Peter Røgen

Automatic classification of protein structure by using Gauss integrals

We introduce a method of looking at, analyzing, and comparing protein structures. The topology of a protein is captured by 30 numbers inspired by Vassiliev knot invariants. To illustrate the simplicity and power of this topological approach, we construct a measure (scaled Gauss metric, SGM) of similarity of protein shapes. Under this metric, protein chains naturally separate into fold clusters. We use SGM to construct an automatic classification procedure for the CATH2.4 database. The method is very fast because it requires neither alignment of the chains nor any chain–chain comparison. It also has only one adjustable parameter. We assign 95.51% of the chains into the proper C (class), A (architecture), T (topology), and H (homologous superfamily) fold, find all new folds, and detect no false geometric positives. Using the SGM, we display a “map” of the space of folds projected onto two dimensions, show the relative locations of the major structural classes, and “zoom into” the space of proteins to show architecture, topology, and fold clusters. The existence of a simple measure of a protein fold computed from the chain path will have a major impact on automatic fold classification.

A new family of global protein shape descriptors

A family of global geometric measures is constructed for protein structure classification. These measures originate from integral formulas of Vassiliev knot invariants and give rise to a unique classification scheme. Our measures can better discriminate between many known protein structures than the simple measures of the secondary structure content of these protein structures.

Sides of the Möbius strip

We present the necessary and suucient conditions for a curve to be the center curve of an analytic and at embedding of the MM obius strip (or an orientable cylinder) into euclidean 3-space. Using these conditions we extend an example by G. Schwarz into a continuous family of analytic and at MM obius strips. This family is split into two connected components. We give a topological argument that explains this behaviour.

Fast large-scale clustering of protein structures using Gauss integrals

MOTIVATION Clustering protein structures is an important task in structural bioinformatics. De novo structure prediction, for example, often involves a clustering step for finding the best prediction. Other applications include assigning proteins to fold families and analyzing molecular dynamics trajectories. RESULTS We present Pleiades, a novel approach to clustering protein structures with a rigorous mathematical underpinning. The method approximates clustering based on the root mean square deviation by first mapping structures to Gauss integral vectors--which were introduced by Røgen and co-workers--and subsequently performing K-means clustering. CONCLUSIONS Compared to current methods, Pleiades dramatically improves on the time needed to perform clustering, and can cluster a significantly larger number of structures, while providing state-of-the-art results. The number of low energy structures generated in a typical folding study, which is in the order of 50,000 structures, can be clustered within seconds to minutes.

Evaluating protein structure descriptors and tuning Gauss integral based descriptors

The general development in the natural sciences from relative comparison to absolute description forces the current relative protein structure comparison, based on similarity measures, to be supplemented, or even replaced, by absolute description of each individual protein structure. This paper addresses the question of what should be required from a good set of protein structure descriptors. As an example a Gauss integral based family of protein structure descriptors, that has been shown to successfully classify the geometry of CATH2.4 connected protein domains, is examined. The CATH2.4 domains are here observed to break a symmetry under reversal of the direction of traversal of the protein backbone that general folded tubes possess. It is thus a challenge for any large scale protein or polymer model to explain this broken symmetry.

Gauss-integral based representation of protein structure for predicting the fold class from the sequence

A representative subset of protein chains were selected from the CATH 2.4 database [C.A. Orengo, A.D. Michie, S. Jones, D.T. Jones, M.B. Swindells, J.M. Thornton, CATH-a hierarchic classification of protein domain structures, Structure 5 (8) (1997) 1093-1108], and were used for training a feed-forward neural network in order to predict protein fold classes by using as input the dipeptide frequency matrix and as output a novel representation of the protein chains in R^3^0 space, based on knot invariant values [P. Rogen, B. Fain, Automatic classification of protein structure by using Gauss integrals, Proceedings of the National Academy of Sciences of the United States of America 100 (1) (2003) 119-124; P. Rogen, H.G. Bohr, A new family of global protein shape descriptors, Mathematical Biosciences 182 (2) (2003) 167-181]. In the general case when excluding singletons (proteins representing a topology or a sequence homology as unique members of these sets), the success rates for the predictions were 77% for class level, 60% for architecture, and 48% for topology. The total number of fold classes that are included in the present data set (~500) is ten times that which has been reported in earlier attempts, so this result represents an improvement on previous work (reporting on a few handpicked folds). Furthermore, distance analysis of the network outputs resulting from singletons shows that it is possible to detect novel topologies with very high confidence (~85%), and the network can in these cases be used as a sorting mechanism that identifies sequences which might need special attention. Also, a direct measure of prediction confidence may be obtained from such distance analysis.

Gauss–Bonnet's Theorem and Closed Frenet Frames

Abstract. Our main result is that integrated geodesic curvature of a (nonsimple) closed curve on the unit two-sphere equals a half integer weighted sum of the areas of the connected components of the complement of the curve. These weights that gives a spherical analogy to the winding number of closed plane curves are found using Gauss–Bonnet’s theorem after cutting the curve into simple closed sub-curves. If the spherical curve is the tangent indicatrix of a space curve we obtain a new short proof of a formula for integrated torsion presented in an unpublished manuscript by C. Chicone and N. J. Kalton. Applying our result to the principal normal indicatrix we generalize a theorem by Jacobi stating that a simple closed principal normal indicatrix of a closed space curve with nonvanishing curvature bisects the unit two-sphere to nonsimple principal normal indicatrices. Some errors in the literature are corrected. We show that a factorization of a knot diagram into simple closed sub-curves defines an immersed disc with the knot as boundary and use this to give an upper bound on the unknotting number of knots.

On Density of the Vassiliev Invariants

The main result is that the Vassiliev invariants are dense in the set of numeric knot invariants if and only if they separate knots.

De novo generation of molecular structures using optimization to select graphs on a given lattice.

A recurrent problem in organic chemistry is the generation of new molecular structures that conform to some predetermined set of structural constraints that are imposed in an endeavor to build certain required properties into the newly generated structure. An example of this is the pharmacophore model, used in medicinal chemistry to guide de novo design or selection of suitable structures from compound databases. We propose here a method that efficiently links up a selected number of required atom positions while at the same time directing the emergent molecular skeleton to avoid forbidden positions. The linkage process takes place on a lattice whose unit step length and overall geometry is designed to match typical architectures of organic molecules. We use an optimization method to select from the many different graphs possible. The approach is demonstrated in an example where crystal structures of the same (in this case rigid) ligand complexed with different proteins are available.

Computing a New Family of Shape Descriptors for Protein Structures

The large-scale 3D structure of a protein can be represented by the polygonal curve through the carbon alpha atoms of the protein backbone. We introduce an algorithm for computing the average number of times that a given configuration of crossings on such polygonal curves is seen, the average being taken over all directions in space. Hereby, we introduce a new family of global geometric measures of protein structures, which we compare with the so-called generalized Gauss integrals.