# What do Mathematical Biology and Puzzles Have in Common?

When I was younger, I used to play puzzle games with objectives like completing a system of pipes so that water would come into the system, flow through an elaborate set of pressure checks, and exit the system. In other words, given a set of observations: a) the water enters through pipe A, b) the water passes through pipes B and C etc., and c) the water exits through pipe Z, the objective was to find a configuration of pipes that satisfied all those observations, with extra points for a simple solution. The chosen configuration was a hypothesis about the proper layout of the pipes, which was then be tested by a cartoon ‘simulation’ of the flow. If all checks were satisfied, then voilà, you could move on to the next level. In many ways, these games provide a useful metaphor for the life of a mathematical biologist.

Maths biology is an inherently multidisciplinary field, which requires expertise from both experimentalists (e.g. biologists, psychologists, experimental physicists) and theoreticians (e.g. mathematicians, statisticians, theoretical physicists). The process starts with an experimentalist, who tests an initial hypothesis by performing a series of protocols and deriving a set of observations which may or may not support that initial hypothesis. In either case, there will be unanswered questions about how those observations come to be realized, which could be because some or all of observations do not fit the hypothesis or because multiple hypotheses could result in the same set of observations. Returning to the puzzles, we now have a set of observations to resolve and perhaps some hints about how to arrange the pieces, but there are always a few segments left to put in place.

The research I did for my PhD involved collaborating with an interdisciplinary group of songbird researchers to develop mathematical models of the zebra finch song system. Much is known about brain areas involved in producing zebra finch song. There is a nucleus that controls timing, which sends signals to another nucleus that coordinates the muscle-controlling nuclei, and more downstream nuclei that directly influence muscle groups such as the syrinx (similar to a voice box) and respiratory system. This hierarchical network works with millisecond precision to produce a set of vocalizations, called syllables, separated by short, silent gaps. In other words, we have a hierarchical system of pipes, with multiple subsystems playing different roles, but at least we know (more or less) what each subsystem does and how each subsystem is connected to and influences the others. Moreover, we know much about how perturbations to the song system (removing and changing pipe segments) affect(s) the song system (flow). Using these pressure checks on the system, we first constructed a mathematical model that could satisfy them, and we were even able to reproduce the changes in flow observed by perturbations of the song system.

In the pipe game, this would have been a success; however, the game of biology has an added twist. Every time a researcher performs a new experiment, a new constraint, or pressure check, is added to the puzzle. Sometimes, the model already satisfies a newly added constraint, further validating the model, and in the best case, the model is used to predict the outcome of the experiment before it is performed. More often, however, a new observation will provide a constraint that causes the configuration to fail. In this case, we have to move some pipes around to find a new configuration that satisfies the new constraint. The good news is that this generally leads to a new set of predictions that can be tested.

The development of our birdsong model was driven by a curious experimental result our group had published, which was lacking a testable hypothesis to explain it. The finding was that removal of different portions of the timing nucleus resulted in different changes to the song. Initially, the bird would sing a syllable sequence such as ABCDEABC etc. Removal of one portion of the nucleus led to syllable omission, so the bird would sing ABC instead of ABCDE, effectively activating the exit pipe prematurely. On the other hand, removal of a different portion led to atypical transitions, so the birds would sing ABABABAB. Prior to development of the mathematical model, it was unclear how this behaviour could be realized in practice, particularly without violating many of the older pressure checks. The model allowed us to test many small changes, and verify which of those could satisfy all the older experimental results as well as the new one. In the end, only minor adjustments were needed to generate a system where removing one pipe led to syllable omission, while removing another led to atypical transitions.

In addition to generating a hypothesis for how these song perturbations might occur, we also used the model to suggest ways of testing the validity of that hypothesis. In other words, we used the model to suggest the locations of some new pressure checks. If our hypothesis is correct, then new experiments will uncover those checks and validate our model, but if not, a different set of checks will be uncovered and we will have to adjust the model to satisfy them. Only by running those experiments can we know how much more plumbing there is to do.

Birdsong modelling paper:

D Galvis, W Wu, RL Hyson, F Johnson, R Bertram, A Distributed Neural Network Model for the Distinct Roles of Medial and Lateral HVC in Zebra Finch Song Production, Journal of Neurophysiology, 118.2:677-692, 2017.

Working with the QBME

As a postdoctoral researcher for the QBME, I have been focused on applying mathematical modelling and network analysis to identify the optimal resection site for epilepsy surgery. In particular, I have been working with the University of Melbourne and St. Vincent’s Hospital to study the extent to which MEG and high-density scalp EEG can be used in the pre-surgical pipeline.

I am also working on two seed corn projects. The first project is to develop mathematical models of cellular ageing, which we used to understand how multiple non-specific markers for cell type (dividing, non-dividing, dying, old) are related to one another. Our paper entitled: A dynamical systems model for the measurement of cellular senescence is under review at the Journal of the Royal Society Interface. The second project uses molecular dynamics simulations to compare structurally similar microscopic appendages, archaellum vs.T4P, and determine which features are necessary contribute to their differing functions, swimming vs. crawling respectively. We will be showing off some of this work at the LSI Open Night on June 20th.

Finally, I will be putting on an Introduction to Mathematical Modelling for Biosciences workshop in July. The goal of the workshop is to introduce the core concepts of mathematical modelling and present some examples of model development, simulation and analysis. Tell your friends!

Cellular ageing paper:

D Galvis, D Walsh, L Harries, E Latorre, J Rankin, A dynamical systems model for the measurement of cellular senescence, Journal of the Royal Society Interface, submitted.