Applying Maths to Movement
IntroductionHi! My name is Helena and I am a PhD student in applied maths at The University of Manchester. What that means is that after finishing my undergraduate degree in Physics, where I was taught a multitude of things about the world surrounding us, I decided I wanted to spend some time actually making discoveries for myself.
In DepthThere are hundreds (or even thousands) of equations out there describing ways movement happens; the movements which people observe all the time in experiments or real life are described by the so-called classical equations. Some of these you're probably already learning about at school.
What I do now is study what we call “anomalous transport”, which basically just means movement that somehow looks odd or unusual. The equations for anomalous transport differ from the classical ones in that they in some way or another require `memory effects' in order to fit experiments. The scientific principles teach us that experiments must always be the starting point of any work we do: we build theories to fit the data, not change the data to fit the theory we already have. And so that's what I do. I try to find mathematical descriptions of the kinds of movements scientists working in e.g. biology see in the lab. Once I manage to find a good fit between my theory and the data they gave me, the experimental scientists can then go away and do more experiments to test the predictions of my models.
Of course it's not just my model, but that of my entire research group. Depending on how difficult a problem is, it can often take several of us to solve it. An example of such a problem is intracellular movement, so movement that happens inside of the cell. For example, researchers in biophysics and biology are interested in how essential nutrients are transported from the nucleus to the cell membrane. This transport happens partly through the work of “motor proteins”, and the movement of these inside the cell are known to be anomalous. An image of how the transport happens is shown below.
Drawing 1: The picture shows a motor protein (brown) moving a cargo (blue) along a microtubule. Microtubules are pathways to transport nutrients across a cell.
When you think about all the different parts of a cell, and the processes that happen in it, it is not very surprising that the equations one would need to describe this kind of transport would have to be rather complex. In particular, what we find is that the movement you see any point in time will likely also depend on what happened a while ago. For example, if there are several motor proteins all moving on a microtubule they might cause some kind of `traffic jam', which will affect the motors for a while until the path becomes clear again. This, and many other things, can be the cause of `memory effects' in our equations so that we may have to account for all movements up until the point we're looking at in order to predict how the movement will continue.
While this makes the work harder, it is very important in understanding what might cause transport in cell to stop happening, leading to cell degeneration. This is linked to various neurodegenerative diseases and could potentially be instrumental in designing better medications.
Other examples of where you might see this kind of anomalous movement include the flights of bumblebees in a field, sharks hunting for prey in the ocean, and even the optimal part a robotic vacuum might take across your living room floor!
Going FurtherIf you're interested in learning more about anomalous transport, our research group has a website with more examples.
Otherwise, if you want to learn more about intracellular transport there is a very useful introduction here:
Finally, if you want to get an idea of all the other amazing areas maths can be applied to you can visit