 0.60). n = 8, c = 0.25, d = 0.5, and k = 3. As opposed to the strong and positive statistical association between genotype distance and f for networks at small D12, this association is considerably weaker at larger distances. Notice the large fraction of unique new phenotypes for almost all network pairs shown (mean f = 0.73). (d) Histogram of f for 1-neighbors (blue), 2-neighbors (red), and 3-neighbors (green) of 2,210 randomly chosen network pairs with Rμ in the interval (0.45, 0.60). Data are shown for n = 8, c = 0.25, and d = 0.5. For one-mutant neighbors, the robustness Rμ of a network w is the fraction of a network's neighbors that has the same gene expression pattern S∞. For k-neighbors with k > 1, we define Rμ as the fraction of all networks that differ from w by no more than k regulatory interactions and that have the same gene expression pattern S∞.
Fig. 3.
Fig. 3.
Conditions for high robustness and the ability to innovate. Each rectangle shows a hypothetical genotype space. Individual genotypes with identical phenotypes (regulatory networks that produce identical gene expression patterns) are shown as circles in this space. Nodes of the neutral network are green. Other colors indicate novel phenotypes. Lines connect genotypes that are nearest neighbors in this space, corresponding in our case to networks that differ in one regulatory interaction. (Left) Genotypes are widely scattered and isolated in this space. (Center) The genotypes are widely scattered but also connected in this space. (Right) The genotypes occur in a small region of the space, and they are connected. We note that this visualization is for expository purposes only. Actual genotype spaces may have hundreds of dimensions, and there may be an astronomical number of genotypes with the same phenotype.
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