Dr. Allen Edmundson and I are looking for some assistance in transforming what seems to be obvious tangible topologic solutions of protein structures done with paper folding to a rigorous mathematical foundation. It appears to us that application of topologic techniques to understanding protein folding has not been ideal. Our beginning observations have been submitted for publication. The observation was that the 13 amino acid protein alpha-conotoxin whose crystal structure has verified to 2.0 angstroms can be modeled with a strip of paper. The rigorous Xray diffraction has taken 2pi years and solutions converged only after enormous calculational times. The concept that a crude simple topologic simulation could exclude a large number of alternative structural solutions or shapes and more rapidly approach meaningful conclusions seems tantalizing to say the least. The conotoxin example has the following features. Sequence analysis indicates that 2 pairs of cystine residues are present in the 13 amino acids that constitute the protein. Chemistry indicates these cystines will form disulfide bridges as well as how they are paired. The disulfide bridges in proteins appear to be able to rotate, flex, and stretch to the degree that they can represent distances 1 to 3 atoms apart.

In any case the protein may be simulated writing the numbers 1 to 13 on a strip of paper. Since it is known that cystine 2 must join or connect with cystine 7, and cystine 3 must join 13, the folding of the paper strip can be done. The folding shows that when position 2 overlies position 7, the back side of 3 remains exposed. Folding the other end of the strip with position 13 over to 3 demonstrates a graphic topologic solution for the protein. If the paper strip that is folded is twisted slightly to permit edges to be side by side, the disulfide bridges can be permitted to connect the slightly opened folds. A triangular structure is the outcome. Most importantly however it is clear that the folds occur at amino acid 5 and 9. The concept of a triangular structure with folds a 5 and 9 took a great deal of time to resolve by crystallographic Xray analysis. At the present time it appears that proteins with many cystine residues may have significant portions of their topologic properties expressed by the locations of these cystine residues. We have considered that the conotoxins may provide a clue in identifying some of the rules that may apply to biologic folding of complex proteins. In that regard I have considered that a linear protein (unfolded) may represent a complex 3 dimensional function that has integer properties with cysteine pairs representing number sets that locate the function as it folds and twists dimensionally. In that regard I have considered that number of cystines relative to the protein length may define the precision of the topologic solution to the reality of a crystallized protein. In essence, I have considered the possibility that a tied ribbon, like a bow, has folds that result from definition of the contact points located by integer representations of the ribbon. With this technique, one might describe very complicated continuous mathematical functions by the integer number sets contained.

Is it possible that Fermat reached his conjecture by the observation of a woman whose necklace was twisted and knotted as it rested on her bodice. As a mathematician did he perceive that the beads had integer properties and the material on which they were strung had properties of a continuous function? Did he consider that an integer transformation of continuous function could lead to an integer representation of the complexity of the function expressed in whole numbers? Well so much for my ramblings. If you know who might have the interest or insight to help Dr. Edmundson and I, we would appreciate it.

Please send comments to Carl V. Manion M.D. at: