Figure 3

Analytical and numerical results of PPI Network evolution under whole genome duplication. A. Phase diagram for the limit degree distribution as a function of network exponential growth rate, γ_o + γ_n, and asymmetric divergence of gene duplicates, γ_o - γ_n. In paricular, network conservation and scale-free topology are found to be intrinsically linked properties of PPI networks under genome duplication. Colored lines correspond to iso-exponent of scale-free degree distribution. All other regions of phase diagram are likely biologically irrelevant (see text). B&C. Comparison with protein direct physical interaction data for Yeast from BIND [38] and MIPS [39] databases: BIND (August 11, 2005 release), 4576 proteins, 9133 physical interactions, $\overline{k}$ = 3.99, $\overline{k^2}$ = 106 (filled symbols) and MIPS (downloaded online April 20, 2006), 4153 proteins, 7417 physical interactions, $\overline{k}$ = 3.57, $\overline{k^2}$ = 78.6 (open symbols). Squares correspond to raw data, while circles and triangles are statistically averaged with gaps in connectivity distribution for large k ≥ 20, due to the finite size of Yeast PPI network. B. One-parameter fit of connectivity distribution data p _k (corresponding to the "X" mark in A., see text). Numerical connectivity distribution averaged over 10,000 network realizations (central green line). Numerical averages plus or minus two standard deviations (±2σ) are also displayed to show the predicted dispersions (upper and lower green lines) [Raw data (squares) do not fit within the mean ± 2σ curves for large k due to the finite size of Yeast PPI network]. The fitting parameter γ = 0.26 corresponds to an effective growth rate of 1 + 2γ = 1.52. C. One-parameter fit of average connectivity of first neighbor proteins g _k [50] (i.e. k.g _k sums connectivities of first neighbors from proteins of connectivity k). Numerical predictions averaged over 10,000 network realizations (central blue line). Numerical averages plus or minus two standard deviations are also displayed (upper and lower blue lines). Same fitting parameter value as in B, γ = 0.26. Note that g _k is rescaled by $\overline{k}$/$\overline{k^2}$ (as $\overline{k{g}_k}$ = $\overline{k^2}$ holds for each network realization); this rescales large g _k fluctuations between network realizations, due to the divergence of $\overline{k^2}$ for p _k ~ k^-_-1 with 2 > α > 0 for the one-parameter model.