Mathematics

By incorporating positive and negative feedback into phosphorylation cycles of proteins such as GTPases (described in Lecture 2), one arrives at a biochemical mini-circuits with properties of a switch or an oscillator. I provide examples from papers by Boris Kholodenko. I also show the connection between bistability and hysteresis and relaxation oscillations (e.g. in the Fitzhugh-Nagumo model). I briefly
discuss applications of such ideas to models of the cell cycle proposed over the years by John Tyson.

This triplet of introductory lectures summarizes a few of the most basic biochemical models with the simple rate equations that they satisfy. I describe production-decay, with Michaelis-Menten and sigmoidal terms, showing how the latter can lead to bistable behaviour and hysteresis. I describe two bistable genetic circuits: the toggle switch by Gardner et al (2000) Nature 403, and the phage-lambda gene by Hasty et al (2000) PNAs 97. The idea of bifurcations is discussed. Finally, I introduce
phosphorylation cycles, and show that sharp responses can arise when the enzymes responsible (kinase and phosphatase) operate near saturation. (This is the so called Goldbeter-Koshland ultrasensitivity).

The mathematical context of the third story, Small Number and the Basketball Tournament, contains some basic principles of combinatorics. The plot of the story and the closing question are structured in a manner that allows the moderator to introduce the notion of permutations and combinations. Since the numbers used in the story are relatively small, this can be used to encourage the young audience to explore on their own. Mathematics is also present in the background. Small Number and his friends do mathematics after school in the Aboriginal Friendship Centre. He loves playing the game of Set and when he comes home his sister is just finishing her math homework. Small Number and his friend would like to participate in a big half-court tournament, and so on.

For more details see http://mathcatcher.irmacs.sfu.ca/content/small-number

This opening lecture lists some of the questions and issues propelling current research in Cell Biology and modelling in this field. I introduce basic features of eukaryotic cells that can crawl, and explain briefly the role of the actin cytoskeleton in cell motility. I also introduce the biochemical signalling that regulates the cytoskeleton and the concept of cell polarization. By simplifying the
enormously complex signalling networks, and applying tools of mathematics (nonlinear dynamics, scaling, bifurcations), we can hope to get some understanding of a few of the basic mechanisms that areresponsible for symmetry breaking, robustness, pattern formation, self-assembly, and other cell-level phenomena.

While Turing is best known for his abstract concept of a "Turing Machine," he did design (but not build) several other machines - particularly ones involved with code breaking and early computers. While Turing was a fine mathematician, he could not be trusted to actually try and construct the machines he designed - he would almost always break some delicate piece of equipment if he tried to do anything practical.

The early code-breaking machines (known as "bombes" - the Polish word for bomb, because of their loud ticking noise) were not designed by Turing but he had a hand in several later machines known as "Robinsons" and eventually the Colossus machines.

After the War he worked on an electronic computer design for the National Physical Laboratory - an innovative design unlike the other computing machines being considered at the time. He left the NPL before the machine was operational but made other contributions to early computers such as those being constructed at Manchester University.

This talk will describe some of his ideas behind these machines.

Turing 2012 - Calgary

This talk is part of a series celebrating The Alan Turing Centenary in Calgary. The following mathtube videos are also part of this series

1. Alan Turing and the Decision Problem, Richard Zach.
2. Turing's Real Machine, Michael R. Williams.
3. Alan Turing and Enigma, John R. Ferris.