Correlation Network Analysis with Bio3D

Background

Bio3D1 is an R package that provides interactive tools for structural bioinformatics. The primary focus of Bio3D is the analysis of bimolecular structure, sequence and simulation data (Grant et al. 2006).

Correlation network analysis can be employed to identify protein segments with correlated motions. In this approach, a weighted graph is constructed where each residue represents a node and the weight of the connection between nodes, i and j, represents their respective cross-correlation value, cij, expressed for example by the Pearson-like form or the linear mutual information. In this example, Normal mode analysis (NMA) is employed for the calculation of cross-correlations2.

Requirements

Detailed instructions for obtaining and installing the Bio3D package on various platforms can be found in the Installing Bio3D Vignette available both on-line and from within the Bio3D package. Note that this vignette also makes use of the IGRAPH R package, which can be installed with the command

install.packages("igraph")

About this document

This vignette was generated using Bio3D version 2.1.0.

Correlation network analysis based on single-structure NMA

In this first example we perform NMA on two crystallographic structures of transducin G protein alpha subunits, an active GTP-analog-bound structure (PDB id 1tnd) and an inactive GDP- and GDI (GDP dissociation inhibitor)-bound structure (PDB id 1kjy), by calling the function nma(). Cross-correlation matrices are calculated with the function dccm(). Correlation networks are then constructed with the function cna(). Network visualization is finally performed with the function plot.cna(). See also help(cna) for more details and example analysis.

library(bio3d)

First we read our selected GTP and GDI bound structures, select chain A, and perform NMA on each individually.

pdb.gtp = read.pdb("1TND")
pdb.gdi = read.pdb("1KJY")

pdb.gtp = trim.pdb(pdb.gtp, inds=atom.select(pdb.gtp, chain="A"))
pdb.gdi = trim.pdb(pdb.gdi, inds=atom.select(pdb.gdi, chain="A"))
modes.gtp = nma(pdb.gtp)
##  Building Hessian...     Done in 0.154 seconds.
##  Diagonalizing Hessian...    Done in 1.226 seconds.
modes.gdi = nma(pdb.gdi)
##  Building Hessian...     Done in 0.148 seconds.
##  Diagonalizing Hessian...    Done in 1.229 seconds.

Residue cross-correlations can then be calculated based on these normal mode results.

cij.gtp = dccm(modes.gtp)
cij.gdi = dccm(modes.gdi)

Correlation networks for both conformational states are constructed by applying the function cna() to the corresponding correlation matrices. Here, the C-alpha atoms represents nodes which are interconnected by edges with weights corresponding to the pairwise correlation coefficient. Edges are only constructed for pairs of nodes which obtain a coupling strength larger than a specified cutoff value (0.35 in this example). The weight of each edge is calculated as -log(|cij|), where cij is the correlation value between two nodes i and j. For each correlation network, betweenness clustering is performed to generate aggregate nodal clusters, or communities, that are highly intra-connected, but loosely inter-connected. By default, cna() returns communities associated with the maximal modularity value.

net.gtp = cna(cij.gtp, cutoff.cij=0.35)
net.gdi = cna(cij.gdi, cutoff.cij=0.35)
net.gtp
## 
## Call:
##   cna.dccm(cij = cij.gtp, cutoff.cij = 0.35)
## 
## Structure: 
##  - NETWORK NODES#:   323     EDGES#: 1427 
##  - COMMUNITY NODES#: 12  EDGES#: 22 
## 
##  + attr: network, communities, community.network,
##         community.cij, cij, call
net.gdi
## 
## Call:
##   cna.dccm(cij = cij.gdi, cutoff.cij = 0.35)
## 
## Structure: 
##  - NETWORK NODES#:   320     EDGES#: 1587 
##  - COMMUNITY NODES#: 8   EDGES#: 11 
## 
##  + attr: network, communities, community.network,
##         community.cij, cij, call

A 3-D visualization of networks can also be performed with the Bio3D function view.dccm() (See help(view.dccm) and the vignette "Enhanced Methods for Normal Mode Analysis with Bio3D" available on-line):

view.dccm(net.gtp$cij, launch=TRUE)

Maximization of modularity sometimes creates unexpected community partitions splitting natural structure motifs, such as secondary structure elements, into many small community 'islands'. To avoid this situation, we look into partitions with modularity close to the maximal value but with an overall smaller number of communities.

mod.select <- function(x, thres=0.005) {
   remodel <- community.tree(x, rescale = TRUE)
   n.max = length(unique(x$communities$membership))
   ind.max = which(remodel$num.of.comms == n.max)
   v = remodel$modularity[length(remodel$modularity):ind.max]
   v = rev(diff(v))
   fa = which(v>=thres)[1] - 1
   ncomm = ifelse(is.na(fa), min(remodel$num.of.comms), n.max - fa)

   ind <- which(remodel$num.of.comms == ncomm)
   network.amendment(x, remodel$tree[ind, ])
}
nnet.gtp = mod.select(net.gtp)
nnet.gtp
## 
## Call:
##   cna.dccm(cij = cij.gtp, cutoff.cij = 0.35)
## 
## Structure: 
##  - NETWORK NODES#:   323     EDGES#: 1427 
##  - COMMUNITY NODES#: 10  EDGES#: 18 
## 
##  + attr: network, communities, community.network,
##         community.cij, cij, call
nnet.gdi = mod.select(net.gdi)
nnet.gdi
## 
## Call:
##   cna.dccm(cij = cij.gdi, cutoff.cij = 0.35)
## 
## Structure: 
##  - NETWORK NODES#:   320     EDGES#: 1587 
##  - COMMUNITY NODES#: 7   EDGES#: 9 
## 
##  + attr: network, communities, community.network,
##         community.cij, cij, call

The resulting networks can be visualized with the Bio3D function plot.cna(), which can generate 2D representations for both full residue-level and coarse-grained community-level networks.

cent.gtp.full = layout.cna(nnet.gtp, pdb=pdb.gtp, full=TRUE, k=3)[,1:2]
cent.gtp = layout.cna(nnet.gtp, pdb=pdb.gtp, k=3)[,1:2]
cent.gdi.full = layout.cna(nnet.gdi, pdb=pdb.gtp, full=TRUE, k=3)[,1:2]
cent.gdi = layout.cna(nnet.gdi, pdb=pdb.gtp, k=3)[,1:2]

The following code plots the four networks.

plot.cna(nnet.gtp, layout=cent.gtp.full, full=TRUE, vertex.label=NA, vertex.size=5)
plot.cna(nnet.gtp, layout=cent.gtp)
plot.cna(nnet.gdi, layout=cent.gdi.full, full=TRUE, vertex.label=NA, vertex.size=5)
plot.cna(nnet.gdi, layout=cent.gdi)

Correlation networks for GTP "active" and GDI "inhibitory"
conformational states of transducin. Networks are derived from NMA
applied to single GTP and GDI crystallographic
structures.

Correlation network analysis based on ensemble NMA

In this example we perform ensemble NMA on 53 crystallographic transducin G-alpha structures, which can be categorized into active GTP-analog-bound, inactive GDP-bound, and inhibitory GDI-bound states. For this analysis we utilize the example transducin structure dataset (further details of which can be obtained via help(transducin) along with the "Comparative Protein Structure Analysis with Bio3D" vignette available on-line). Briefly, this dataset includes aligned PDB coordinates (pdbs), structural invariant core positions (core) and annotations for each PDB structure (annotation). Cross-correlation matrices for all structures in the ensemble are calculated with the function dccm(). State-specific ensemble average correlation matrices are then obtained with the function filter.dccm(). Correlation networks are finally constructed with the function cna(), and visualization of the networks performed with the function plot.cna():

attach(transducin)
modes <- nma(pdbs, ncore=8)
cijs0 <- dccm(modes)

Below we perform optional filtering and averaging per state of the individual structure cross-correlation matrices. In this example, we also utilize a cutoff for correlation (0.35 in this example) and a cutoff for C-alpha distance (10 angstrom) (See help(filter.dccm) and main text for details).

cij <- filter.dccm(cijs0, xyz=pdbs, fac=annotation[, "state3"], cutoff.cij=0.35, 
                dcut=10, scut=0, pcut=0.75, ncore=8)
nets <- cna(cij, cutoff.cij = 0)
net1 = mod.select(nets$GTP)
net2 = mod.select(nets$GDI)
ref.pdb <- pdbs2pdb(pdbs, inds=1, rm.gaps=TRUE)[[1]]
cent.gtp.full = layout.cna(net1, pdb=ref.pdb, full=TRUE, k=3)[,1:2]
cent.gtp = layout.cna(net1, pdb=ref.pdb, k=3)[,1:2]
cent.gdi.full = layout.cna(net2, pdb=ref.pdb, full=TRUE, k=3)[,1:2]
cent.gdi = layout.cna(net2, pdb=ref.pdb, k=3)[,1:2]
plot.cna(net1, layout=cent.gtp.full, full=TRUE, vertex.label=NA, vertex.size=5)
plot.cna(net1, layout=cent.gtp)
plot.cna(net2, layout=cent.gdi.full, full=TRUE, vertex.label=NA, vertex.size=5)
plot.cna(net2, layout=cent.gdi)

Correlation networks for GTP "active" and GDI "inhibitory"
conformational states of transducin. Networks are derived from ensemble
NMA of available GTP and GDI crystallographic
structures.

Document Details

This document is shipped with the Bio3D package in both R and PDF formats. All code can be extracted and automatically executed to generate Figures and/or the PDF with the following commands:

library(rmarkdown)
render("Bio3D_cna-transducin.Rmd", "all")

Information About the Current Bio3D Session

sessionInfo() 
## R version 3.1.1 (2014-07-10)
## Platform: x86_64-redhat-linux-gnu (64-bit)
## 
## locale:
##  [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C              
##  [3] LC_TIME=en_US.UTF-8        LC_COLLATE=en_US.UTF-8    
##  [5] LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_US.UTF-8   
##  [7] LC_PAPER=en_US.UTF-8       LC_NAME=C                 
##  [9] LC_ADDRESS=C               LC_TELEPHONE=C            
## [11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C       
## 
## attached base packages:
## [1] stats     graphics  grDevices utils     datasets  methods   base     
## 
## other attached packages:
## [1] igraph_0.7.1    bio3d_2.1-0     rmarkdown_0.3.3
## 
## loaded via a namespace (and not attached):
## [1] digest_0.6.3    evaluate_0.5.5  formatR_1.0     htmltools_0.2.6
## [5] knitr_1.6       stringr_0.6.2   tools_3.1.1     yaml_2.1.13

References

Grant, B.J., A.P.D.C Rodrigues, K.M. Elsawy, A.J. Mccammon, and L.S.D. Caves. 2006. “Bio3d: An R Package for the Comparative Analysis of Protein Structures.” Bioinformatics 22: 2695–96. doi:10.1093/bioinformatics/btl461.


  1. The latest version of the package, full documentation and further vignettes (including detailed installation instructions) can be obtained from the main Bio3D website: http://thegrantlab.org/bio3d/

  2. This vignette contains executable examples, see help(vignette) for further details. 

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