Research on Approximate Symmetries in Biology

After passing my qualifying requirements this semester, I am thrilled to have more time to focus on my research, which aims to investigate the formation of symmetry in biology. Symmetry is a vital aspect of organisms, from the rotational symmetry of flowers to the fractal symmetry of trees and even bilateral symmetry in humans. Although symmetries are often indicative of survivability in biology, their imperfect nature makes it challenging to apply mathematical definitions of idealized symmetry. As such, I plan to explore a looser definition of symmetry and extend bifurcation theory tools to understand how these symmetries form.

In 2021, Gandhi et al. proposed a real-valued operator, known as transformation information (TI), which measures approximate symmetries by evaluating an object’s change under a parameterized transformation. When plotted on the parameter space, the object’s symmetries appear as local minima. For simple systems, the extrema of the graph undergo changes that are qualitatively similar to changes in equilibria for classical bifurcations.

I am interested in exploring whether this phenomenon holds for more complicated models that accurately represent biological systems. Specifically, I intend to experiment with Turing patterns, which are reaction-diffusion PDEs that can explain pattern formation, to observe when symmetry bifurcations correspond to changes in extrema of the TI. Since Turing patterns lack analytic solutions, I will rely on numerical methods for my experiments. If there is evidence of a connection between TI and bifurcation theory, I will attempt to prove some theoretical results.

Gandhi, P., Ciocanel, M. V., Niklas, K., & Dawes, A. T. (2021). Identification of approximate symmetries in biological development. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 379(2213). https://doi.org/10.1098/rsta.2020.0273