Magnetotactic bacteria optimally navigate natural pore networks

Animals have developed a variety of strategies to detect direction, from specialised cells and organs to complex brain structures. This can be particularly difficult for microorganisms, such as bacteria, which often live in microenvironments where gravity cues are generally negligible.

For example, magnetotactic bacteria – bacteria that sense and orient along magnetic fields – live in waterlogged environments and instead use the geomagnetic field, the Earth’s magnetic field, to navigate. They contain magnetic crystals that act like a compass needle, aligning the cells with magnetic field lines. Because geomagnetic field lines are rarely parallel to the sediment surface, these bacteria move vertically through sediments. This motion allows them to position themselves at precise depths, follow changing gradients, and move between layers with distinct chemical conditions.

However, moving vertically through the sediment is challenging. First, a microbe must couple its motion to some external field with up-down asymmetry. Several options exist, including the gravitational field and gradients of chemicals or light. It must then balance motion along field lines with the need to move around obstacles.

Despite their similar behaviors, magnetotactic bacteria show a large diversity in their morphological and motility traits. Petroff et al. sought to understand how natural selection has shaped these traits to balance vertical motion with efficient obstacle avoidance.

Using a combination of theory and experiment, the researchers explored how swimming speed, magnetic moment and size of an organism affect navigation through porous environments. They measured the average speed of the magnetotactic bacterium Magnetoglobus through an artificial pore space in a controlled magnetic field.

The results showed that efficient navigation requires an optimal combination of swimming speed, magnetic moment and body size. Movement along magnetic lines was fastest when the distance a microbe swims before realigning with the magnetic field matches the pore size. Microbes that reorient too quickly get trapped; those that align too slowly wander. Additionally, because the efficiency of magnetotaxis only depends on the ratio of environmental factors and bacterial phenotype, there is no single characteristic that can be tuned to allow optimal navigation. Rather, optimality is achieved when different phenotypic characteristics balance one another. For example, small cells with weak magnetic moments perform as well as large cells with strong ones.

Magnetotactic bacteria have found a remarkable solution to navigating complex environments. Natural selection has shaped their phenotypes so that the environmental randomness is mirrored in their trajectories. Similar strategies may also apply to artificial navigation in complex settings, such as robots designed to move through rubble. To this end, it could be useful to design magnetotactic robots.

Magnetotactic bacteria represent a phylogenetically diverse group—including species from Desulfobacterota, several branches of Proteobacteria, and Nitrospira—that use the geomagnetic field to navigate water-saturated sediment (Lefèvre et al., 2014; Delong et al., 1993). The dynamics of magnetotaxis are similar across all known bacterial examples (Klumpp et al., 2019; Frankel et al., 2007). A cell, or consortium of cells (Abreu et al., 2007; Schaible et al., 2024), uses flagella to exert a force that is parallel (Frankel, 1984; Esquivel and De Barros, 1986) to its permanent magnetic moment, which is produced by biologically precipitated inclusions of magnetite or greigite called magnetosomes (Faivre and Schüler, 2008). The magnetic torque on a bacterium turns it like a compass needle to align its magnetic moment with magnetic field lines. This motion is purely physical and requires no behavioral adaptations by the organism (Frankel, 1984; Esquivel and De Barros, 1986). As magnetotactic bacteria swim parallel to their magnetic moments, they move along field lines. Because the geomagnetic field is rarely parallel to the sediment surface, magnetotaxis moves bacteria vertically through the sediment, along the dominant chemical gradients that shape microbial life in the top several centimeters of sediment (Blakemore and Frankel, 1981; Lefèvre et al., 2014). This biased motion allows magnetotactic bacteria to position themselves in changing chemical gradients (Bazylinski and Frankel, 2004; Smith et al., 2006) and to shuttle between vertically stratified chemical environments (Li et al., 2020).

Despite the similarity of their motion, a comparison of different species reveals tremendous diversity (Lefèvre et al., 2014). Magnetotactic bacteria may be cocoidal (Keim et al., 2007; Acosta-Avalos et al., 2019), rod-like (Spring et al., 1993), or spiral (Bazylinski and Frankel, 2004) with lengths between 1 μm and 20 μm (Esquivel and De Barros, 1986). While most species are unicellular, multicellular consortia are also common (Lefèvre et al., 2014). Swimming speeds vary by an order of magnitude, from 12 μm/s (Esquivel and De Barros, 1986) to more than 140 μm/s (Bente et al., 2020). Magnetic moments vary by two orders of magnitude, from ≈0.5 fAm² to 54 fAm² (Esquivel and De Barros, 1986). A table of the typical phenotypic characteristics of diverse magnetotactic bacteria is available in Appendix 1.

Here, we investigate how magnetotactic bacteria tune their phenotypes in order to balance motion along magnetic field lines with the necessity to move around obstacles such as sand grains. We have previously argued (Petroff et al., 2022) that there is an optimal combination of magnetic moment, cell size, and swimming speed that maximizes the average speed of magnetotactic bacteria through a pore space. This hypothesis was recently independently proposed and tested by Codutti et al., 2024, who found that the average speed of magnetotactic bacteria through an artificial pore space is indeed maximized if the magnetic torque on a bacterium is tuned. This analysis is closely related to the work of Dehkharghani et al., 2023, who examined the effect of obstacle geometry on motion of magnetotactic bacteria through a complex pore space. In this paper, we first show that the speed of a specific type of magnetotactic bacteria through an artificial pore space is maximized if the distance it swims while aligning with the ambient magnetic field is tuned to the pore size. Proceeding from a literature review, we then argue that a variety of unrelated magnetotactic bacteria from around the world are all optimally efficient navigators of their native pore spaces.

We choose to study members of the Desulfobacterota genus Magnetoglobus, which are generally referred to as multicellular magnetotactic bacteria (MMB) (Abreu et al., 2007). MMB are the only type of bacteria that are known to lack a unicellular stage in their life cycle (Keim et al., 2004). Groups of several tens of non-clonal cells (Schaible et al., 2024) are physically attached together and coordinate their motion and growth to behave like a single multicellular organism, which is called a consortium. Individual bacteria are not observed to join or leave existing consortia. Rather, new consortia form when existing consortia divide symmetrically (Keim et al., 2004). MMB consortia are spherical with a typical diameter $a=3-8$ μm and composed of a monolayer of cells (Leão et al., 2017). Each cell in a consortium has tens of magnetosomes and its outer surface is covered in about 30 flagella (Rodgers et al., 1991). Cells within a consortium coordinate their activity to exert a net force that is parallel to the consortium’s net magnetic moment (Almeida et al., 2013), which varies among individuals from 3.9 fAm² to 10.9 fAm², enabling them to swim at speeds ranging from 45 μm/s to 134 μm/s (Petroff et al., 2022).

MMB provide an apt system for the broader study of magnetotaxis and its evolution for three reasons. As these consortia are significantly larger than those studied in previous investigations (Codutti et al., 2024), it is comparatively simpler to track their motion through a pore space. Additionally, as these bacteria are taken directly from their natural habitat, their phenotypes do not reflect the process of domestication. These bacteria are readily enriched from tide pools in Little Sippewissett Salt Marsh (Simmons and Edwards, 2007; Shapiro et al., 2011; Schaible et al., 2024), near Falmouth, Massachusetts. Finally, the physical characteristics of their habitat can be directly measured. These bacteria live in the spaces between (Martins et al., 2009; Simmons et al., 2007; Keim et al., 2007), which we measure to have a mean diameter of 0.6 ± 0.16 mm. A random packing of such grains forms pores with a typical radius (Andreotti et al., 2013) of $r=0.2\pm 0.05\mathrm{m}\mathrm{m}$. The local geomagnetic field (NOAA, 2024) is 50.9 μT, with an inclination of 65.6°.

Figure 1a shows the two-dimensional artificial pore space in which we track the motion of consortia. It is composed of a square lattice of convex pores. When a magnetotactic bacterium collides with a convex barrier, it must temporarily swim against the applied magnetic (which points from left to right in Figure 1) in order to find a passageway leading forward. Because the geometry of these convexities can be precisely controlled, the study of locomotion through this pore space provides a useful experimental system in which to study the dynamics of backtracking. While convex pores cannot be formed by convex particles (e.g. spheres), there is good reason to believe that backtracking is necessary for efficient navigation in natural sediment where clogging of pores creates pore-scale dead ends and convexities. Biofilms growing between arrays of circular posts in a microfluidic device are observed to connect neighboring posts (Kurz et al., 2023; Hassanpourfard et al., 2016; Gaol et al., 2021), thus forming barriers that include both convex and concave sections and create dead ends in the pore space. Similar features are found in three-dimensional simulations of biofilm growth between spherical particles (Peszynska et al., 2016; von der Schulenburg et al., 2009). Additionally, muddy pore water contains many small cohesive particles. When these particles move through narrow passageways, they clog (Dressaire and Sauret, 2017; Yin et al., 2024) and create boundaries similar to those shown in Figure 1a. Air bubbles can also create pore-scale dead ends that would require backtracking (Wang et al., 2021). The concentration of convex traps present in the pore network analyzed here is almost certainly much greater than in any natural habitat. Nonetheless, as magnetotactic bacteria constantly swim through the pore space, they will inevitably encounter convexities and pore-scale dead ends. Efficient navigation thus requires the ability to backtrack. In order to understand the phenotypic traits that allow efficient backtracking, we choose to analyze a pore space with many convex traps.

Figure 1

Download asset Open asset

An applied magnetic field directs MMB through the pore network. The field is oriented 45° off the primary axis of the lattice (i.e. from left to right in Figure 1) to ensure that every field line intersects the walls of a pore. Note that a square lattice of circular pores would align the shortest path through the pore, from the entrances on the left to the exits on the right in Figure 1, with the magnetic field. To remove this unrealistic artifact of a highly symmetric pore space, we add small rectangular barriers to the top and bottom of each circular pore.

We observe that as the applied magnetic field is reduced, the consortia explore an increasingly larger fraction of the pore. At very high magnetic fields (Figure 1b, Video 1), consortia cannot turn away from a field line and so cannot move around an obstruction from one pore to the next. As they become trapped against pore boundaries for prolonged periods, magnetotaxis is slowed. At an intermediate magnetic field (Figure 1c, Video 1), consortia concentrate near the connections that lead northward and rapidly move through the network. At low magnetic field (Figure 1d, Video 1), consortia are uniformly distributed through the pore and often escape southward. In the limit of vanishing magnetic field, the motion is unbiased and magnetotaxis is obviously impossible. Consequently, we expect that maximizing the average speed $U_{\mathrm{d}\mathrm{r}\mathrm{i}\mathrm{f}\mathrm{t}}$ of a consortium through a given pore space requires a balance between magnetotaxis and obstacle avoidance. If consortia align with the magnetic field too quickly or too slowly, this balance is broken, and the organism becomes trapped or wanders randomly.

Video 1

Download asset

Dimensional analysis (Bridgman, 1922) gives insight into how this balance is reflected in the phenotype of an arbitrary magnetotactic bacterium and its environment. For pores of a given shape, the important aspects of the environment are described by the ambient magnetic field intensity $B$ and the pore radius $r$. Five aspects of the phenotype may effect $U_{\mathrm{d}\mathrm{r}\mathrm{i}\mathrm{f}\mathrm{t}}$. They are as follows: the swimming speed $U_0$, the magnetic moment $m$, the length $a$ of the organism, its rotational diffusion coefficient $D_{\mathrm{r}\mathrm{o}\mathrm{t}}$, and its rotational hydraulic mobility ${\mu }_{\mathrm{r}\mathrm{o}\mathrm{t}}$. The mobility ${\mu }_{\mathrm{r}\mathrm{o}\mathrm{t}}\propto 1/(\eta a^3)$—where $\eta$ is the viscosity of water and the proportionality constant is determined by the shape of the organism—relates to torque on the organism to its angular velocity. As these eight variables share four physical dimensions (charge, length, time, and mass), it follows that the efficiency $U_{\mathrm{d}\mathrm{r}\mathrm{i}\mathrm{f}\mathrm{t}}/U_0$ can be expressed as a function of three dimensionless combinations of phenotypic and environmental variables.

We call the first, and most important, of these dimensionless numbers the scattering number

where $\gamma ={\mu }_{\mathrm{r}\mathrm{o}\mathrm{t}}mB$ is the rate at which a magnetotactic bacterium aligns with the ambient magnetic field. Sc has two useful and equivalent interpretations. As bacteria are rapidly scattered by collisions with surfaces (Drescher et al., 2011; Petroff and McDonough, 2024; Lauga, 2016), it compares the rate $U_0/r$ at which an organism is randomly reoriented by collisions, to the rate at which it aligns with the magnetic field. Alternatively, a magnetotactic bacterium swims a distance of order $U_0/\gamma$ in the time required for it to realign with the magnetic field after a random reorientation. Consequently, if $\text{Sc}<1$, Sc is the typical fraction of the pore that is explored by a magnetotactic bacterium. As described in Materials and methods, we measure Sc for each enrichment of consortia immediately before it enters the pore space.

The next dimensionless number is $a/r$. One expects that cell lengths $a=1-20\mu \mathrm{m}$ are small compared to the size of pores they swim through. While papers rarely report the grain size of the sediment from which magnetotactic bacteria are enriched, it is reasonable to assume that the tidal ponds, marshes, estuaries, and rivers are composed of grains similar to medium or coarse sand. Such is the case in the ponds of Massachusetts, where we enrich MMB and measure $r=0.2\pm 0.05\mathrm{m}\mathrm{m}$. As $a/r\approx 0.05$, we conclude that the excluded volume of bacteria does not meaningfully affect their ability to navigate. In what follows, we assume that unreported pore radii are within a factor of 2 of $0.2\pm 0.05\mathrm{m}\mathrm{m}$.

The final dimensionless number is $D_{\mathrm{r}\mathrm{o}\mathrm{t}}r/U_0$. This ratio compares the rate that a magnetotactic bacterium is reoriented by rotational diffusion to the rate it is reoriented by collisions. The importance of rotational diffusion on the motion of Magnetococcus marinus through a pore space has been previously examined (Dehkharghani et al., 2023). In natural sediment, $D_{\mathrm{r}\mathrm{o}\mathrm{t}}r/U_0$ takes values of 0.4 ± 0.1 for MMB and, we estimate (Dehkharghani et al., 2023), 0.08–0.3 for M. marinus. In the experiments presented here, the pore sizes are smaller than in nature and $0.15<D_{\mathrm{r}\mathrm{o}\mathrm{t}}r/U_0<0.25$. Figure 1d and Video 2 show roughly straight trajectories at low magnetic field. As $D_{\mathrm{r}\mathrm{o}\mathrm{t}}r/U_0$ is reasonably small, we conclude that the largest changes in orientation are the result of collisions with walls rather than rotational diffusion.

Video 2

Download asset

Thus, we find that magnetotactic bacteria can be treated as point-like swimmers that turn deterministically to align with the ambient magnetic field and are randomly scattered by collisions with the pore boundaries. In this limit

where $f$ is an unknown function.

We proceed to measure $f(\text{Sc})$ from the rates at which MMB transition between pores. To better visualize these pore-scale dynamics, we reduce the full pore network (Figure 1a) to the miniature network shown in Figure 2a. We define $k_{+}$ as the rate at which consortia move in the direction of the magnetic field from one pore to another. The corresponding rate for motion against the field is $k_{-}$. As shown in Appendix 2, the continuum limit of this pore network yields

Figure 2 with 2 supplements see all

Download asset Open asset

Figure 2—source data 1

Experimental data for transition rates of multicellular magnetotactic bacteria (MMB) between chambers.

The file includes the scattering number, the dimensionless transition rates $a{k}_{\pm }/U_0$, the dimensionless drift velocities $U_{\mathrm{D}\mathrm{r}\mathrm{i}\mathrm{f}\mathrm{t}}/U_0$ for each experiment, and the associated uncertainty in all of these quantities.

https://cdn.elifesciences.org/articles/104797/elife-104797-fig2-data1-v1.csv

Download elife-104797-fig2-data1-v1.csv

Figure 2—source data 2

Theoretical data for the transition rates between chambers.

The file includes the scattering number at which the simulation was conducted, the predicted dimensionless drift velocity, and the numerical uncertainty in this quantity.

https://cdn.elifesciences.org/articles/104797/elife-104797-fig2-data2-v1.csv

Download elife-104797-fig2-data2-v1.csv

where the geometric prefactor $\alpha =0.8344$ is determined by the ratio of the lattice spacing to pore size. We measure $k_{+}$ and $k_{-}$ from the number $N(t)$ of consortia in a given pore as a function of time $t$ and the times that consortia move to neighboring pores. If we observe $n_{+}$ transitions to a northward pore in a certain amount of time $T$, the best estimate of $k_{+}=n_{+}({\int }_0^{T}N(t)\mathrm{d}t{)}^{-1}$. We estimate $k_{-}$ in a similar manner. We experimentally vary the scattering number by adjusting the magnitude of the applied magnetic field.

Figure 2b shows how the transition rates vary as Sc increases from 0.14 ($B=5000$ μT) to 9.3 ($B=75$ μT). The rate $k_{+}$ of northward motion is non-monotonic and is maximized at a value of Sc where consortia concentrate near the connectors to the northward pores (see Figure 1c). The rate of southward migration increases monotonically with Sc as consortia explore an ever-growing fraction of the pore space. The drift velocity (Figure 2c) is non-monotonic and reaches its maximal value of $\approx 0.25U_0$ at a critical scattering number of ${\text{Sc}}_{\mathrm{c}}\approx 0.4$.

One can crudely estimate the critical scattering number directly from the pore geometry. As the scattering number gives the typical fraction of the pore that a consortium explores, we expect that ${\text{Sc}}_{\mathrm{c}}\sim \delta /r$, where δ (see Figure 2a) is the distance between the northernmost pore wall and the exit to the next pore. In this pore network, $\delta /r=1-2^{-1/2}\approx 0.29$, which is similar to ${\text{Sc}}_{\mathrm{c}}$.

Next, we compare these measurements to the motion of simulated consortia. In light of the dimensional analysis presented above, we approximate the consortia as points that swim parallel to their magnetic moments at a constant speed $U_0$. The angle θ between the direction of motion and the magnetic field evolves in time $t$ as $\mathrm{d}\theta /\mathrm{d}t=-\gamma \sin (\theta )$. We assume that, after a collision with a boundary, the angle at which consortia escape the wall is uniformly distributed. Non-dimensionalizing time by $1/\gamma$ and distances by $r$ reveals the dimensionless swimming speed to be Sc. We simulate 10³ consortia in the pore geometry shown in Figure 2a and calculate the time it takes each to escape to a neighboring pore. As discussed in Materials and methods, these first passage times are exponentially distributed and the decay constants give $k_{+}$ and $k_{-}$ for any particular choice of Sc. Figure 2—video 2 shows a representative simulation. Repeating this process for 100 choices of Sc provides predictions for $k_{+}$, $k_{-}$, and $f(\text{Sc})$ (see Figure 2c) with no free parameters. In the particular geometry shown in Figure 2a, we predict that the drift velocity is maximized at a critical scattering number of ${\text{Sc}}_{\mathrm{c}}=0.40$, which is found by fitting the maximum of the smooth curve in Figure 2c to a parabola. Details of these simulations are provided in Materials and methods.

The good agreement (Figure 2d) between the predicted transition rates and our measurements leads us to accept the hypothesis that the magnetotactic efficiency $U_{\mathrm{d}\mathrm{r}\mathrm{i}\mathrm{f}\mathrm{t}}/U_0$ is maximized at a finite value of the scattering number. We next ask if natural consortia are optimized for their native environments.

Answering this question requires an estimate of the critical scattering number for natural pore geometries. The shape of the smooth function $f(\text{Sc})$ shown in Figure 2c is generic but not universal. It necessarily goes to zero at very high scattering number (e.g. $B=0$, see Figure 1d), where magnetotactic organisms do not align with the magnetic field. It similarly vanishes if $\text{Sc}=0$ (e.g. $B=\mathrm{\infty }$, see Figure 1b), where organisms cannot swim around obstructions. These limits imply a global maximum at a critical value of ${\text{Sc}}_{\mathrm{c}}$, where the random fluctuations in swimming orientation due to collisions with boundaries regularly direct consortia toward northward pores, but not so great that consortia routinely escape to southward pores.

We estimate that, in natural sediment, $0.1⪅{\text{Sc}}_{\mathrm{c}}⪅1$. As Sc gives the typical fraction of the pore that a consortium explores and passageways leading north are most likely concentrated along the north-facing pore wall, we expect that the critical scattering number should be slightly less than 1. Moreover, a consortium cannot explore a pore if the distance $U_0/\gamma$ it is scattered after collisions is smaller than its body size $a$ (Petroff et al., 2022). It follows that ${\text{Sc}}_{\mathrm{c}}>a/r\sim 0.05$. Appendix 3 derives a similar estimate of ${\text{Sc}}_{\mathrm{c}}$ proceeding from the assumption that the locations of connections between pores are randomly distributed.

In their natural habitat, consortia swim through pores with a typical radius $r\approx 0.2\mathrm{m}\mathrm{m}$ and align with a local geomagnetic field of magnitude $B_{\mathrm{g}\mathrm{e}\mathrm{o}}=50.9\mu \mathrm{T}$. As described in Materials and methods, we measure the average geomagnetic turning rate $⟨{\gamma }_{\mathrm{g}\mathrm{e}\mathrm{o}}⟩=1.1\pm 0.2\mathrm{s}$ and average swimming speed $⟨U_0⟩=132\pm 7\mu \mathrm{m}/\mathrm{s}$ of 31 individual consortia. These values correspond to a population-averaged scattering number of 0.6±0.2, which is consistent with optimal navigation. There is substantial variability across individuals. While the majority of individuals fall in the predicted range and the smallest scattering number is 0.2, five individuals display scattering numbers between 2 and 3.2. Our theory cannot explain the tail of this distribution. It is plausible that a minority of consortia are close to division (Keim et al., 2004) and are unusually large and thus slowly turning. Alternatively, this variability could be a reflection of the polydisperse grains in which these consortia live.

Several years ago, we described (Petroff et al., 2022) the motion of consortia that were enriched from the same pond as those described here. These consortia were systematically slower, with a typical swimming speed of just 75 μm/s. Nonetheless, the measured values for $U_0$, $m$, and $a$ of these consortia correspond to $\text{Sc}=0.4\pm 0.3$. We conclude that, despite the substantial phenotypic variability of this population across time, the scattering number is nearly constant and remains consistent with optimal navigation.

Finally, we turn our attention to the phylogenetic diversity of magnetotactic bacteria. A literature review (Petersen et al., 1989; Spring et al., 1993; Pan et al., 2009; Nadkarni et al., 2013; Bahaj et al., 1996; Esquivel and De Barros, 1986; Acosta-Avalos et al., 2019; Petroff et al., 2022; Carvalho et al., 2021) reveals tremendous phenotypic variability across taxa and environment. As described in Appendix 1, species from three phyla are included from latitudes between 22° S and 54° N. The geomagnetic field strengths across these latitudes range from $B_{\mathrm{g}\mathrm{e}\mathrm{o}}=23$ μT with an inclination of -42°, in Rio de Janeiro, Brazil, to $B_{\mathrm{g}\mathrm{e}\mathrm{o}}=50.4$ μT with an inclination of 69° near Grimmen, Germany. The sizes (1–18 μm) and speeds (12–141 μm/s) of magnetotactic bacteria both vary by an order of magnitude. The magnetic moments (0.3–54 fAm²) of the species vary by two orders of magnitude.

Despite this enormous variability across species, nearly all of these magnetotactic bacteria display scattering numbers that are within the expected range of $0.1⪅{\text{Sc}}_{\mathrm{c}}⪅1$. Figure 3a shows the scattering numbers for these species. These estimates use the local geomagnetic field and a pore size of medium to coarse sand. The average scattering number is 0.58, and individual values vary from 0.1 to 1.9.

Figure 3

Download asset Open asset

As the scattering number is determined by the product of three phenotypic variables that all vary by more than an order of magnitude, it is surprising that the scattering numbers are as narrowly distributed as they are. To quantify this peculiarity, we consider an unbiased null model of phenotypic diversity. We assume that collision rates $U_0/r$, hydrodynamic mobilities ${\mu }_{\mathrm{r}\mathrm{o}\mathrm{t}}$, and geomagnetic torques $m{B}_{\mathrm{g}\mathrm{e}\mathrm{o}}$ are independently tuned to the local environment. The distributions of these traits reflect the range of biologically accessible phenotypes and the relative abundances of ecological conditions that select for the particular value of each trait. We randomly generate ecologically plausible phenotypes through a bootstrapping procedure; we sample from the measured phenotypic variables with replacement.

When compared to random phenotypes, the scattering numbers of real magnetotactic bacteria are anomalously small and narrowly distributed. The average scattering number of randomly generated phenotypes is 9.5 with 95% of these values falling between $2\times {10}^{-3}$ and 72.8. Thus, in the absence of selective pressure, we would expect to find substantially more variability in the scattering numbers between species. Most of these species would move diffusively through the pore space. The probability that 15 random phenotypes fall within the measured range of naturally occurring scattering numbers is $2.5\times {10}^{-6}$.

We conclude that natural selection has tuned the scattering numbers of magnetotactic bacteria in a way that is consistent with optimal navigation. This tuning is reflected in correlations between the speeds, sizes, shapes, and magnetic moments of magnetotactic bacteria, which are absent in the random phenotypes.

The scaling analysis also predicts the specific correlation between phenotypic variables. Optimal navigation requires the scattering rate $U_0/r$ of each species to be proportional to its geomagnetic alignment rate ${\gamma }_{\mathrm{g}\mathrm{e}\mathrm{o}}$. As shown in Figure 3b, these rates are indeed proportional. The correlation coefficient is 0.77. The probability that the scattering numbers of 15 random phenotypes show correlations that are this large is $4\times {10}^{-4}$. Note that if flagella and magnetosomes compete for resources or energy, then $U_0$ and ${\gamma }_{\mathrm{g}\mathrm{e}\mathrm{o}}$ would be negatively correlated. As discussed in Appendix 4, correlations between these phenotypic parameters do not reflect the dependence of each upon body size.

There is clearly substantial scatter in Figure 3. This scatter likely reflects two aspects of magnetotactic motility that do not generalize well from the study of MMB. First, the orientation of swimming MMB is decorrelated primarily by collisions with boundaries, and the effect of rotational diffusion can be ignored. It is likely that rotational diffusion may be more important for smaller magnetotactic bacteria. In such cases, the critical scattering number becomes a function of $D_{\mathrm{r}\mathrm{o}\mathrm{t}}r/U_0$. Additionally, when an MMB collides with a surface, the angle at which it escapes can be approximated as uniformly distributed. If microbes swim along surfaces (Ostapenko et al., 2018; Codutti et al., 2022) for a distance $\ell$ and escape at typical angles (Kühn et al., 2017; Petroff and McDonough, 2024), then the critical scattering number also depends on $\ell /r$ and the moments of the escape angle distribution. There is evidence of these effects in Figure 3. The Desulfobacterota live at scattering numbers of 0.9±0.3 that are somewhat greater than those of the α-Proteobacteria, which live at scattering number 0.37 ±0.03. It is plausible that this difference reflects taxonomic differences in rotational diffusion and cell-wall interactions. However, as the scattering numbers of these bacteria remain within the expected range of values, it seems that these corrections are of secondary importance.

We conclude that all magnetotactic bacteria discussed here are, to a good approximation, equally and optimally efficient navigators of their native pore spaces.

We have found that in an ecologically relevant limit, the efficiency of magnetotaxis is a function of a single dimensionless combination of environmental and phenotypic parameters. The optimal phenotype swims a distance that is similar to the pore radius before aligning with the magnetic field. Doing so causes magnetotactic bacteria to concentrate in the fraction of the pore where escapes are most likely to be found. Because the efficiency of magnetotaxis depends only on the ratio of phenotypic and environmental parameters, it can be carried out equally well by magnetotactic bacteria with different phenotypes. This degeneracy is found to accommodate the natural diversity of magnetotactic bacteria.

As the magnetic moment $m=U_0/({\text{Sc}}_{\mathrm{c}}r{\mu }_{\mathrm{r}\mathrm{o}\mathrm{t}}{B}_{\mathrm{g}\mathrm{e}\mathrm{o}})$, the results presented in Figure 3 are equivalent to showing that this analysis predicts the magnetic moments of unrelated magnetotactic bacteria across two orders of magnitude. This rescaling is difficult in practice as the rotational hydraulic mobility ${\mu }_{\mathrm{r}\mathrm{o}\mathrm{t}}$ of the organisms can only be roughly estimated from their shapes. The magnitude of the magnetic moment is not determined by the rotational diffusion of the magnetotactic bacteria, which we find to be of secondary importance. Rather, as has been previously suggested (Dehkharghani et al., 2023), the orientation of a swimmer is decorrelated primarily by collisions with the pore boundaries. Our results agree well with previous work (Codutti et al., 2024; Petroff et al., 2022), which shows that too great a magnetic moment ($\text{Sc}\ll {\text{Sc}}_{\mathrm{c}}$, see Figure 1b) is as detrimental to magnetotaxis as a magnetic moment that is too small ($\text{Sc}\gg {\text{Sc}}_{\mathrm{c}}$, see Figure 1d).

Insofar as the ability to quickly move through the pore space aids bacteria in reproducing (Smith et al., 2006), $f(\text{Sc})$ can be considered a one-dimensional slice of a fitness landscape (Bank, 2022). Figure 2 shows that this slice is smooth and displays a single global maximum. Reaching this maximum requires natural selection to tune a species’ phenotype to reflect its environment such that $U_0/{\mu }_{\mathrm{r}\mathrm{o}\mathrm{t}}m={\text{Sc}}_{\mathrm{c}}{B}_{\mathrm{g}\mathrm{e}\mathrm{o}}r$. This result suggests that if a bacterium acquires the genes for magnetosome formation and alignment—either de novo or through horizontal gene transfer—then its magnetotactic efficiency is immediately nonzero. The smoothness of $f(\text{Sc})$ suggests that natural selection can quickly guide the phenotypic variability in size, swimming speed, and magnetic moment to greater efficiencies. Studies in which the genes for magnetosome formation are added to non-magnetotactic bacteria (Kolinko et al., 2014; Dziuba et al., 2024) could provide a strong test of this prediction.

Efficient magnetotaxis imposes a trade-off between size (through ${\mu }_{\mathrm{r}\mathrm{o}\mathrm{t}}$), speed, and magnetic moment. A magnetotactic bacterium can be fast swimming and large only if it produces a very large number of magnetosomes. The constraint that $U_0/r{\gamma }_{\mathrm{g}\mathrm{e}\mathrm{o}}={\text{Sc}}_{\mathrm{c}}$ defines a two-dimensional surface in a three-dimensional space of phenotypes. Any species with phenotypic variables laying on this surface is optimally efficient. This degeneracy allows a species to remain optimally efficient at magnetotaxis while tuning any two of its three phenotypic parameters to other constraints imposed by its ecology. The phenotypic flexibility provided by this degeneracy gives insight into how magnetotaxis became so widely distributed across bacteria with diverse phylogenies, morphologies, and ecological roles.

The experimental data in figure 2 is available in Figure 2—source data 1, the corresponding data from the simulations is available in Figure 2—source data 2. The data for figure 3 are recorded in Appendix 1.

1. 1. Abreu F
   2. Martins JL
   3. Silveira TS
   4. Keim CN
   5. de Barros HGPL
   6. Filho FJG
   7. Lins U

   (2007) “Candidatus Magnetoglobus multicellularis”, a multicellular, magnetotactic prokaryote from a hypersaline environment

   International Journal of Systematic and Evolutionary Microbiology 57:1318–1322.

   https://doi.org/10.1099/ijs.0.64857-0

   • PubMed
   • Google Scholar
2. 1. Acosta-Avalos D
   2. de Figueiredo AC
   3. Conceição CP
   4. da Silva JJP
   5. Aguiar K
   6. de Lima Medeiros M
   7. do Nascimento M
   8. de Melo RD
   9. Sousa SMM
   10. de Barros HL
   11. Alves OC
   12. Abreu F

   (2019) U-turn trajectories of magnetotactic cocci allow the study of the correlation between their magnetic moment, volume and velocity

   European Biophysics Journal 48:513–521.

   https://doi.org/10.1007/s00249-019-01375-2

   • PubMed
   • Google Scholar
3. 1. Almeida FP
   2. Viana NB
   3. Lins U
   4. Farina M
   5. Keim CN

   (2013) Swimming behaviour of the multicellular magnetotactic prokaryote “Candidatus Magnetoglobus multicellularis” under applied magnetic fields and ultraviolet light

   Antonie van Leeuwenhoek 103:845–857.

   https://doi.org/10.1007/s10482-012-9866-0

   • PubMed
   • Google Scholar
4. Book

   1. Andreotti B
   2. Forterre Y
   3. Pouliquen O

   (2013) Granular Media: Between Fluid and Solid

   Cambridge University Press.

   https://doi.org/10.1017/CBO9781139541008

   • Google Scholar
5. 1. Bahaj AS
   2. James PAB
   3. Moeschler FD

   (1996) An alternative method for the estimation of the magnetic moment of non-spherical magnetotactic bacteria

   IEEE Transactions on Magnetics 32:5133–5135.

   https://doi.org/10.1109/20.539514

   • Google Scholar
6. 1. Bank C

   (2022) Epistasis and adaptation on fitness landscapes

   Annual Review of Ecology, Evolution, and Systematics 53:457–479.

   https://doi.org/10.1146/annurev-ecolsys-102320-112153

   • Google Scholar
7. 1. Bazylinski DA
   2. Frankel RB

   (2004) Magnetosome formation in prokaryotes

   Nature Reviews Microbiology 2:217–230.

   https://doi.org/10.1038/nrmicro842

   • Google Scholar
8. 1. Bente K
   2. Mohammadinejad S
   3. Charsooghi MA
   4. Bachmann F
   5. Codutti A
   6. Lefèvre CT
   7. Klumpp S
   8. Faivre D

   (2020) High-speed motility originates from cooperatively pushing and pulling flagella bundles in bilophotrichous bacteria

   eLife 9:e47551.

   https://doi.org/10.7554/eLife.47551

   • PubMed
   • Google Scholar
9. 1. Blakemore RP
   2. Frankel RB

   (1981) Magnetic Navigation in Bacteria

   Scientific American 245:58–65.

   https://doi.org/10.1038/scientificamerican1281-58

   • Google Scholar
10. Book

    1. Bridgman PW

    (1922)

    Dimensional Analysis

    Yale university press.

    • Google Scholar
11. 1. Carvalho AL
    2. Abreu F
    3. Acosta-Avalos D

    (2021) Multicellularity makes the difference: multicellular magnetotactic prokaryotes have dynamic motion parameters dependent on the magnetic field intensity

    The European Physical Journal Plus 136:203.

    https://doi.org/10.1140/epjp/s13360-021-01187-4

    • Google Scholar
12. 1. Codutti A
    2. Charsooghi MA
    3. Cerdá-Doñate E
    4. Taïeb HM
    5. Robinson T
    6. Faivre D
    7. Klumpp S

    (2022) Interplay of surface interaction and magnetic torque in single-cell motion of magnetotactic bacteria in microfluidic confinement

    eLife 11:e71527.

    https://doi.org/10.7554/eLife.71527

    • PubMed
    • Google Scholar
13. Preprint

    1. Codutti A
    2. Charsooghi MA
    3. Marx K
    4. Cerdá-Doñate E
    5. Munoz O
    6. Zaslansky P
    7. Telezki V
    8. Robinson T
    9. Faivre D
    10. Klumpp S

    (2024) Escape problem of magnetotactic bacteria - physiological magnetic field strengths help magnetotactic bacteria navigate in simulated sediments

    bioRxiv.

    https://doi.org/10.1101/2023.12.08.570788

    • Google Scholar
14. 1. Dehkharghani A
    2. Waisbord N
    3. Guasto JS

    (2023) Self-transport of swimming bacteria is impaired by porous microstructure

    Communications Physics 6:1136.

    https://doi.org/10.1038/s42005-023-01136-w

    • Google Scholar
15. 1. Delong EF
    2. Frankel RB
    3. Bazylinski DA

    (1993) Multiple evolutionary origins of magnetotaxis in bacteria

    Science 259:803–806.

    https://doi.org/10.1126/science.259.5096.803

    • PubMed
    • Google Scholar
16. 1. Drescher K
    2. Dunkel J
    3. Cisneros LH
    4. Ganguly S
    5. Goldstein RE

    (2011) Fluid dynamics and noise in bacterial cell–cell and cell–surface scattering

    PNAS 108:10940–10945.

    https://doi.org/10.1073/pnas.1019079108

    • Google Scholar
17. 1. Dressaire E
    2. Sauret A

    (2017) Clogging of microfluidic systems

    Soft Matter 13:37–48.

    https://doi.org/10.1039/C6SM01879C

    • Google Scholar
18. 1. Dziuba MV
    2. Müller FD
    3. Pósfai M
    4. Schüler D

    (2024) Exploring the host range for genetic transfer of magnetic organelle biosynthesis

    Nature Nanotechnology 19:115–123.

    https://doi.org/10.1038/s41565-023-01500-5

    • PubMed
    • Google Scholar
19. 1. Esquivel SDM
    2. De Barros HGPL

    (1986) Motion of magnetotactic microorganisms

    Journal of Experimental Biology 121:153–163.

    https://doi.org/10.1242/jeb.121.1.153

    • Google Scholar
20. 1. Faivre D
    2. Schüler D

    (2008) Magnetotactic bacteria and magnetosomes

    Chemical Reviews 108:4875–4898.

    https://doi.org/10.1021/cr078258w

    • Google Scholar
21. 1. Frankel RB

    (1984) Magnetic guidance of organisms

    Annual Review of Biophysics and Bioengineering 13:85–103.

    https://doi.org/10.1146/annurev.bb.13.060184.000505

    • PubMed
    • Google Scholar
22. Book

    1. Frankel RB
    2. Williams TJ
    3. Bazylinski DA

    (2007) Magneto-aerotaxis

    In: Frankel RB, Williams TJ, editors. Magnetoreception and Magnetosomes in Bacteria. Springer. pp. 1–24.

    https://doi.org/10.1007/7171_2006_036

    • Google Scholar
23. 1. Gaol CL
    2. Ganzer L
    3. Mukherjee S
    4. Alkan H

    (2021) Investigation of clogging in porous media induced by microorganisms using a microfluidic application

    Environmental Science 7:441–454.

    https://doi.org/10.1039/D0EW00766H

    • Google Scholar
24. 1. Greenberg M
    2. Canter K
    3. Mahler I
    4. Tornheim A

    (2005) Observation of magnetoreceptive behavior in a multicellular magnetotactic prokaryote in higher than geomagnetic fields

    Biophysical Journal 88:1496–1499.

    https://doi.org/10.1529/biophysj.104.047068

    • PubMed
    • Google Scholar
25. 1. Hassanpourfard M
    2. Ghosh R
    3. Thundat T
    4. Kumar A

    (2016) Dynamics of bacterial streamers induced clogging in microfluidic devices

    Lab on a Chip 16:4091–4096.

    https://doi.org/10.1039/c6lc01055e

    • PubMed
    • Google Scholar
26. 1. Keim CN
    2. Martins JL
    3. Abreu F
    4. Rosado AS
    5. de Barros HL
    6. Borojevic R
    7. Lins U
    8. Farina M

    (2004) Multicellular life cycle of magnetotactic prokaryotes

    FEMS Microbiology Letters 240:203–208.

    https://doi.org/10.1016/j.femsle.2004.09.035

    • PubMed
    • Google Scholar
27. Book

    1. Keim CN
    2. LopesMartins J
    3. Lins de Barros H

    (2007)

    Structure, behavior, ecology and diversity of multicellular magnetotactic prokaryotes

    In: Keim CN, editors. Magnetoreception and Magnetosomes in Bacteria. Springer. pp. 103–132.

    • Google Scholar
28. 1. Klumpp S
    2. Lefèvre CT
    3. Bennet M
    4. Faivre D

    (2019) Swimming with magnets: From biological organisms to synthetic devices

    Physics Reports 789:1–54.

    https://doi.org/10.1016/j.physrep.2018.10.007

    • Google Scholar
29. 1. Kolinko I
    2. Lohße A
    3. Borg S
    4. Raschdorf O
    5. Jogler C
    6. Tu Q
    7. Pósfai M
    8. Tompa E
    9. Plitzko JM
    10. Brachmann A
    11. Wanner G
    12. Müller R
    13. Zhang Y
    14. Schüler D

    (2014) Biosynthesis of magnetic nanostructures in a foreign organism by transfer of bacterial magnetosome gene clusters

    Nature Nanotechnology 9:193–197.

    https://doi.org/10.1038/nnano.2014.13

    • PubMed
    • Google Scholar
30. 1. Kühn MJ
    2. Schmidt FK
    3. Eckhardt B
    4. Thormann KM

    (2017) Bacteria exploit a polymorphic instability of the flagellar filament to escape from traps

    PNAS 114:6340–6345.

    https://doi.org/10.1073/pnas.1701644114

    • Google Scholar
31. 1. Kurz DL
    2. Secchi E
    3. Stocker R
    4. Jimenez-Martinez J

    (2023) Morphogenesis of biofilms in porous media and control on hydrodynamics

    Environmental Science & Technology 57:5666–5677.

    https://doi.org/10.1021/acs.est.2c08890

    • Google Scholar
32. 1. Lauga E

    (2016) Bacterial Hydrodynamics

    Annual Review of Fluid Mechanics 48:105–130.

    https://doi.org/10.1146/annurev-fluid-122414-034606

    • Google Scholar
33. 1. Leão P
    2. Chen YR
    3. Abreu F
    4. Wang M
    5. Zhang W-J
    6. Zhou K
    7. Xiao T
    8. Wu L-F
    9. Lins U

    (2017) Ultrastructure of ellipsoidal magnetotactic multicellular prokaryotes depicts their complex assemblage and cellular polarity in the context of magnetotaxis

    Environmental Microbiology 19:2151–2163.

    https://doi.org/10.1111/1462-2920.13677

    • PubMed
    • Google Scholar
34. 1. Lefèvre CT
    2. Bennet M
    3. Landau L
    4. Vach P
    5. Pignol D
    6. Bazylinski DA
    7. Frankel RB
    8. Klumpp S
    9. Faivre D

    (2014) Diversity of magneto-aerotactic behaviors and oxygen sensing mechanisms in cultured magnetotactic bacteria

    Biophysical Journal 107:527–538.

    https://doi.org/10.1016/j.bpj.2014.05.043

    • PubMed
    • Google Scholar
35. 1. Li J
    2. Liu P
    3. Wang J
    4. Roberts AP
    5. Pan Y

    (2020) Magnetotaxis as an adaptation to enable bacterial shuttling of microbial sulfur and sulfur cycling across aquatic oxic‐anoxic interfaces

    Journal of Geophysical Research 125:2020JG006012.

    https://doi.org/10.1029/2020JG006012

    • Google Scholar
36. 1. Martins JL
    2. Silveira TS
    3. Silva KT
    4. Lins U

    (2009)

    Salinity dependence of the distribution of multicellular magnetotactic prokaryotes in a hypersaline lagoon

    International Microbiology 12:193–201.

    • PubMed
    • Google Scholar
37. 1. Nadkarni R
    2. Barkley S
    3. Fradin C

    (2013) A comparison of methods to measure the magnetic moment of magnetotactic bacteria through analysis of their trajectories in external magnetic fields

    PLOS ONE 8:e82064.

    https://doi.org/10.1371/journal.pone.0082064

    • PubMed
    • Google Scholar
38. Website

    1. NOAA

    (2024) Noaa national centers for environmental information magnetic field calculator

    Accessed October 24, 2024.

    https://www.ngdc.noaa.gov/geomag/calculators/magcalc.shtml
39. 1. Ostapenko T
    2. Schwarzendahl FJ
    3. Böddeker TJ
    4. Kreis CT
    5. Cammann J
    6. Mazza MG
    7. Bäumchen O

    (2018) Curvature-guided motility of microalgae in geometric confinement

    Physical Review Letters 120:068002.

    https://doi.org/10.1103/PhysRevLett.120.068002

    • PubMed
    • Google Scholar
40. 1. Pan Y
    2. Lin W
    3. Tian L
    4. Zhu R
    5. Petersen N

    (2009) Combined approaches for characterization of an uncultivated magnetotactic coccus from Lake Miyun near Beijing

    Geomicrobiology Journal 26:313–320.

    https://doi.org/10.1080/01490450902748633

    • Google Scholar
41. 1. Peszynska M
    2. Trykozko A
    3. Iltis G
    4. Schlueter S
    5. Wildenschild D

    (2016) Biofilm growth in porous media: Experiments, computational modeling at the porescale, and upscaling

    Advances in Water Resources 95:288–301.

    https://doi.org/10.1016/j.advwatres.2015.07.008

    • Google Scholar
42. Book

    1. Petersen N
    2. Weiss DG
    3. Vali H

    (1989) Magnetic bacteria in lake sediments

    In: Petersen N, editors. Geomagnetism and Palaeomagnetism. Springer. pp. 231–241.

    https://doi.org/10.1007/978-94-009-0905-2_17

    • Google Scholar
43. 1. Petroff A
    2. Rosselli-Calderon A
    3. Roque B
    4. Kumar P

    (2022) Phases of active matter composed of multicellular magnetotactic bacteria near a hard surface

    Physical Review Fluids 7:053102.

    https://doi.org/10.1103/PhysRevFluids.7.053102

    • Google Scholar
44. 1. Petroff AP
    2. McDonough S

    (2024) Trapping and scattering of a multiflagellated bacterium by a hard surface

    Physical Review. E 109:034403.

    https://doi.org/10.1103/PhysRevE.109.034403

    • PubMed
    • Google Scholar
45. Book

    1. Rodgers FG
    2. Blakemore RP
    3. Blakemore NA

    (1991) Intercellular junctions, motility and magnetosome structure in a multicellular magnetotactic prokaryote

    In: Rodgers FG, editors. Iron Biominerals. Springer. pp. 231–238.

    https://doi.org/10.1007/978-1-4615-3810-3_16

    • Google Scholar
46. 1. Schaible GA
    2. Jay ZJ
    3. Cliff J
    4. Schulz F
    5. Gauvin C
    6. Goudeau D
    7. Malmstrom RR
    8. Ruff SE
    9. Edgcomb V
    10. Hatzenpichler R

    (2024) Multicellular magnetotactic bacteria are genetically heterogeneous consortia with metabolically differentiated cells

    PLOS Biology 22:e3002638.

    https://doi.org/10.1371/journal.pbio.3002638

    • PubMed
    • Google Scholar
47. 1. Sepulchro AGV
    2. de Barros HL
    3. de Mota HOL
    4. Berbereia KS
    5. Huamani KPT
    6. da S Lopes LC
    7. Sudbrack V
    8. Acosta-Avalos D

    (2020) Magnetoreception in multicellular magnetotactic prokaryotes: a new analysis of escape motility trajectories in different magnetic fields

    European Biophysics Journal 49:609–617.

    https://doi.org/10.1007/s00249-020-01467-4

    • PubMed
    • Google Scholar
48. 1. Shapiro OH
    2. Hatzenpichler R
    3. Buckley DH
    4. Zinder SH
    5. Orphan VJ

    (2011) Multicellular photo‐magnetotactic bacteria

    Environmental Microbiology Reports 3:233–238.

    https://doi.org/10.1111/j.1758-2229.2010.00215.x

    • Google Scholar
49. 1. Simmons SL
    2. Bazylinski DA
    3. Edwards KJ

    (2007) Population dynamics of marine magnetotactic bacteria in a meromictic salt pond described with qPCR

    Environmental Microbiology 9:2162–2174.

    https://doi.org/10.1111/j.1462-2920.2007.01330.x

    • Google Scholar
50. 1. Simmons SL
    2. Edwards KJ

    (2007) Unexpected diversity in populations of the many-celled magnetotactic prokaryote

    Environmental Microbiology 9:206–215.

    https://doi.org/10.1111/j.1462-2920.2006.01129.x

    • PubMed
    • Google Scholar
51. 1. Smith MJ
    2. Sheehan PE
    3. Perry LL
    4. O’Connor K
    5. Csonka LN
    6. Applegate BM
    7. Whitman LJ

    (2006) Quantifying the magnetic advantage in magnetotaxis

    Biophysical Journal 91:1098–1107.

    https://doi.org/10.1529/biophysj.106.085167

    • PubMed
    • Google Scholar
52. 1. Spring S
    2. Amann R
    3. Ludwig W
    4. Schleifer K-H
    5. van Gemerden H
    6. Petersen N

    (1993) Dominating role of an unusual magnetotactic bacterium in the microaerobic zone of a freshwater sediment

    Applied and Environmental Microbiology 59:2397–2403.

    https://doi.org/10.1128/aem.59.8.2397-2403.1993

    • Google Scholar
53. 1. Tinevez JY
    2. Perry N
    3. Schindelin J
    4. Hoopes GM
    5. Reynolds GD
    6. Laplantine E
    7. Bednarek SY
    8. Shorte SL
    9. Eliceiri KW

    (2017) TrackMate: An open and extensible platform for single-particle tracking

    Methods 115:80–90.

    https://doi.org/10.1016/j.ymeth.2016.09.016

    • PubMed
    • Google Scholar
54. 1. Velho Rodrigues MF
    2. Lisicki M
    3. Lauga E

    (2021) The bank of swimming organisms at the micron scale (BOSO-Micro)

    PLOS ONE 16:e0252291.

    https://doi.org/10.1371/journal.pone.0252291

    • Google Scholar
55. 1. von der Schulenburg DAG
    2. Pintelon TRR
    3. Picioreanu C
    4. Van Loosdrecht MCM
    5. Johns ML

    (2009) Three‐dimensional simulations of biofilm growth in porous media

    AIChE Journal 55:494–504.

    https://doi.org/10.1002/aic.11674

    • Google Scholar
56. 1. Wang C
    2. Mehmani Y
    3. Xu K

    (2021) Capillary equilibrium of bubbles in porous media

    PNAS 118:2024069118.

    https://doi.org/10.1073/pnas.2024069118

    • Google Scholar
57. 1. Whitesides GM
    2. Ostuni E
    3. Takayama S
    4. Jiang X
    5. Ingber DE

    (2001) Soft lithography in biology and biochemistry

    Annual Review of Biomedical Engineering 3:335–373.

    https://doi.org/10.1146/annurev.bioeng.3.1.335

    • PubMed
    • Google Scholar
58. 1. Yang X
    2. Prakash M
    3. Brumley DR

    (2024) Escape motility of multicellular magnetotactic prokaryotes

    Journal of the Royal Society, Interface 21:20240310.

    https://doi.org/10.1098/rsif.2024.0310

    • PubMed
    • Google Scholar
59. 1. Yin Y
    2. Cui Y
    3. Jing L

    (2024) Clogging and unclogging of fine particles in porous media: micromechanical insights from an analog pore system

    Water Resources Research 60:e2023WR034628.

    https://doi.org/10.1029/2023WR034628

    • Google Scholar

Author details

1. Alexander P Petroff

   Department of Physics, Clark University, Worcester Massachusetts, United States

   Contribution

   Conceptualization, Data curation, Formal analysis, Supervision, Funding acquisition, Validation, Investigation, Methodology, Writing – original draft, Project administration, Writing – review and editing

   For correspondence

   apetroff@clarku.edu

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-9612-6593

2. Julia Hernandez

   Department of Physics, Clark University, Worcester Massachusetts, United States

   Contribution

   Investigation

   Competing interests

   No competing interests declared

3. Vladislav Kelin

   Department of Physics, Clark University, Worcester Massachusetts, United States

   Contribution

   Investigation

   Competing interests

   No competing interests declared

4. Nina Radchenko-Hannafin

   Department of Physics, Clark University, Worcester Massachusetts, United States

   Contribution

   Investigation

   Competing interests

   No competing interests declared

• 885

  views

• 16

  downloads

• 0

  citations

Views, downloads and citations are aggregated across all versions of this paper published by eLife.