Explicit analysis of magnetotactic bacteria motion reveals the length scaling of magnetic moments

Abstract

Magnetotactic bacteria (MTB) are a diverse group of microorganisms whose movement can be directed via a magnetic field, which makes them attractive for applications in medicine and microfluidics. One of their key properties is the magnetic moment $m$, which is challenging to measure. We perform optical imaging experiments with MSR-1 MTB, and derive both the $m$ statistics and the scaling of $m$ with the MTB size using an explicit and fully automated method to determine $m$ from the MTB trajectories via the U-turn protocol, which measures $m$ based on the U-shaped trajectories exhibited by the MTB in an applied alternating magnetic field. The proposed method is an alternative to the standard U-turn time-based moment calculation and uses the theoretical U-turn shape function we have derived. This directly accounts for the U-turn geometry and determines the moment from the U-turn branch width. We couple this approach with a robust U-turn decomposition algorithm that detects U-turns from MTB tracks regardless of their orientations. We report a linear dependence of $m$ on the size of the bacteria, accounting for the bacteria velocity variations during the U-turns. We also show that the new U-turn shape-based and the conventional time-based methods produce significantly different results. The proposed method can be used to differentiate between various types of MTB within the same population based on their velocity and magnetic moments, and to precisely characterize the magnetic properties of a culture.

Significance

Magnetotactic bacteria (MTB) are microorganisms suitable for biomedical and microfluidic applications, since they are controllable with a magnetic field. However, accurately measuring their magnetic moment, a key biophysical property, remains a challenge. We thus introduce a fully automated and rigorous approach to extract magnetic moment from MTB motion in an applied magnetic field. We obtain the magnetic moment statistics for the widely used MSR-1 MTB, and clearly show that the magnetic moment scales linearly with the cell size. Our results enable differentiating between various MTB types within populations based on velocity and magnetic moment. Full automation enables magnetic moment extraction from much larger data sets than achieved previously.

Introduction

Magnetotactic microorganisms are a diverse group of single- and multicellular organisms that are able to biomineralize magnetic nanoparticles, allowing them to passively align with an external magnetic field (MF), allowing their magnetic control (1). They also propel themselves using a flagellar apparatus (2,3). This group includes magnetotactic bacteria (MTB) (4,5), multicellular magnetotactic prokaryotes (6,7,8), as well as magnetotactic holobionts, a host organism that lives in symbiosis with MTB (9). Magnetic microswimmers are of particular interest due to potential applications in drug delivery and cancer therapies, as MTB cells and magnetosomes are biocompatible and nontoxic (10,11,12,13,14). It is possible to form ensembles of swimmers and control their emergent behaviors, with a wide range of prospective applications in microrobotics (1,3,15,16,17,18,19,20,21,22,23,24,25). For example, there is a clear potential for general-purpose object manipulation applications (19,20,21,22).

An important parameter of MTB and other magnetic microorganisms is their magnetic moment $m$, which can be determined by several methods, directly measuring the properties of the magnetic crystal or indirect measurements where $m$ is calculated based on how cells behave in the MF (26). For example, by observing the motion of the MTB in the rotating MF (16), studying the fluctuations of the cell orientation (27), etc. Measurement of $m$ for a single cell can be performed by determining its velocity in an inhomogeneous MF. For instance, by studying the motion of the MTB in a rotating MF, m can be calculated from fluctuations in cell orientation (28). Kalmijn measured the magnetic moment by applying a strong pulsed MF and reversing the dipole moment of the bacterium—since thermal fluctuations alter the magnetic moment orientation, the mean bacterium velocity is MF dependent, which allows determining the magnetic moment from the velocity statistics over the bacterium trajectory (29). However, it should be noted that intrinsic noise due to biochemically induced velocity fluctuations could affect the measurements using this method. The most common approach is the U-turn method, an indirect measurement proposed by Esquivel and Lins De Barros in 1985 (30), which derives the magnetic moment from the trajectory of microorganisms in a pulsed MF (9,31,32). Bacteria are oriented and swim along the MF direction, and change the swimming direction upon MF reversal, creating the distinctive U-shaped trajectories. By determining the time $\tau$ it takes the microorganism to complete a U-turn, $m$ can be estimated as follows (30):

$$\tau =\frac{8\pi \eta R^3}{mB}·\ln \left(\frac{2mB}{kT}\right)$$ (1)

where $\eta$ is the environment (sample) viscosity, $R$ is the radius of a microorganism (assuming spherical shape), $B$ is the applied MF induction, $k$ is the Boltzmann constant, and $T$ is the sample temperature. The above equation can be interpreted as the dependence of the U-turn time on the ratio of torques due to rotational drag (a spherical body is assumed) and realignment with reversed MF, with a correction term due to thermal fluctuations.

However, the issue is that the underlying derivation is based on physical assumptions that are not sufficiently general. Typically, $\tau$ is measured based on a fraction of a U-turn, which is arbitrarily selected and can be subjected to considerable noise and the typically observed cell movement deviation from the MF direction. That is, this method does not explicitly determine $m$ from the shape of the U-turn, which would allow obtaining a more accurate estimate for $m$ based on the entire U-turn. Measurements of $\tau$ are most commonly performed semimanually (e.g., manual selection of eligible trajectories and U-turns, and U-turn diameter estimates without shape fitting) (33) or, as shown in (31), by detecting U-turn boundaries from extrema in U-turn trajectory coordinate derivatives (8,9,26). However, derivatives in only one direction are computed (vertical/horizontal), which is prone to low U-turn endpoint contrast in the coordinate time series. Errors due to data noise are generally problematic in the case of trajectories with an arbitrary configuration of U-turns, and do not account for U-turns tilted at an angle to the horizontal/vertical axis of the image and/or at an angle to applied MF. The latter is important because significant misalignment between the MTB magnetic moment and its body axis has been reported for Magnetospirillum magneticum (AMB-1) (34), but also because the MTB trajectories may be perturbed by other means. In some cases, trajectory and U-turn analysis is not sufficiently documented (35), making reproducibility an issue.

We propose an explicit and fully automated method to determine $m$ from the trajectories of microorganisms with U-turns in alternating MF. We present a theoretical shape function for MTB U-turns, which enables computing $m$ explicitly from the width of the U-turn, circumventing $\tau$ calculations, as well as an algorithm to decompose MTB trajectories into U-turns and determine $m$ for each instance (Algorithm 1). We demonstrate the effectiveness of our approach on optically imaged MSR-1 U-turn motion in an alternating MF. We detect MTB from bright-field images, recover their trajectories, and apply the new approach to determine the $m$ statistics and the $m$ scaling with the MTB size, then compare the results with the magnetic moment calculated using $\tau$. We also show the effects of different estimates for characteristic U-turn velocity values on the resulting $m$ statistics. The proposed data analysis pipeline is shown in Fig. 1.

Algorithm 1. Determining MTB magnetic moment m from trajectory data.

Input: MBT trajectories reconstructed with MHT-X

 Trajectory filtering

1 Threshold by temporal length $N_t$, $N_t<N_{t1}$ cases eliminated

2 Threshold by oriented aspect ratio $\chi$, $\chi <{\chi }_{c1}$ cases eliminated

3 Median and mean filtering, 1 point kernel width

 U-turn decomposition

4 Compute second Gaussian derivatives $\left({\sigma }_c\right)$ of coordinate time series

5 Threshold extrema points by relative magnitude $M$, removing $M<M_c$ cases, after Gaussian smoothing $\left({\sigma }_p\right)$

6 Compute the U-turn curvature with a Gaussian kernel $\left({\sigma }_{k1}\right)$

7 Remove extrema points with $M<M_{k1}$ after Gaussian smoothing $\left({\sigma }_{t1}\right)$—define as turning points

8 Combine the coordinate derivative extrema points with the turning points, resolve duplicates and proximity conflicts—define as partition points

9 Partition tracks into U-turns at the partition points, keep partitions containing the turning points

10 Prune the U-turns by $N_{\text{cut}}$ points from either endpoint

 U-turn filtering

11 Eliminate instances with $N_t<N_{t2}$

12 Eliminate the self-intersecting U-turns

13 Eliminate instances with $\chi <{\chi }_{c2}$

14 Threshold by oriented minimum area $S$, $S<S_c$ cases eliminated

15 Refine the turning points via steps 6 and 7 with parameters ${\sigma }_{k2}$, $M_{k2}$, and ${\sigma }_{t2}$, performing refinement on U-turns upsampled by a factor $K_1$

16 Eliminate U-turns with multiple turning points

 Determining U-turn $L$ values

17 Symmetrize U-turn branches by physical length

18 Translate the turning points to $\left(x,y\right)=\left(0,0\right)$

19 Eliminate instances with $N_t<N_{t3}$

20 Eliminate U-turns with longest/shortest branch temporal length ratio $>{\chi }_b$

21 Resample “secondary” turning points via steps 6 and 7 with parameters ${\sigma }_{k3}$, $M_{k3}$, and ${\sigma }_{t3}$ with the upsampling factor $K_2$, eliminate instances with multiple secondary points

22 Eliminate U-turns with MF alignment factor $A<A_c$

23 Run the $L$ optimization routine Eq. 2

24 Keep U-turns and $L$ values with $g<g_c$ Eq. 4 and $f+\beta g<f_c$ Eq. 2

 Computing $m$ values

25 Propagate MTB property data from tracks to eligible U-turns

26 Compute $\alpha$ Eq. 6

27 Compute the mean-filtered (1 point-wide kernel) velocity time series

28 Compute Gaussian-smoothed $\left({\sigma }_{k4}\right)$ curvature for U-turns

29 Compute mean velocity $v$ for U-turn points with relative curvature $>M_{k4}$

30 Compute $m$ via Eq. 23 from $\alpha$, $v$, and $L$

 Output:

• • MTB magnetic moment m values with uncertainties
• • Corresponding filtered U-turns

Figure 1.

An overview of the algorithm used to determine $m$ from MTB trajectories.

Materials and methods

Cell cultivation

The flask standard medium established by Heyen and Schüler (36) was used to cultivate Magnetospirillum gryphiswaldense (MSR-1) MTB. One hundred microliters of cell suspension was inoculated in 16 $\text{mL}$ plastic centrifuge tubes filled with 15 $\text{mL}$ flask standard medium, ensuring microaerophilic conditions. Cells were incubated at $29°\mathrm{C}$ for 3 days. Bacterium suspension added directly from the growth tube was used for microscopy.

Optical microscopy and image acquisition

Microscopy samples were prepared by filling a glass capillary $\left(0.2\times 2.0\times 16\text{mm}\right)$ with a suspension of bacteria. Both ends of the capillary were sealed with thermopaste, preventing the formation of an oxygen gradient and the formation of a microaerobic bacteria band. The total experiment time, starting with the removal of the culture from the incubator, was 90 min.

The capillary was placed on an optical microscope stage (Leica DMI3000B) equipped with a $40\times$ magnification objective. The microscope stage has three pairs of built-in coils, which generate a homogeneous MF in the sample (Fig. 2). The MF is generated using an AC power supply (Kepco BOP 2010M), and controlled via a data acquisition card and a LabVIEW program. A Basler AC1920155UM camera was used to record the cell motion.

Figure 2.

The microscopy setup: a microscope stage equipped with three coil pairs generating a homogenous MF.

The sample was placed in an alternating MF ($B=2.0\text{mT}$, switching frequency $f=0.7\text{Hz}$, square wave), causing the most motile and magnetic cells to change their swimming direction when the MF direction is reversed, thus inducing the characteristic U-turn motion (Fig. 3). The image acquisition frame rate is set to 20 frames per second (FPS). The image recording is triggered by the MF direction change. The duration of every constant MF direction interval is equal. The MF strength is calculated from the coil current measurements at the moment when camera exposure for each frame starts.

Figure 3.

An MSR-1 bacterium trajectory in an alternating MF. The dashed line indicates the trajectory, and the white outlines are cell shapes captured over sequential frames. The shown cell was tracked for $\sim 1.35$ s at 20 FPS. The cell entered the FOV from the bottom and exited at the top left. To see the MTB in motion and more of the reconstructed tracks, please refer to Videos S1 and S2.

MTB detection and tracking

To remove static background artifacts and features from the raw images, a flat-field correction (FFC) is applied to images. This FFC procedure is adaptive and detects intervals of constant field of view (FOV) and image focus, splits the image sequence into these intervals, and then performs FFC with mean interval images for each sequence interval. This is because the FOV and focus are sometimes adjusted for a better view with more active and MF-compliant MTB, and to assess MTB behavior within different fluid layers. Constant FOV or focus intervals are detected as intervals between extrema in the normalized second derivative magnitude of the mean-squared error time series for consecutive pairs in the image sequence. To avoid overfitting insignificant background variations, the mean-squared error time series is filtered with the Gaussian total variation (TV) filter (37), and the extrema are thresholded by relative magnitude. Other static artifacts and stuck MTB are removed by detecting them from the temporal standard deviation projection image computed from the corrected image sequence. This is done by applying color tone mapping (CTM) (38), normalizing the image, binarizing with a manual threshold or using the Kapur’s method (39), slightly inflating the artifact masks via morphological dilation (circular kernel) (40), and afterward the artifacts are removed via texture synthesis inpainting (41). Then, to improve the contrast/noise ratio, local adaptive CTM and referenceless FFC are applied. Next, the Poisson model TV filter (42) is used for denoising. The Poisson TV filter is chosen because it effectively counteracts the additional luminance-dependent noise introduced during the above image correction stages. To segment MTB from filtered images, the contrast/noise ratio is further enhanced by applying soft CTM masking (43) (used for the detection of MTB swarms and individual MTB in (44)) and the maximum filter, and then Otsu binarization (45) is performed, followed by size thresholding to remove fine artifacts. Finally, any remaining segments of the ringing artifacts due to optical imaging are removed using the luminance-based false positive elimination method developed in (43,46) and applied previously to the detection of MTB in (44). MTB segment coordinates and shape parameters are then determined, and the trajectory reconstruction is performed using the MHT-X object tracking code (46,47,48). The resulting trajectories are then analyzed (Algorithm 1). The parameters for the image processing and trajectory tracking codes are provided with the code, which is linked in the data and code availability. Wolfram Mathematica was used for image and data analysis, as well as visualization, and the MTB tracking code is implemented in Python.

Determining MTB magnetic moment from experimental data

Generally, MTB tracks can have arbitrary shapes despite consistent alternating MF, and may have different orientations due to magnetic moment and cell body misalignment (34), moment misalignment in magnetosome chains (49), etc., with many U-turns per track (Fig. 3). Therefore, the new method we have developed is to decompose the tracks into U-turns and fit the track points to Eq. 22, then to determine $L$, $v$, and $\alpha$, and then determine $m$ from Eq. 23, as derived in the results and discussion section.

Before performing the U-turn decomposition, tracks are filtered by temporal length to exclude undersampled MTB motion, then by minimum-oriented bounding box aspect ratio (oriented aspect ratio, excludes trajectories of MF that do not respond to MF changes, since they have very high aspect ratio values), and then filtered using median and mean filters (one point-wide kernel).

The eligible filtered tracks are decomposed into U-turns by finding peaks in the absolute values of the second derivative of their coordinate time series and thresholding the peaks by the relative values, and extrema of curvature (their turning points) within the Gaussian-filtered track (thresholded by the relative curvature value). Trajectories are then split into fragments at the coordinates of these points, resolving overlaps/proximity cases, if any. U-turns are then found among the resulting track fragments as those that contain the curvature extrema points, and the branches of the U-turns (points on either side of the turning points) are pruned to remove artifacts from the tip of the branch of the U-turn.

Among the U-turns generated from the trajectories, the eligible U-turns are found through a sequence of filtering procedures. After filtering U-turns by temporal length, instances with self-intersections are eliminated by converting the sequence of U-turn points into a graphics line object, then to a raster image with boundary padding; the image is then binarized via the Otsu method, Gaussian-filtered, and binarized again to ensure that the rasterized line is unbroken, and afterward the number of domains in the image is counted—for a U-turn without self-intersection, there is only a single domain (albeit not simply connected), whereas self-intersection instances generate additional domains. After removing self-intersecting U-turns, the remainder are filtered by minimum bounding box aspect ratio and area (this removes instances with stuck MTB and straighter fragments of tracks), and then the turning point locations are refined (turning points are detected for uniformly upsampled versions of U-turns, B-spline order 2), and U-turns with multiple turning points post-refinement are eliminated.

The surviving U-turns are symmetrized by physical length with respect to their turning points—this is to minimize bias during theoretical shape fitting. Then the U-turns are translated such that their turning points are at $\left(x,y\right)=\left(0,0\right)$ coordinates, and then subjected to another round of filtering by temporal length, by the ratio of position samples in branches (further eliminating bias from overfitting one of the branches), and another stage of multiple turning point instance elimination. Finally, the misalignment of the orientation of the U-turn with the alternating MF axis is checked—the U-turns and the MF axis are rotated to align the MF along the $x$ axis, and then, for all U-turns, the ratio of the coordinate sums of $x$ and $y$ over both branches of the U-turn is computed. If the ratio is too low, the U-turn does not correspond to a motion compliant with the MF. The remaining U-turns are eligible for fitting to Eq. 22.

To fit U-turn coordinate sets to Eq. 22, U-turn turning points, previously translated to initial coordinates (0,0), are used as anchors for shape fitting. The U-turn is fitted to the offset version of the theoretical shape function Eq. 22 $y_{\text{offset}}\left(x,L,x_0,y_0\right)=y_0+y\left(x-x_0,L\right)$ given by the translation of the shape function $\left(x_0,y_0\right)$ and the parameter value $L$ by finding the optimal U-turn coordinate set ${\overrightarrow{r}}_k\left(\varphi \right)=\left(x_k,y_k\right),k\mathbb{\in }\mathbb{N}$ generated by rotating the initial U-turn by an angle $\varphi$, such that the following optimization problem is solved:

$$L:\arg \underset{\left(x_0,y_0,\varphi ,L\right)}{\min }f\left(x_0,y_0,\varphi ,L\right)+\beta ·g\left(\varphi \right)$$ (2)

where $\beta \ge 0,L>0,\varphi \in [0;2\pi ),x_0\in I_1\left({\overrightarrow{r}}_k\right),y_0\in I_2\left({\overrightarrow{r}}_k\right)$. The functions $f$ and $g$ are defined as follows:

$$f\left(x_0,y_0,\varphi ,L\right)=\frac{1}{{\sum }_k\Vert {\overrightarrow{r}}_k\left(\varphi \right)\Vert }\sum _k{\left(y_{\text{offset}}\left(x_k\left(\varphi \right),L,x_0,y_0\right)-y_k\left(\varphi \right)\right)}^2$$ (3)

$$g\left(\varphi \right)=\frac{1}{{\sum }_k{y}_k^2\left(\varphi \right)}\sum _{y_k\left(\varphi \right)>0}{y}_k^2\left(\varphi \right)$$ (4)

The solution minimizes the discrepancy between the U-turn data points and the theoretical shape function Eq. 3 while penalizing U-turn rotations resulting in U-turn points with $y_k>0$ (Eq. 4). The latter is implemented because Eq. 22 has the property $y\left(x,L\right)<0\forall x\ne 0$. Furthermore, to enforce $f+\beta g:\mathbb{R}\to \mathbb{R}\mathbb{\forall }\left\{x_k\left(\varphi \right),L\right\}$, the $y_{\text{offset}}\left(x_k\left(\varphi \right),L,x_0,y_0\right)$ values are sampled for every $L$ instance $N_{\epsilon }$ times uniformly in the $x_k\in \left[-L+\epsilon ;L-\epsilon \right]$ interval. This is due to $y\left(x,L\right)\mathbb{\in }\mathbb{C}\mathbb{\forall }\left|x\right|>L$. As a result, ${\left(y_{\text{offset}}\left(x_k\left(\varphi \right),L,x_0,y_0\right)-y_k\left(\varphi \right)\right)}^2$ in Eq. 3 are instead computed as

$$\underset{s\mathbb{\in }\mathbb{N}}{\min }{\Vert {\overrightarrow{r}}_s-{\overrightarrow{r}}_k\left(\varphi \right)\Vert }^2;{\overrightarrow{r}}_s=\left(y_s,x_s\right):y_s=y_{\text{offset}}\left(x_s,L,x_0,y_0\right),x_s\in \left[-L+\epsilon ;L-\epsilon \right]$$ (5)

where ${\overrightarrow{r}}_s$ are the sampling points for $y_{\text{offset}}$. Note that Eq. 2 penalizes temporally and spatially longer tracks slightly less, which is intentional—this way, better reconstructed U-turns are prioritized.

After all U-turn fits are complete and $L$ are determined, the $L$ values and the U-turn data are filtered by their $f+\beta g$ and $g$ values. Then $\alpha$ is computed from the MTB and flow medium properties assuming spherically shaped MTB:

$$\alpha =8\pi \eta R^3$$ (6)

Note, however, that any other rotational drag model can be used based on the morphology of cells—for example, cylindrical (25), ellipsoidal (50), the sphere chain model presented in (51), or an experimental spirillum drag model as in (33).

We then estimate the characteristic U-turn $v$ to compute $m$ from Eq. 23. This is done by computing the U-turn curvature over the arc length and computing the mean velocity over the segment with curvature exceeding a relative threshold, that is, velocity is not sampled from the straighter segments of the track.

All of the above procedures and the parameters are summarized in Algorithm 1 in appendix A, and are also provided with the code, which is linked in the data and code availability.

Results and discussion

MTB U-turn trajectory model

Let the orientation of an MTB and its magnetic moment be $\overrightarrow{n}=\left(\cos \left({\theta }^{\prime }\right),\sin \left({\theta }^{\prime }\right),0\right)$, where ${\theta }^{\prime }$ is the angle between $\overrightarrow{n}$ and the $x>0$ direction, and the MF strength before/after reversal is $\overrightarrow{H}=\left(0,H,0\right)$ and $\overrightarrow{H}=\left(0,-H,0\right)$, respectively. Since in our experiment $H\ll 2\pi M$, we assume that the magnetic moment of the MTB is along its body axis (16). The MTB moves in the $XY$ plane (Fig. 2), and its thermal fluctuations are negligible. The MTB angular velocity is $\overrightarrow{\mathrm{\Omega }}=\left[\overrightarrow{n}\times \dot{\overrightarrow{n}}\right]=\stackrel{.}{{\theta }^{\prime }}{\overrightarrow{e}}_z$. According to the Bean model (30,51), the balance of hydrodynamic and magnetic torques is as follows:

$$-\alpha \overrightarrow{\mathrm{\Omega }}+\left[\overrightarrow{m}\times \overrightarrow{H}\right]=0$$ (7)

where $\alpha$ is the rotational drag coefficient and $\overrightarrow{m}=m\overrightarrow{n}$ is the MTB magnetic moment. Projected onto the $z$ axis, this becomes $\alpha \stackrel{.}{{\theta }^{\prime }}=-mH\cos \left({\theta }^{\prime }\right)$. Introducing $\theta ={\theta }^{\prime }-\pi /2$, the angle between the MTB and MF before its reversal (the MF direction is $y>0$), one has

$$\alpha \stackrel{.}{\theta }=mH\sin \left(\theta \right)$$ (8)

The velocity of the MTB translational motion is $\overrightarrow{v}=v\overrightarrow{n}$ and a constant $v$ is assumed. Introducing the trajectory arc length $l$ and velocity $v=dl/dt$, Eq. 8 may be rewritten as follows:

$$\frac{d\theta }{dl}=\frac{mH}{\alpha v}·\sin \left(\theta \right)$$ (9)

The asymptotic U-turn conditions are

$$\underset{l\to -\infty }{\lim }\theta \left(l\right)=0;\underset{l\to +\infty }{\lim }\theta \left(l\right)=\pi$$ (10)

at the U-turn start and the U-turn end, respectively. Then, integrating Eq. 9, the following is obtained (28):

$$\tan \left(\theta /2\right)=\exp \left(\frac{mHl}{\alpha v}\right)$$ (11)

Solution Eq. 11 allows calculating the trajectory by integrating the equations of motion

$$\frac{dx}{dl}=-\sin \left(\theta \right);\frac{dy}{dl}=\cos \left(\theta \right)$$ (12)

using

$$\sin \left(\theta \right)=\frac{2\tan \left(\theta /2\right)}{1+{\tan }^2\left(\theta /2\right)}$$

and obtaining

$$\frac{dx}{dl}=-\frac{1}{\cosh \left(\beta l\right)}$$ (13)

where $\beta =mH/\alpha v$. Integrating Eq. 13 yields

$$x\left(l\right)=-\frac{2}{\beta }·\arctan \left(\tanh \left(\beta l/2\right)\right)+C_1$$ (14)

where $x\left(0\right)=0\Rightarrow C_1=0$ and

$$x=-\frac{2}{\beta }·\arctan \left(\tanh \left(\beta l/2\right)\right)$$ (15)

Similarly, using

$$\cos \left(\theta \right)=\frac{1-{\tan }^2\left(\theta /2\right)}{1+{\tan }^2\left(\theta /2\right)}$$

an equation for the $y$ coordinate can be written as

$$\frac{dy}{dl}=-\tanh \left(\beta l\right)$$ (16)

Integration over $l$ results in

$$y\left(l\right)=-\frac{1}{\beta }·\log \left(\cosh \left(\beta l\right)\right)+C_2$$ (17)

and, since $y\left(0\right)=0\Rightarrow C_2=0$. Using

$$\cosh \left(\beta l\right)=\frac{1+{\tanh }^2\left(\beta l/2\right)}{1-{\tanh }^2\left(\beta l/2\right)}$$

and combining Eq. 15 with Eq. 17, the trajectory equation is

$$y=-\frac{1}{\beta }·\ln \left(\frac{1+{\tan }^2\left(\beta x/2\right)}{1-{\tan }^2\left(\beta x/2\right)}\right)$$ (18)

Since

$$\underset{\beta x/2\to \pm \pi /2}{\lim }y\left(x\right)=-\infty$$ (19)

the trajectory can be parametrized by introducing the asymptotic distance $L$ between the branches of the U-turn trajectory. This distance also corresponds to the first-order asymptotes of the tangent function in Eq. 18, e.g., given by $\beta x/2=\pi /2$. This means $x=\pi /\beta =L$. Then

$$y=-\frac{L}{\pi }\ln \left(\frac{1+{\tan }^2\left(\frac{\pi x}{2L}\right)}{1-{\tan }^2\left(\frac{\pi x}{2L}\right)}\right)$$ (20)

with

$$\underset{x\to \pm L/2}{\lim }y\left(x\right)=-\infty$$ (21)

which means $L$ is exactly the full asymptotic width of the U-turn, as it should be. Finally, we can simplify the above expression using the following expressions:

$$2{\sin }^2\left(\frac{u}{2}\right)=1-\cos u;2{\cos }^2\left(\frac{u}{2}\right)=1+\cos u$$

to a more concise form

$$y\left(x,L\right)=-\frac{L}{\pi }\ln \left(\sec \left(\frac{\pi x}{L}\right)\right)$$ (22)

Given $L$ determined from an experimental trajectory, the MTB magnetic moment can be computed via

$$m=\frac{\pi \alpha v}{HL}$$ (23)

if the rotational drag coefficient $\alpha$ and velocity of MTB $v$ over the U-turn are known. In our case, we assume the MTB are spherical and use the Stokes rotational drag, but in general any drag model can be used here—for example, elongated bacteria can be approximated as a chain of spheres, for which the rotational drag coefficient has been derived (51). Alternatively, MTB can be modeled as cylinders (33) or ellipsoids (50), or an experimentally determined spiral/spirillum drag coefficient can be used (33).

MTB magnetic moment statistics and scaling

We calculate $m$ using the new U-turn method for a sample population of MTB. We imaged the motion of MSR-1 MTB in alternating MF using an optical microscope and then recovered the cell trajectories from the acquired images. We then analyzed $\sim 18\mathrm{K}$ trajectories retrieved from $1\mathrm{K}$ images that yielded 304 eligible U-turns (i.e., from trajectories and subsequently U-turns that survived the filtering stages in Algorithm 1) for which $m$ was calculated. An example of a U-turn segmentation and processing from a trajectory is shown in Fig. 4), and the radii and velocity of population cells can be seen in Figs. 5 and 6. The resulting overall $m$ statistics are shown in Fig. 7, and the relationship between $m$ and the effective cell radius $r$ is presented in Fig. 8. $m$ data obtained using the proposed theoretical U-turn shape function Eq. 22 were then compared with the results obtained using the calculations based on the U-turn time $\tau$ using Eq. 1 (Figs. 13 and 14). Figs. 7 and 8 provide a direct visual comparison.

Figure 4.

An example of U-turn decomposition and MTB moment calculation: (a) a MSR-1 track in an alternating MF, where points are color coded chronologically from blue to red. (b) Track (a) decomposed into individual U-turns (1–4), before symmetrization/translation (four out of five detected U-turns shown). (c) U-turn (4) filtered and fitted to Eq. 22; the coordinate system differs from (a and b) due to Eq. 3. (d) Evaluating U-turn velocity (color coded by magnitude from blue to red) $v$ for Eq. 23: U-turn $v$ is computed as the mean for the U-turn interval where the relative curvature (inset) is above a threshold. Please see Algorithm 1 for details and parameters.

Figure 5.

Normalized probability $\rho$ (Freedman-Diaconis binning) for the MSR-1 MTB velocity with MF applied in the $x$ direction (horizontal image axis). The most probable MTB velocity magnitude is $\Vert \overrightarrow{v}\Vert \sim 51.3\mu \mathrm{m}/\mathrm{s}$.

Figure 6.

A histogram relating the effective radii $r$ measured for the cell population and cell velocity magnitude $\Vert \overrightarrow{v}\Vert$. The relative bin size is 0.015 of the data range, and value display relative threshold 0.02.

Figure 7.

MSR-1 magnetic moment $m$ probability $\rho$ histograms (Freedman-Diaconis binning) for $m$ calculated using the proposed U-turn shape function Eq. 22 and U-turn analysis algorithm (Algorithm 1), colored light red, and using the existing U-turn time $\tau$-based formula Eq. 1 and the proposed U-turn decomposition algorithm (Algorithm 1), colored blue.

Figure 8.

Magnetic moment $m$ versus MTB effective radius $r$: linear fits and $q=0.95$ uncertainty regions Eq. 22 for $m$ calculated using the proposed U-turn shape function Eq. 22 and U-turn analysis algorithm (Algorithm 1), colored light red, and using the existing U-turn time $\tau$-based formula Eq. 1 and the proposed U-turn decomposition algorithm (Algorithm 1), colored blue. Error bands for the linear fits are indicated with transparent margins around the dashed lines.

Figure 13.

MSR-1 magnetic moment histogram (count $N$ versus moment $m$, Freedman-Diaconis binning), calculated using the existing U-turn time $\tau$-based formula Eq. 1 and the proposed U-turn decomposition algorithm (Algorithm 1), $N_{\text{total}}=264$.

Figure 14.

Magnetic moment $m$ versus MTB effective radius $r$: 264 values determined using the U-turn time $\tau$-based formula Eq. 1 and the proposed U-turn decomposition (Algorithm 1), with a $m\left(r\right)=C_1+C_2·r$ linear fit $\left(R^2=0.773\right)$. Here, $C_2=\left(1.290\pm 0.129\right)·{10}^{-10}\mathrm{A}·{\mathrm{m}}^3$ and $C_1=-\left(0.706\pm 0.086\right)·{10}^{-16}\mathrm{A}·{\mathrm{m}}^2$, with $m\left(r_c\right)=0$ for $r_c=0.55\mu m$.

Furthermore, we tested the sensitivity of our proposed method with respect to different estimates of the U-turn velocity $v$ for the $m$ equation Eq. 23. $m$ statistics for three $v$ estimates are compared in Fig. 9. The simplest assumption is that the U-turn $v$ is the most probable population velocity, determined from the velocity statistics derived from the trajectory data, as seen in Figs. 5 and 6. The maximum probability population velocity is $51.3\mu \mathrm{m}/\mathrm{s}$ with a standard deviation $\sim 20.3\mu \mathrm{m}/\mathrm{s}$, which is consistent with the results reported in (33), but differs from the $30\mu \mathrm{m}/\mathrm{s}$ reported in (52), although it is arguably within the error margin. The other estimates are obtained via our proposed method (Algorithm 1), which estimates the value of $v$ for each U-turn as seen in Fig. 4, which also illustrates the U-turn decomposition process. We use both the individual $v$ and the mean U-turn $v$ (not to be confused with the entire population mean).

Figure 9.

MSR-1 magnetic moment $m$ probability histograms (Freedman-Diaconis binning) for $m$ calculated using the proposed U-turn shape function Eq. 22 and U-turn analysis algorithm (Algorithm 1). Characteristic velocity used in Eq. 22: individual U-turn values (light red), mean value for all U-turns (blue) and the most probable velocity magnitude for the observed MTB population (light green; Fig. 5).

Using our proposed $m$ calculation algorithm and the individual estimate of the U-turn $v$ (Fig. 4), we obtained the $m$ histogram as seen in Fig. 10 where most cells have $m\in \left(0.4;1.0\right)·{10}^{-16}\mathrm{A}·{\mathrm{m}}^2$. However, viewing $m$ statistics this way only does not provide the full picture—therefore, $m$ scaling with $r$ is presented in Fig. 11. Note the positive correlation between the cell $m$ and $r$ values. To assess how $m$ scales with $r$, the data is assigned an error-weighed linear fit $m\left(r\right)=C_1+C_2·r$, where $C_2=dm/dr=\left(2.83\pm 0.13\right)·{10}^{-10}\mathrm{A}·{\mathrm{m}}^3$ and $C_1=-\left(1.33\pm 0.09\right)·{10}^{-16}\mathrm{A}·{\mathrm{m}}^2$. This implies a critical $r=r_c$ for which cell size is insufficient to host magnetosomes that generate a significant $m$; here $m\left(r_c\right)=0$ for $r_c\sim 0.47\mu \mathrm{m}$. The linear fit slope in Fig. 11 is also quite well aligned with the inclination of the quantile uncertainty region $q=0.95$. In Fig. 12, it can be observed that the most probable effective radii among the MTB corresponding to eligible U-turns are $r\sim 0.66\mu \mathrm{m}$ with $m\sim 0.6·{10}^{-16}\mathrm{A}·{\mathrm{m}}^2$, and $r\sim 0.72\mu \mathrm{m}$ with $m\sim 0.9·{10}^{-16}\mathrm{A}·{\mathrm{m}}^2$, respectively.

Figure 10.

MSR-1 magnetic moment histogram (count $N$ versus moment $m$, Freedman-Diaconis binning), calculated using the proposed U-turn shape function Eq. 22 and U-turn analysis algorithm (Algorithm 1), $N_{\text{total}}=304$.

Figure 11.

Magnetic moment $m$ versus MTB effective radius $r$: 304 values determined using the proposed U-turn shape function Eq. 22 and Algorithm 1 are represented by gray dots, with the $q=0.95$ quantile uncertainty region indicated as the light gray area with a green boundary, and the error-weighed $m\left(r\right)=C_1+C_2·r$ linear fit given by the dashed red line with light red slope error margins $\left(R^2=0.611\right)$. Here, $C_2=\left(2.83\pm 0.13\right)·{10}^{-10}\mathrm{A}·{\mathrm{m}}^3$ and $C_1=-\left(1.33\pm 0.09\right)·{10}^{-16}\mathrm{A}·{\mathrm{m}}^2$, with $m\left(r_c\right)=0$ for $r_c=0.47\mu \mathrm{m}$. Fit weights are inverse squares of the total uncertainty due to $m$ and $r$ uncertainties (error propagation). The uncertainty region is derived by computing 250 directional quantiles (53,54), connecting the quantiles with a smooth curve, and applying curve evolution polyline simplification (55) to the resulting quantile envelope.

Figure 12.

Magnetic moment $m$ versus MTB effective radius $r$: a smooth density histogram (Sheather-Jones bandwidth estimator, Epanechnikov kernel) computed for $m$ determined via proposed U-turn shape function Eq. 22 and Algorithm 1.

To determine the U-turn $\tau$ values (the characteristic U-turn completion time) and compare the values of $m$ between the standard equation Eq. 1 and the proposed ones, Eq. 22 and Eq. 23, we compute $\tau$ and $m\left(\tau \right)$ for the same U-turns we used to estimate $m$ based on the proposed approach. To explicitly determine $\tau$ without relying on finding coordinate derivative extrema, we record the coil voltage time series. Voltage sign switches correspond to MF direction reversals. Therefore, using the MF switching data, we detect the MF switch timings from the coil voltage time series by thresholding the time series with the 0.95 relative voltage values and finding time instances with nonzero differences in the resulting signal. We then match the MF switch times to the U-turn timestamps and find the intersection between the two MF switch timestamps nearest to the U-turn temporal endpoints, and the U-turn time interval. The $\tau$ values are then the temporal lengths of the U-turns truncated by the MF switch timestamps multiplied by the image acquisition frame duration. The resulting statistics based on $\tau$-based $m$ are shown in Fig. 13, where most cells have $m\in \left(0.05;0.25\right)·{10}^{-16}\mathrm{A}·{\mathrm{m}}^2$. The $m$ scaling with $r$ is shown in Fig. 14, where the experimental limitation clearly appears when using $\tau$ as a parameter that characterizes the U-turn motion of the cell—its precision is strongly limited by the frame rate used for image acquisition. As a result, Fig. 14 shows banding artifacts in the $m\left(r\right)$ data set computed from $\tau$. This is because the $\tau$ values are multiples of the frame duration, and therefore the differences in $m\left(r\right)$ are in the form of clearly visible steps and distinct curves. As frame rate tends to infinity, these artifacts should disappear, but for lower frame rates the effect is very significant. While $L$ is also limited by the frame rate, $L\mathbb{\in }\mathbb{R}$, which means that the continuum of $\left(m,r\right)$ values is filled much more uniformly by sampling the MTB population $m$ values from $L$ values for U-turns. The calculation of $m$ through $L$ is more limited by image resolution, which in this case is $1920\times 1200\text{pixel}$ ($\sim 28\text{pixels}$ per MTB length), which is usually less of an issue than a lower frame rate.

Importantly, Fig. 14 is not the first instance in which the $m\left(r\right)$ data point banding can be observed—$m\left(r\right)$ correlation was also considered in (31), although for a wild sample of magnetic cocci, not MSR-1. Instead of $r$, the $m$ versus the MTB volume $V$ correlation was produced (31) (Fig. 10), which exhibited banding within the plot region with a higher data point number density. We therefore argue that our observations are not a coincidence. Additionally, the frame rate in (31) was 82 FPS (versus our 20 FPS) while the most probable MTB population velocity was estimated at $\sim 80\mu \mathrm{m}/\mathrm{s}$ ((31)[Fig. 6] versus our $\sim 51.3\mu \mathrm{m}/\mathrm{s}$), meaning that the corresponding temporal sampling should be better than in our case by a factor of $\sim 2.6$, but the artifacts in the $m\left(V\right)$ correlation are still present.

Similarly to the $m\left(L\right)$ case, a trend in $m\left(\tau \right)$ can be observed where $m$ increases with $r$, with the most probable MTB $r$ among the considered U-turns $r\sim 0.68\mu \mathrm{m}$ with $m\sim 0.15·{10}^{-16}\mathrm{A}·{\mathrm{m}}^2$, as seen in Fig. 15. Although the $m\left(r\right)$ data for $\tau$-based $m$ in Fig. 14 exhibit a slight nonlinearity, it is arguably appropriate to assign a linear fit $m\left(r\right)=C_1+C_2·r$, yielding $C_2=\left(1.290\pm 0.129\right)·{10}^{-10}\mathrm{A}·{\mathrm{m}}^3$ and $C_1=-\left(0.706\pm 0.086\right)·{10}^{-16}\mathrm{A}·{\mathrm{m}}^2$, with $m\left(r_c\right)=0$ for $r_c\sim 0.55\mu \mathrm{m}$.

Figure 15.

Magnetic moment $m$ versus MTB effective radius $r$: a smooth density histogram (Sheather-Jones bandwidth estimator, Epanechnikov kernel) computed for $m$ determined via the U-turn time $\tau$-based formula Eq. 22 and U-turn decomposition proposed in Algorithm 1.

The results obtained using both methods are compared side by side in Figs. 7 and 8. The $\tau$-based measurement method yields systematically lower $m$ values, and the probability distribution of $m$ is skewed toward larger values, more so than in the case of estimates based on $L$. In the latter, $m$ is characterized by both a larger range and larger values. However, we would like to reiterate that reducing the overall $m$ statistics is not representative of the diversity of the MTB properties in the population, which is better characterized by the scaling of $m\left(r\right)$. In Fig. 8 the slopes for $m\left(r\right)$ between the $L$- and $\tau$-based $m$ estimation cases differ very significantly, even though the zero-moment effective radii $r_c$ are not that different. Since our method captures the spatial configuration of the U-turns, there is a much more diverse set of $m$ values at every $r$. However, because of the banding artifacts in the $\tau$-based data and, arguably, the more precise/physical treatment of trajectories and U-turns with Eq. 22 and Eq. 23, we posit that our method yields results more representative of the observed MTB population. Note that both the $m\left(r\right)$ data sets and the linear fits indicate a similar $r_c$ value, meaning that no magnetic cells smaller than $r_c$ were captured (that is, MTB whose $r<r_c$ did not exhibit a significant U-turn motion). It is possible that the difference between our $L$- and $\tau$-based results may decrease as the imaging frame rate increases, but it is likely that the difference is the explicit physical modeling of the U-turn motion.

Since the $L$-based $m$ measurement method contains the characteristic U-turn velocity $v$ as a parameter, we tested the same data set as shown in Figs. 10 and 11 using three different approaches to determine cell velocity—the effects of different $v$ estimates on the $m$ probability statistics are shown in Fig. 9. Here, $m$ is calculated using the individual U-turn $v$ computed using Algorithm 1, the mean $v$ for all U-turns, and a simpler estimate based on the maximum probability velocity for the entire MTB population. Individual and mean U-turn $v$ estimates yield similar $m$ statistics, with the mean U-turn $v$ producing slightly less probable peaks at $m\sim 0.6·{10}^{-16}\mathrm{A}·{\mathrm{m}}^2$ and $m\sim 0.9·{10}^{-16}\mathrm{A}·{\mathrm{m}}^2$. However, using the most probable population velocity, the $m$ statistics are significantly different: $m$ distribution is more dispersed and with a significantly shifted mean and most probable values, indicating that one cannot neglect the $v$ calculations for individual U-turns in favor of a coarser, more readily available estimate.

Finally, it is also important to compare our results with the data available in the literature for MSR-1 MTB. Several approaches have been used to measure and calculate $m$. Tracking and magnetic tweezer-based methods reported in (33) and (28), respectively, yield lower $m$ values than those calculated from TEM images (28). A study by Nadkarni et al. (26) indicates that, for AMB-1, also a spirillum and arguably similar to MSR-1, the $m$ values calculated using the $\tau$-based U-turn method were close to those obtained by magnetosome volume measurement. However, it is known that the growth conditions of the MTB population (e.g., oxygen concentration (36) and iron content of the medium can influence the characteristics of the cell population, including cell size and magnetosome formation (56). Therefore, it is reasonable to expect a significant variation of $m$ values between different cultures and experiments.

Table 1 summarizes how our results compare with those reported for MSR-1. The first thing to note is that our mean values for $m$ using both $L$- and $\tau$-based methods are within the expected order of magnitude, $m\sim {10}^{-16}\mathrm{A}·{\mathrm{m}}^2$. However, they are also significantly lower than the rest of the presented $m$ values. There could be a number of reasons for this, including inherently lower $m$ for our MSR-1 population due to the way magnetosomes developed under the conditions defined by our culture preparation. While the velocimetry results (Figs. 5 and 6) for the population indicate that the observed MTB move with velocity close to what is reported by Pichel et al. (33), there were relatively many bacteria in the lower end of the velocity range.

Table 1.

Comparison of different MSR-1 magnetic moment values

$\mathrm{m}({10}^{-16}\mathrm{A}·{\mathrm{m}}^2$)	Method	Source	MTB size $(\mu \mathrm{m}$)
$0.58\pm 0.03$; $N=304$	U-turn, $\tau$ based Eq. 1	This paper	$l=3.14\pm 0.34$
	Drag correction from (33)		$d=0.24\pm 0.06$
$1.37\pm 0.05$; $N=304$	U-turn, $L$ based Eq. 22	This paper	$l=3.14\pm 0.34$
	Drag correction from (33)		$d=0.24\pm 0.06$
$2.5\pm 0.5$; $N=174$	U-turn diameter based	Pichel et al. (33)	$l=5.0\pm 0.2$
	Semi-manual, no shape fitting		$d=0.24\pm 0.01$
$2.4\pm 1.1$; $N=265$	Rotation in alternating MF	Zahn et al. (28)	$l=3.7\pm 0.7$
	Magnetic tweezers		$d=0.42\pm 0.03$
$7.7\pm 3.4$; $N=86$	Translation in constant MF	Zahn et al. (28)	$l=3.8\pm 0.7$
	Magnetic tweezers		$d=0.42\pm 0.03$
$9.9\pm 2.6$; $N=50$	Magnetosome volumes via TEM	Zahn et al. (28)	N/A
$3.54\pm 2.65$; $N=102$	Magnetosome volumes via TEM	Codutti et al. (32)	N/A

Importantly, Pichel et al. performed measurements indicating that the spiral shape of the body of an MTB has a $64\pm 5\%$ higher (rotational) drag $\alpha$ than a spheroid with equivalent length and diameter (33). However, even factoring in this observation and noting that $m\sim \alpha$ in Eq. 23, and that

$$\frac{m}{\ln m}\sim \frac{\alpha }{\tau }$$ (24)

in Eq. 1, our expected $m$ values from $L$-based and $\tau$-based calculations still do not exceed $\sim 1.37·{10}^{-16}\mathrm{A}·{\mathrm{m}}^2$ and $\sim 0.58·{10}^{-16}\mathrm{A}·{\mathrm{m}}^2$, respectively (also recorded in Table 1). Therefore, this correction does not explain all the difference between our $L$- and $\tau$-based results, and does not account for a significant part of the difference between our data and the reported MSR-1 results.

This means that spontaneous loss of magnetic properties of our MSR-1 population during the growth process must be considered (57,58), especially since the U-turn decomposition and $m$ calculation procedure (Algorithm 1) filtered out very many tracks corresponding to MTB that do not respond to MF. These include MTB that are stuck, moving in patterns uncorrelated to MF direction or MF switching, traveling straight through the FOV, exhibiting only minor large-wavelength trajectory perturbations during MF switching, etc. This is reflected in the fact that the $\sim 18\mathrm{K}$ trajectories retrieved from the $1\mathrm{K}$ images yielded $N=304$ U-turns that Algorithm 1 determined as viable based on the criteria summarized in appendix A. We have therefore analyzed another population of MSR-1 – $1\mathrm{K}$ images were acquired and filtered to $\sim 3.3\mathrm{K}$ MTB trajectories, resulting in $N=45m$ values with the mean value $m=0.49\pm 0.4·{10}^{-16}\mathrm{A}·{\mathrm{m}}^2$ ($L$-based calculation Equation 23), corresponding to $r=0.70\pm 0.02\mu \mathrm{m}$ (our MTB length is $l=3.14\pm 0.34$ and width is $d=0.24\pm 0.06$, as seen in Table 1). Here, the $r$ statistics match what we show in Table 1 and the $m$ values are of the same order of magnitude, indicating consistency despite the perhaps lower viability of the control batch. However, we have chosen not to combine the data with the population data in Table 1 since the velocity statistics were significantly different, therefore only using the smaller $m$ value batch as a reference. Note that the number of $m$ measurement instances we have acquired exceeds the size of the data set from any other source for MSR-1, which allowed us to assess the $m\left(r\right)$ dependency, so data sparsity is not the issue here. In addition, converting the MTB dimensions specified in Table 1 from the other sources on MSR-1 indicates that the observed MTB were of similar size. Given all of the above, it is reasonable to assume that the differences in the values of $m$ are probably mainly due to the different growth conditions of MSR-1 prior to the experiments.

It is important to point out that our approach, which treats the MTB trajectory and U-turn shapes explicitly based on a physical model, is also fully automated from start to finish: image processing, MTB tracking, eligible trajectory selection, U-turn decomposition, U-turn selection, and $m$ calculations. This serves to minimize or eliminate human error and bias. This is, for instance, in contrast to what is reported in (33), where a much less robust image MTB detection algorithm is presented that may not work for the number density of MTB in images as considered here, or in the presence of diffraction artifacts at MTB locations or other dynamic artifacts in the FOV. Moreover, nearest-neighbor tracking is used (33), which is known to be very unstable at higher MTB number densities, and when multiple MTB are in proximity and/or their paths are spatiotemporally similar or are crossing. This reduced data acquisition efficiency (less viable MTB tracks are reconstructed) and track reconstruction errors are more frequent and significant. More advanced methods are required for general-purpose MTB tracking, and for the U-turn method case in particular—for instance, approaches such as shake-the-box (59) (robust, thoroughly validated and very widely used in the hydrodynamics community—can be used for long track reconstruction) and multiple hypothesis tracking (MHT), specifically its offline version MHT-X used here (46,47,48) with motion modeling based on weak conservation laws, and velocimetry-assisted motion prediction with boundary condition support. Other relevant methods are reviewed in (60).

In addition, the eligible track selection process is reported as semimanual, which precludes larger-scale data analysis, such as that reported here, and is inherently biased. The U-turn diameter $D$ used in the $m$ calculations (33) is determined manually—the trajectory fragments corresponding to the U-turn motion are segmented manually and $D$ is determined as the U-turn endpoint distance, not the asymptotic branch distance as in Eq. 22 or, simpler, by an ellipse or circle fit. Alternatively, this could have been done via curvature calculation and thresholding, as in Algorithm 1 for the calculation of U-turn $v$. Interestingly, despite a higher reported $m$ value and similar population velocity statistics, the U-turn diameters $D$ in (33) at $\sim 1.85\text{mT}$ are on average $\sim 32\pm 4\mu \mathrm{m}$ (derived from the data reported in (33, Figs. 24 and 25), whereas our statistics for $L$ at $2mT$ indicate an average value of $\sim 7.7\pm 3.7\mu \mathrm{m}$. This seems to be more in line with U-turn numerical simulations performed in (33), where the U-turn width at $2\text{mT}$ is $\sim 11\mu \mathrm{m}$. However, the properties of MTB used for these simulations are specified only as “realistic” (33), so direct comparison is difficult. We would like to stress again that comparing our $L$ values against $D$ can only be done very roughly, since the calculation procedures are vastly different and $D$ is not strictly defined.

We have also noticed that the MTB average (characteristic) velocity $v$ for $m$ calculations in (33) is computed for the entire U-turn (i.e., the turning arcs and branches), whereas our $m$ values are computed from $v$ sampled from the turning arcs (Fig. 4 D). This might be important, as we have shown in Fig. 9. In our case, the mean U-turn velocity is $\sim 20\%$ smaller than the mean population velocity. If this is also true for the population considered in (33), then, with everything else unchanged, $m\sim \left(2.0\pm 0.4\right)·{10}^{-16}\mathrm{A}·{\mathrm{m}}^2$. This is still greater than our drag-corrected value $m=\left(1.37\pm 0.05\right)·{10}^{-16}\mathrm{A}·{\mathrm{m}}^2$, but given the error margins, the results are arguably close.

Among the $m$ measurements listed in Table 1 for MSR-1, the U-turn method is only used in (33). It is then also important to consider other $m$ measurement cases based on U-turn methods, although they are not relevant to MSR-1. In simpler cases, methods deriving U-turns from trajectories via the extrema in the U-turn trajectory coordinate derivatives (8,9,26,31) may be sufficient (although errors and lower precision can be expected). In cases with longer and more complex trajectories, as well as with higher MTB number density in images, these methods should not be considered reliable and precise.

Although the U-turn method is not used in (32), the data presented in (32, Fig. 5B) and in the repository linked to the article can still be used to estimate $m$. As shown in Table 1, the $m$ value due to TEM measurements is $\left(3.54\pm 2.65\right)·{10}^{-16}\mathrm{A}·{\mathrm{m}}^2$, and this value is used as a reference for calculations and data analysis in (32). At the same time, using the available data, we recovered the MSR-1 MTB trajectory turning radii (not exactly the U-turns as defined here or in (33)) within circular traps, and the respective MTB velocity values. Since MTB length is not provided explicitly in (32), but rather a body length scale is specified $l\sim 3\mu \mathrm{m}$, we can only provide a very rough estimate via Eq. 23, assuming the same mean MTB size as in our case, and taking the $0.5\text{mT}$ MF as in (32). With this, we estimate $m\sim \left(2.08\pm 0.07\right)·{10}^{-16}\mathrm{A}·{\mathrm{m}}^2$ with a spherical drag model, and $m\sim \left(3.44\pm 0.12\right)·{10}^{-16}\mathrm{A}·{\mathrm{m}}^2$ if the correction from (33) is taken into account, although the error margins cannot be rigorously estimated (e.g., velocity and turning radii uncertainties not specified in (32)), and are likely much wider. Note that for MTB with different dimensions, the estimate will differ significantly (52). This and the strongly confined motion (turning radius affected by trap boundary conditions) make comparisons with our results and the data from (33) problematic, but validates the order of magnitude for the measured $m$ values.

Arguably, given the current state of the literature on MTB $m$ measurements (relatively few measurements per research paper (9,28,31,32,33) and little to no discussion of $m\left(r\right)$ scaling (31)), and the diversity of MTB types and their properties (which depend on the growth conditions), there is a need for an extensive measurement campaign where many different cultures of various MTB are analyzed systematically and automatically, to obtain likely thousands of magnetic moment measurement instances. The U-turn method is promising in this regard, as it can be used for simultaneous $m$ measurements for many MTB, and long image sequences can be acquired for processing. The methodology we have presented here, unlike the time-based calculations for the U-turn, works directly with the U-turn shapes, and thus more precisely, and is readily scalable to measuring $m$ from many thousands of images with little or no parameter adjustment. Most of the underlying code is efficiently parallelized and can benefit from high-performance computing hardware.

Conclusions and outlook

Using the proposed methodology, we performed optical microscopy experiments with high number density motion of MSR-1 MTB in alternating MF, and determined the magnetic moment for 304 MTB. Our results show that computing $m$ from time-based methods underestimates the magnitude of $m$ compared with our method based on the distance between the trajectory branches $L$. Our results are within the order of magnitude of the data reported in the literature (closer after adjusting for a helical MTB drag model), and the differences are possibly due to spontaneous mutations that can cause the culture to lose magnetic properties. We have also shown the scaling of MTB magnetic moment with MTB size, which is roughly linear for our data—to our knowledge, such data are currently missing for widely used MSR-1 MTB.

To achieve the above, we have derived a theoretical shape function for the U-turn motion of magnetotactic microorganisms, including MTB, in alternating MF, and used it to improve the precision of the U-turn method used to determine the magnetic moment for magnetotactic microorganisms. We have also presented an alternative expression for computing the magnetic moment, which is based on the asymptotic width of the shape function, the MTB rotational drag coefficient, alternating MF strength, and the characteristic U-turn velocity. This method, unlike the U-turn time-based $\left(\tau \right)$ moment calculation, explicitly determines the magnetic moment from the U-turn geometry and uses the entire U-turn length, without the need to compute $\tau$.

We have coupled the new shape function with a fully automated method for MTB trajectory decomposition into U-turns, which also determines the characteristic U-turn velocity and then computes the magnetic moment by fitting the theoretical U-turn shape function to the experimentally observed U-turns.

We have reported significant differences in the magnetic moment statistics and scaling with MTB size between the established $\tau$-based moment calculation and the shape function-based method proposed here, and highlighted two important limitations of the $\tau$-based method—its much stronger dependence on higher image acquisition frame rate to sample the magnetic moment statistics, and its lower precision since it does not model the U-turn geometry. This could explain the difference in the observed $m$ mean values and statistics, but the discrepancy could decrease with a higher frame rate. We have also shown that it is important to determine the characteristic U-turn velocity values explicitly for each U-turn, since the magnetic moment statistics based on these differ significantly from the calculation based on a cruder population-based estimate.

We argue that the U-turn method is very promising for simultaneous measurement of magnetic moment for many MTB from a sequence of images, given that the method is paired with sufficiently robust, precise, and automated methods for MTB detection, tracking, and U-turn analysis. We posit that our U-turn shape function-based magnetic moment calculation, combined with the proposed fully automated U-turn decomposition and analysis approach, will enable convenient property determination with many more data points than currently reported (magnetic moment distribution, magnetic moment scaling with MTB size, etc.) for populations of different magnetotactic microorganisms. The proposed methods are scalable to automatically analyze thousands of images and tens of thousands of trajectories, all the relevant code is open source, and the parameters are provided either in this paper or with the code.

Lastly, we propose to apply the methods developed in this paper to process a large data set of images with MSR-1 motion in alternating MF to refine the current results. This should be continued by a systematical study how the utilized rotational drag model affects the magnetic moment statistics and scaling with MTB size by considering the spirillum drag model experimentally established by Pichel et al. (33), as well as the cylindrical (25), ellipsoidal (50), and the sphere chain (51) MTB shape models.

Data and code availability

Experimental data are available on reasonable request; please contact the corresponding author.

The image analysis, object tracking, and trajectory analysis code for $m$ calculations are available on GitHub:

• •MTB detection from images: Mihails-Birjukovs/MTB_detection_tracking.
• •MTB tracking: Peteris-Zvejnieks/MHT-X.
• •MTB magnetic moment retrieval from trajectories: Mihails-Birjukovs/MTB_magnetic_moment_from_U-turn_trajectories.

Acknowledgments

The authors acknowledge funding from the Latvian Council of Science, project A4Mswim, project lzp-2021/1-0470. The authors are grateful to Dr. Paul Janmey from the University of Pennsylvania for his feedback and editing of the manuscript.

Author contributions

M.S. prepared the MTB culture and set up and performed the experiments. M.S. and M.B. performed data analysis and interpretation. M.B. developed the image processing, trajectory/U-turn analysis, and magnetic moment calculation methods and code. P.Z. contributed the trajectory reconstruction software and assisted with its application. I.D. developed the real-time MF recording setup. A.C. derived the theoretical U-turn shape function. G.K. provided funding and resources, and contributed to result interpretation. M.S., M.B., A.C., and G.K. contributed to the first draft. All authors reviewed and edited the final version of the manuscript.

Declaration of interests

The authors declare no competing interests.

Editor: Paul Janmey.

Appendices

Appendix A. MTB magnetic moment retrieval from trajectories

Parameters for Algorithm 1:

• 1.$N_{t1}=10$, ${\chi }_{c1}=10$
• 2.${\sigma }_c=5$, $M_c=0.85$, ${\sigma }_p=5$
• 3.${\sigma }_{k1}=0.05·N_t$, $M_{k1}=0.95$, ${\sigma }_{t1}=2.5$
• 4.$N_{\text{cut}}=\max \left(1;∟n_{\text{cut}}·N_t\right),n_{\text{cut}}=7.5·{10}^{-2}$
• 5.$N_{t2}=N_{t1}$, ${\chi }_{c2}={\chi }_{c1}/2$
• 6.$S_c=3.5·{\left(z_x·z_y\right)}^{1/2}$, where $z_k$ are image dimensions
• 7.$\left({\sigma }_{k2},M_{k2},{\sigma }_{t2}\right)=\left(0.85,1·{10}^{-3},0.25\right)\ast \left({\sigma }_{k1},M_{k1},{\sigma }_{t1}\right)$, $K_1=10$
• 8.$N_{t3}=N_{t1}$, ${\chi }_b=1.75$
• 9.$\left({\sigma }_{k3},M_{k3},{\sigma }_{t3}\right)=\left(0.25,1.75·{10}^{-3},0.25\right)\ast \left({\sigma }_{k1},M_{k1},{\sigma }_{t1}\right)$, $K_2=K_1$
• 10.$A_c=2$
• 11.$\beta =1$, $g_c=1·{10}^{-5}$, $f_c=1·{10}^{-2}$
• 12.${\sigma }_{k4}=0.25·N_t$, $M_{k4}=0.25$