Protein-Protein interaction networks: why static MpK model works and preferential attachment does not

Various approaches have been proposed to explain the observed scale free structure $p(k) \sim k^{-\gamma}$ of protein-protein interaction networks. We argue that the preferential attachment coming from gene duplication[1] is questionable. A static ``MpK'' model produces the scale free structure via computer simulations[2] for unexplained reasons. On the other hand, it was analytically proved[3] that deterministic threshold models produce scale free networks (with $\gamma\equiv 2$) if fitness distributions are exponential. We study the static MpK model further and find the above analytical proof applicable with extensions, and $\gamma$ dependent on the threshold parameter. This work not only predicts the dependence of $\gamma$ on protein concentrations, but also provides a generic mechanism of scale free networks. The clustering coefficient distribution in the model is interpreted by a simple picture. \newline \newline [1] A.-L. Barab\'asi and Z. N. Oltvai, Nature Reviews Genetics \textbf{5}, 101 (2004). \newline [2] E. J. Deeds, O. Ashenberg, E. I. Shakhnovich, Proc. Natl. Acad. Sci. USA \textbf{103}, 311 (2006). \newline [3] G. Caldarelli, A. Capocci, P. De Los Rios, and M. A. Mu\~noz, Phys. Rev. Lett. \textbf{89}, 258702 (2002).