A simple physical model for scaling in protein-protein interaction networks - PubMed (original) (raw)

A simple physical model for scaling in protein-protein interaction networks

Eric J Deeds et al. Proc Natl Acad Sci U S A. 2006.

Abstract

It has recently been demonstrated that many biological networks exhibit a "scale-free" topology, for which the probability of observing a node with a certain number of edges (k) follows a power law: i.e., p(k) approximately k(-gamma). This observation has been reproduced by evolutionary models. Here we consider the network of protein-protein interactions (PPIs) and demonstrate that two published independent measurements of these interactions produce graphs that are only weakly correlated with one another despite their strikingly similar topology. We then propose a physical model based on the fundamental principle that (de)solvation is a major physical factor in PPIs. This model reproduces not only the scale-free nature of such graphs but also a number of higher-order correlations in these networks. A key support of the model is provided by the discovery of a significant correlation between the number of interactions made by a protein and the fraction of hydrophobic residues on its surface. The model presented in this paper represents a physical model for experimentally determined PPIs that comprehensively reproduces the topological features of interaction networks. These results have profound implications for understanding not only PPIs but also other types of scale-free networks.

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Figures

Fig. 1.

Fig. 1.

Correlation between PPI networks. (a) The correlation between the network degree of a given protein in the Ito (13) and Uetz (12) data sets. Each point corresponds to a particular protein that exhibited interactions in both experiments. (b) A plot similar to a but comparing the ItoCore data set with Uetz.

Fig. 2.

Fig. 2.

A physical model for PPI measurements. (a) A schematic of the model described in the text. Association free energies are largely the result of desolvation of the two protein surfaces. The overall burial of hydrophobic groups is represented by the sum of the contributions from each protein. (b) The distribution of surface hydrophobicities in yeast proteins. The fraction of surface residues that are hydrophobic (defined as residues AVILMFYW) is calculated according to the description in the

supporting information

. This distribution is taken from proteins in the Ito experiment (13). The red squares represent the model hydrophobicities sampled from a Gaussian distribution with the same mean and standard deviation as the Ito proteins themselves. (c) A degree distribution for the realization of the model used in b. The cutoff was chosen such that the power-law fit gives an exponent of approximately-2.0, close to that of Ito graph. The degrees in this plot are shifted by +1 to allow for orphans (nodes of degree 0) to be displayed on a log-log plot. Note that the fraction of orphans in the graph is very high.

Fig. 3.

Fig. 3.

Degree distributions and correlations for model PPI networks. (a) Comparison of degree distributions for the Ito data set (13) and a realization of the random linking model. In this case, all orphans from the model graph of 3,200 nodes are connected to one node that does exhibit connections randomly. The line represents a power-law with an exponent of -2. The degrees in this plot are not shifted as they are in Fig. 2_c_. (b) The correlation between degrees for in the model of Ito (13) compared with the model of Uetz (12). In this case, the different experiments are represented as independent sampling of values of K from a population of proteins with the distribution of p values shown in Fig. 2_b_. The Ito model is equivalent to the random linking results in a, and the Uetz degrees are taken from its random linking model (the degree distribution for that graph may be found in the

supporting information

). The linear correlation is 0.04 in this case.

Fig. 4.

Fig. 4.

Correlations between hydrophobicity and connectivity. The dependence of the correlation between 〈log(k) 〉 and 〈p 〉 as a function of the bin size in p used to define the populations over which p and log(k) are averaged. The dependence for hydrophobic residues is very similar between the ItoCore data and the physical model for ItoCore. In the case of the charged data set, p is taken to be the percentage of charged residues on the surface, and the correlation dependence is calculate exactly as for the hydrophobic residues. None of the correlations for the charged data set are statistically significant.

Fig. 5.

Fig. 5.

Standard deviation in connectivity. As predicted by the MpK model, in the model and the ItoCore network, the standard deviation of log(k) in a given bin in p increases with increasing 〈p 〉 for that bin. The bin size is set at 0.05 for the model and experimental networks. The lack of a maximum at p = 0.5 (as predicted by Eq. 3) is due to the fact that very few proteins exist in those bins with large 〈p 〉, decreasing the standard deviation for the most hydrophobic bin in each case.

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