Sequence analysis is often the first guide for the prediction of residues in a protein family that may have functional significance. A few methods have been proposed which use the division of protein families into subfamilies in the search for those positions that could have some functional significance for the whole family, but at the same time which exhibit the specificity of each subfamily ("Tree-determinant residues"). However, there are still many unsolved questions like the best division of a protein family into subfamilies, or the accurate detection of sequence variation patterns characteristic of different subfamilies. Here we present a systematic study in a significant number of protein families, testing the statistical meaning of the Tree-determinant residues predicted by three different methods that represent the range of available approaches. The first method takes as a starting point a phylogenetic representation of a protein family and, following the principle of Relative Entropy from Information Theory, automatically searches for the optimal division of the family into subfamilies. The second method looks for positions whose mutational behavior is reminiscent of the mutational behavior of the full-length proteins, by directly comparing the corresponding distance matrices. The third method is an automation of the analysis of distribution of sequences and amino acid positions in the corresponding multidimensional spaces using a vector-based principal component analysis. These three methods have been tested on two non-redundant lists of protein families: one composed by proteins that bind a variety of ligand groups, and the other composed by proteins with annotated functionally relevant sites. In most cases, the residues predicted by the three methods show a clear tendency to be close to bound ligands of biological relevance and to those amino acids described as participants in key aspects of protein function. These three automatic methods provide a wide range of possibilities for biologists to analyze their families of interest, in a similar way to the one presented here for the family of proteins related with ras-p21.
We study in detail the bound state spectrum of the generalized Morse potential (GMP), which was proposed by Deng and Fan as a potential function for diatomic molecules. By connecting the corresponding Schrödinger equation with the Laplace equation on the hyperboloid and the Schrödinger equation for the Pöschl-Teller potential, we explain the exact solvability of the problem by an so(2, 2) symmetry algebra, and obtain an explicit realization of the latter as su(1, 1) ⊕ su(1, 1). We prove that some of the so(2, 2) generators connect among themselves wave functions belonging to different GMP's (called satellite potentials). The conserved quantity is some combination of the potential parameters instead of the level energy, as for potential algebras. Hence, so(2, 2) belongs to a new class of symmetry algebras. We also stress the usefulness of our algebraic results for simplifying the calculation of Frank-Condon factors for electromagnetic transitions between rovibrational levels based on different electronic states.
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