Abstract: In this piece, I summarise some of the challenges I faced in writing my SRS report ‘Dynamic Stiffening of Flecible Microswimmers’, and the methods that I used to overcome them. In particular, I highlight the role of simplifying assumptions in the craft of mathematics, both as a tool for making problems easier, and as a route to novel insights that could be obscured by more complicated systems.
Mathematics can be very difficult. I am sure that I am not the only person who has noticed this. My goal with this piece, however, is to convince you that simplifying away some of this difficulty is an effective way to gain insights into hard problems, mathematical or otherwise.
My summer research project for the AMSI concerned the way that marine bacteria change direction. I spent roughly the first week of this project reading a recent paper that considered the movement a bacterium in flowing water and, in short, I did not get very much out of it. Lacking any exposure to the high level fluid mechanics I was faced with, I was not yet mathematically prepared to understand the ways that the flowing water reflected back and forth between the cell body and the spiral shaped flagellum that acted as its propellor.
As such, I decided to make a simplification. Instead of considering bacteria in flowing fluid, I imagined the fluid completely still. Because the bacteria in question are so small (in mathematical terms, they have low Reynolds number) the effect of their movement on the surrounding water is minimal, and the water can be effectively modelled as completely still—even as the bacteria move through it.
Previous work suggested that the turning behaviour was caused by a flexible joint between the cell body and flagellum, that varied its flexibility over time. As such, to learn about this particular feature, I made all other aspects of this problem as simple as possible. Instead of a 3D cell body attached to a spiral propellor, I imagined a 2D ellipse (an oval) attached to a rectangle, with a simple thrust force tacked on for propulsion.
Despite the extent of my assumptions, the simplified model was in fact able to recreate the behaviour observed in experiments, and give some key relationships between physical parameters and the behaviour of the bacterium. Because the model did not include external flow or complicated cell geometry, this research provides more evidence that the varying flexiblity of the connective joint, rather than these other factors, is in fact the mechanism that the bacteria use to turn!
Furthermore, once I had built up a base of understanding, I read back over the original paper and found it much more accessible than I had originally thought. The prospect of examining my model in progressively more complicated scenarios, and determining exactly how the solution changes, no longer sounds daunting but rather very exciting. I look forward to examining the ways that three dimensions of movement, a spiral-shaped flagellum and even external flow govern the behaviour of the real-world bacteria.
I hope that the next time you are faced with a problem you cannot solve—mathematical or otherwise—you start again and try the simplest possible version of the problem, and are able to glean some useful insight! I know that I will.
Joshua Borsky
The University of Melbourne
