Figure 1.
Cell orientation with respect to the magnetic field and the optical axis.
Two different spherical coordinates systems were used to describe cell orientations (assimilated to the direction of their magnetic moment). In the first coordinate system (blue), the zenith direction is the direction of the magnetic field (x-axis), α is the polar angle (and the actual angle between and
) and β is the azimuth angle. In the second coordinate system (orange) the zenith direction is set along the optical axis (z-axis), φ is the complementary angle to the polar angle (and the inclination of the cell out of the focal plane) and θ is the azimuth angle (and the apparent angle between
and
when trajectories are projected in the focal plane).
Figure 2.
Dimensions of M. magneticum cells and magnetosomes.
A) Distribution of cell lengths as measured from 118 cells observed under the light microscope. The reported length of each cell is an average of five measurements done in different movie frames. B) Transmission electron micrograph of a typical cell. Magnetosome dimension estimate from the image returns a value of the magnetic moment μ = 4.3×10−16 A⋅m2 for this cell.
Figure 3.
Cell trajectories in periodically reversing magnetic fields.
A) Trajectories of four cells moving in response to a periodically reversing magnetic field of frequency 700 mHz and amplitude indicated by trace colour. Subsequent analysis of trajectory 1 is presented in panels B, C & D. B) Position of the cell in the direction perpendicular to that of the magnetic field. C) Sine of the angle between the direction of movement of the cell and the magnetic field. D) Square of the sine of the angle plotted in panel C. Fit to Eq. 12 yields a magnetic moment μ = 3.2×10−16 A⋅m2 for this cell, as explained in the text.
Figure 4.
Cell trajectories in constant magnetic fields.
A) Trajectories of 25 cells in a zero (left) and non-zero (right, B = 3.3 mT) magnetic field. B) Displacement of the cells after t = 46 ms (upper panels) and t = 139 ms (lower panels). Only regular trajectories used in subsequent analyses are presented here, with data for B = 0 mT (left panels, 10 cells) and B = 3.3 mT (right panels, 51 cells). Each different colour represents a different cell, with the displacement of each cell measured over multiple time intervals.
Figure 5.
Rotational bias of cells placed in constant magnetic fields.
A) Average angular velocity of cells as a function of their initial orientation, plotted for magnetic fields with different magnitudes (mean ± standard error). Fit was performed according to the rotational bias model (Eq. 9). B) Ratio of magnetic torque to rotational drag coefficient for cells placed in magnetic fields of various magnitudes. Plotted values are obtained from fit of data such as those shown in panel A. Linear fit to the data measured at magnetic fields B<2 mT yields a magnetic moment μ = 5.7±0.2×10−16 A⋅m2, as explained in the text.
Figure 6.
Cell orientation distributions in constant magnetic fields.
A) Mean orientation of the cells evaluated as the mean sine of the angle between the observed direction of movement in the focal plane and that of the magnetic field, <<sinθ>>. B) Variance of <sinθ>. Lines correspond to the expectation of the paramagnetic model: The solid line is a fit assuming 2D trajectories (Eq. 7, yielding μ = 1.9×10−16 A⋅m2), while the dashed line is a fit assuming 3D trajectories (Eq. 5, yielding μ = 2.1×10−16 A⋅m2). C) Distribution of cell orientations for three different values of the magnetic field. Fits are Boltzmann distributions for a monodisperse cell population according to the paramagnetic model for 2D trajectories (Eq. 6). D) Ratio of magnetic energy to thermal energy of the cells obtained from the fit of orientation distributions such as those shown in panel C, assuming either 2D trajectories (black symbols), or 3D trajectories (grey symbols, not visible on the graph as they overlap with the previous ones). Linear fit of the data measured at magnetic fields B<2 mT yields a magnetic moment μ = 0.90±0.05×10−16 A⋅m2 assuming 2D trajectories (continuous line) and μ = 0.92±0.05×10−16 A⋅m2 assuming 3D trajectories (dashed line).
Figure 7.
Orientation correlation function of cells placed in constant magnetic fields.
A) Correlation of cell orientation over time for different values of the magnetic field, (mean ± standard error). Fit lines correspond to a modified worm like chain model (Eq. 15). B) Long time limit of the orientation correlation function. Fits correspond to the expectation of the paramagnetic model assuming either 2D trajectories (Eq. 17, continuous line), yielding a magnetic moment μ = 1.2×10−16 A⋅m2, or 3D trajectories (Eq. 16, continuous line), yielding μ = 1.5×10−16 A⋅m2. C) Persistence time of the orientation correlation function. The line represents a linear fit of the data for B<2 mT.
Table 1.
Average magnetic moment of M. magneticum AMB-1 cells measured by six different methods.
Figure 8.
Distribution of magnetic moments.
A) Experimental distribution of magnetic moment (black solid line), obtained by combining the values obtained by measurement of magnetosome size and those obtained using the U-turn method. Colored lines are fits of the data with either a single Gaussian distribution (purple line, μ0 = 3.3×10−16 A.m2, σ = 1.4×10−16 A.m2) or the sum of two Gaussian distributions (green line, μ1 = 3.3×10−16 A.m2, σ1 = 1.1×10−16 A.m2, μ2 = 7.6×10−16 A.m2 and σ2 = 0.9×10−16 A.m2). B) Orientation distributions expected for cell populations with normal distributions of magnetic moment values, with μ0B/kT = 5 (brown lines) or μ0B/kT = 15 (red lines), and with σ/μ0 = 0 (continuous lines), σ/μ0 = 0.5 (dashed lines) or σ/μ0 = 1 (dotted lines). C) Orientation distributions expected for a cell population with normally distributed magnetic moment values, μ0B/kT = 10 and different values of σ/μ0. D) Measured values of the magnetic moment, μ, obtained from fits of the orientation distributions assuming a monodisperse population of cells (i.e. using Eq. 6), normalized by the actual average value of the magnetic moment, <μ> (which is different from μ0 for distributions with large standard deviation since the distribution does not extend to negative values) as a function of the actual normalized standard deviation of the magnetic moment distribution, σ/μ0. E) and F) Experimental orientation distribution measured at B = 0.3 mT (solid symbols). Lines indicate the best fit assuming 3D trajectories and a monodisperse population (Eq. 4, dashed grey line, μ = 1.23×10−16 A.m2) or 2D trajectories and either a monodisperse cell population (Eq. 6, solid black line, μ = 1.16×10−16 A.m2), a population made of cells with two different values of the magnetic moment (dashed green line, μ1 = 0.43×10−16 A.m2 and μ2 = 1.90×10−16 A.m2, <μ> = 1.33×10−16 A.m2), or a population with a normal distribution of magnetic moment values (Eq. 7, solid red line, returning μ0 = 1.11×10−16 A.m2 and σ = 0.97×10−16 A.m2, corresponding to <μ> = 1.34×10−16 A.m2).