Fitting a geometric graph to a protein–protein interaction network

Abstract

Motivation: Finding a good network null model for protein-protein interaction (PPI) networks is a fundamental issue. Such a model would provide insights into the interplay between network structure and biological function as well as into evolution. Also, network (graph) models are used to guide biological experiments and discover new biological features. It has been proposed that geometric random graphs are a good model for PPI networks. In a geometric random graph, nodes correspond to uniformly randomly distributed points in a metric space and edges (links) exist between pairs of nodes for which the corresponding points in the metric space are close enough according to some distance norm. Computational experiments have revealed close matches between key topological properties of PPI networks and geometric random graph models. In this work, we push the comparison further by exploiting the fact that the geometric property can be tested for directly. To this end, we develop an algorithm that takes PPI interaction data and embeds proteins into a low-dimensional Euclidean space, under the premise that connectivity information corresponds to Euclidean proximity, as in geometric-random graphs. We judge the sensitivity and specificity of the fit by computing the area under the Receiver Operator Characteristic (ROC) curve. The network embedding algorithm is based on multi-dimensional scaling, with the square root of the path length in a network playing the role of the Euclidean distance in the Euclidean space. The algorithm exploits sparsity for computational efficiency, and requires only a few sparse matrix multiplications, giving a complexity of O(N 2 ) where N is the number of proteins. Results: The algorithm has been verified in the sense that it successfully rediscovers the geometric structure in artificially constructed geometric networks, even when noise is added by re-wiring some links. Applying the algorithm to 19 publicly available PPI networks of various organisms indicated that: (a) geometric effects are present and (b) two-dimensional Euclidean space is generally as effective as higher dimensional Euclidean space for explaining the connectivity. Testing on a high-confidence yeast data set produced a very strong indication of geometric structure (area under the ROC curve of 0.89), with this network being essentially indistinguishable from a noisy geometric network. Overall, the results add support to the hypothesis that PPI networks have a geometric structure.

Spatial Models for Virtual Networks

“…In this model, nodes are embedded in a metric space according to a given probability distribution, and two nodes are linked precisely when their distance is smaller than a given threshold value θ. The random geometric graph was proposed as a model for spatial networks, wireless multi-hop networks, and for biological networks, see for example [12,13,25]. In [19], a variant of the model is presented, where the metric space is the hyperbolic space.…”

Topological Network Alignment Uncovers Biological Function and Phylogeny

“…Thus, we search for a well-fitting theoretical null model. Arguably the best currently known theoretical model for PPI networks, requiring the fewest tunable parameters, is the geometric random graph model ("GEO"), 37,39,65 in which proteins are modeled as existing in a metric space and are connected by an edge if they are within a fixed, specified distance of each other.…”

“…[67][68][69] In the light of new PPI network data, several studies 37,39,65 have presented compelling evidence that the structure of PPI networks is closer to geometric than to scale-free networks. This was done by comparing frequencies of graphlets in real-world and model networks 37 and by measuring a highly-constraining agreement between "graphlet degree distributions."…”

“…39 Finally, it has been shown that PPI networks can be successfully embedded into a low-dimensional Euclidean space, thus directly confirming that they have a geometric structure. 65 The superior fit of the GEO model to PPI networks over other models may not be surprising, since it can be biologically motivated. In particular, the currently accepted paradigm for evolution is based on a series of gene duplication and mutation events.…”

Some Typical Properties of the Spatial Preferred Attachment Model

“…vertices whose information entities are closely related will be at a short distance from each other in the metric space. A number of spatial models have been proposed up to date [10,11,[17][18][19]26]. We direct the reader to the recent survey for more details [20].…”