A Living Ising Model: Bacterial Vortex Lattices

Today I read an interesting paper in Nature Physics by Hugo Wioland and friends, called "Ferromagnetic and antiferromagnetic order in bacterial vortex lattices." A lot of Nature Physics is devoted to solid state physics which I personally don't find too interesting, so I almost glossed over until I saw the "bacterial vortex" at the end of the title. In the paper, they grew bacterial colonies in circular cavities that spontaneously rotated, and showed that each colony vortex can behave the way atoms do in magnetic solids, and use it as a jumping-off point to model complex living systems with lattice physics.

A bacterial vortex, taken from the supplemental material of the paper. The graininess is due to me trying to convert from mov to gif and is not part of the paper. The colonies are 50 microns in diameter.

The bacteria, Bacillus subtilis, is covered in flagella which are constantly waving around. The bacteria cannot occupy the same space as one another and so organize themselves in such a way as to avoid that, and influence each other through hydrodynamic interactions of the beating flagella. When they are packed into these circular cavities, they fill the space, and when the flagella beat coherently the colonies start to rotate. One rotating colony they call a bacterial vortex, and they can rotate clockwise or counterclockwise with varying magnitude.

Four connected colonies. The top left and bottom right spin counterclockwise, and the top right and bottom left spin clockwise. (You can see this in the movies from the paper) 

The circular cavities are arranged in a lattice, either square or triangular, with a gap of a certain size connecting each one. Tuning the size of the gaps tunes the interactions between cavities, which are mediated by the row of bacteria on the edge of each circle next to the walls. If the gaps are narrow, the bacteria do not move through and they interact hydrodynamically through the flagella beats, and want to move in the same direction as their neighbour across the gap, which makes the vortices spin in opposite directions. If the gaps are wide, bacteria tend to line up along the walls of the gap, such that a row of bacteria will do a "180" going from one cavity to the next, meaning neighbours will move in opposite directions and thus the vortices will spin in the same direction.

Diagram from the paper of inter-vortex interactions. If the gaps are small, they bacteria at the gaps interact hydrodynamically and move in the same Cartesian direction. If the gaps are wide the bacteria move along the edges, making adjacent cavities rotate the same way. I can foresee my explanation being confusing and unsatisfactory.

This is cool and all, but at this point I should take a step back and actually explain why they are doing this experiment.

Physics is hard. There are very few complex problems that can be exactly solved, but there are computational methods that can get approximate solutions. One of these is called lattice field theory, where space and time are broken into finite-size steps (sites on a lattice), and interactions between adjacent lattice sites are considered and the system is simulated with a computer.

One of the simplest but most ubiquitous lattice models is called the Ising Model*, which is used to understand magnetic materials. In the Ising model, there is a lattice of "spins" that can either be "up" (+1) or "down" (-1), and the total energy of the system depends on whether each spin is pointing the same direction or the opposite direction as its neighbor**. (Imagine two adjacent wire loops with electrical current going around them, and consider the torques they exert on each other if the current is going in opposite directions. Then try to consider a thousand loops.)  Materials where the spins want to point in the same direction are ferromagnetic (like iron) and materials where the spins want to point in opposite directions as their neighbor are called antiferromagnetic (like chromium). Solving the Ising model can be complicated, but it's much simpler than considering the interactions of $10^{23}$ interacting atoms.

A two dimensional Ising lattice, showing ferromagnetic order (left) and antiferromagnetic order (right).

The authors wanted to see they could apply the Ising model to a living system, so the created these bacterial vortices to see if they obeyed Ising-like behaviour. From a thermodynamic standpoint, living matter is substantially more complicated than inert matter: it's constantly producing its own energy and is never in equilibrium. There is a whole relatively new branch of physics just dedicated to studying the thermodynamics of active matter. A network of colonies, each a network of bacteria, each a network of interacting proteins of incredible complexity, would be essentially impossible to model from a "bottom up" approach, but mapping it onto the Ising model would allow its large scale behaviour to be predicted and studied, and may open the door to more generally studying the physics of living systems.

Back to the results of the paper. You may recall that I said that for lattices with narrow gaps, the adjacent bacterial vortices spin oppositely, and for wide gaps adjacent vortices spin in the same direction. The former case corresponds to antiferromagnetism, and the latter to ferromagnetism. By changing the size of the gaps, they can ordain what kind of "magnet" these bacterial colonies will be. The critical size where the behaviour crosses over is about 8 microns.

Left: An antiferromagnetic bacterial vortex lattice, where adjacent cavities tend to spin in opposite directions (alternating green and purple). Right: A ferromagnetic lattice, where neighbors tend to spin in the same direction, with domains of green and purple.

To make the whole thing slightly cooler, they explain the spin-spin interactions between adjacent vortices in terms of an "edge current" of the outer layer bacteria moving along the walls in the opposite direction as all the bacteria in the middle. This is analogous to certain quantum materials such as a quantum hall state in a two dimensional electron gas (a "thin cold semiconductor" doesn't sound as neat), where the electrons propagate in the opposite direction along the outside of the material. Many-body quantum mechanics and bacterial fluid mechanics do not have much in common, but both can be described by this lattice model. I generally think it's neat that these colonies can be modelled as a lattice interacting spins the same way that atoms in a magnet can, even though the first kind of spin is the net motion of the bacteria and the second kind is the intrinsic angular momentum of an electron.

People invariably ask what the practical applications of a given paper are. I will quote the how-we-will-save-the-world-if-we-get-more-funding section from the last paragraph, where the authors state: "Improved prevention strategies for pathogenic biofilm formation, for example, will require detailed knowledge of how bacterial flows interact with complex porous surface structures to create the stagnation points at which biofilms can nucleate."

Overall, very cool paper.

*Named after Ernst Ising and also a good name for a physics hockey team.
**Still haven't decided to go with American or Canadian spelling on this one.