Residues chain, conformational partition

The necessity for a proper averaging over a multitude of conformations remains when the protein is partially ordered. The power of the matrix methods can still be utilized if the dominant ordering interactions are of relatively short range, as they are in helix formation (6). The conformational partition function, Z, for a chain of n residues is formulated as shown in Equation 1. 

Two of the hydrophihcity scales in Table 2 were derived from experimental measures of the behavior of amino acids in various solvents, namely partitioning coefficients [K-D index of Kyte and Doolittle (30)] or mobility in paper chromatography [Rf index of Zimmerman et al. (31)]. By contrast, the Hp index was obtained from quantum mechanics (QM) calculations of electron densities of side chain atoms in comparison with water (32). The Hp index is correlated highly with these two established hydrophobicity scales (Table 4). Therefore, like the polarizability index, it is possible to represent fundamental chemical properties of amino acids (hydrophUicity, Hp) with parameters derived from ab initio calculations of electronic properties. However, in contrast to polarizabihty (steric effects), hydrophihcity shows significant correlation with preference for secondary structure. Thus, hydrophobic amino acids prefer fi-strands (and fi-sheet conformations) and typically are buried in protein structures, whereas hydrophilic residues are found commonly in turns (coil structure) at the protein surface. 

In the noncooperative model (Figure 9.1), the biomacromolecule converts stepwise from a to b, with an increase in the fraction of b with each transition. Each step j represents a different state j with wy being the statistical weight in which j also represents the total number of residues in b conformation with the probability, s = [h]/[a]. There are N unique combinations, i.e. degeneracy with a single residue b for a chain of N residues, therefore wi = Ns and Wy = gys. The general form of the partition function for N number of residues is the polynomial expression ... 

The statistical weight matrix constructed from the relative statistical weights permits the calculation of the partition function of a given sequence by matrix multiplication. The partition functions contribute to the evaluation of the conformational sequence probabilities P (i/n/ q> /), i.e. the probability to find a sequence of n residues in a specific conformational state q> starting at the i-th position of the chain. The results of these calculations can graphically be presented in the form of conformational probability profiles. 

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