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A Run-and-Tumble Model for Phytoplankton Aggregation

By Jillian Kunze

Plankton comprise a large collection of organisms that are vital to Earth’s ecosystems, as they form the basis for many aquatic food chains and produce much of the oxygen we breathe. However, nutrient pollution can cause phytoplankton reproduction to explode into harmful algae blooms, which produce damaging toxins and deplete oxygen in the water. Scientists wish to gain a better understanding of why these devastating algae blooms are happening and what can be done to stop them.

During a minisymposium presentation at the 2020 SIAM Conference on the Life Sciences, Nicholas Russell of the University of Delaware presented his model of phytoplankton aggregation. He first observed the natural behavior of Heterosigma akashiwo plankton by placing them in a thin layer of water under a microscope and taking a series of pictures. The resulting images indicated the plankton’s immediate aggregation; they seemed very ready to clump together. 

Russell monitored the aggregation patterns of plankton over time.
Phototaxis—the movement of an organism in response to light—appeared to play some sort of role in the plankton’s motion since the type of lighting changed the pattern of aggregation. However, it was clear that the plankton were not solely responding to light because they also aggregated in the dark. It thus seemed that the main mechanism at play was actually autochemotaxis; the plankton were moving towards a chemical that they produced themselves. 

Russell also tried monitoring plankton in a deeper liquid and found that the aggregation patterns in that three-dimensional space were far more complex than in the thin, two-dimensional (2D) layer. He therefore focused his modeling efforts on the 2D system, which can serve as an analogue to plankton in shallow water, and planned to mathematically investigate the role of chemical signaling in 2D plankton aggregation. 

Russell used the run-and-tumble method, which is similar to a random walk, to model the plankton’s movements. Each plankton moves in a certain direction and then tumbles—changes direction with a specific probability, depending on its motion relative to the chemical field—before repeating the process. The equation that models that movement was a better match to the plankton’s real-life behavior with the addition of a smoothing parameter. 

Russell tried out several different functions—including a constant function, a switch function, and a linear switch function—to represent the chemical field in the model. Because the amount of chemicals released by the plankton depended on the number of other plankton they sensed, the plankton’s density was also an important factor. In Russell’s model, density depended on a Fokker-Planck function. 

It was easiest for Russell to begin testing the model in one dimension, which meant that he only had to consider the density of right- and left-moving plankton. To determine the way in which perturbations affected the modelled plankton, Russell perturbed the density and chemical fields, plugged these altered parameters into the system, and then found the cubic characteristic equation of their behavior. He also experimented with altering the smoothing parameter and realized that it had a hugely important effect; moving it from just 0.01 to 0.03 resulted in a major change in the system’s stability regions. By plotting the plankton density over time from his one-dimensional (1D) simulation, Russell found that the model did show plankton aggregations building. 

One could then modify the 1D model of plankton aggregation to represent a plankton system in two dimensions. Russell began with a constant solution to the model, then applied a perturbation to the steady state. After identifying the relevant evolution equations, he performed a 2D Fourier transform that led to a linear system of ordinary differential equations. This resulting system can get fairly large, so Russell sometimes had to truncate it to an adequate approximation. 

Based on this model, Russell found that similar stability regions for plankton exist in both a 1D and 2D space. He used these results to model the time evolution of a 2D system of 160,000 plankton. By trying out various chemical field functions, he observed the different aggregation patterns they produced and compared them to the plankton’s real activity. 

In the future, Russell hopes to further extend his research by developing a run-and-tumble model of autochemotaxis in three dimensions. He would also like to incorporate additional parameters—such as photosynthesis, fluid motion, and toxins—into the model. Extending the model’s parameter space could increase scientists’ understanding of this plankton’s behavior, thus advancing the search for practical solutions to dangerous algae blooms. 

   Jillian Kunze is the associate editor of SIAM News.