[Paper Review] Asymptotic Evolution of Protein-Protein Interaction Networks for General Duplication-Divergence Models

Kirill Evlampiev, Hervé Isambert|ArXiv.org|Nov 22, 2006

Bioinformatics and Genomic Networks1 references3 citations

TL;DR

This paper proposes a general duplication-divergence (GDD) model to study the asymptotic evolution of protein-protein interaction (PPI) networks, incorporating stochastic selection of duplication-derived interactions through six γ parameters. It analytically derives the long-term network topology and shows that only conserved, non-dense networks—necessarily scale-free—emerge under biologically relevant conditions.

ABSTRACT

Genomic duplication-divergence events, which are the primary source of new protein functions, occur stochastically at a wide range of genomic scales, from single gene to whole genome duplications. Clearly, this fundamental evolutionary process must have largely conditioned the emerging structure of protein-protein interaction (PPI) networks, that control many cellular activities. We propose and asymptotically solve a general duplication-divergence model of PPI network evolution based on the statistical selection of duplication-derived interactions. We also introduce a conservation index, that formally defines the statistical evolutionary conservation of PPI networks. Distinct conditions on microscopic parameters are then shown to control global conservation and topology of emerging PPI networks. In particular, conserved, non-dense networks, which are the only ones of potential biological relevance, are also shown to be necessary scale-free.

Motivation & Objective

• To develop a general theoretical framework for modeling the long-term evolution of protein-protein interaction (PPI) networks through gene duplication and divergence.
• To identify the conditions under which PPI networks remain evolutionarily conserved and topologically non-dense, as required for biological relevance.
• To formally define and analyze a conservation index for PPI networks based on statistical selection of duplication-derived interactions.
• To resolve the asymptotic behavior of PPI networks under exponential duplication dynamics, contrasting with prior time-linear models.
• To establish a theoretical baseline for incorporating additional evolutionary processes such as domain shuffling and horizontal gene transfer.

Proposed method

• Proposes a general duplication-divergence (GDD) model with a constant fraction $ q $ of genes duplicated at each time step, followed by asymmetric divergence into 'old' (o) and 'new' (n) duplicates.
• Introduces six γ parameters ($ \gamma_{ij} $) to model the stochastic conservation of interactions between gene partners based on their duplication state (s: singular, o: old, n: new).
• Derives recurrence relations for network features (e.g., degree distribution, clustering, triangles) using generating functions $ H^{(n)}(x,y) $, $ T^{(n)}(x,y,z) $, and $ F^{(n)}(x) $, tracking node types and connectivity.
• Applies ensemble averaging $ \langle Q^{(n)} \rangle $ over all evolutionary realizations to study asymptotic network properties.
• Reduces the full 7-parameter model to three-parameter limits for local ($ q \ll 1 $) and whole-genome ($ q = 1 $) duplications.
• Solves the recurrence relations asymptotically to determine the long-term growth and topology of PPI networks, including exponential growth of features like triangle counts.

Experimental results

Research questions

• RQ1 Under what conditions on the γ parameters does the PPI network remain conserved and non-dense, as required for biological plausibility?
• RQ2 How does the asymptotic topology of PPI networks—particularly scale-free degree distributions—emerge from duplication-divergence dynamics?
• RQ3 What is the role of exponential duplication dynamics in shaping PPI network evolution, compared to time-linear models?
• RQ4 How do different duplication regimes (local vs. whole-genome) affect the conservation and structural properties of PPI networks?
• RQ5 Can a formal conservation index be defined for PPI networks based on statistical selection of duplication-derived interactions?

Key findings

• Conserved, non-dense PPI networks are only possible when the γ parameters satisfy specific constraints, ensuring that only a subset of duplication-derived interactions are retained.
• The model predicts that conserved, non-dense networks must necessarily be scale-free, with a power-law degree distribution emerging asymptotically.
• The mean number of triangles in the network grows exponentially with time, governed by a growth rate determined by a combination of γ parameters and duplication fraction $ q $, as shown in Eq. (84).
• The time-linear duplication-divergence model of Ispolatov *et al.* is recovered as a special limit of the GDD model when $ q \ll 1 $ and specific γ parameters are set.
• For whole-genome duplication ($ q = 1 $), the network evolution is governed by three γ parameters: $ \gamma_{oo}, \gamma_{on}, \gamma_{nn} $, which control the conservation of interactions among old, new, and mixed duplicates.
• The recurrence relations for network features preserve symmetry and are derived by tracking the contribution of each node type (s, o, n) to new connections and motifs, enabling exact asymptotic analysis.