The Protein Folding Problem

The protein folding problem is one of the most challenging problems in current biochemistry and has kept scientist busy for decades. It arises, when one wants to predict the three-dimensional structure of a protein under physiological conditions by looking solely at its underlying sequence of amino acids.

The aim of this article is, to present an introduction into the mathematical formulation of the protein folding problem. For this purpose, it is necessary to consider the chemical structure and abstract geometry of proteins and the properties of potential energy functions of molecules.

To get a rst understanding of the problem, along with possible solutions and the di culties that come with them, one has to take a close look at the polypeptide chains, which appear in living organisms, and at the di erent energy states of large, exible molecules.
All living cells contain giant amounts of proteins every protein performs a speci c biological task within the organism. The well-known molecule hemoglobin, for example, serves as an oxygen transporter within the cardiovaskular system of many animals and causes the characteristic red colour of blood. Other proteins support signal transmission within the brain or are used to build skin or muscle cells. Some proteins also function as part of the immune system to protect the body against infections and diseases. These examples already indicate, that proteins are extremely multifunctional biochemical tools, that are essential for a broad variety of biological functions. The function of a speci c protein, however, is strongly determined by the three-dimensional shape (tertiary structure) in which it naturally appears in the body of the organism. The geometrical shape of the protein is, for example, responsible for its ability to dock on certain receptors or to bond with other substances. Therefore biochemists, pharmacists, doctors, and other scientists are eager to develop theoretical tools, that can help to accurately predict what the shape of a certain protein is going to look like. The need for these tools invites mathematicians into the eld.

Let us now take a closer look on the basic building blocks of proteins a protein consists of a (often long) chain of amino acid residues. The amino acids are processed within the body by little, round cell organelles, the ribosomes. Within the ribosomes, amino acid residues are connected to long strings under seperation of water by so-called peptide bonds. These strings are sometimes also referred to as peptide chains and are exactly what one calls proteins.

Small proteins consist of 25-100 amino acid residues, whereas large peptide chains can have up to 3000 residues or more. The exact sequence of amino acids used for the synthesis is encoded in the DNA of the organism and is called the primary structure of the protein. Due to the fact that amino acid residues are asymmetric, a chain of residues does not form the same protein as the chain in reversed order.

In spite of the strictly linear amino acid sequence, atomic forces twist and bend the peptide chain in a very characteristic way, causing the protein to fold and take its natural shape within the organism. Thus the question arises, how the sequence of amino acids (the primary structure) gives rise to the folded shape of the protein (the natural tertiary structure). This, however, is precisely what is known as the protein folding problem.
The underlying hypothesis is, that the information content, that one can observe in the three-dimensional structure of a protein under physiological circumstances, can be found completely within the linear sequence of the involved amino acid residues. But what is the folding code?

A satisfying solution for the folding problem is not only interesting from a theoretical point of view, but also promises some highly valuable applications, for example in modern pharmacy and in the eld of drug design. Suppose that, for medical purposes, one wants to synthesize a protein, which is supposed to ful ll a certain biological task and therefore needs to have a speci c shape. If one could decide via computer simulation, which amino acid sequence corresponds with the needed shape, then one could take this sequence and implement it into the DNA of a strain of living bacteria to make it produce the exact wanted protein. This way, the development of new drugs could be accelerated and scientists would be able to systematically design the biologically active substances they need. Another medical motivation to understand the folding process of proteins is the fact, that the wrong folding of certain proteins in an organism can be the cause of serious illnesses like Alzheimer s or Creutzfeldt - Jakob disease. If the understanding of the underlying mechanisms of the folding of peptide chains gets deeper, then future scientists will have a better chance to precisely identify the factors that lead to the wrong folding and might even be able to develop better therapies.

The protein folding is more than just another fascinating theoretical secret of nature like most important problems, it combines high relevance for mankind with immense complexity, and its multidisciplinarity encourages scientists of di erent branches to pull together. Due to its signi cance for basic research, pharmacy and medicine, it will be further deeply investigated and will continue to be a great motivator to push the limits of what is scienti cally possible.