Age-dependent evolution of the yeast protein interaction network suggests a limited role of gene duplication and divergence

Abstract

Proteins interact in complex protein-protein interaction (PPI) networks whose topological properties-such as scale-free topology, hierarchical modularity, and dissortativity-have suggested models of network evolution. Currently preferred models invoke preferential attachment or gene duplication and divergence to produce networks whose topology matches that observed for real PPIs, thus supporting these as likely models for network evolution. Here, we show that the interaction density and homodimeric frequency are highly protein age-dependent in real PPI networks in a manner which does not agree with these canonical models. In light of these results, we propose an alternative stochastic model, which adds each protein sequentially to a growing network in a manner analogous to protein crystal growth (CG) in solution. The key ideas are (1) interaction probability increases with availability of unoccupied interaction surface, thus following an anti-preferential attachment rule, (2) as a network grows, highly connected sub-networks emerge into protein modules or complexes, and (3) once a new protein is committed to a module, further connections tend to be localized within that module. The CG model produces PPI networks consistent in both topology and age distributions with real PPI networks and is well supported by the spatial arrangement of protein complexes of known 3-D structure, suggesting a plausible physical mechanism for network evolution.

Figure 1. Interaction density (D) patterns depend upon the attachment rule.

The protein age groups G1, G2, and G3 emerge at times t = 1, 2, and 3, respectively. In all cases, the first age group, G1, makes intra-group connections at t = 1. (A) In the random attachment (RA) model, G2 makes connections to G1 and within G2 with an equal probability at t = 2, showing that D_1,1 = D_1,2. Similarly, G3 makes connections to G1, G2, and within G3 (D_1,3 = D_2,3 = D_3,3). The interaction densities between protein age groups are shown in the right panel. (B) In the preferential attachment (PA) model, G2 attaches more frequently to G1 than within G2 because, on average, G1 is more connected (D_1,2>D_2,2). At t = 3, G3 is preferentially connected to older groups in the order of G1>G2>G3 (D_1,3>D_2,3>D_3,3). (C) In anti-preferential attachment (AP), the interaction density shows the reverse pattern to PA. Because a new node prefers less-connected nodes or younger groups, the density pattern shows D_1,2<D_2,2 and D_1,3<D_2,3<D_3,3. Therefore, the interaction density (D) decreases in AP but increases in PA from top to bottom in the right panel.

Figure 2. The network properties of the yeast PPI network are compared with the different models for network evolution.

None of the canonical models (PA, DD, and AP) were compatible with the real PPI_yeast in terms of both topology and the age-dependency of interaction density. Only the CG model shows similar characteristics to the PPI_yeast for all the network properties tested. The plots in each row, I-IV, indicate (I) the degree distribution P(k), (II) the clustering coefficient C(k), (III) the average degree of nearest neighbors <k_nn>(k), and (IV) the interaction density pattern (ΔD) between protein age groups. In the yeast PPI, the network shows a scale-free degree distribution, hierarchical modularity, and dissortative mixing properties (negative correlation in rows I-III, respectively). In row IV, the interaction density tends to be dense within the same group (diagonal) and sparse between different age groups (off-diagonal) in each column with positive ΔD, similar in pattern to the anti-preferential attachment (AP) in Figure 1C. In the PA model, the resulting network is scale-free (I) and slightly dissortative (III), similar to the PPI_yeast. However, it is not hierarchically modular (II) and shows an inverse pattern of negative ΔD. In the DD model, the resulting network is scale-free (I), dissortative (III), and also hierarchically modular but not as highly as the PPI_yeast (II). It shows an inverse pattern of negative ΔD as the PA model. In the AP model, the resulting network is highly different from the PPI_yeast, showing non scale-free, non hierarchically modular, and non dissortative structure (I-III), although the interaction density pattern (ΔD>0) is similar (IV). In the CG model, the network shows highly similar network characteristics to PPI_yeast in both topology (I-III) and interaction density (IV). The number of nodes is N = 3,000 in all cases. The average degree is <k> = 8 in the PA, AP, and CG models, and in the DD model the parameters are set as p = 0.1 and q = 0.6, where the resulting average degree is <k>≈4.

Figure 3. A schematic diagram (A) and a flowchart (B) show the process of network growth by the CG model.

(A) The CG model mimics sequential incorporation of new proteins to crystals grown in solution. In stage I, the initial set of proteins (red) form seeds of new crystals. In stage II, a new protein is added, which either forms a new seed crystal (n) or attaches to an existing crystal (e). In the latter case, the protein e attaches to one protein in the crystal (solid arrow) and then further interacts with nearby proteins (dotted arrow). In stages III and IV, the second- (orange) and third- (yellow) generation proteins repeat the process of stage II, with the result that the early generation tends to be located at the core of each crystal and the late generation at the periphery. (B) Similarly, the CG model starts with a small number of seed nodes (N₀). In each cycle, modules are defined and a new node is added that makes a fixed number of connections (ΔE). A new node creates a new module at a probability P_new and makes connections to any other node in accordance with the AP rule. Otherwise, one module (crystal) is randomly selected and the new node is connected exclusively to the nodes in the selected module. After ΔE connections are made, modules are redefined and the cycle is repeated.

Figure 4. The comparison of network property indices between the yeast PPI networks and the models tested.

(A) PPI_yeast, (B) the PA and AP models, (C) the DD model at p = 0.1, q = 0.5∼0.7, and (D) the CG model. In (B), the scale-free index, γ, of the AP model is not shown because the resulting network is not scale-free. The properties of the CG model are more similar to PPI_yeast than those of the PA, AP, and DD models. Index values are normalized so that the average indexes of LC and HTP are zero, calculated as I_norm = (I_raw−I_yeast)/I_range, where I_norm is the normalized index and I_raw is the index value of each model. I_yeast is the average index between LC and HTP except for <k>, where I_yeast is set to the average degree of LC because <k>_LC is similar to <k> = 8.0 in the PA, AP, and CG models. The denominator I_range is set to max(I_raw) observed in LC, HTP, and the models, except for δ and ΔD showing both negative and positive values. In the case of ΔD and –δ, the denominator I_range is set to max(I_raw)−min(I_raw) because these indexes range from negative to positive values in LC, HTP, and the models. The sign of δ is reversed to −δ to give the index positive values for LC and HTP.

Figure 5. The frequencies of homodimers are age-dependent.

The ratio between the observed (Obs.) and the expected (Exp.) number of homodimers is plotted for each age group, calculating for each age group the fraction of homodimeric proteins divided by the fraction of total yeast proteins accounted for by that age group.

Figure 6. The spatial subunit arrangement of known multi-protein complexes is consistent with the CG model.

Subunits of all 12 known multi-protein complexes with at least three proteins and two-age groups are colored according to their age groups: The most ancient group, ABE, is colored in red tones (yellow, pink, magenta, orange, red). The AB, AE, or BE groups are in green tone, and the most recent A, B, and E groups are in blue. For visual clarity, the older group(s) is presented in cartoon models and the youngest group in space-filling models in each complex. The age group assignments in (K) and (L) were ambiguous because the chains assigned AE could be assigned to ABE if the BLAST hit cut-off was slightly relaxed to 25% instead of 30% sequence identity (the “twilight zone” for homology detection). Therefore, (K) and (L) may, in fact, consist of the subunits of the ABE group only. The subunits are in various configurations. In (A) and (L), the younger subunits are spatially separated, but the older subunits are aggregated. In (B–E), two old subunits (three in (E)) and one young subunit are linearly connected. In all four cases, the older subunits are all connected without insertion of the younger subunit in the middle. In (F–H), the subunits form a trans-membrane helix bundle, where young subunits are always located at the periphery while old subunits are at the center. In (I) and (J), all the subunits are in contact with each other. In the case of (K), there are two modules—the clamp (upper homo-trimeric ring) and the clamp loader (lower hetero-pentameric ring). Considering the clamp loader alone, both younger and older subunits are separated. (A) APRIL and TACI (TNF receptor) complex (Protein databank code 1xu2). (B) Urokinase receptor, urokinase, and vitronectin complex (3bt1). (C) Factor Xa/NAP5 complex (2p3f). (D) Thrombin-PAR4 complex (2pv9). (E) Complexin/SNARE complex. (F) Cytochrome b6f complex (2e76). (G) Cyanobacterial photosystem I (1jb0). (H) Photosynthetic Oxygen-Evolving Center. A cross-section of the trans-membrane helix bundle is shown (1s5l). (I) APPBP1-UBA3-NEDD8 complex (1r4m). (J) Cytochrome ba3 Oxidase (1xme). (K) DNA clamp–clamp loader complex (1sxj). (L) Cythochrome bc complex (1ezv).