Targeting functional motifs of a protein family

The structural organization of a protein family is investigated by devising a method based on the random matrix theory (RMT), which uses the physiochemical properties of the amino acid with multiple sequence alignment. A graphical method to represent protein sequences using physiochemical properties is devised that gives a fast, easy, and informative way of comparing the evolutionary distances between protein sequences. A correlation matrix associated with each property is calculated, where the noise reduction and information filtering is done using RMT involving an ensemble of Wishart matrices. The analysis of the eigenvalue statistics of the correlation matrix for the $\beta$-lactamase family shows the universal features as observed in the Gaussian orthogonal ensemble (GOE). The property-based approach captures the short- as well as the long-range correlation (approximately following GOE) between the eigenvalues, whereas the previous approach (treating amino acids as characters) gives the usual short-range correlations, while the long-range correlations are the same as that of an uncorrelated series. The distribution of the eigenvector components for the eigenvalues outside the bulk (RMT bound) deviates significantly from RMT observations and contains important information about the system. The information content of each eigenvector of the correlation matrix is quantified by introducing an entropic estimate, which shows that for the $\beta$-lactamase family the smallest eigenvectors (low eigenmodes) are highly localized as well as informative. These small eigenvectors when processed gives clusters involving positions that have well-defined biological and structural importance matching with experiments. The approach is crucial for the recognition of structural motifs as shown in $\beta$-lactamase (and other families) and selectively identifies the important positions for targets to deactivate (activate) the enzymatic actions.

Figure 1

Plot of the first 40 residues of three sequences with four different properties for the $\beta$-lactamase family.

Figure 2

Fluctuations in hydrophobicity values at each position for the $\beta$-lactamase family.

Figure 3

Histogram of elements of $C_{i,j}$ for the $\beta$-lactamase family for (a) string, (b) hydrophobicity, (c) polarity, and (d) volume property.

Figure 4

Variation in median of all correlations for the first two properties with sequences for the $\beta$-lactamase and random system.

Figure 5

Eigenvalue distributions. Insets show EVs outside the lower bound for hydrophobicity. (a) EV distribution for $\beta$-lactamase family. (b) EV distribution for the shuffled system.

Figure 6

Nearest-neighbor eigenvalue spacings distribution for the unfolded eigenvalues of the correlation matrix based on the hydrophobicity for the $\beta$-lactamase family. The solid line is the density plot for the $\beta$-lactamase family and the dash line is the nearest-neighbor spacings distribution for correlation matrix-created GOE ensemble given by Eq. (5).

Figure 7

Nearest-neighbor eigenvalue spacings distribution for the unfolded eigenvalues of the correlation matrix based on the other properties for the $\beta$-lactamase family. The solid is the density plot for the $\beta$-lactamase family and the dash line is the nearest-neighbor spacings distribution for correlation matrix-created GOE ensemble given by Eq. (5).

Figure 8

Nearest-neighbor eigenvalue spacings distribution for the unfolded eigenvalues of the coevolution matrix calculated by treating amino acid as string for the $\beta$-lactamase family. The solid is the density plot for the $\beta$-lactamase family and the dash line is the nearest-neighbor spacings distribution for correlation matrix-created GOE ensemble given by Eq. (5).

Figure 9

Number variance ${\mathrm{\Sigma }}^2\left(l\right)$ for the $\beta$-lactamase family calculated from the unfolded eigenvalues of the hydrophobicity-based correlation matrix and string-based coevolution matrix. The observations are compared with the results of the GOE matrices and the uncorrelated series.

Figure 10

The eigenvalue components of eigenvalues of the correlation matrix for the hydrophobicity property and compared with the GOE and string-based approach. The components of the eigenvectors inside the bulk behaves as the RMT prediction, whereas the eigenvalue outside the RMT bound shows significant deviation from the RMT predictions both on the lower as well as upper side of the spectra. Solid line represents the Porter-Thomas distribution.

Figure 11

Comparison of the inverse participation ratio for the $\beta$-lactamase family for the hydrophobicity-based eigenvectors with string-based analysis and random system. RMT bounds are marked in the figure. Eigenvalues inside the RMT bounds are random and constitute the bulk part of eigenvalue spectrum. IPR for low eigenvectors shows significant contribution compared to large eigenvectors for the hydrophobicity property.

Figure 12

Eigenvector entropy of the string-based, property-based correlation (only first two properties), and shuffled system for the $\beta$-lactamase family.

Figure 13

The eigenvector entropy of properties studied. Each curve represents a different property.

Figure 14

Comparing the EVC square of the smallest and largest eigenvectors (a) hydrophobicity-based correlation matrix and (b) string-based MI for $\beta$-lactamase.

Figure 15

Plot of the first 40 residues of three sequences with four different properties.

Figure 16

Nearest-neighbor eigenvalue spacings distribution for the unfolded eigenvalues of the correlation matrix based on the hydrophobicity for the Serine protease family. The solid line is the density plot for the Serine protease family and the dash line is the nearest neighbor spacings distribution for correlation matrix-created GOE ensemble given by Eq. (5).

Figure 17

Entropy of eigenvectors of property-based correlation for the Serine protease family for three different properties (hydrophobicity, polarity, and polarizability).

Figure 18

Comparing the square of components of the smallest and largest eigenvectors of property-based correlation matrix for Serine protease family [for two different properties (hydrophobicity and polarity)].

Figure 19

Plot of the first 40 residues of three sequences with four different properties.

Figure 20

Nearest-neighbor eigenvalue spacings distribution for the unfolded eigenvalues of the correlation matrix based on the hydrophobicity for the HSP70 family. The solid line is the density plot for the HSP70 family and the dash line is the nearest-neighbor spacings distribution for correlation matrix created GOE ensemble given by Eq. (5).

Figure 21

Entropy of eigenvectors of property-based correlation for the HSP70 family for three different properties (hydrophobicity, polarity, and polarizability).

Figure 22

Comparing the square of components of the smallest and largest eigenvectors of property-based correlation matrix for HSP70 family [for two different properties (hydrophobicity and polarity)].

Figure 23

Plot of the first 40 residues of three sequences with four different properties.

Figure 24

Nearest-neighbor eigenvalue spacings distribution for the unfolded eigenvalues of the correlation matrix based on the hydrophobicity for the G-protein family. The solid line is the density plot for the G-protein family and the dash line is the nearest-neighbor spacings distribution for correlation matrix created GOE ensemble given by Eq. (5).

Figure 25

Entropy of eigenvectors of property-based correlation for the G-protein family for three different properties (hydrophobicity, polarity, and polarizability).

Figure 26

Comparing the square of components of the smallest and largest eigenvectors of property-based correlation matrix for G protein family [for two different properties (hydrophobicity and polarity)].

Figure 27

Plot of the first 40 residues of three sequences with four different properties.

Figure 28

Nearest-neighbor eigenvalue spacings distribution for the unfolded eigenvalues of the correlation matrix based on the hydrophobicity for the PF00126 family. The solid line is the density plot for the PF00126 and the dashed line is the nearest-neighbor spacings distribution for correlation matrix created GOE ensemble given by Eq. (5).

Figure 29

Entropy of eigenvectors of property-based correlation for the PF00126 family for three different properties (hydrophobicity, polarity, and polarizability).

Figure 30

Comparing the square of components of the smallest and largest eigenvectors of property-based correlation matrix for PF00126 family [for two different properties (hydrophobicity and polarity)].