Would you like to explore and research the latest scientific ideas?
Do you enjoy mathematics and/or science?
If so, have you thought about joining the University of Queensland’s elite Advanced Studies Program (ASP)? One of the highlights of the ASP is the opportunity to talk with and learn from researchers working at the frontiers of science. In this program second year students are placed with leading research teams and actively participate in research projects – sounds exciting, doesn’t it!
You can find out more general information about the ASP on page 4. Below we have a brief description of some of the mathematical research projects current ASP students Jabin, Tracy and Marissa, are involved in.
Jabin Kirk, working with Diane Donovan, is looking for evidence of hidden codes within protein-coding genes.
Cells store the instructions for the production of proteins in fragments of DNA/RNA, known as genes. More precisely, RNA is made up of sequences of nucleotides, G, A, U and C, and amino acids are subsequences of RNA of length three; that is amino acids are encoded by triplets of nucleotides. To produce a protein, cells locate specific regions or strands of RNA. The cell reads the nucleotides on the strand in groups of three, with each such triplet identifying an amino acid needed for the production of the protein. An interesting aspect of this process is the fact that there are essentially only 20 different types of amino acids, but 64 = 4³ = 4 x 4 x 4 different triplets (as there are four different nucleotides for each of the three positions of a triplet). So two or more triplets are often used to encode the same amino acid. The assignment of the 64 triplets to different types of amino acids is set out in the table above and is known as the universal genetic code. Scientists, including Jabin, are eager to know if this redundancy follows a strict pattern and whether through evolution the universal genetic code has developed (mathematical) coding theory properties, which minimizes the impact of mutations. A mutation in a DNA/RNA strand occurs when one nucleotide is replaced by another, and may result in the incorrect production of proteins because the amino acid is changed.

Universal Genetic Code

** (the notation UAC (tyr) implies that the triplet of nucleotides UAC codes for, or corresponds to, the amino acid tyrosine.)

To study the universal genetic code we can begin by studying a simpler code and then try to extend the knowledge we gain to explain the more complicated code. So consider a binary code, over just two symbols 0 and 1; that is, 2x2x2= 23 = 8 triplets, 000, 001, 010, 011, 100, 101, 110, 111. A single mutation in a triplet means a 0 will change to a 1 or vice versa.

Jabin has represented this concept of mutation by the mathematical diagram shown above. Jabin uses dots to represent triplets, and two triplets are joined by a line if one triplet can be obtained from the other by a single mutation. Now let a triplet say, 001, code for an amino acid. A single mutation in 001 gives 000, 101 or 011, and so if a cell reads any one of the three triplets 000, 101 or 011 it should recognize that they all correspond to the same amino acid as 001 and so should “decode” back to 001. This leaves four triplets and if we choose 110 for a second amino acid, then a single mutation gives 010, 100 or 111, and a cell should “decode” each of these words back to the same amino acid as 110. In the diagram the big dots represent the code words and if a triplet corresponding to a small dot is received it should be corrected back to the closest big dot. If you study the diagram, you will find other similar “codes”. (For instance, let 000 represent one amino acid; then 111 should represent the other amino acid, and we leave you to apportion the remaining triples to these amino acids, so that a cell can always correct a single mutation.) However all these “codes” have three special properties: exactly two amino acids, say X and Y, can be encoded; the triplets corresponding to X and Y are separated by at least two dots; and the triplets joined to X by a line also code for amino acid X and similarly for Y. This is the best possible code for correcting single mutations. What is interesting is that Jabin’s approach can be extended to studying codes based on four different symbols G, A, U and C. However, his analysis shows that 64 triplets is not enough to have the facility to correct single mutations for 20 code words of length 3 and that the “code words” would need to be longer than triplets. Jabin’s ideas differ from research published in the Journal of Theoretical Biology, which suggests this modeling approach might provide evidence of an underlying code. What is exciting about Jabin’s research is that he has produced mathematical results which challenge us to question existing theory. Hopefully Jabin’s ideas will lead to further discussion and a more rigorous analysis of the structure of the universal genetic code.

Tracy Rout, a third year student, has been working for the last two years with Hugh Possingham’s research team designing better marine ecology reserves.
In a world where economic and political agendas compete with conservation goals, it is important to ensure that terrestrial and marine reserves are designed to maximise and conserve biodiversity while minimising and compacting the area to be maintained. Tracy has been using a computer program called Marxan (created by Ian Ball and Hugh Possingham) to design marine reserve systems for South Australia. Marxan uses simulated annealing to maximise the ecological aspects while minimising the “cost” – the weighted sum of area and boundary length – of the reserve system. So Tracy has constructed cost functions based on the boundary length and area of the reserve and then used mathematical techniques to find the minimum value for this function. For zones currently being planned by the South Australian Department of Environment and Heritage, Tracy has been studying and designing reserves on three different scales: for the area as a whole; by bioregion; and by marine sections. By comparing the efficiency, compactness and differences in representation between these reserves, Tracy hopes to uncover the effect of planning at divergent scales. She is currently writing up her results and hopes to have them published in a paper.

Marissa McBride also worked with Hugh Possingham’s research team in 2003.
Her brief was to set up a demographic (age structured) population transition matrix and use Markov chain and Monte Carlo methods to determine the optimal parameters (eg birth/death/growth rates at each stage) for a model of a snail population. Marissa then tested the performance of her model by comparing it against data taken from field experiments. Marissa wrote a basic Matlab program for a simplified version of the model, and generated the necessary data. Her supervisor, Chris Wilcox, is currently collaborating with statisticians and conducting a statistical analysis of Marissa’s results.

© Discipline of Mathematics, The University of Queensland.