Performance of a Helical Microswimmer Traversing a Discrete Viscoelastic Network with Dynamic Remodeling

By Rudi Schuech

The ability to swim is as ubiquitous in the microbial world as it is in the more familiar macroscopic world of fish and other animals. Perhaps the most well-known example of microbial motility is that of human sperm cells, which beat a thin, flexible flagellum to push themselves through the surrounding fluid environment. Bacteria are smaller still, yet recent discoveries indicate that the micron-sized bacteria in the human gut microbiome—which may outnumber a human's own cells [8]—significantly affect human health [5]. Like sperm, the majority of bacterial species also propel themselves with one or more flagella. Yet despite their superficial similarities to sperm flagella, bacterial flagella are actually fairly rigid corkscrews that rotate through the fluid due to an incredibly complex bacterial flagellar motor in the cell membrane [4]. Over the last several years, researchers have constructed artificial bacteria-inspired microrobots that are propelled by helical corkscrews but driven by external magnetic fields [10]. In the future, swimming microrobots might even navigate through the human body and enable treatment from within, much like a real-life version of the 1966 film Fantastic Voyage.

Researchers have been studying microswimmer fluid mechanics both experimentally and theoretically for decades, generally with Newtonian fluids (like water) as the medium. These fluids exhibit a simple linear relationship between an applied shear stress and the resulting velocity gradient. However, many real fluids—such as mucus—are not Newtonian and instead exhibit complex properties like viscoelasticity. A lot of biological environments also do not appear as continuum media from the perspective of a single cell; instead, microorganisms often navigate tangled webs of polymeric filaments that are suspended in water, where the space between filaments is comparable to cell size [6]. Furthermore, some microbes can actively remodel the local environment. For instance, the ulcer-causing bacterium Helicobacter pylori secretes enzymes that chemically alter the surrounding stomach mucus and allow it to swim largely unhindered through what is otherwise an extremely viscous, gel-like substance [1].

Mathematically modeling the movement of microswimmers through heterogeneous viscoelastic environments is no easy task. Despite the popularity of continuum models of swimming activity in complex fluids, adapting these models to account for spatially varying fluid properties is quite challenging. We therefore decided to utilize an approach known as the method of regularized Stokeslets to explicitly model a network of viscoelastic elements that are suspended in a conventional Newtonian fluid. A Stokeslet—named after physicist and mathematician George Gabriel Stokes—is the fluid velocity field that results from the application of an infinitely concentrated (singular) force at a point in the fluid. Scientists can model the flow around a microswimmer via the superposition of many Stokeslets, but their singular nature is a practical obstacle. In response, Ricardo Cortez introduced the concept of Stokeslet regularization (blurring the force over a small volume) in 2001 [2]. We modeled the fluid dynamic interactions between a model microswimmer and a grid of points that represent the viscoelastic network by distributing regularized Stokeslets over the surface of the swimmer and at each network node [7]. The bacterium-inspired microswimmer was driven by an imposed torque between the spherical body and helical tail, much like real bacteria (see Figure 1).

We initialized our simulations by placing the microswimmer just behind a discrete viscoelastic network in a rectangular lattice configuration (see Animation 1). The impending obstacle immediately causes the swimmer to slow down slightly, and it continues to get progressively slower as it penetrates the network. As the swimmer pushes the nearby network nodes out of the way, the links that join these nodes simultaneously stretch and exert forces that resist the motion. Despite this obstacle, the swimmer eventually exits the network. Due to its ability to push back against the nodes behind it, the swimmer then undergoes a small but conspicuous boost in swimming speed that is higher than it would otherwise experience in the absence of a network (which we refer to as the Newtonian case). Nonetheless, the network ultimately always slows down the microswimmer in our simulations.

The viscoelastic properties of our discrete network arise from the links between the network’s nodes, which we assume consists of a spring and a dashpot in series. Each spring’s stiffness constant \(E\) describes the elastic behavior of the links, while the dashpot constant \(\eta\) describes their viscous damping or permanent stretching. First, we quantified the effects of these two parameters based on the ease with which the microswimmer traversed the network. While one might expect that increasingly stiffer links would cause the microswimmer to penetrates the network more slowly, we found that this trend only holds up to moderate stiffness. In fact, our system’s geometry allowed the microswimmer to pass through the stiffest networks nearly as quickly as through a purely Newtonian fluid. We opted to initially align the microswimmer with the grid-like lattice network, which allowed it to swim largely unimpeded through lattice channels in cases of minimal network deformation (see Animation 2). Unexpectedly, the microswimmer was most hindered at an intermediate network link stiffness \(E\) (see Animation 1). 

We encountered another surprise when we varied the viscous damping parameter \(\eta\). Small \(\eta\) signifies that the resting lengths of the network springs rapidly adapt to stretching (like taffy), meaning that the forces exerted by the network are small. However, small \(\eta\) also drew a large number of network nodes toward the microswimmer (see Animation 3). A portion of the network then became wrapped around the rotating tail; while this circumstance had little effect on swimming speed in the short term, the material eventually inhibited the microswimmer in the long term. 

Finally, we returned to the issue of microswimmers that chemically remodel the viscoelastic environment, as is the case with H. pylori. We leveraged the flexibility of our discrete network modeling approach by assuming that any links that come within a small distance of the microswimmer body dissolve (as if due to secreted enzymes). Doing so allowed the microswimmer to effectively clear a tunnel around itself and traverse the network nearly unhindered, regardless of the stiffness of the network links (see Animation 4).

Looking forward, we could certainly extend our model in several ways. We simulated a simple model bacterium with a spherical body and a single flagellum, but real bacteria exhibit an enormous diversity of shapes that might interact differently with a viscoelastic network [3]. As such, we could easily simulate other shapes with our method. Suspension of many bacteria, which is also common in nature, might involve more frequent viscoelastic interactions than is currently assumed [9]; exploring this possibility would hence be another straightforward (albeit computationally expensive) extension of our approach. 

Finally, the properties of the viscoelastic network itself deserve further study. We assumed that all links in the discrete network have the same stiffness and damping constants, but varying these parameters across the network would permit us to model smoother interfaces between Newtonian and viscoelastic regions. Networks with less geometric order than the simple rectangular lattice that we use here could also reveal entirely different dynamics, i.e., the microswimmer might potentially become trapped instead of swimming through a channel in the lattice.

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Rudi Schuech presented this research during a minisymposium presentation at the 2021 SIAM Annual Meeting, which took place virtually in July 2021.

References
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[7] Schuech, R., Cortez, R., & Fauci, L. (2022). Performance of a helical microswimmer traversing a discrete viscoelastic network with dynamic remodeling. Fluids, 7(8), 257.
[8] Sender, R., Fuchs, S., & Milo, R. (2016). Revised estimates for the number of human and bacteria cells in the body. PLOS Biol., 14(8), e1002533. 
[9] Sretenovic, S., Stojković, B., Dogsa, I., Kostanjšek, R., Poberaj, I., & Stopar, D. (2017). An early mechanical coupling of planktonic bacteria in dilute suspensions. Nat. Commun., 8, 213.
[10] Wang, Q., Du, X., Jin, D., & Zhang, L. (2022). Real-time ultrasound doppler tracking and autonomous navigation of a miniature helical robot for accelerating thrombolysis in dynamic blood flow. ACS Nano, 16(1), 604-616.

Rudi Schuech is a postdoctoral fellow at Tulane University. His research focuses on the fluid dynamics of microorganisms and other small-scale biological systems.