Innovation and robustness in complex regulatory gene networks

Ciliberti et al. 10.1073/pnas.0705396104.

Supporting Information

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SI Text
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SI Figure 4

Fig. 4. (a) Randomly chosen network pairs with identical gene expression pattern have vastly different organization. The horizontal axis shows genotype distance D. For each genotype distance D, the vertical axis shows the number of randomly chosen network pairs with identical gene expression patterns in that distance category. Three values of the number N of network genes are used for illustration. (Inset) The mean (95% confidence intervals) genotype distance of these distributions as a function of N. c = 0.25; Hamming distance between S(0) and S¥ is d = 0.5; sample size n = 5 ´ 10⁵ networks pairs, except for n = 36 where n = 1.79 ´ 10⁵. (b) The fraction of all possible 2^N gene expression patterns (phenotypes, vertical axis) that can be found in genotypes at distance D (horizontal axis) from a network with a given phenotype. Data for different numbers of genes are based on exhaustive enumeration (n = 6, all distances; n = 8 and D < 0.25; n = 10 and D < 0.12) or on random sampling (all other values of N and D). For n = 6 (n > 6), the data are based on 50 (25) randomly chosen networks around which a neighborhood of increasing radius is explored. Where random sampling was used to explore a neighborhood around a given network, the sample size was at least n = 10⁸; c = 0.25; d = 0.5. One sees that only neighborhoods of radius D > 1/2 are assured to contain nearly all possible phenotypes.

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Fig. 5. Randomly chosen network pairs with identical gene expression pattern have vastly different organization. The horizontal axis shows genotype distance D' of two genotypes w and w' chosen at random from the neutral network. This distance is defined as

, where the function d[x,y] = 1 iff x = y, and d[x,y] = 0 otherwise. For each genotype distance D', the vertical axis shows the number of randomly chosen network pairs with the same phenotype in that distance category. Three values of the number N of network genes are used for illustration. (Inset) The mean (and standard deviation) genotype distance of these distributions as a function of N. c = 0.25; Hamming distance between S(0) and S¥ is d = 0.5; sample size n = 5 ´ 10⁵ networks, except for n = 36 where n = 1.79 ´ 10⁵. In contrast to D, D' simply corresponds to the fraction of regulatory interactions that differ between two networks w and w', regardless of their sign.

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Fig. 6. Graphical representation of network neighborhoods of increasing distance from a reference network w. Each node corresponds to a network of a given topology. Two nodes are connected by an edge if they differ in one regulatory interaction. Green, yellow, and gray nodes, respectively, indicate networks that show the same stable gene expression pattern as w, networks that show a different stable gene expression pattern, and networks that show no stable gene expression pattern. (a) The nearest neighbors (1-neighbors) of a randomly chosen genotype for the neutral network with n = 6, c = 0.25, and d = 0.5. Nearest neighbors are genotypes that differ from w in one regulatory interaction. (b) All 2-neighbors of w. (c) All 3-neighbors of w. (d) All 4-neighbors of w that arrive at some stable gene expression pattern. Qualitatively, one observes an explosive increase in the number of genotypes as one goes to larger k, and a clustering of viable networks: If a 1-neighbor w' of w is viable, then the 1-neighbors of w' neighbors also tend to be viable, and vice versa. In addition, the number of networks with new gene expression patterns in the neighborhood of any viable network w shows a very broad distribution: Some viable networks have few neighbors that produce a new gene expression pattern, whereas others have many such neighbors. Graph rendering was performed using AGD (http://aragorn.ads.tuwien.ac.at/AGD).

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Fig. 7. Genotype-phenotype relations. (a and b) Histogram (a) and contour plot (b) of the distance of phenotypes (gene expression patterns) S¥ for network pairs of a given genotype distance D, where each member of a pair has the same initial gene expression state S(0); n = 12 genes, c = 0.25. (b Inset) The mean phenotype distance as a function of the genotype distance. The histogram was generated with at least n > 10⁶ network pairs sampled for each value of D and d. The increase of P_D(d) for large d is due to the global symmetry of the model: Equilibrium states come in pairs, related by changing the sign of each gene's expression state. (c) The horizontal axis shows an upper bound for the minimum genotype distance D_min of two networks belonging to two random typical neutral networks. (See SI Text for definition of typical neutral networks.) The vertical axis shows the number of pairs observed in each upper bound distance category. Different histogram colors correspond to different network sizes, as shown in the legend. (c Inset) The mean (and standard deviation) of the distribution of upper bounds as a function of the number of genes. Note the monotonic decrease of the mean D_min with increasing N; c = 0.25, d = 0.5, sample size n = 1600 neutral network pairs. We also find that the mean D_min decreases with increasing N and increasing c (data not shown).

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Fig. 8. Fixed point networks become rarer with increasing distance from a network w. The mean fraction of fixed point networks at a given distance D = k/k_max of a reference network for several values of N (c = 0.25, d = 0.5).

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Fig. 9. Size of the space of all networks. The total number of networks at a given distance D = k/k_max of a reference network for several values of N (c = 0.25, d = 0.5).

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Fig. 10. Fixed point networks are rare, viable network even rarer. The fraction of fixed point networks is exponentially small in N, although it is always much larger than the fraction of viable networks for a given pair of initial and equilibrium states. Thus the great majority of networks exhibit limit cycles and no fixed points (c = 0.25, d = 0. 5).

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Fig. 11. Old and new phenotypes as a function of genotypic distance. The probability of reaching

(a) and the probability of reaching a new equilibrium state (b) as a function of the genotypic distance from a network w with a given .

Table 1. Maximum genotype distances D and D' found among a sample of randomly chosen network pairs with the same gene expression pattern

Sample sizes are n = 5×10⁵ network pairs, except for N = 36, where n = 1.79 × 10⁴, and N = 20 (c = 0.5, d = 0.5), where n = 5×10⁷. N is the number of network genes, c is the expected fraction of regulatory interactions per gene (M ˜ cN²), and d is the expected (Hamming) distance of gene expression patterns S(0) and S_∞ for any one network in the sample.

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