## 2 Materials and Methods
### Magnetic Measurements and Data Quantifying
The simultaneous fitting of the real and imaginary parts of the measured signals using Equation (7) yields the following quantities: (i) the sample saturation magnetization \(\widetilde{M}\), (ii) the mean magnetic moment \(M_{C}\) and the width of the\(f_{M}\) distribution \(\sigma\), (iii) the mean anisotropy energy \(E_{a}\) and field \(H_{a}\), (iv) the average angle \(\Psi\) between \(\mathbf{n}\) and \(\mathbf{H}\) (Figure 1), and (v) the parameters defining magnetization dynamics, viz., the damping constant \(\alpha\) and the Neel relaxation time \(\tau_{N}\). Given the concentration of iron, additional information can be extracted, namely, the mass and volume saturation magnetizations of the sample, the average number of iron ions in the particles, the mean particle size, and some others. This is a rather general approach. In particular, the requirement \(h\ll H\) on the alternating magnetic field is not mandatory, thereby keeping the formalism valuable even for quite small \(H\). The formalism was applied to study a colloidal aqueous solution of SPM iron oxide nanoparticles in the dextran shell (SPIONs) [26]. The criterion of applicability of the procedure is the absence of, or the presence of only a small of, magnetic hysteresis in the \(H\) dependences of the signals, which points on the SPM nature of the response. The computational resources of the PIK Data Centre of NRC \"Kurchatov Institute\" (PNPL, Gatchina, Russia) were used, with the in-house provided software. Prior to the fitting, the raw data were averaged between the direct (\(l_{dir}\)) and reverse (\(l_{rev}\)) scans and the part that was antisymmetric relative to \(H\) = 0 was extracted, as required by the model, as follows:  
\[I_{av}=\frac{1}{4}[I_{dir}(H)+I_{rev}(H)-I_{dir}(-H)-I_{rev}(-H)]\,. \tag{8}\]  
This averaging removed a possible constant bias to obey the condition \(\mathrm{Re}M_{2}(H=0)=0\) required by the formalism and reduced the data to the region \(H\) > 0 to use in the data fit procedure. In addition, as the magnetosomes in the specimens were disoriented, the isotropic distribution of the anisotropy axes directions was implied. A set of parameters characterizing the magnetosome ensembles in the bacteria under study was obtained, describing their magnetic properties, such as the magnetic moments, the magnetic anisotropy energy, and the saturation magnetization. The most relevant parameter for the assessment of the magnetosome state is the number of magnetically active centers in each bacteria line derived as the ratio of the saturation magnetization and the magnetic moment. To obtain additional information on the magnetic state of the magnetosomes, we used EMR. We applied a special homemade X-band spectrometer operating at the frequency \(F=\omega/2\pi=8.54\) GHz [38]. The spectrometer was supplied with a cylindrical two-mode balanced microwave resonator with the TE\({}_{111}\) type of electromagnetic oscillations. The _dc_ magnetic field \(\mathbf{H}\) was directed along the cylinder \(z\) axis. The sample (MTB in the cultivation medium with the addition of formalin for the bacteria fixation) was placed at the bottom of the resonator in the special cylindrical cavity, where it was affected by the _ac_ field \(\mathbf{h}\) linearly polarized along the \(x\) axis perpendicular to \(\mathbf{H}\) (the excitation \(xz\) plane). The detection \(yz\) plane was perpendicular to the excitation one, and thus the detected signal was proportional to the off-diagonal component of the susceptibility tensor \(\chi_{yx}(\omega)\) corresponding to the \(y\) component of the induced magnetic moment \(M_{y}(\omega)=\chi_{yx}(\omega)h_{x}(\omega)\). Deep frequency-independent decoupling between the excitation and detection modes was achieved by ensuring the frequency degeneracy, viz., the independence of the eigenfrequencies of the resonator from the rotation of the excitation plane around the \(z\) axis in the \(xy\) plane of the resonator.