Correlation network representation and analysis

(Isoelectric point value characteristic)

First, we have detected clusters of positions that are highly correlated. A set of the protein positions, each position displaying a significant correlation with at least one position from the set, was considered a cluster of correlated positions.
To detect such clusters of correlated positions, we applied the hierarchical clustering approach. We clustered all the positions basing on the following measure of correlation for a pair of positions i, j:

,

Robust estimates for partial correlation coefficients were used.
This is tree representation of the correlation networks between homeodomen positions. Red vertical lines corresponds to critical values of the correlation coefficients.

Two networks (clusters)  were selected.

Cluster I

Localization of residues in a spatial structure of DNA-homeodomain complex [ PDB ID: 1HDD; Kissinger et al [3]). Green lines correspond to sald bridges (distance between charged atoms of opposite sign < 4 A):

Another representation of this cluster of residues:

Schematic representation of the correlation network for cluster 1 (blue arrows - significantly negatively correlating pairs):

Proposed conserved characteristic: net isoelectric point value (net charge) Q_I=pI₁₅+pI₁₈+pI₁₉+pI₂₃+pI₃₀+pI₃₇ +pI₃₃

Analysis of the Q_I characteristic dispersion

M=10⁵ random samples of normal independent variables were generated with parameters corresponding mean and dispersion estimates in real homeodomain sample. The distribution parameters of Q_I dispersion in random samples were estimated as follows:

- number of random samples with the dispersion value of the  Q_I greater than in original sample.
Distributions of the dispersion values of the characteristics Q_I  in simulated random samples with independent changes in pI values at individual positions within homeodomain: x axis, D_rand; y axis, fraction of samples with the corresponding D_rand value; arrows indicate the dispersion values at homeodomain positions.

Therefore it may be concluded  that characteristic Q_I is conserved with the confidence greater than 99.99%.

Cluster  II

Localization of residues in a spatial structure of DNA-homeodomain complex [ PDB ID: 1HDC; Kissinger et al [3]):

Another representation of this cluster of residues:

Schematic representation of the correlation network for cluster 1 (blue arrows - significantly negatively correlating pairs, red arrows - significantly positively correlating pairs):

Analysis of the Q_II characteristic dispersion

M=10⁵ random samples of normal independent variables were generated with parameters corresponding mean and dispersion estimates in real homeodomain sample. The distribution parameters of Q_I dispersion in random samples were estimated as follows:

Distributions of the dispersion values of the characteristics   Q_II in simulated random samples with independent changes in pI values at individual positions within homeodomain: x axis, D_rand; y axis, fraction of samples with the corresponding D_rand value; arrows indicate the dispersion values at homeodomain positions.

Therefore it may be concluded  that characteristic Q_II is conserved with the confidence greater than 95%.