By Jonathon Liu, The University of Melbourne

In highschool chemistry, we learn about an important class of molecules called polymers – long, chain like molecules which are composed of repeating subunits. Examples of polymers include DNA, proteins, and hydrocarbons. The behaviour of these molecules is of significant interest in many sciences, ranging from medicine to engineering, and the study of their behaviour is a major focus of modern research.

Polymers can be very large molecules which undergo lots of complicated physical interactions – as such, modelling their behaviour is often difficult, even with extremely powerful computers. One way to simplify the picture whilst keeping some important details is to represent a polymer as a self-avoiding walk. These models of polymers were first studied by chemists and have since become a classic model for long polymer chains.

To roughly understand what a self-avoiding walk (SAW) is, imagine the path traced out on the ground by a person who walks around randomly but can only take steps of equal size in the four directions of forwards, backwards, to the left, and to the right. If we add the condition that this person can never return to a point she has previously occupied, then the traced-out path will be self-avoiding. This kind of model ignores the detailed interior structure of the molecule and focuses on its overall connectivity and shape. The attached picture shows a series of Dyck paths, which are essentially a kind of self-avoiding walk that are related to the SAWs investigated in this project. Specifically, we specifically investigated the case where a semi-stiff polymer propagates down a slit, undergoing interactions such as adsorption with the walls.

Mathematically speaking, we have embedded the polymer into a discretised space. This brings our complicated problem into the realm of maths known as combinatorics, which is all about counting finite structures. We can now bring to a bear a range of powerful tools from combinatorics. Using these tools, we managed to find generating functions that ‘count’ every possible configuration of the polymer, as well as prove some interesting physical properties of the system.

 

Image retrieved from http://mathworld.wolfram.com/DyckPath.html

 

Jonathon Liu was a recipient of a 2018/19 AMSI Vacation Research Scholarship.

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