Identifying Hierarchical and Overlapping Protein Complexes Based on Essential Protein-Protein Interactions and “Seed-Expanding” Method

Algorithm 1

The description of algorithm MCSE.

Input: PPI network G (V, E, W), Essential PPI set S,

   parameter 𝜆_th

Output: Identified Clusters

Process:

//1. Generate the weighted PPI network 𝐺^𝑊(𝑉,𝐸,𝑊)

 (1) for each edge (𝑣_𝑖,𝑣_𝑗)∈𝐸  do

    calculate its weight 𝑤(𝑣_𝑖,𝑣_𝑗) by formula (2);

//2. Seed Selecting

 (2) 𝛽= the average value of 𝑊(𝐺^𝑊);

 (3) 𝐸𝑠=𝜙;

 (4) for each edge (𝑣_𝑖,𝑣_𝑗)∈𝐸 do

     if  𝑤(𝑣_𝑖,𝑣_𝑗)≥𝛽 then 𝐸𝑠←(𝑣₁,𝑣₂);

 (5) 𝐸𝑠=𝐸𝑠∪𝑆;

 (6) sort all edges in Es to queue Sq in non-increasing order

   of edge’s weight first and essentiality second;

//3. Seed Expanding

 (7) 𝐶=𝜙;

 (8) 𝑀𝑎𝑟𝑘𝑒𝑑=𝜙;

 (9) while 𝑆𝑞≠𝜙 do

    (𝑣₁,𝑣₂)←𝑆𝑞;

    𝐻={𝑣₁,𝑣₂};

    𝜆_𝐻 = the 𝜆 value of the cluster 𝐻;

    if 𝜆_𝐻<𝜆_th then flag1 = 1; else flag1 = 0;

    𝛾_𝐻 = the percentage of marked vertices in 𝐻

    if 𝛾_𝐻<0.5 then flag2 = 1; else flag2 = 0;

    while flag1 = 1 and flag2 = 1 do

     for each neighbor vertex 𝑣_𝑖 of 𝐻 in 𝐺^𝑊 do

      𝑓(𝑣_𝑖,𝐻)=∑_{𝑣𝑗∈𝐻,⟨𝑣𝑖,𝑣𝑗⟩∈𝐸}𝑤(𝑣_𝑖,𝑣_𝑗);

     sort all neighbor vertex of H to queue 𝑉_𝑞 in

     non-increasing order by their f value;

     if 𝑉_𝑞≠𝜙 then

      𝑣_𝑎←𝑉_𝑞;

      𝐻=𝐻+{𝑣_𝑎};

      recalculate 𝜆_𝐻;

      if 𝜆_𝐻<𝜆_th then flag1 = 1; else flag1 = 0;

      recalculate 𝛾_𝐻

      if 𝛾_𝐻<0.5 then flag2 = 1; else flag2 = 0;

    if flag1 = 0 then

     𝐶=𝐶∪{𝐻};

     put all vertices of H in Marked;

     remove edges include vertices of H from Sq;

 (10) Output 𝐶