Introduction

Magnetotactic bacteria represent a phylogentically 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; S. 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 Schuler, 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; S. 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 (S. 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 (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 (S. Esquivel and De Barros, 1986).

Here we investigate how magnetotactic bacteria tune their phenotypes in order to efficiently follow geomagnetic field lines through a labyrinthine network of pores.

Results

We begin by investigating how a single type of magnetotactic bacterium navigates an artificial pore space under an applied magnetic field. We choose to study a species of Desulfobacterota that is generically called a 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 thirty 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 between individual from 3.9 − 10.9 fAm²—to swim at speeds of between 45 μm/s to 134 μm/s (Petroff et al., 2022). Both reported ranges of values reflect 95% confidence intervals.

MMB provide an apt system for the broader study of magnetotaxis and its evolution for three reasons. The size and magnetic moment magnitude of MMB are similar to the median values found in a selection of diverse magnetotactic bacteria (see Appendix 1). 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 sand grains (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 form pores with a typical radius (Andreotti et al., 2013) of r = 0.2 ± 0.05 mm. 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 pores. The applied magnetic field is oriented 45° off the primary axis of the lattice 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 Fig. 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. While pores formed by convex particles (e.g., spheres) cannot produce convex pore spaces, such as those shown here, convex pores could form if biofilm or clay coats sections of the pore walls.

Figure 1.

We observe that as the applied magnetic field is reduced, the consortia explore an increasingly larger fraction of the pore. At very high magnetic field (Figure 1b, supplemental video SV1), 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, supplemental video SV2), consortia concentrate near the connections that lead northward and rapidly move through the network. The representative trajectory shown depicts the motion of a consortium that escaped the pore in less than a third of the time required for those at higher or lower magnetic fields. At low magnetic field (Figure 1d, supplemental video SV3), 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_drift 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.

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_drift. They are as follows: the magnetic moment m, the swimming speed U₀, the length a of the organism, its rotational diffusion coefficient D_rot, and its rotational hydraulic mobility μ_rot. The mobility μ_rot ∝ 1/(ηa³)—where η 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_drift /U₀ 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 γ = μ_rot 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 U₀/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 U₀/γ in the time required for it to realign with the magnetic field after a random reorientation. Consequently, if Sc < 1, Sc is the typical fraction of the pore that is explored by a magnetotactic bacterium. As described in Methods and Materials, 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 μm 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 ± 0.05 mm. As a/r ≈ 0.05, we conclude that the excluded volume of bacteria does not meaningfully effect their ability to navigate. In what follows, we assume that unreported pore radii are within a factor of 2 of 0.2 ± 0.05 mm.

The final dimensionless number is D_rot r/U₀. 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_rot r/U₀ 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_rot r/U₀ < 0.25. Figure 1d and supplementary video SV4 show roughly straight trajectories at low magnetic field. As D_rot r/U₀ is reasonably small, we conclude that largest changes in orientation are the result of collisions with walls rather than rotational diffusion.

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 (Sc) from the rates at which MMB transition between pores. To better visualize these pore-scale dynamics, we reduce the full pore network (Fig 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₋. The continuum limit of a one-dimensional network of pores yields

Figure 2.

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 . 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 Fig 1c). The rate of southward migration increases monotonically with Sc as consortia explore an ever growing fraction of the pore space. The drift velocity (Fig. 2c) is non-monotonic and reaches its maximal value of ≈ 0.25U₀ at a critical Scattering number of Sc_c ≈ 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 Sc_c ∼ δ/r, where δ (see Fig. 2a) is the distance between the northernmost pore wall and the exit to the next pore. In this pore network δ/r = 1 − 2^−1/2 ≈ 0.29, which is similar to Sc_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 U₀. The angle θ between the direction of motion and the magnetic field evolves in time t as dθ/dt = −γ sin(θ). We assume that, after a collision with a boundary, the angle at which consortia escape the wall is uniformly distributed. Nondimensionalizing time by 1/γ 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 Methods and Materials, these first passage times are exponentially distributed and the decay constants give k₊ and k₋ for any particular choice of Sc. Video SV6 shows a representative simulation. Repeating this process for 100 choices of Sc provides predictions for k₊, k₋, and f (Sc) with no free parameters. In the particular geometry shown in Fig.2a, we predict that the drift velocity is maximized at a critical Scattering number of Sc_c = 0.401 ± 0.001, which is found by fitting the maximum of the smooth curve in Fig.2c to a parabola. Details of these simulations are provided in Materials and Methods.

The good agreement between the predicted transition rates (shown as solid lines Fig. 2b–c) and our measurements leads us to accept the hypothesis that the magnetotactic efficiency U_drift /U₀ 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 (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 Fig. 1d), where magnetotactic organisms do not align with the magnetic field. It similarly vanishes if Sc = 0 (e.g., B = ∞, see Fig. 1b), where organisms cannot swim around obstructions. These limits imply a global maximum at a critical value of Sc_c, where the random fluctuations in swimming orientation due to collisions with boundaries regularly direct consortia towards northward pores, but not so great that consortia routinely escape to southward pores. In a random packing of grains, a typical pore is connected to other pores on all its sides of it. It is consequently rare that all of the connections are concentrated on the south-facing boundary. Thus, Sc_c ⪅ 1. Sc_c is also bounded from below. A consortium cannot explore a pore if the distance U₀/γ it is scattered after collisions is much smaller than its body size a (Petroff et al., 2022). Thus, Sc_c ⪆ a/r ≈ 0.05.

A simple geometric argument suggests that the critical value of Sc_c in natural sediment is more similar to the value found in the experiments presented here than is suggested by the preceding upper and lower bounds. Consider the distance δ (see Figure 2a) that a magnetotactic bacterium must swim to find an escape from a spherical pore with ν connections to neighboring pores. If connections are randomly and independently distributed over the boundary, ⟨δ/r⟩ = 2/(1+ν). Thus, the pore fraction that an MMB must explore before it escapes varies only from ∼ 0.4 to ∼ 0.2 as the coordination number of pores varies from 4 (local tetrahedral grain packing) to 8 (local octohedral grain packing) and remains similar to the value of ≈ 0.29 that we prescribe in our experiments. We conclude that 0.05 ⪅ Sc_c ⪅ 1 and likely differs only slightly from the critical value in the experiments. Since adapting our numeric simulations to three dimensions and solving for Sc_c in realistic geometries is beyond the scope of this paper, we proceed by assuming that Sc_c ≈ 0.401.

In their natural habitat, consortia swim through pores with a typical radius r ≈ 0.2 mm and align with a local geomagnetic field of magnitude B_geo = 50.9 μT. The phenotypes of the consortia measured in the experiments presented here differ considerably from previously reported values. The typical swimming speed of consortia in the pores is 150 μm/s, which is faster than 98% of the consortia taken directly from the sediment. We believe that this discrepancy is a result of how we load consortia in the pores. We use a magnetic field to direct consortia to swim 3 cm through the microfluidic device from the point of inoculation to the section of device where we measure their Scattering number. Although this process produces a highly enriched population of consortia it naturally selects the fastest swimming consortia. As these fastest swimming consortia are not representative of the entire population, we take the values for U₀, m, and a that we previously reported (Petroff et al., 2022) for MMB, which were collected from the same tide pool at a similar time of year, and we find Sc = 0.4 ± 0.26. This value is quite similar to the optimal value of 0.401 found in our artificial pores.

Finally, we turn our attention to the natural diversity of magnetotactic bacteria. Our results predict that optimal magnetotactic efficiency requires the typical scattering rate U₀/r to be proportional to the alignment rate γ_geo in the local geomagnetic field. Furthermore, the proportionality constant is predicted to be approximately 0.401. These predictions provide a strong test of the hypothesis that magnetotactic bacteria have evolved to optimally balance magnetotaxis with obstacle avoidance. Null models in which flagella and magnetosomes compete for resources or energy predict a negative correlation between U₀ and γ_geo. As discussed in Appendix 3, correlations between these phenotypic parameters do not reflect the dependence of each upon body size.

Figure 3 shows that the scattering rates and alignment rates are indeed proportional for a variety of magnetotactic bacteria collected from around the world. These data are taken from literature review (Petersen et al., 1989; Spring et al., 1993; Pan et al., 2009; Nadkarni et al., 2013; Bahaj et al., 1996; S. Esquivel and De Barros, 1986; Acosta-Avalos et al., 2019; Petroff et al., 2022; Carvalho et al., 2021) in which the distance a magnetotactic bacteria swims as it aligns with the magnetic field is directly measured. The plot includes both unicellular and multicellular species and representatives from three phyla enriched from four continents. Species are included from latitudes between 22° S and 54° N. The geomagnetic field across these latitudes ranges from B_geo = 23 μT with an inclination of −42°, in Rio de Janeiro, Brazil, to B_geo = 50.4 μT with an inclination of 69° near Grimmen, Germany. The average Scattering Number 0.47 ± 0.09 of these bacteria is indistinguishable from the predicted optimal Scattering number in our experiments. This value has a simple physical interpreta-tion: Regardless of size, swimming speed, and magnetic moment, any north-seeking magnetotactic bacteria from this data set tends to concentrate within a distance Sc_cr = 90 ± 20 μm of the northfacing wall of a pore, where passages that lead northward are presumably most likely to be found.

Figure 3.

The same argument applies to south-seeking bacteria. We conclude that all magnetotactic bacteria discussed here are equally and optimally efficient navigators of their native pore spaces.

Discussion

We have found that in the 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 about half a 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 = U₀/(Sc_crμ_rot B_geo), 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 μ_rot 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 go further by showing that too great a magnetic moment (Sc ≪ Sc_c, see Fig. 1b) is as detrimental to magnetotaxis as a magnetic moment that is too small (Sc ≫ Sc_c, see Fig. 1d).

Insofar as the ability to quickly move through the pore space aids bacteria in reproducing (Smith et al., 2006), f (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 section to tune a species’ phenotype to reflect its environment such that U₀/μ_rot m = Sc_cB_geor. This result suggests that if a bacterium acquires the genes for magnetosome formation and alignment—either de novo or through horizontal gene transfer—then it’s magnetotactic efficiency is immediately non-zero. The smoothness of f (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 could provide a strong test of this prediction.

Efficient magnetotaxis imposes a trade-off between size (through μ_rot), speed, and magnetic moment. A magnetotactic bacteria can be fast swimming and large only if it produces a very large number of magnetosomes. The constraint that U₀/rγ_geo = Sc_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.

Methods and Materials

Geometry of the Pore Space

The pore spaces shown in Figures 1 and 2 are produced using standard photolithography techniques (Whitesides et al., 2001). We use the negative epoxy photoresist SU-8 2050 (Kayaku Advanced Materials) to form a mold for the chambers. PDMS (Dow Sylgard 184) is poured over the photomask to form the microfluidic chip, which is attached to a microscope slide. Figure 4 shows a typical photomask used in our experiments.

Figure 4.

All pore spaces have a height of 40 μm. We found that this height keeps all consortia in the focal plane of the Zeiss 5x objective. Because the height of the chamber is significantly greater than the diameter of a consortium, the consortia remain fairly dilute even at high magnetic field. All pore networks are laid out on a square lattice of pores. We experimented with several pore geometries. We attempted to vary the shape of a pore from circular (as shown in Figures 1 and 2) to make the north and south facing pore boundaries slightly more or less curved. These changes were found to be too subtle as their effect on k_± is small relative to the measurement error in the rates. The rates measured in these slightly different geometries are averaged together. We varied the pore radius from 25 μm to 150 μm. We found that chamber radii between 100 μm and 125 μm provided the greatest experimental control. Smaller chambers quickly become densely packed with consortia, especially in the Summer months when consortia are most abundant. Larger chambers require very weak magnetic fields to probe high Scattering numbers (B = U₀/rμ_rot mSc). As we seek to precisely tune the Scattering number, the maximum chamber radius is limited by the spatial inhomogenities (∼ 5 μT) in the magnetic field produced by the objective.

Loading the Microfluidic Chamber

MMB are loaded into the mircofluidic pore space in five steps. These steps ensure that: (1) the pores are initially free of all microbes and chemical gradients, (2) the consortia entering the pores are at very high purity, and (3) the Scattering number of a consortia is directly measured, moments before the start of the experiment. The microfluidic device shown in figure 4 makes it possible to meet all three of these criteria.

We reduce the time between these steps to about seven minutes to minimize the number of consortia that arrive to the pore space. Doing so provides a dilute concentration within the pore space and limits interactions between consortia. However, this procedure selects strongly for the fastest swimming consortia, which are not representative of the natural community.

Step 1: building the vestibule

Before pouring the PDMS over the photoresist, we place a 7 mm tall right triangular prism, which we 3D print, on the wafer (see figure 4). After baking the PDMS, this prism is removed to form a 1 ml void space, which slopes towards the chambers. Following Step 2, we inject the enrichment of consortia into this void space, which we call the “vestibule.” Its large size and distance from the chambers reduces the number of non-magnetotactic microbes that make their way into the pore network.

Step 2: filling the pore space

To prevent the introduction of microbes into the pore space before the start of the experiment, the microfluidic device is first filled with filtered (0.2 μm) sea water, which is collected from the salt marsh where the consortia grow.

Step 3: secondary enrichment

Consortia are first enriched using the same method as described by Schaible et al. (2024). This procedure produces a reasonably high purity enrichment of MMB, however a small number of ciliates and other sediment bacteria often remain in the fluid, which complicate image processing and tracking. This enrichment is loaded into the vestibule of the microfluidic chamber. A 300 μT field directs consortia from the vestibule to accumulate near the wall facing the pore space. The consortia begin accumulating a minute or two after the magnetic field is applied.

Step 4: measurement of Sc

The concentration of consortia decays exponentially from the wall with a decay length of λ = 1.5U₀/γ (Petroff et al., 2022). We measure λ and, knowing r for the particular choice of pore space, calculate the Scattering number Sc_load = λ/1.5r at B_load = 300 μT. This measured Scattering number is rescaled to the particular experiment at field B as Sc_loadB_load/B.

To measure λ, we take three images in quick succession (75 frames/s). We track the motion of consortia between these frames to measure their swimming speed and to distinguish consortia from the background. Repeating this process once every five seconds for three minutes provides several thousand instantaneous positions. We fit the histogram of positions relative to the wall to an exponential to find λ, which varies only slightly between experiments.

Step 5: start of the experiment

Immediately following the collection of images needed to measure λ, the microscope is focused on the pore space and several images of the empty pore space are recorded. The magnetic field is adjusted to select a particular value of Sc. Reversing the direction of the magnetic field directs consortia into the pore network, where they are tracked. Additionally, the field reversal directs consortia that are not in the pore space back towards the vestibule. This design prevents consortia from arriving in the pore space partway through an experiment.

Control of the Magnetic Field

A uniform magnetic field is produced by a three-axis system of Helmholtz coils which were custom designed with Woodruff Engineering to fit on a Zeiss Axio Observer inverted microscope. Two sets of Helmholtz coils produce uniform magnetic fields that are parallel to the table top. The experiment sits in the center of a solenoid, which produces a magnetic field that is normal to the table top. The coils are powered by three Kepco bipolar operation amplifiers, which supply constant current.

The magnetic field B at the center of the experiment is an affine transformation of the current supplied to each coil. We define B = MI + b where I_j is the current passing through coil j, M is a 3 × 3 matrix with constant coefficients that are determined by the geometry of the coils and microscope, and b_i is the i^th component of the ambient magnetic field. To measure M and b, we focus the microscope on a small (0.82mm× 0.82mm) triple-axis magnetometer (Adafruit MMC5603) that is centered in the field of view of the microscope. We apply a current I_i(t) = A_i sin(2πf t), where A_i is chosen to be as large as possible without saturating the magnetometer and f = 1 s⁻¹, through each of the coils (one at a time) and record B(t). This procedure provides several thousand measurements of the magnetic field for various currents. We measure M and b as

where ⁺ represents the pseudoinverse and 1 is a row vector of ones. Finally, we multiply M and b by a rotation matrix R, which rotates the measured magnetic field about three orthogonal axes by small angles. We choose these angles to minimize the off-diagonal components of the matrix RM. This rotation corrects for any misalignment between the magnetometer and the system of coils. For example, the magnetometer chip is rotated by about 1° relative to the edges of the board it is mounted on. This calibration procedure allows one to cancel out the magnetic field due to the Earth and the microscope and produces a uniform magnetic field with a precision of a few μT in any direction to within less than one degree.

We found that the coils must be recalibrated after several experimental runs as the magnetic field produced by the microscope objective drifts. At the start of each experiment, we measure the angle between the walls of the microfluidic chamber and the axes of the Helmholtz coils. We rotate the applied magnetic field such that the magnetic field is oriented 45° relative to the principal axis of the pore space.

Tracking of MMB

We track the MMB consortia moving through the pores using TrackMate (Tinevez et al., 2017) in ImageJ. Before directing the consortia into the pore space, we average five images of the pore space, which we take as the background. During the experiment, we record the positions of consortia at either 20 frames/s (Fig. 1) or 75 frames/s (Fig. 2). We subtract the background image from each of the frames and convolve the image with the Laplacian of a Gaussian filter, the width of which is chosen to match the typical diameter of a consortium. Finally, we specify regions of interest in the connections between the pores and, using TrackMate, reconstruct the trajectories.

We identify a transition from one pore to the next when a trajectory crosses a line that equally divides the connection between neighboring pores. We linearly interpolate the positions of consortia along their trajectories to resolve the crossing times with sub-frame rate temporal resolution. This tracking provides the times that consortia leave and enter each pore.

This tracking method is reasonably robust. However, when two consortia swim close together, their IDs may switch. We estimate from the movies that this mislabeling happens a few times in each experiment. Importantly, as these mislabeling errors do not effect the number of trajectories crossing the midpoint line of the connectors, these errors do not effect the measurements of k_±.

Measurement of transition rates

The rates k_±(Sc) are measured in the miniature pore network shown in Figure 2 rather than in the full network of Figure 1. The choice to move to a small network to measure the pore scale dynamics solves several experimental difficulties.

The first problem is that, at relatively high magnetic fields, consortia accumulate so tightly together that it is difficult to distinguish individuals. As we measure k_± from the number N(t) of consortia within a pore and number of times that consortia transition between pores, it is vital that N(t) be accurately known. Rather than tracking the consortia within a pore, we only track them in the connections between pores. Starting from an empty pore space, we measure the N(t) by integrating the fluxes into and out of it. Recording data only within this reduced pore network allows us to increase the frame rate from 20 s⁻¹ to 75 s⁻¹. We double the dimensions of the pores in order to extend the time that consortia remain in the passageways and also to ensure that consortia are dilute. With these modifications, consortia remain in the passages between pores for tens of frames and move less than a one third of a consortium diameter between frames. This reasonably high temporal resolution allows us to stitch instantaneous positions into trajectories with little ambiguity.

The second experimental challenge results from the the phenotypic variability among the consortia. It appears that consortia move through the full pore space (Fig. 1) with slightly different drift velocities. This effect is apparent in Supplemental video SV1, which shows a front of consortia moving through the pore space at the highest magnetic field. Consortia are trapped for longer durations at the tail than at the leading edge. We expect that the phenotypic variability within the population leads to a diversity of Scattering numbers and thus a diversity of drift velocities. Consequently, in the full network, the rates k_± measured in a given pore vary in time as different subpopulations enter it. The rates at which diverse consortia move between two pores better reflects the population averaged Scattering number, which we directly measure.

Simulations of MMB motion

MMB consortia are approximated as points that swim at a constant speed U₀ in the direction of their magnetic moment. As the microfluidic pores are quasi two-dimensional, the trajectories of consortia are assumed to be two-dimensional. The trajectory of such a swimmer has an analytic solution

where τ = γt is dimensionless time and θ₀ is the initial angle between the magnetic moment and the magnetic field. We nondimensionalize distances by the pore radius r. The dimensionless position x(τ) along the direction of the magnetic field is

The dimensionless position y(τ) in the orthogonal direction is

These simulations do not include the so-called “escape motility” trajectories that have been observed in MMB (Greenberg et al., 2005; Sepulchro et al., 2020; Yang et al., 2024). Instances of this motion, where MMB briefly swim against the applied magnetic field, are visible in supplemental videos SV1–3, however they do not lead consortia from one pore to another.

This analytic solution allows one to simulate the motion of a swimmer in a confined geometry without discreet time steps. For a given initial position, we solve equations (6) and (7) for the time τ at which the swimmer comes in contact with a boundary. We move the swimmer to the point of contact and randomly select a new value of θ₀. We assume the distribution of escape angles is uniform over the range of values that prevent swimmers from moving though walls. We repeat these two steps for as long as is necessary. We record the times and locations of contacts between the swimmer and the boundaries.

We calculate k_±(Sc) in two steps. First, we simulate the motion of a swimmer that is confined to a single pore, which is identical in shape to those shown in the experiment and scaled to have radius 1. We run this simulation until the probability that the swimmer strikes any given section of the boundary ceases to evolve in time. This steady state distribution P (x, y) gives the probability that a themalized swimmer collides with any given section of the pore boundary. Next, we randomly choose an initial position (x₀, y₀) on the pore boundary from P (x, y). We start a swimmer from this position with a random initial orientation. We solve for the collisions between swimmer and pore boundary until the swimmer collides with a section of the boundary that would lead it to a neighboring pore. We record the times T₊ and T₋ that swimmers escape to northward and southwards pores, respectively. Repeating this process a thousand times provides the distribution of first passage times, which is found to be exponential. Figure 5a shows the distribution of first passage times in a representative experiment. We report transition rates (Fig. 2) that are nondimensionalized by the time r/U₀ for a swimmer to cross a pore radius. We measure these values in the simulations as rk_±/U₀ = ⟨T_±(Sc)⟩/Sc. At very low Scattering number, swimmers do not escape to southward pores in computationally accessible time scales. We extrapolate these rates from our simulations assuming that rk₋/U₀ ∝ Sc for Sc ≪ 1. We exactly solve for the scattering rates in the limit of infinite Sc from similar code in which swimmers move in straight trajectories between collisions.

Figure 5.

We repeat this process of choosing a specific value of Sc and calculating k₊, k₋, and U_drift for 100 different values, which are shown in Figure 5b. Because we are able to simulate these dynamics at Sc = ∞, it is convenient to choose values of Sc⁻¹ that are uniformly spaced from 0 to a large value. We find that U_drift /U₀ ≈ 0.16Sc⁻¹ if Sc⁻¹ ≪ 1. In the limit of Sc⁻¹ ≫ 1, U_drift /U₀ ≈ 0.58e^−0.22/Sc.

Assuming that an analytical solution for the transition rates is smooth, large differences between rates that are simulated at similar values of Sc⁻¹ result from the finite number of swimmers simulated. The blue curve in Figure 5b shows an approximate function that passes close to results of each simulation, which we take to be f (Sc).

Acknowledgements

We would like to thank A. Libchaber, A. Kudrolli, A. Flamholz, A. Goyal, M. Houssais, and O. Devauchelle for their comments and assistance. E. Imiya Mudiyanselage measured the pore size distribution. This work was supported National Science Foundation (NSF PHY-2042150). Much of this reasoning was clarified at the BIRS meeting 24w5315 “Formation of Looping Networks - from Nature to Models.” We thank the organizers and the Banff International Research Station. Bill Huang at Woodruff Engineering designed the magnetic coils.

Appendix 1

Phenotypic Diversity of magnetotactic bacteria

Here we discuss the selection of species used in Figure 3. Appendix 1 Figure 1 provides a key to this figure. Because the Scattering number represents the ratio of the distance a magnetotactic bacteria swims as it aligns with the ambient magnetic field to the pore size, we only considered papers that directly measure and report the swimming speed U_o and either U₀/γ or γ. This choice produces systematic errors in the estimates of the magnetic moments of the bacteria. Inverting measured values of γ for the magnetic moment requires a measurement of μ_rot, which depends on the shape of the magnetotactic bacteria and the distribution of flagella on its surface. Generally, only a crude estimate of μ_rot can be made. To limit the effect of these errors, wherever possible we rescale the raw data that authors report.

Two classes of experiments met these criteria. The first class (Petersen et al., 1989) analyze the motion of a magnetotactic bacteria in a rotating magnetic field of magnitude B₀. If the field rotates slowly, magnetotactic bacteria swim in circular paths. At a critical period of rotation, the field rotates faster than magnetotactic bacteria can turn and their trajectories begin to drift. The critical period T_c = 2π/γ. Our reported value of γ_geo = 2πB_geo/T_cB₀. The ratio of the diameter of the circular trajectory to the period of rotation gives U₀. The second class of experiments (Frankel, 1984; S. Esquivel and De Barros, 1986) analyze the trajectory of a magnetotactic bacteria after a field reversal. The magnetotactic bacteria makes a “U-turn” of width W = πU₀/γ. For this class of experiments, our reported values of γ_geo = πU₀B_geo/W B₀.

Additionally, to limit the effect of domestication on the phenotypes, the data presented is heavily biased towards experiments that extracted magnetotactic bacteria directly from the sediment, rather than relying on pure cultures. The reported values of B_geo are found by looking up the magnetic field (NOAA, 2024) as close as possible to the sampling location.

Appendix 1 Figure 1.

Appendix 1 Table 1.

Appendix 2

Relationship of Scattering number and body size

Figure 3 shows that the rate U₀/r that magnetotactic bacteria collide with pore boundaries is proportional to the rate γ_geo that they align with the geomagnetic field. It seems plausible that this correlation reflects an anatomical relationship that is unrelated to magnetotaxis and obstacle avoidance. Perhaps larger organisms swim more quickly and produce larger magnetic moments. However, this null hypothesis can be quickly rejected.

Consider first the relationship between γ_geo = μ_rot mB_geo and body size a. Recall that the rotational hydraulic mobility μ_rot ∝ 1/ηa³, where η is the viscosity of water and the proportionality is determined by the shape of the organism. Thus, γ_geo ∝ m₀cB_geo/η, where m₀ is the magnetic moment of a single magnetosome and c is the concentration of magnetosomes. It follows that γ_geo is an intensive quantity and does not scale with a. From the data collected in Appendix 1 Table 1, we find that the magnetic moment is uncorrelated with body size.

The swimming speed does not scale simply with the body size. Balancing the forces exerted by flagella with drag on the organism yields a swimming speed U₀ ∝ f₀σa/η, where f₀ is the force exerted by a single flagellum and σ is the surface density of flagella. While the swimming speed does scale with a, the variability in σ and shape overwhelm any correlation. For example, consider MTB-2 and MTB-8 from Appendix 1 Table 1. These organisms swim at similar speeds despite differing in size by almost an order of magnitude. Moreover, it is the smaller organism that swims faster. A comparison of all of the organisms in this table shows a weak negative correlation (−0.57) between size and swimming speed that is of marginal significance (p-value of 0.04). A comparison of 87 non-magnetotactic bacteria undertaken by Velho Rodrigues et al. (2021) reveals no simple relationship between swimming speed and size, suggesting that there is no biomechanical reason for U₀ to scale with body size.

We conclude that neither the scattering rate nor the alignment rate of bacteria scale with the body size. Rather, natural selection adapts the Scattering number of a species primarily by acting upon its shape, concentration of magnetosomes, and surface density of flagella.

Appendix 3

Supplementary Videos

SV1 Multicellular Magnetotactic Bacteria move through an artificial pore space. Each white dot shows a spherical consortium composed of tens of individual bacteria, which grow and move like a single multicellular organism. The red pillars show the separations between neighboring pores. The magnetic field is 3500 μT. The Scattering number Sc = 0.08 is small compared to the critical Scattering number Sc_c = 0.401, which implies that consortia align too quickly with the magnetic field to move efficiently between pores. Consortia become trapped near pore boundaries for prolonged times, which slows their progress through the pore space. This video corresponds to the experiment shown in Figure 1b. The video is slowed to 80% of the true speed. Clickable link: https://youtu.be/Icz5X3vC4s4

SV2 Multicellular Magnetotactic Bacteria (white dots) move through an artificial pore space. The red pillars show the separations between neighboring pores. The magnetic field is 940 μT. The Scattering number Sc = 0.31 is similar to critical Scattering number Sc_c = 0.401. These consortia effectively balance obstacle avoidance with magnetotaxis and move quickly through the pore space. This video corresponds to the experiment shown in Figure 1a and c. The video is slowed to 80% of the true speed. Clickable link: https://youtu.be/qfQksS2IzAQ

SV3 Multicellular Magnetotactic Bacteria (white dots) move through an artificial pore space. The red pillars show the separations between neighboring pores. The magnetic field is 75 μT. The Scattering number Sc = 3.85 is large compared to the critical Scattering number Sc_c = 0.401, which implies that the motion is dominated by randomizing collisions with pore boundaries. The motion is approximately diffusive. This video corresponds to the experiment shown in Figure 1d. The video is slowed to 80% of the true speed. Clickable link: https://youtu.be/biFjEz3JvxA

SV4 Video of a single representative pore from SV3. The trajectories of consortia (bright lines) are roughly straight between collisions. As changes in swimming direction are primarily due to collisions, we conclude that rotational diffusion is of secondary importance. The video is slowed to 80% of the true speed. Clickable link: https://youtu.be/Bh172VZDOC0

SV5 Video of a small pore space in which k_± are measured. Figure 2a shows a still of from this video. Consortia appear as white dots. Trajectories appear as colored lines. The blue line highlights a consortium that moves between pores in the direction of the magnetic field. The red trajectory moves between pores twice, once with the applied magnetic field and then against it. The applied magnetic field is B = 207 μT and Sc = 1.98 ± 0.36. The video is slowed to 80% of the true speed. Clickable link: https://youtu.be/SibUoIuCc2c

SV6 A simulation of point-like consortia (red dots) moving between pores shows the equilibrium distribution and fluctuations at Sc = 1.25. The magnetic field points to the right. As k₊ > k₋, simulated consortia accumulate mainly in the northernmost pore. These dynamics and the associated transition rates depend only on the Scattering number. Simulations that calculate the first passage times between pores at 100 different values of Sc are used to produce the smooth curves in Figure 2b and c. Clickable link: https://youtu.be/a897UT0hZ2I