It is an essential movement that helps propagate life—as it turns out human life and bacterial life. It is how sperm move through viscous liquids and how bacteria move through mucus. It is based on the mechanical motions of a high-angle helix, and how it works is anything but intuitive.

Researchers at Brown University, Providence, R.I., and the Univ. of Wisconsin, Madison, recently have shown exactly how a high-angle helix helps organisms like sperm and bacteria swim through mucus and other visco-elastic fluids. The work, which was recently published Physical Review Letters, helps clear up some conflicting findings about how microorganisms swim using flagella, the helical appendages that provide propulsion as they rotate.

The physics of helical swimming turns out to be “a really interesting fluid dynamics problem,” says Thomas Powers, a professor of engineering and physics at Brown and one of the study’s authors.

Size matters

To understand how cells move, you first must understand the scale you are talking about and how everything is experienced at that scale. At the scale of a single cell, fluids become much more viscous than on larger scales, Powers says. A bacterium swimming through water “would be like us trying to swim in tar,” he adds.

“Scale is critical. For example the question of whether or not a fluid is viscous is not so much a question about material properties of the fluid but rather a question about scale,” Powers adds. “Water, which we think of as a fluid that is not very viscous at the human scale, is very viscous at the scale of a bacterium.”

In addition, tiny helical swimmers rely exclusively on drag to move forward. The turning flagellum creates an apparent wave that propagates out from behind the creature. The drag force against that wave pushes the creature in the opposite direction.

In recent years there has been theoretical work aimed at fully understanding the physics of this kind of swimming, much of it by modeling how helical swimmers behave in water. But bacteria and sperm spend a lot of time in fluids like mucus and cervical fluid—fluids that are not only more viscous than water, but also elastic since they are full of springy polymers. Because a rotating helix might be able to push against the polymers, it could be that viscoelastic fluid makes swimming easier.

“It’s better to think of viscosity as a property of a flow rather than a property of a fluid,” Powers explains. “So with cells we are always in a situation in which cells must use drag to propel themselves.”

There have been several different approaches to the question—theoretical (analytic calculations valid for small amplitude waves), numerical (computer calculations valid for high-amplitude waves) and experimental with model systems and real swimmers.

“Some of these say the swimmer should speed up when you add polymers to the fluid, some say it should slow down. These results are not in contradiction,” Powers says. “For example, theory says small amplitude swimmers should be slowed, but numerical calculations show that for the right beating frequency, large-amplitude waves can go faster. The situation was that many different groups reported different answers because they were looking at the question in different regimes.”

“What our most recent work does is take a specific, well-defined system (rotating helix in a dilute polymer solution) and shows that as you increase the ‘amplitude’ of the helix—the helical radius—you go from a regime where the swimmer is always slowed to one in which the swimmer can go faster at just the right range of rotation frequencies,” Powers explains.

In the Brown study Powers, and Bin Liu (a post doctoral associate) worked with Saverio Spagnolie, a professor of mathematics at the University of Wisconsin.

Using what Powers describes as “some clever numerical methods and a lot of hard work” Spagnolie was able to computationally show that the pitch angle of the helix – the degree to which the helix is coiled – matters in how well it performs in viscoelastic fluids.

At a low-pitch angle (like what you might have with a stretched phone cord), helices move more slowly in viscoelastic fluids. When the pitch angle increases (relaxed phone cord) swimming performance improves, Powers explains.

The findings reconcile the experimental and earlier theoretical work. Much of the theoretical work, which suggested more viscosity would slow down swimming, assumed a small pitch angle for the sake of keeping the computations manageable. The experimental work, which showed viscosity sped up swimming, involved higher pitch angles. By showing numerically that a higher pitch angle increases swim speed, the researchers were able to explain the discrepancy.

The researchers know much work remains.

“We don’t really understand the result because it is so hard to visualize the 3-D configuration of all of the forces involved,” Powers says. “The next big step is to study these questions with real swimmers.”

“My colleague, professor Kenneth Breuer, working with Bin Liu, has built a tracking microscope that will follow individual bacteria as they swim. Thus, they will be able to observe the bacteria for long times” instead of just catching a glimpse as they swim across the field of view in a conventional microscope.

“Right now they are looking at swimmers in water but the plan is to add polymers” to simulate viscous fluids, Powers says.