The CRASP package uses multiple alignments of protein sequences as input data and performs analysis of three kinds. First, it allows estimating correlations between residue substitutions at pairs of protein positions. Second, it allows estimating the Pearson's correlation coefficient and the linear functional regression equation for the physicochemical value at a position pair. Third, it analyzes integral physicochemical protein characteristics that involve groups of protein positions. The package enables to use the information on pairwise positional correlations for deducing conserved (or extremely variable) integral protein characteristics. Below, we describe the methods and algorithms implemented by the CRASP package. The calculation parameters are given in the Help Information files.

(We also refer to the statistics textbooks: Anderson, 1958; Kendall and Stuart, 1967).

The first step of analysis performed by the CRASP package is the estimation of the pairwise correlation of residue substitutions at protein positions. The approach is based on estimation of the linear correlation coefficients between the amino acid physicochemical property values. 

In a sample of N aligned sequences of the length L, a value of the user-selected property f is attributed to every amino acid in the alignment. As a result, a matrix, whose elements f_ki are the values of a property at the i-th position of the k-th sequence, is calculated. 

The mean value of a physicochemical property at the position i is estimated as:

where w_k is the weight of a sequence (see Data weighting),.

where s_ij are the elements of the covariation matrix S. The positive (negative) correlation coefficients for the position pair of i,j suggest that if the f value increases due to mutation at the i position, a substitution causing an increase (or a decrease) in the f value is likely to occur at the other position j.

where a_ij are the elements i,j of the matrix inverse to the covariation matrix. For detailed information on the partial correlation coefficients estimation see Anderson, 1958; Kendall and Stuart, 1967.

The results of CRASP's estimation of the pairwise positional correlations are represented as an LxL matrix of the corresponding values (correlation or partial correlation). The user can change the Type of the matrix parameter to specify the type of the calculated matrix.

The t value follows the Student's t-distribution with m=N-2 degrees of freedom (Kendall and Stuart, 1967), where N is the number of sequences in the alignment. For the partial correlation coefficient (4), the degree of freedom in equation (5) is equal to m = N-L. This equation allows estimating the critical value of the correlation coefficient r_c by applying the t_P value of Student's statistics at the given significance level P. In the CRASP package, the values r_c for the representation of highly correlated pairs are chosen by the Significance level parameter.

With the CRASP package approach, information on correlation coefficients for the amino acids physicochemical properties is used for detecting the conserved (or extremely variable) integral characteristics of a protein.

Integral characteristics of proteins

  Homologous proteins possess various conserved features, physicochemical characteristics that remained constant in an evolving protein family. For example, such characteristics might have be protein hydrophobic core volume, net charge of a spatially close group of residues in an active site, hydrophobic moment of an alpha-helix, etc. These characteristics might have been of functional or structural importance for the family of homologous proteins. Therefore, information on such characteristics would shed light on the specificities of functional interactions between amino acid residues of the queried proteins. For example, it would facilitate predictions of the functional motifs in protein (Eisenhaber et al, 1998). 

Two possible mechanisms might have provided the constancy of these characteristics. One is the conservation of residues at certain protein positions. This mechanism is frequently used to infer functionally important protein residues from their conservation estimates. 

The second is of particular interest and may be an "extension" of the first, acting when residues at certain positions vary, but the protein characteristic related to the positions varies slightly, if at all. This mechanism implies correlated amino acid substitutions: the constancy of characteristic here is provided by a coordinated residue substitutions (the so-called "compensatory mutations"). The characteristic constancy here means that its variation value in a sample of homologous sequences is significantly less than expected for the independent amino acid substitutions. The feasibility of the second mechanism has been examined for the constancy of the protein hydrophobic core volume (Lim and Ptitsyn, 1970; Ptitsyn and Volkenstein, 1986; Gerstein et al., 1994).

In the CRASP package the case of the integral characteristic F is considered, the linear. It is defined as a linear combination of a physicochemical property value at protein positions, for example, for the alignment sequence with the index k:

,

where c_i are arbitrary numbers. This definition represents a broad class of protein physicochemical characteristics, such as protein hydrophobic core volume (total volume values of the residues that form the hydrophobic core), net charge of the protein residues in an active site; projection of the alpha- helical hydrophobic moment, etc. In the CRASP package the Characteristic Description parameter describes the integral characteristics the user provides. The parameter allows to specify the values c_i for protein positions.

As a measure of the conservation of the value F for a set of protein family sequences, the variation D(F) is used (Lim and Ptitsyn, 1970; Gerstein et al., 1994):

,

where the mean value is defined as:

.

The CRASP enables to use the information on the pairwise positional correlations to detect conserved (or extremely variable) integral characteristics of proteins. Currently, this step of analysis is not automated and the integral characteristic is the user's choice. The underlying idea is described below.

Let us consider two protein positions i and j. Let D_i and D_j, be the variations in a physicochemical property f at these positions, respectively, and r_ij be the correlation coefficient between the values f_i and f_j. The variation in the sum of these values can be calculated as:

.

Hence, if the correlation coefficient is negative, the variation in the sum of the values f_i and f_j, decreases. Similarly, if the correlation coefficient is positive, the variation in the difference between the values f_i and f_j decreases. As for the group of positions that mainly yield negative pairwise correlation coefficients, the total value f over these positions is expected to be less than in the case of random amino acid substitutions at these positions.

After the inference of the physicochemical integral characteristic, its constancy relative to the independent substitution model can be compared. The CRASP package enables the user to compare the D(F) values for a queried protein set with those for the independent substitution model. This makes it possible to estimate directly the impact of co-ordinated residue substitutions on the constancy maintenance of the integral physicochemical protein characteristics.

The variance of an integral characteristic F is defined as the sum of two components

,

where D(f_i) is the variation in  the property f at position i, r_ij  is the correlation coefficient for the property for the position pair i,j. It is easy to see that D_var is dependent on the variability in  protein positions and D_cov is dependent on the coordinated substitutions. Note, D_var is always 0 or greater, whereas D_cov can be positive, negative or zero. The latter can be used as the null hypothesis for testing the significance of the contribution of coordinated substitutions to D(F). In this case (for all r_ij=0), the expected variance of the physicochemical characteristic is

.

Coordinated substitutions contribute to the stability of the integral characteristic F, if D(F)<D_exp(F). To test this inequality for significance, the variance ratio l= D_exp(F)/D(F) is used, thus, l serves as a characteristic of the contribution of coordinated substitutions to the variation in the integral characteristic F. At l > 1, the contribution narrows the variation range (increases conservation); at l < 1 it widens the variation range (increases variability). At l»1, the contribution of coordinated substitutions is insignificant. It is known that l obeys the Fisher's F distribution with the L(N-1) and N-1 degrees of freedom, where N is the number of sequences (Selvin,1998). Therefore, the percentile points of the F distribution can be used for the estimation of the significance of the deviation of l from 1. 

In case of several integral characteristics F_i , the software CRASP allows estimating the parameters of their mutual distributions. The correlation coefficient r(F_i,F_j) is estimated from the expressions (1-3) using  F_ki instead of f_ki (where i is the index of the integral characteristic). The regression parameters for the linear relationships between F_i and F_j,

are calculated, too. 

In our case, estimation of the linear regression parameters differs from the commonly used regression y=a* x+b. The latter defines the relation between the independent variable õ (is not subject to random errors) and the dependent variable ó. In this model, the regression line minimizes the sum of the standard deviations of the y values from the observed values.

f(F_i ,F_j )=const,

where f(F_i,F_j)  is a linear function. Therefore, the regression parameters in (6) minimize the sum of the square distances d from the observed points to the regression line, that is, the lengths of perpendiculars dropped from the points on the line (Kendall and Stuart, 1967), see difference in the figure below for common (A) and functional regression (B). For the parameter a of a linear regression (6), we also estimate its 95% confidence limits (Kendall and Stuart, 1967).

,

where w_i is the weight of the i-th sequence and D_ik is the distance between the i- th and the k-th sequences. The distance measure is simply the number of different amino acids in two sequences. All the weights are normalized to the N - number of sequences, i.e..

Henikoff and Henikoff (1994) have suggested weights based on the diversity of each position in alignment.

Networks ( clusters) of correlated positions

The relationship between the protein positions relative to their pairwise correlations is also described by a binary tree. We clustered all the positions basing on the following measure of closeness for a pair of positions i,j:

i.e., the stronger the correlation between two positions in a protein, the smaller this distance. The nearest neighbor method was used for clustering. The results were described by a binary tree. Each node of the tree unites two sets of positions (the vertical branches from the node) and the position of a node on the dendrogram corresponds to the maximum correlation coefficient between all the possible pairs. Thus, both the pairs of correlated positions and the position groups where substitutions occur dependently can be identified. This diagram, in principle, allows identifying sets of the positions within a protein, each significantly correlated to one at least position from the set. Therefore, these sets of positions are conceived as networks of correlated positions. We suggest that the linear combinations of the amino acid physicochemical properties values in networks of highly correlated positions can be good candidates for the conserved characteristics of proteins.

where q is the proportion of significant correlation pairs for the entire matrix, n is the number of matrix elements in the window, m is the number of significant matrix element in a window. The prevalence of significant correlation of different sign (positive only, negative only or both) can be estimated. Thus, this region was considered as significant if p < 1% (at a P=99%). The regions of higher prevalence of significant correlation are shown in a separate diagram as matrix elements with magenta coloured borders.

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