The size and shape of macromolecules such as proteins and nucleic acids play an important role in their functions. Prior efforts to quantify these properties have been based on various discretization or tessellation procedures involving analytical or numerical computations. In this article, we present an analytically exact method for computing the metric properties of macromolecules based on the alpha shape theory. This method uses the duality between alpha complex and the weighted Voronoi decomposition of a molecule. We describe the intuitive ideas and concepts behind the alpha shape theory and the algorithm for computing areas and volumes of macromolecules. We apply our method to compute areas and volumes of a number of protein systems. We also discuss several difficulties commonly encountered in molecular shape computations and outline methods to overcome these problems.

Analytical shape computation of macromolecules: I. molecular area and volume through alpha shape

In this paper we introduce a novel method of deriving a pairwise potential for protein folding. The potential is obtained by optimization procedure, which simultaneously maximizes the energy gap for {\it all} proteins in the database. To test our method and compare it with other knowledge-based approaches to derive potentials, we use simple lattice model. In the framework of the lattice model we build a database of model proteins by a) picking randomly 200 lattice chain conformations; b) designing sequences which fold into these structures with some arbitrary ``true'' potential; c) use this database for extracting a potential; d) fold model proteins using the extracted potential. This test on the model system showed that our procedure is able to recover the potential with correlation $r \approx 91\%$ with the ``true'' one and we were able to fold all model structures using the recovered potential. Other statistical knowledge-based approaches were tested using lattice models and the results indicate that they also can recover the ``true'' potential with high degree of accuracy. When applied to real protein structures with energy function taken in contact pairwise approximation, our potential scored somewhat better than existing ones. However, the discrimination of the native structure from decoys is still not strong enough to make the potential useful for {\em ab initio} folding. We argue that more detail of protein structure and energetics should be taken into account to achieve better energy gaps. The suggested method is general to allow to systematically derive parameters for more sophisticated energy function. The internal control of validity of the potential derived by our method, is converegency to a unique solution upon addition of new proteins to the database.

How to derive a protein folding potential? A new approach to the old problem

Voronoi and Voronoi-related tessellations in studies of protein structure and interaction.

Escherichia coli cytidine deaminase provides a molecular model for ApoB RNA editing and a mechanism for RNA substrate recognition

PROSPECTOR (PROtein Structure Predictor Employing Combined Threading to Optimize Results) is a new threading approach that uses sequence profiles to generate an initial probe-template alignment and then uses this "partly thawed" alignment in the evaluation of pair interactions. Two types of sequence profiles are used: the close set, composed of sequences in which sequence identity lies between 35% and 90%; and the distant set, composed of sequences with a FASTA E-score less than 10. Thus, a total of four scoring functions are used in a hierarchical method: the close (distant) sequence profiles screen a structural database to provide an initial alignment of the probe sequence in each of the templates. The same database is then screened with a scoring function composed of sequence plus secondary structure plus pair interaction profiles. This combined hierarchical threading method is called PROSPECTOR1. For the original Fischer database, 59 of 68 pairs are correctly identified in the top position. Next, the set of the top 20 scoring sequences (four scoring functions times the top five structures) is used to construct a protein-specific pair potential based on consensus side-chain contacts occurring in 25% of the structures. In subsequent threading iterations, this protein-specific pair potential, when combined in a composite manner, is found to be more sensitive in identifying the correct pairs than when the original statistical potential is used, and it increases the number of recognized structures for the combined scoring functions, termed PROSPECTOR2, to a total of 61 Fischer pairs identified in the top position. Application to a second, smaller Fischer database of 27 probe-template pairs places 18 (17) structures in the top position for PROSPECTOR1 (PROSPECTOR2). Overall, these studies show that the use of pair interactions as assessed by the improved Z-score enhances the specificity of probe-template matches. Thus, when the hierarchy of scoring functions is combined, the ability to identify correct probe-template pairs is significantly enhanced. Finally, a web server has been established for use by the academic community (http://bioinformatics.danforthcenter.org/services/threading.html).

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Defrosting the frozen approximation: PROSPECTOR--a new approach to threading.

We propose new algorithms for sequence-structure compatibility (fold recognition) searches in multi-dimensional sequence-structure space. Individual amino acid residues in protein structures are represented by their C alpha atoms; thus each protein is described as a collection of points in three-dimensional space. Delaunay tessellation of a protein generates an aggregate of space-filling, irregular tetrahedra, or Delaunay simplices. Statistical analysis of quadruplet residue compositions of all Delaunay simplices in a representative dataset of protein structures leads to a novel four body contact residue potential expressed as log likelihood factor q. The q factors are calculated for native 20 letter amino acid alphabet and several reduced alphabets. Two sequence-structure compatibility functions are computed as (i) the sum of q factors for all Delaunay simplices in a given protein, or (ii) 3D-1D Delaunay tessellation profiles where the individual residue profile value is calculated as the sum of q factors for all simplices that share this vertex residue. Both threading functions have been implemented in structure-recognizes-sequence and sequence-recognizes-structure protocols for protein fold recognition. We find that both profile and total score based threading functions can distinguish both the native fold from incorrect folds for a sequence, and the native sequence from non-native sequences for a fold.

A new approach to protein fold recognition based on Delaunay tessellation of protein structure.

The topology of folded proteins from the representative dataset of well-defined three-dimensional protein structures is studied using a statistical geometry approach. Amino acid residues in protein chains are represented by C alpha atoms, thus reducing the protein three-dimensional structure to a set of points in three dimensional space. The Delaunay tessellation of a protein structure generates an aggregate of space-filling irregular tetrahedra, or Delaunay simplices. Each simplex objectively defines four nearest neighbor C alpha atoms, i.e. four nearest neighbor residues. The statistical analysis of residue composition of Delaunay simplices reveals nonrandom preferences for certain quadruplets of amino acids. These nonrandom preferences are used to develop a fitness function that evaluates sequence-structure compatibility. Using this fitness function, several tested native proteins score the highest among 100,000 random sequences with average protein amino acid composition. The statistical geometry approach, based solely on first principles, provides a unique means for protein structure analysis and has direct implications for inverted protein structure prediction.

Statistical geometry analysis of proteins: implications for inverted structure prediction.

A strategy for rational design of targeted combinatorial libraries is described. The aim of this approach is to select a subset of available building blocks for the library synthesis that are most likely to be present in the active compounds. Building blocks that are used in the underlying combinatorial chemical reaction are randomly assembled to produce virtual combinatorial library compounds, which are represented by various chemical descriptors. Stochastic algorithms (simulated annealing, genetic algorithms, neural net methods) are used to search the potentially large structural space of virtual chemical libraries in order to identify compounds similar to lead compound(-s). The selection of a virtual molecule as a candidate for the targeted library is based either on its chemical similarity to a biologically active probe or on its biological activity predicted from a pre-constructed QSAR equation. Frequency analysis of building block composition of the selected virtual compounds identifies building blocks that can be used in combinatorial synthesis of chemical libraries with high similarity to the lead compound(-s). This method is applied to rational design of the library with bradykinin potentiating activity. Twenty eight bradykinin potentiating pentapeptides were used as a training set for the development of a QSAR equation, and, alternatively, two active pentapeptides, VEWAK and VKWAP, were used as probe molecules. In each case, the frequency distribution of amino acids in the top 100 peptides suggested by the method resembles the frequency distribution of amino acids found in the active peptides. The results obtained after GA optimization also compared favorably with those obtained by the exhaustive analysis of all possible 3.2 millions pentapeptides.

Zheng W

Papers

A new approach to protein fold recognition based on Delaunay tessellation of protein structure.

Statistical geometry analysis of proteins: implications for inverted structure prediction.

Focus-2D: a new approach to the design of targeted combinatorial chemical libraries.