Amyloid-beta Folding Algebra
Amyloids are insoluble fibrous protein aggregates sharing specific structural traits. They arise from at least 18 inappropriately folded versions of proteins and polypeptides present naturally in the body. These misfolded structures alter their proper configuration such that they erroneously interact with one another or other cell components forming insoluble fibrils. They have been associated with the pathology of more than 20 serious human diseases in that, abnormal accumulation of amyloid fibrils in organs may lead to amyloidosis, and may play a role in various neurodegenerative disorders.
The Amyloid-Beta protein is involved in the Alzheimer's disease process. A misfolding of this protein causes the disease. The transition from the native state to the misfolded state involves unkown numbers of intermittent intermediate states. These kinetic states have an algebraic relationship. Studying these intermediate states may lead to new therapies.
The RCSB PDB contains files of protein images. On occasion they have images of the same protein attached to different small molecules. Similar images of misfolded proteins involved in diseases would allow insight into these properties.
Twenty Elementary Algebras
The above image is a Moncznik (Perry Moncznik) multiplication table (in the Abstract Algebra sense) where the elements being "multiplied" are the different possible conformational states of a single amino acid. The twenty amino acids can each be represented by such a table and thus form twenty elemental algebras, which it could be argued, may in some sense form the basis from which a large portion of mathematics arises.