How Math Helps Us Understand How Proteins Fold: Knot theory

Knot theory is a branch of mathematics that studies the properties of knots and links, which are essentially strings or loops that can be intertwined in different ways. Imagine tying a knot in a shoelace – that's a simple example of a mathematical knot. A knot is a closed loop of string that cannot be continuously deformed into an unknotted loop without cutting the string. A link is two or more knots that are linked together.

How does knot theory relate to proteins?

Knot theory may seem like a far cry from biology, but it turns out that knots and links can be used to describe the topology of proteins. The topology of a protein is its overall shape, independent of its size or position. By studying the topology of proteins, we can learn more about how they fold and function.

By analyzing the knots and links formed by proteins, scientists can gain insights into the folding process and the stability of the folded protein. This information can be crucial for developing new drugs and therapies that target specific proteins.

 Traditional folding algorithms represent proteins as simple chains of amino acids. However, this representation does not capture the complex topology of knotted proteins. These "knotted proteins" are relatively rare, but they play important roles in various biological processes. For example, a protein called thioredoxin plays a crucial role in protecting cells from damage, and it exists in a knotted form.

Knot theory can provide a more accurate way to represent protein topology, which can help folding algorithms to make better predictions. This is achieved by identifying new search strategies: Folding algorithms explore a vast space of possible protein structures to find the one with the lowest energy. This search can be very time-consuming, especially for knotted proteins. Knot theory can help folding algorithms to identify more efficient search strategies, which can lead to faster and more accurate predictions.

Proving polynomial time in knot theory and its impact on structural predictions 

One of the holy grails of knot theory is to prove that the decision problem of whether a given knot is equivalent to another knot can be solved in polynomial time. This means that there would be an algorithm that can solve the problem in a time that is proportional to the size of the input, rather than exponential time, which is currently the best-known algorithm.

If such proof were found, it would have a major impact on the field of structural biology. It would allow us to develop much faster and more accurate algorithms for predicting the 3D structure of proteins, including knotted proteins. This would have a wide range of applications, from drug discovery to protein engineering.

While a polynomial time solution for the knot equivalence problem remains elusive, there have been significant advances in recent years. In 2013, mathematicians discovered a new way to represent knots, called Khova-Jones theory, which has led to new algorithms for solving knot theory problems. These algorithms are still not polynomial time, but they are much faster than traditional algorithms.

The continued development of knot theory and its applications to protein folding is a promising area of research with the potential to revolutionize our understanding of proteins and their role in biology.

In conclusion, knot theory is a fascinating branch of mathematics that has found unexpected applications in the field of biology. By studying the knots and links formed by proteins, scientists are gaining new insights into the folding process and the stability of the folded protein. This information has the potential to lead to new drugs, therapies, and other advances in human health.