Amino Acid Sequential Matrix

One of the present authors arrived at the exact solution of the protein alignment problem [34] by modifying the amino acid adjacency matrix [24]. The amino acid adjacency matrix (AAA), as mentioned before, is a 20 x 20 matrix, in which each row and column belong to one of the 20 natural amino acids. The matrix elanent (AAA)ij counts how many times in the primary sequence of a protein is amino acid i followed by amino acid j. The 20 amino acids have been ordered as follows  

Amino acids of the ND6 proteins of human and mouse selected for comparison are shown in Table 13.9. 

The AAA matrix for theND6 human protein of Table 13.9 is shown in Matrix 13.3. The first row of the amino acid adjacency matrix for the ND6 human protein thus is 

It registers the presence of the pairs AR, AG, AI, and AW occurring once and the pairs AL and AM occurring twice in the primary sequence of the ND6 human protein. The zeros indicate that in the sequence of this protein there are no adjacent amino acids AA, AN, AD, AC, AQ, and so on. Similarly, the first row of the amino acid adjacency matrix for the ND6 protein of mouse, the second protein of Table 13.9, is 

Note Each group of 10 amino acids is separated for easier count. The first 45 amino acids used for construction of the superposition AAA matrix are shown in bold. 

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In the case of the AAA matrix, in order to recover the lost information on the location of individual amino acids in the sequence of proteins, one should record the sequential numbers of adjacent amino acids rather than their abundance. This is possible by modifying the amino acid adjacency matrix and using the sequential labels of amino acids as input so that the resulting matrix has full information on protein sequences. The so-modified AAA matrix allows, if required, reconstruction of the protein sequence. Observe that now the matrix elements are not numbers, but sets of numbers, the special case of which are numbers that can also be viewed... 

A mean-square helical hydrophobic moment,

, is defined for polypeptides in analogy to the mean-square dipole moment, , for polymer chains. For a freely jointed polymer chain, is given by X rr , where mi denotes the dipole moment associated with bond /. In the absence of any correlations in the hydrophobic moments of individual amino acid residues In the helix,

is specified by X Wj2, where H denotes the hydrophobicity of residue /, Matrix-generation schemes are formulated that permit rapid evaluation of

and . The behaviour of

I

is illustrated by calculations performed for model sequential copolypeptides. 

Example of a protected amine. The protected amino acid V -ferf-butoxy carbonyl (Boc)-l-lysine, when used as the amine to synthesize the triazine-based ligands, is deprotected prior to screening with protein. Protected immobilized ligands, after washing with the appropriate solvent in a sintered funnel, are immersed in 99% v/v trifluoroacetic acid (TFA) and stirred for 1 h in the fume hood. Methanol is added to dilute the TFA, and the waste solution is carefully disposed of. The matrixes are washed sequentially with methanol and distilled water (3 10 gel... 

Feller [50] has presented an approach which is very similar in concept to the Markov chain approach described above. He works in terms of a matrix M, of sequential probabilities which are different from, but related to, the Markov chain transition probabilities. He shows that for the system A,-B, copolymerizing with A2-B2 (representative of two different hydroxy or amino acids) the CLD may be calculated from ... 

By comparing only the two first rows of the ND6 proteins, one can already see that besides being similar they also show dissimilarity. Although the AAA matrix is accompanied with a loss of information on the sequential locations of pairs of adjacent amino acids, it nevertheless offers a useful characterization of proteins. The sequential labels, sites, or locations here indicate the exact position of an amino acid in the sequence of the protein. For example, the sequential labels for alanine are 4, 72, 74, 81, 97,142,144, and 171. 

Amino acid adjacency matrices, which in addition to the information on abundance of individual amino acids also give the count of successive pairs of amino acids, are accompanied by some loss of information, in the view that information on the location of various pairs is not known. A remedy for this loss of information is to record the locations of pairs of adjacent amino acids rather than just their occurrence. In this way, the AAA matrix of Matrix 13.13 has a new form shown in Matrix 13.4, in which we show only the portion of the matrix involving elements G-V. The modified AAA matrix shown in Matrix 13.5 is associated with the sequence of the first 45 amino acids of the human ND6 protein. Observe the elements (G, L) and (G, F) that have in the corresponding AAA matrix the abundance count 2, are now replaced by two sequential numbers. Although many matrix elements of the modified AAA matrix may remain to be single numbers, in general, the elements of the modified AAA matrix are not numbers, but sets, that is, collections of numbers. In Matrix 13.5, we show the same portion of the AAA matrix for the second protein of Table 13.10. 

We have referred to Matrices 13.4 and 13.5 as a sequential AAA matrix because their entries are the sequential labels of amino acids in proteins. These matrices have all the information that is contained in the corresponding AAA matrices, but in addition, they also include information on the locations of amino acids in the corresponding sequences. The elements of such matrices, as already mentioned, are sets, not numbers (sets, of course, can have a single element, including zero, which are numbers). Whether the sequential AAA matrices here outlined are the first matrices in mathematical literature whose elements are sets, we don t know. That is less important, in our view, than the fact that they are the r t matrices that offered the exact solution to the over 45-year-old open problan on pairwise protein alignment. Observe that they have not only information on individual amino acids but on all pairs of adjacent amino acids. 

Here we will briefly comment only on the Basic program, the output of which is the sequential numbers for amino acids having the same shift. For more information on the basic program and other versions, one should consult [37]. Let us return to Table 13.11, the exact solution for the aligmnent human-mouse proteins of Matrix 13.3. We would like to emphasize three things ... 

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