Israel M. Gelfand

This work evaluates the hypothesis that proteins with an identical supersecondary structure (SSS) share a unique set of residues-SSS-determining residues-even though they may belong to different protein families and have very low sequence similarities. This hypothesis was tested on two groups of sandwich-like proteins (SPs). Proteins in each group have an identical SSS, but their sequence similarity is below the "twilight zone.'' To find the SSS-determining residues specific to each group, a unique structure-based algorithm of multiple sequences alignment was developed. The units of alignment are individual strands and loops rather than whole sequences. The algorithm is based on the alignment of residues that form hydrogen bonds between corresponding strands. Structure-based alignment revealed that 30-35% of the positions in the sequences in each group of proteins are "conserved positions'' occupied either by hydrophobic- only or hydrophilic-only residues. Moreover, each group of SPs is characterized by a unique set of SSS-determining residues found at the conserved positions. The set of SSS-determining residues has very high sensitivity and specificity for identifying proteins with a corresponding SSS: It is an "amino acid tag'' that brands a sequence as having a particular SSS. Thus, the sets of SSS-determining residues can be used to classify proteins and to predict the SSS of a query amino acid sequence.

From a computer analysis of the spatial organization of the secondary structures of beta-sandwich proteins, we find certain sets of consecutive strands that are connected by hydrogen bonds, which we call "strandons." The analysis of the arrangements of strandons in 491 protein structures that come from 69 different superfamilies reveals strict regularities in the arrangements of strandons and the formation of what we call "canonical supermotifs." Six such supermotifs account for approximately 90% of all observed structures. Simple geometric rules are described that dictate the formation of these supermotifs.

For a large class of proteins called sandwich-like proteins (SPs), the secondary structures consist of two β-sheets packed face-to-face, with each β-sheet consisting typically of three to five β-strands. An important step in the prediction of the three-dimensional structure of a SP is the prediction of its supersecondary structure, namely the prediction of the arrangement of the β-strands in the two β-sheets. Recently, significant progress in this direction was made, where it was shown that 91% of observed SPs form what we here call “canonical motifs.” Here, we show that all canonical motifs can be constructed in a simple manner that is based on thermodynamic considerations and uses certain geometric structures. The number of these structures is much smaller than the number of possible strand arrangements. For instance, whereas for SPs consisting of six strands there exist a priori 900 possible strand arrangements, there exist only five geometric structures. Furthermore, the few motifs that are noncanonial can be constructed from canonical motifs by a simple procedure.

We investigate the supersecondary structure of a large group of proteins, the so-called sandwich proteins. The analysis of a large number of such proteins has led us to propose a set of rules that can be used to predict the possible arrangements of strands in the two beta-sheets forming a given sandwich structure. These rules imply the existence of certain invariant supersecondary substructures common to all sandwich proteins. Furthermore, they dramatically restrict the number of permissible arrangements. For example, whereas for proteins consisting of three strands in each beta-sheet 180 possible strand arrangements exist a priori, our rules imply that only 15 of them are permissible. Five of these predicted arrangements describe all currently known sandwich proteins with six strands.

In this paper we present a definition of oriented Lagrangian symplectic matroids and their representations. Classical concepts of orientation and this extension may both be thought of as stratifications of thin Schubert cells into unions of connected components. The definitions are made first in terms of a combinatorial axiomatisation, and then again in terms of elementary geometric properties of the Coxeter matroid polytope. We also generalise the concept of rank and signature of a quadratic form to symplectic Lagrangian matroids in a surprisingly natural way.