Swimming microbes steer themselves into mathematical order

Freeing thousands of microorganisms to swim in random directions in an infinite pool of liquid may not sound like a recipe for order, but eventually the swarm will go with its own flow.

Theoretical modeling led by University of Wisconsin–Madison applied mathematician Saverio Spagnolie shows that the forces generated by different kinds of tiny swimmers will sweep them all up in predictable ways.

“When each individual particle experiences the flows created by all the other particles, it’s known that really surprising effects can naturally emerge,” says Spagnolie. “The flows and orientations of the swimmers become coherent on a length scale much longer than any individual particle, resulting in huge flocks of organisms swimming in the same direction and, perhaps unintentionally, working together.”

When a computer simulation of a sheet of swimming particles (yellow marks their highest concentration, blue empty liquid) is nudged into action, the forces and flows bend, disperse and gather the organisms in what looks like coherent flocks. Video courtesy of Saverio Spagnolie

The movement of crowds of things too small to easily see — like single-celled organisms and filaments inside individual cells responsible for cell-division — is critically important to research in materials science, engineering and biochemistry.

By simulating the interactions of large groups of particles which each create a flow, Spagnolie and Arthur Evans of UW–Madison, University of Michigan physicist Christopher Miles and mathematician Michael Shelley of the Flatiron Institute and New York University found that when the particles are confined to a thin sheet and allowed to expand into an empty fluid, the collective motion can be described by equations already used in entirely different classical problems in fluid mechanics. The group published its findings today in the journal Physical Review Letters.

“If you’re solving for the trajectory of 10,000 or 100 or even 10 things bouncing around, it’s hard to see what’s going on. You can lose sight of deep structure,” says Spagnolie, whose work is supported by the National Science Foundation. “But if there are enough particles, they can be seen themselves as a type of active fluid, with equations describing the velocity and density of a local group of particles — just like how we think about deriving equations to describe flowing water or air.”

The researchers worked out the relevant equations for particles that move by various means — swimmers that actively push or pull themselves through fluid, and types (like microtubules inside a cell) that push or pull themselves through molecular means without active appendages like flagella — and goosed them into motion.

“From that perturbation there’s this explosion of motion,” Spagnolie says. “And then we watch how the different forces play out on different types of particles.”

While a tight colony of pulling swimmers, for example, stretches itself out in a line perpendicular to the direction they’re headed, a colony of pushers stretches quickly in the direction of motion, and then bends on itself over and over in a cascade of shrinking folds.

The movement of crowds of things too small to easily see — like single-celled organisms and filaments inside individual cells responsible for cell-division — is critically important to research in materials science, engineering and biochemistry.

“That these individuals can group together passively due to their fluid interactions alone, and that this results in large-scale events and effects they can’t achieve as independent particles, is relevant to many biological functions — like nutrient mixing and bacterial resistance to antibiotics in bacterial swarms and biofilms,” Spagnolie says.

The researchers believe their theoretical description of the rapid growth of active sheets — which unexpectedly resembled well-known equations like those used to describe the movement of fluids trapped between plates or dispersed through soil — will be of use to others working at the point where fluids interact with miniature movers like bacteria and microtubules.

“This is one of the first theoretical considerations of concentrated particles invading a bulk fluid,” Spagnolie says. “The hope is that this will be a case of theory leading experiment, offering predictions that can be validated or invalidated by researchers who are very much on the edge of carrying such an experiment out.”