Introduction

Magnetotactic bacteria (MTB) are microorganisms that contain magnetosomes that are magnetic nanoparticles of magnetite (Fe3O4) or greigite (Fe3S4) with a membrane and, in general, arranged in linear chains (Abreu and Acosta-Avalos 2018). The magnetosome chain confers a magnetic moment to MTB that permits them to interact passively with the geomagnetic field through the magnetic torque, which aligns the bacterial body and its swimming direction to the magnetic fields lines (Kalmijn 1981). That property is known as magnetotaxis. In optical microscopy MTB can be recognized because they start to swim when the local magnetic field direction is changed, following the new field direction. A challenge in the study of MTB is the measurement of the bacterial magnetic moment in vivo. Several techniques have been proposed to achieve this goal: estimate directly from the magnetosome chain by electron microscopy (Frankel and Blakemore 1980), the Langevin function (Kalmijn 1981), optical scattering and birefringence (Rosenblatt et al. 1982a, b), magnetic tweezers (Zahn et al. 2017), SQUID magnetometry (Wajnberg et al. 1986), analysis by U-turn trajectories (Esquivel and Lins de Barros 1986; De Melo and Acosta-Avalos 2017) and through rotating magnetic field trajectories (Petersen et al. 1989) among others. From those techniques, the U-turn analysis is one of the simplest techniques that permits the estimation of the magnetic moment for individual MTB. This technique is based on the analysis of the movement in the Low Reynolds number regime (Esquivel and Lins de Barros 1986). In this case, the total torque is null and the following must be satisfied:

$$T_{{{\text{mag}}}} \; + \;T_{{{\text{viscous}}}} \; = \;0,$$
(1)

where \(T_{{{\text{mag}}}} \; = \;mB\sin (\gamma )\) is the magnetic torque and \(T_{{{\text{viscous}}}} \; = \; - \;8\pi \eta R^{3} (d\gamma /dt)\) is the viscous drag torque for a spherical microorganism. γ is the angle between the magnetic moment m and the magnetic field B, and is assumed that the magnetic moment follows the trajectory direction. In the above expressions, η is the water viscosity (about 10–3 Pa s) and R is the microorganism radius. 8πηR3 is known as the viscous coefficient and its expression is different for other body geometries. So, Eq. (1) can be rewritten as

$$d\gamma /dt\; = \;[mB/8\pi \eta R^{3} ]\;\sin (\gamma ).$$
(2)

Interestingly, Eqs. (1) and (2) do not include the effect of the flagellar movement. In some cases, to include the effect of the flagella, it has been proposed a correction to the viscous coefficient by a numerical factor (Chen et al. 2018) or the inclusion of an extra torque in Eq. (1) (Pichel et al. 2018). In the present study, we will ignore the flagellar effect. Figure 1 shows a U-turn observed in an optical microscope. Esquivel and Lins de Barros (1986) calculated two parameters from the U-turn curve using Eq. (2): the U-turn radius and time τ. The U-turn radius is difficult to be measured because it is necessary that the MTB swims in the focal plane during all the U-turn trajectory, and that is not the case in general. The U-turn time τ is easier to measure from the U-turn trajectory (e.g., De Melo and Acosta-Avalos 2017). As the U-turn trajectory starts from a sudden inversion in the magnetic field direction and the trajectory is symmetrical, Eq. 2 can be rewritten as \(d\gamma /\sin (\gamma )\; = \;[mB/8\pi \eta R^{3} ]dt\) and integrated from 0 to τ/2 in time and from γi ≠ 0 to π/2 in the angle γ to avoid indeterminations found at γ = 0 and γ = π. Esquivel and Lins de Barros (1986) understand γi as a perturbation in the initial MTB orientation and calculate a value for it using the expression for the average MTB orientation to a magnetic field B through the Langevin equation (Kalmijn 1981): \(< \cos (\gamma ) > \; = \;L\left( {mB/kT} \right),\) where \(L\left( x \right)\; = \;\coth x + 1/x\). From that expression, it is possible to show that \(\gamma_{i} \; \approx \;\left( {2kT/mB} \right)^{1/2}\). The U-turn time τ becomes

$$\tau \; = \;[8\pi \eta R^{3} /mB]\;\ln \;\left( {2mB/kT} \right),$$
(3)
Fig. 1
figure 1

U-turn trajectory example. Each point represents the position of the MTB coccus observed by videomicroscopy. The drop border is to the right and the trajectory starts in the point near to coordinates (90,47). Coordinates were obtained frame to frame and the units were converted from pixel to μm using a calibration ruler. Observe that in this example it is difficult to define a U-turn radius

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where k is the Boltzmann constant and T is the temperature (about 300 K). From Eq. (3) it is possible to estimate the value of m for each MTB observed in the microscope performing U-turn trajectories.

As far as we know, in the literature, there are no studies of simultaneous movement and magnetic measurements in individual MTB. Therefore, the aim of the present paper is to estimate the magnetic moment, the velocity and the body volume of natural MTB from a lagoon and observe their distribution and possible correlations among them.

Materials and methods

Uncultured magnetotactic cocci were collected in Rodrigo de Freitas lagoon (22°58′S, 43°12′W), an urban lagoon located in Rio de Janeiro city. The sediments were collected in 2007 and have been maintained in a glass aquarium since then, completing the aquarium water level from time to time using tap water. The local geomagnetic parameters in Rio de Janeiro are horizontal component = 18 μT, vertical component = − 15 μT, and total intensity = 23 μT.

To concentrate MTB for the experiments, a sub-sample was transferred to a specially designed flask containing a lateral capillary aperture and a small magnet generates a magnetic field aligned to the capillary aperture (Lins et al. 2003). The studied uncultured magnetotactic cocci were south-seeking and swam towards the capillary facing the north pole of a magnet. After 5 min, samples were collected with a micropipette.

Samples of MTB were prepared for scanning electron microscopy (SEM) as follows: MTB in a drop were concentrated in the border using a magnet, then MTB were fixed with 2.5% glutaraldehyde in 0.1 sodium cacodylate buffer, washed with fresh water and dehydrated in acetone series, later gold sputtered and observed in a JEOL JSM-6490LV. These magnetically enriched cells were transferred to formvar/carbon-coated copper grids and observed by transmission electron microscopy (Tecnai Spirit, FEI Company).

To estimate the magnetic moment of MTB, the U-turn method was used to calculate the U-turn time τ. On the stage of an inverted microscope (Nikon Eclipse TS100) was set a pair of coils connected to a DC power supply and fixed to a glass microscope slide where the collected drop with MTB was placed (Fig. 2). The used lens had magnification of 40× and numerical aperture of 0.55 allowing the measurement of the MTB radius R. The magnetic field generated by the coils was of about 280 μT. An electric circuit for changing the voltage polarity (reversal of current) was connected between the power supply and coils, leading to the inversion of the magnetic field direction when the button was turned on or off. After two magnetic field inversions, the MTB perform U-turn trajectories (Fig. 1). The magnetic moment m can be estimated using Eq. (3). To calculate τ, the following procedure was performed: U-turn trajectories were recorded, at a rate of 82 fps, in the inverted microscope with a digital camera (Lumera Infinity 1). The coordinates of the U-turn trajectories were obtained using the software ImageJ (NIH—USA). The coordinates were in pixel units and the conversion to μm was done using a calibration ruler that consists of a 1 mm line divided into 100 parts. In the experimental setup, the external magnetic field is applied in the x-direction, meaning that the x-coordinate of the U-turn in Fig. 1, as function of time, must be two straight lines with different slopes (Fig. 3a). The U-turn time τ must be the time necessary for the change of slope and can be calculate through the derivative dx/dt (Fig. 3b). As τ depends on the radius of the microorganism, a table of theoretical values for τ/R3 using Eq. (3), as a function of m and maintaining η, B and T constant, permits the determination of the value of m for every MTB through the comparison with the experimental values of τ/R3.

Fig. 2
figure 2

Experimental setup for observing MTB. A pair of coils was attached to microscope glass slide in such a way that the whole structure fits in the microscope stage over the objective lens in an inverted microscope Nikon TS100. The objective lens was 40×. The wires to the right are connected to a DC power supply to generate a magnetic field between the coils

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Fig. 3
figure 3

a x-coordinate of the U-turn trajectory (Fig. 1) as function of time. MTB starts to move from the border of the drop when the magnetic field inverts its direction. Another sudden inversion in the magnetic field direction produce the U-turn curve observed in Fig. 1 and the x-coordinate shows that V shape. b Derivative of x. The transition time between the negative and positive velocities is identified as the U-turn time: τ = tf − ti, where ti is the initial time and tf the final time of the turn. In this case, τ = 0.036 s. c x coordinate, corresponding to Fig. 1, before the second inversion in the magnetic field direction. The inclination of the straight line corresponds with Vx. In this case, Vx = − 84.3 μm/s. d y-coordinate, corresponding to Fig. 1, before the second inversion in the magnetic field direction. The inclination of the straight line corresponds with Vy. In this case Vy = 4.9 μm/s. From Vx and Vy values, the axial velocity becomes Vax = 84.5 μm/s

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The axial velocity was measured analyzing the movement before the magnetic field inversion associated with the U-turn, assuming that the MTB trajectory is similar to a cylindrical helix. In this case, if the helix axis is aligned to the magnetic field direction then the coordinates must have the following parametrization:

$$x^{\prime } \left( t \right)\; = \;x_{0} \; + \;V_{{{\text{ax}}}} \; \times \;t,$$
(4a)
$$y^{\prime } \left( t \right)\; = \;y_{0} \; + \;r\; \times \;\cos (2\pi ft\; + \;\phi_{0} ),$$
(4b)

where x0 and y0 are initial values of the coordinates, Vax is the axial velocity, r is helix radius, f is the helix frequency and ϕ0 is the phase constant. However, the trajectory is not fully parallel to the magnetic field direction because of thermal perturbations that disorient the bacterial swimming (Kalmijn 1981). In the case that the trajectory is tilted relative to the magnetic field by an angle θ, the coordinates must have the following expressions:

$$x\;\left( t \right)\; = \;x^{\prime } \cos \theta \; + \;y^{\prime } \sin \theta \; = \;r\sin \theta \cos (2\pi ft\; + \;\phi_{0} )\; + \;V_{{{\text{ax}}}} \; \times \;t \times \;\cos \theta ,$$
(5a)
$$y\;\left( t \right)\; = \; - x^{\prime } \sin \theta \; + \;y^{\prime } \cos \theta \; = \;r\cos \theta \cos (2\pi ft\; + \;\phi_{0} )\; - \;V_{{{\text{ax}}}} \; \times \;t\; \times \;\sin \theta .$$
(5b)

The observed coordinates x and y as function of time must be oscillating functions with linear tendencies. If θ is near 0° then x(t) must be similar to a straight line and y(t) must be an oscillating function with an inclination. An example of that can be observed in Fig. 3c, d. From that figures, the inclinations correspond to \(V_{{\text{x}}} \; = \;V_{{{\text{ax}}}} \; \times \;\cos \theta\) and \(V_{{\text{y}}} \; = \;V_{{{\text{ax}}}} \; \times \;\sin \theta\), respectively, and \(V_{{{\text{ax}}}} \; = \;\left( {V_{{\text{x}}}^{2} \; + \;V_{{\text{y}}}^{2} } \right)^{1/2}\). The inclination for each coordinate is calculated through a linear fit as a function of time for the curve before the magnetic field inversion (Fig. 3c, d). In general, it is not easy to calculate r and f because the coordinate y(t) has a complex oscillating behavior, similar to the sum of two oscillating functions with two frequencies and radiuses. Even with that complex oscillating behavior in y(t), the axial velocity can be calculated from the time derivatives shown in Eqs. 5a, 5b and in Fig. 3c, d.

All graphics and analysis were done using the software Microcal Origin and the statistics with the software GraphPad InStat and Oriana for circular statistics.

Results and discussion

Table 1 shows the results for the size of the uncultured magnetotactic cocci measured by SEM and by otical microscopy (OM). The mean values apparently are different (t test, p < 0.0001) but observing the histograms for both distributions (Fig. 4), it is clear that by OM two size groups were measured, one with maximum at about 0.75 μm and the other at about 1.15 μm. The size distribution observed by SEM (Fig. 4b) has a peak at about 0.65 μm and is similar to the first distribution observed by OM (Fig. 4a). Figure 5a shows TEM images of cocci with both sizes, being the majority in the first group. Video 1 (supplementary material) shows an example of swimming cocci where bacteria from both size groups can be observed. This shows that the size estimation using OM is trustable when compared with a well-established technique.

Table 1 Statistics for MTB radius, velocity, trajectory angle θ, cos(θ), U-turn time and magnetic moment calculated using the U-turn time method
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Fig. 4
figure 4

a Histogram of MTB radiuses measured by optical microscopy (OM). Those values were obtained from the videos recorded for the U-turn analysis. As can be seen, two well-defined distributions are present, one with maximum at about 0.75 μm and another maximum at about 1.15 μm. b Histogram of MTB radiuses measured by scanning electron microscopy (SEM) images. In this case, only one distribution is observed with maximum at about 0.65 μm

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Fig. 5
figure 5

Transmission electron microscopy images of uncultured magnetotactic cocci. a Uncultured magnetotactic cocci showing different sizes. In the figure, the diameter is indicated by a dashed line. For the bacterium marked with asterisk, the diameter is 2 μm and for the bacterium marked with double asterisk it is 1.5 μm. b Magnetotactic coccus showing two non-parallel magnetosome chains

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As magnetotaxis is related to the presence of the intracellular magnetosome chain, because it confers a magnetic moment to the magnetotactic microorganism, it is interesting to discuss their distribution in the uncultured magnetotactic cocci. Figure 5b shows magnetotactic cocci with two magnetosome chains. Each magnetosome chain has an average number of seven nanoparticles (SD = 2, N = 16). Assuming magnetite as the magnetic mineral, it is possible to estimate the average magnetic moment for the uncultured magnetotactic cocci, as done by Frankel and Blakemore (1980). Considering each nanoparticle as a parallelepiped, it is possible to calculate its magnetic moment as V × MV, where V is the volume and MV is the magnetic moment per unit volume of magnetite (0.48 Am2/cm3). Each magnetosome has an average length of 115 nm (SD = 22 nm, N = 89), width of 93 nm (SD = 23 nm, N = 89) and width to length ratio of 0.79 (SD = 0.11, N = 89) which is characteristic for single-domain magnetite nanoparticles. Each magnetosome presents an average magnetic moment of 5.3 × 10–16 Am2 (SD = 2.7 × 10–16 Am2, N = 89). Each magnetotactic coccus has an average magnetic moment of 6.8 × 10–15 Am2 (SD = 4.3 × 10–15 Am2, N = 7) assuming that both magnetosome chains are parallel. Unfortunately, the flagella were not observed in TEM images. In other magnetotactic cocci, it is common to observe one tuft of flagella (Nogueira and de Barros 1995) or two parallel tufts of flagella as in Magnetococcus marinus strain MC-1 (Felfoul et al. 2016) or in Magnetofaba autralis strain IT-1 (Araujo et al. 2016). As both magnetosome chains are not parallel in general, the resultant magnetic moment must present an inclination relative to the orientation of the tufts of flagella present in the magnetotactic coccus.

The statistics for the axial velocity Vax are shown in Table 1 and the corresponding histogram in Fig. 6. The value of the mean velocity is in agreement with the velocity measured in other MTB (see, for example, Kalmijn 1981; Zhang et al. 2012). Removing the highest velocity outliers, Fig. 7 shows that there is no correlation between the MTB volume and velocity (Spearman rank correlation r = − 0.21, significance of r: p = 0.08).

Fig. 6
figure 6

Histogram of MTB axial velocities Vax. Two local maxima are observed, being the great majority of the individuals in the first distribution with maximum at about 80 μm/s. The second distribution has less individuals and its maximum is at about 130 μm/s

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Through the axial velocity components, Vx and Vy, it is possible to calculate the angle θ between the magnetic field and the MTB trajectory as discussed in Materials and Methods. Table 1 shows the mean angle and Fig. 8 shows their circular histogram. The applied magnetic field was directed in the 0° direction, and Fig. 8 shows that MTB swam following the magnetic field direction as expected for magnetotaxis. Also in Table 1 is shown the statistics for cos(θ). Its mean value is 0.993 meaning a good MTB orientation related to the applied magnetic field.

Fig. 7
figure 7

Axial velocity Vax as function of the MTB volume. There is no correlation between both variables (Spearman rank correlation r: − 0.21, probability p = 0.08 of r be significantly different from zero)

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Fig. 8
figure 8

Circular histogram of the orientation angles θ. Each value of θ is represented by a triangle. The straight line represents the mean angle (3.2°) and the bar at the end of the line represents the standard deviation (5.7°). The magnetic field is directed in the 0° direction. The water drop border is in the line 90°–270°. The insert inside the circle represents the meaning of the angle θ: it is the angle between the axial velocity Vax and the magnetic field B. Both are not parallel because of thermal perturbations that disorient the bacterial swimming (Kalmijn 1981)

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Kalmijn (1981) analyzed the statistics for cos(θ) assuming that MTB behave as paramagnetic particles. In this way, a Boltzmann probability distribution can be applied to calculate the mean values of cos(θ): \(< \;\cos (\theta )\; >\) and its variance σ2. The combination of \(< \;\cos (\theta )\; >\) and σ2 permits the determination of the energy quotient X = mB/kT, where the product mB is the magnetic energy and kT represents the thermal energy (being k the Boltzmann constant and T the absolute temperature). Assuming that the MTB magnetic moment is parallel to its motility axis, Kalmijn (1981) showed that \(< \cos (\theta ) > \; = \;\coth \left( X \right)\; - \;1/X\) and \(\sigma^{2} \; = \;1\; - \;\coth^{2} \left( X \right)\; + \;1/X^{2}\). Simple math permits to show that

$$X\; = \;(2\; < \;\cos (\theta )\; > )/(1 - < \cos (\theta )\; >^{2} \; - \;\sigma^{2} ).$$
(6)

Using \(< \;\cos (\theta )\; >\)  = 0.993 and σ2 = 0.000107, we obtain X = 288.1. As kT ≈ 4.1 × 10–21 J for T = 300 K and B = 2.8 × 10–4 T, m = 4.2 × 10–15 Am2, which is a good value for natural samples (e.g., Petersen et al. 1989). That value of m represents an average for all the analyzed sample and is in good agreement with the average value of the bacterial magnetic moment measured by TEM and to the median value of the magnetic moment distribution (5.4 × 10–15 Am2).

Table 1 shows the statistics for the U-turn time τ that has a mean value of 0.07 s and a standard deviation of 0.04 s, meaning that the distribution has a large amplitude (minimum of 0.02 s and maximum of 0.3 s). As the experimental measurement of τ does not depend on the measurement of the MTB volume it is interesting asking for the correlation between them. There is a positive correlation among both parameters (Spearman rank correlation r = 0.275, significance of r: p = 0.022). The value of r is low, meaning a weak correlation between the U-turn time and the volume, and a tendency to bigger MTB have higher U-turn times. From that tendency, it is difficult to infer the tendency of the magnetic moment as function of the volume, because the U-turn time is a function of the magnetic torque (that must decrease the value of τ when it increases) and of the viscous torque (that must increase the value of τ when it increases). Using the U-turn technique, it is possible to measure the individual value of m for each MTB observed. Figure 9 shows the histogram for MTB's magnetic moments. As is shown, the distribution has a large tail to the side of larger magnetic moments, similar to a log-normal distribution. The distribution shows a peak at about 1.4 × 10–15 Am2, which is a common value measured by several techniques (Abreu and Acosta-Avalos 2018). The higher values observed can be associated with the presence of bigger cells as shown in the size histogram (Fig. 4). In Table 1 is shown the statistics for the whole set of magnetic moment values. The standard deviation is high because of the non-symmetrical distribution observed in the histogram (Fig. 9).

Figure 10 shows the magnetic moment as a function of volume. There is a positive correlation between them (Spearman rank correlation r = 0.649, significance of r: p < 0.0001). So, bigger MTB have bigger magnetic moments. This must be related to a continuous magnetosome production during the bacterium life cycle, in such a way that in the moment of cellular division it has enough magnetosomes to share between the new bacteria. A similar observation has been done in Candidatus Magnetoglobus multicellularis (Perantoni et al. 2009). Figure 11 shows the magnetic moment as function of velocity, and as happens with the velocity as function of the volume, there is no correlation between both variables (Spearman rank correlation r = − 0.17, significance of r: p = 0.16). The correlation among the magnetic moment and cos(θ) is not significant (Spearman coefficient r = 0.022, significance of r: p = 0.85).

Fig. 9
figure 9

Histogram of MTB magnetic moments obtained using the U-turn technique. As can be seen, the distribution is complex and a first Gaussian-like distribution is observed until about 6 × 10–15 Am2, with maximum at about 1.4 × 10–15 Am2. Other maxima are observed, as one at 8.5 × 10–15 Am2 and another at 12.5 × 10–15 Am2. This complex distribution produces a mean value at 8.2 × 10–15 Am2 and median value at 5.4 × 10–15 Am2. Interestingly, the median value is similar to the average magnetic moment calculated using the expressions for \(< \;\cos (\theta )\; >\) and σ2 (Eq. 6): 4.2 × 10–15 Am2

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Fig. 10
figure 10

Magnetic moment m as function of MTB volume. It is observed a significant positive correlation (Spearman rank correlation r = 0.649, significance of r: p < 0.0001)

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Fig. 11
figure 11

Magnetic moment m as a function of the axial velocity Vax. In the same way as happens with Vax and the volume (Fig. 7) there is no correlation between both parameters (Spearman rank correlation r = − 0.17, significance of r: p = 0.16)

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The U-turn technique allows the measurement of the magnetic moment of individual MTB and also the measurement of parameters associated with the relative orientation (angle θ) and kinematics (the velocity Vax). The magnetic moment histogram (Fig. 9) shows that in natural samples its distribution is non-symmetrical and can be complex. The determination of an average magnetic moment for the studied sample, using the statistical method of Kalmijn (1981), hides that complexity. The value of 4.2 × 10–15 Am2 is a good value compared with the average magnetic moment calculated from the magnetosome chains, but Fig. 9 shows a peak at about 1.4 × 10–15 Am2, a value that has been measured as the MTB magnetic moment in natural samples and cultures [Abreu and Acosta Avalos 2018]. In our study, the three methods used to estimate the average magnetic moment of the magnetotactic cocci sample (U-turn, paramagnetic model and magnetosome observation) provided similar estimates of the average MTB magnetic moment.

The magnetic moment does not correlate with the velocity, in agreement with our understanding of magnetotaxis: the magnetic field only produces a torque to align the magnetosome chain and in consequence the bacterial body, and not produce any force to move the MTB through the water. Also, the magnetic moment shows a positive correlation with the volume. So, bigger MTB carry higher values of m. Also, the magnetic moment does not correlate with cos(θ), considered to be a parameter related to the degree of orientation to the magnetic field. It means that MTB with higher magnetic moment are not better aligned to the magnetic field. This can mean that the magnetic moment in our natural samples is so high in each MTB that all of them are sufficiently aligned and the deviation from cos(θ) = 1 (perfect alignment) can be due to thermal perturbations or (an)other kind(s) of perturbation(s).

Conclusion

In conclusion, the U-turn technique was used for the measurement of the individual magnetic moments in a population of the uncultured magnetotactic cocci, together with their velocity and size. The use of the statistical technique based on the Langevin function produced a good value for the mean magnetic moment, similar to the average value obtained by TEM observations. With the measurement of the magnetic moment in individual MTB it was possible to analyze the correlation between the magnetic moment and the volume, the velocity and the coefficient of alignment cos(θ). The magnetic moment distribution obtained is similar to a log-normal distribution which implies that the average value is not a good representative value for the magnetic moment when compared to the median value. The magnetic moment shows a strong positive correlation with the volume, and a null correlation with velocity and cos(θ). The null correlations can be explained by the current understanding of magnetotaxis: the magnetic field only orients the bacterial body through the magnetic torque.