## Abstract

Concentrated active agents can exhibit turbulent-like flows reminiscent of hydrodynamic turbulence. Despite its importance, the influence of external fields on active turbulence remains largely unexplored. Here we demonstrate the ability to control the swimming direction and active turbulence of *Bacillus subtilis* bacteria using external magnetic fields. The control mechanism leverages the magnetic torque experienced by the non-magnetic, rod-shaped bacteria in a magnetizable medium containing superparamagnetic nanoparticles. This allows aligning individual bacteria with the magnetic field, leading to a nematically aligned state over millimetric scales with minute transverse undulations and flows. Turning off the field releases the alignment constraint, leading to directly observable hydrodynamic instability of the dipole pushers. Our theoretical model predicts the intrinsic length scale of this instability, independent of the magnetic field, and provides a quantitative control strategy. Our findings suggest that magnetic fields and torques can be excellent tools for controlling non-equilibrium phase transitions in active systems.

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## Introduction

Collections of autonomous elements that convert energy locally into mechanical force and motion, active matter, tend to display a wealth of collective behaviors and self-organizations ranging from schooling fish and flocking birds down to bacterial swarming^{1,2,3}. The non-equilibrium nature of active systems allows for diverse and nontrivial phase transition phenomena not seen in equilibrium systems, including motility-induced phase separation^{4,5,6} and the emergence of true long-range order in two dimensions^{1,7,8,9,10}.

In the presence of ambient fluid, the active agents exert active stress on the fluid, resulting in flows that play a crucial role in the collective states^{11,12,13,14,15}. As for self-propelled particles that exert extensile forces such as pusher-type microswimmers^{16}, the presence of the inherent hydrodynamic instability, the so-called Simha-Ramaswamy hydrodynamic instability^{11,12}, prevents the spontaneous emergence of long-range orientational order. This has been experimentally demonstrated in the mixture of swimming bacteria and liquid crystals^{17,18,19,20} and kinesin-driven microtubules’ assembly^{21,22}. The activity causing the instability can constantly stir the suspension of the active particles in the surrounding fluid, giving rise to a turbulent collective state consisting of transient vortices and jets. This active turbulence is recognized as another hallmark of active matter systems^{23}.

Although in the regime of low Reynolds numbers, active turbulence is reminiscent of classical hydrodynamic turbulence, featuring self-sustainment by local energy injection from their constituent elements. Paradigmatic examples range from biological systems, e.g., swimming bacteria^{24,25,26,27,28,29}, eukaryotic cells^{30,31}, sperm cells^{32}, and cytoskeletal extracts^{21,33} to non-biological systems such as synthetic colloidal particles^{34}. Due to its ubiquity, the active turbulence has attracted great attention during the last two decades. However, because of the Simha-Ramaswamy instability leading to the turbulent-like flow, the active turbulence has turned out to be difficult to control, and hence what physical properties are inherent in active turbulence and how it can be controlled have not yet been fully elucidated.

To address this issue, several recent efforts have been devoted to controlling the active turbulence. For example, imposing mesoscale boundary conditions using static geometric walls on active fluids has enabled the stabilization of turbulent flows into individual vortices^{35}, interacting ordered vortices^{36,37,38,39}, and even directed coherent flows^{40,41,42}. In a different approach, Guillamat et al. have demonstrated that a magnetically aligned layer of 8CB molecules can streamline the active turbulence of a living liquid crystal into a laminar flow^{43}. Another promising approach to externally control individual active agents – uniformly even in the bulk volume of the active fluid – is the use of active matter that is able to respond to external fields or stimuli, such as magnetic^{44,45} or photokinetic^{46,47} bacteria. Yet, this approach requires the use of specific species of bacteria or special techniques such as genetic engineering. A general strategy for direct control of the active agents via external fields would be even more highly desired.

In this article, we propose and demonstrate a general method for exerting well-defined torques on large populations of non-spherical active agents by immersing them in a magnetic liquid medium and applying an external magnetic field^{48}. We demonstrate this using *Bacillus subtilis* 3610 strain – the archetypical bacteria forming active turbulent states^{24,25,26,27,28,29}. We show that the swimming direction of the bacteria can be controlled, by applying an external magnetic field, to nematic alignment in a dilute bacterial suspension. In a dense bacterial suspension, this microscopic alignment constraint enforces the turbulent state into long-range nematic ordering over a length scale of millimeters – despite the hydrodynamic instability due to the active stresses from the bacteria. We show that the nematic ordered phase is accompanied by the transverse flows, which induce a minute orientational undulation. Interestingly, even in the nematic phase, there is an undulating orientation structure with a characteristic length scale that is independent of magnetic field strength. When the magnetic field is switched off, the undulation rapidly grows and leads to the active turbulent state, providing conclusive experimental evidence of the hydrodynamic instability inherent in bacterial turbulence and allowing the direction measurement of the characteristic length scales. We quantitatively corroborate our experimental results using a two-field continuum model with magnetic torque, consisting of the bacterial orientation and fluid flow fields. Our findings suggest that controlling the formation of ordered structures of active matter with external fields can deepen the fundamental understanding of the physics of active matter, also opening opportunities for designing new programmable active materials.

## Results

### Magnetic control of the alignment of individual bacteria

The magnetically controllable bacterial suspensions were prepared by mixing *B. subtilis* cultured in Terrific Broth (TB) medium with polyethylene glycol (PEG) stabilized ferrofluid (Ferrotec, PBG300). The volumetric concentration *ϕ* of the ferrofluid was varied from 0.02 to 0.10 (i.e., from 2% to 10%). The addition of the magnetic nanoparticles (together with the associated stabilizing agents and small dilution of the TB medium) slightly affected the mean swimming velocity of the bacteria particularly for the highest ferrofluid concentration of *ϕ* = 0.1 (Supplementary Fig. 1), but the reduction was at most about 20%. Furthermore, as desired, the nanoparticles were observed not to attach to the bacteria (Supplementary Fig. 2). The suspensions were studied under a uniform magnetic field generated by a horizontal Helmholtz coil (Fig. 1a).

At low bacterial densities (*c*_{0} ≈ 1 ~ 2 × 10^{7} cells/cm^{3}) and in the absence of magnetic field, the bacteria swim at speeds of ~ 20 *μ*ms^{−1} in an isotropic manner as expected. When a uniform magnetic field is applied, the bacteria align with the field in a nematic manner, and the nematic order increases with increasing magnetic field strength (Fig. 1b, Supplementary Movie 1, 2), which can be characterized by the velocity orientation distribution of tracked bacteria (Fig. 1c). The degree of the nematic alignment can be quantified by using the nematic order parameter

where *θ*_{j}(*t*) denotes the angle of the velocity vector (**v**_{b}) of *j*th bacterium with respect to the axis of the magnetic field at time *t*, and 〈⋅〉_{j,t} indicates ensemble average over all tracked bacteria and temporal durations of the tracks (10 s) measured after the new steady-state has been reached after turning on the magnetic field. As shown in Fig. 1d, the nematic order increases with the magnetic field strength and approaches unity at *ϕ* = 0.1 already under a modest field strength of 10 mT.

To clarify the mechanism of the magnetic orientation, we analyzed the bacterial body orientations, \({{{{{{{{\bf{n}}}}}}}}}_{b}=(\cos {\theta }^{{\prime} },\sin {\theta }^{{\prime} })\) satisfying \({\theta }^{{\prime} }\in [-\pi /2,\pi /2]\), such that the head and tail are indistinguishable, by using ellipsoidal fits (Fig. 1e). The cell body orientation distribution is well fitted by \(A\exp [-\beta {\sin }^{2}{\theta }^{{\prime} }]\) where *A* is a normalization factor and *β* is a fitting parameter. This suggests that the bacterial body orientations follow the alignment mechanism described by the nematic potential proportional to \(-{\sin }^{2}{\theta }^{{\prime} }\) (see Methods for details). This nematic reorientation mechanism can be explained as follows. As shown for rod-shaped inanimate colloidal particles^{48}, non-magnetic bacteria create rod-shaped voids in the magnetic fluid, where anti-parallel magnetic moments are induced in the presence of the magnetic field, resulting in a nematic torque along the magnetic field in the bacterial body due to the interaction between the uniform magnetic field and this effective magnetic moment. In Fig. 1e, the fitting parameter corresponds to the inverse of the orientational fluctuation, i.e., \(\beta =1/\langle \delta {\theta }^{{\prime} 2}\rangle\) that we found to increase almost linearly with the magnetic field strength (Fig. 1f)^{48}.

At the highest magnetic field strengths and longest durations of observation, the magnetic nanoparticles start to form chains as expected from dipolar forces (Fig. 2b and Supplementary Movie 2). However, the chaining occurs much more slowly than the directional change in bacterial orientation. Once the magnetic field is turned off, the nanoparticle chains redisperse quickly into a homogeneous isotropic dispersion.

### Magnetic control of bacterial turbulence

At high bacterial densities (*c*_{0} ≈ 6 × 10^{10}cells/cm^{3}), the active turbulence appeared in the magnetic bacteria suspensions similarly as in the regular non-magnetic suspension (Fig. 2a–c), which is characterized by the bacterial orientation field **n**(**r**, *t*) and velocity field **v**(**r**, *t*) from particle image velocimetry (PIV) (see Methods for details). The addition of the magnetic nanoparticles did not significantly change the velocity from PIV or the intrinsic vortex structure, compared to the control experiments done without magnetic nanoparticles (Supplementary Fig. 3a, b). In contrast to the dilute samples studied in glass capillaries, the dense suspensions were investigated as thin films ~ 60 *μ*m high with a large liquid-air interface to allow the *B. subtilis* bacteria to access oxygen to maintain their motility (Fig. 2a). Due to the high demand for oxygen by *B. subtilis*, the bacterial motility is highly maintained only within a few tens of microns from the liquid-air interface, thus allowing us to consider the system to be quasi-two-dimensional. When a magnetic field was applied, the disordered turbulent state was maintained at low field strengths, while in stronger magnetic fields the bacteria aligned with the direction of the applied field (Fig. 2b and Supplementary Movie 3). This is in contrast to the dilute system (Fig. 1), where the nematic alignment begins to increase even at the lowest magnetic fields (Fig. 1d), suggesting that the active turbulent states can resist some degree of external torque.

Peculiarly, additional transverse flows appeared in high magnetic fields (Fig. 2c and Supplementary Movie 4). In the same way as Eq. (1), the nematic order parameters for the bacterial orientation **n**(**r**, *t*) and velocity **v**(**r**, *t*) from PIV are defined by \({S}_{t}={\langle \cos 2\theta ({{{{{{{\bf{r}}}}}}}},t)\rangle }_{{{{{{{{\bf{r}}}}}}}}}\) where *θ*(**r**, *t*) is the angle of **n**(**r**, *t*) and **v**(**r**) with respect to the magnetic field, respectively, and 〈⋅〉_{r} denotes an ensemble average in space. The corresponding time series are plotted in Fig. 2d. When the magnetic field is off, both order parameters are close to zero. However, at high field strengths, the order parameter for the bacterial orientation sharply increases up to about unity, whereas that for the velocity from PIV drops down to about − 0.3. When the field is turned off again, both values approach zero, and in turn, the turbulent state is recovered – demonstrating the ability to switch the system between nematic order and active turbulence magnetically. The time-averaged nematic order parameter, \(S={\langle {S}_{t}\rangle }_{t}\) where the average is taken over 10 s under the application of the magnetic fields, shows that the tendencies of nematic ordering and transverse flows increase monotonically with the magnetic field strength (Fig. 2e).

To further clarify the effect of the magnetic field on the velocity magnitude, we analyzed the spatially averaged magnitude of the velocity field and its components in the direction parallel and perpendicular to the magnetic field, respectively. Their time series are plotted for the case of the strongest magnetic field (*B* = 28.2 mT) in Fig. 2f. The velocity was significantly suppressed by applying the magnetic field, particularly for the parallel component. This is because in the nematically aligned state, bacteria swim in the magnetic field direction and pass each other, easily canceling out the parallel component. The time-averaged velocity magnitudes depending on the magnetic field strength show a sharp decrease in velocity above *B* = 20 mT, with the perpendicular component approaching the velocity magnitude, which is indicative of a state that differs significantly from an active turbulent state due to the emergence of a dominant flow perpendicular to the magnetic field (Fig. 2g). This result indicates that the properties of material transport, including the degree of transport and its anisotropy, can be controlled by the magnetic field.

In order to quantify the ordered structures in the emergent patterns, we define the correlation function of bacterial orientation as follows:

where Δ**r** is a distance, and 〈⋅〉_{r,t} denotes an ensemble average in space and time (averaged over 10 s). To clarify the anisotropic structural details in the orientational ordering, we decompose the correlation into components that are parallel (*x*) and perpendicular (*y*) to the magnetic field (Fig. 3a,b). Depending on the magnetic field, these correlations display extension of correlated area in both parallel and perpendicular directions. The orientational correlation can be fitted by

where *l* denotes the coherence length that corresponds to the average nematic domain length, *a* stands for the degree of order, and *b* is the distribution width of *l*^{49}. The parameter *l* is plotted for each of the directions parallel and perpendicular to the applied field as a function of the magnetic field strength in Fig. 3c. While the coherence length in the magnetic field direction is almost independent of the magnetic field, that in the transverse direction shows a gradual increment with the increasing magnetic field strength. The nematic order characterized by *a* monotonically increases in both directions parallel and perpendicular to the magnetic field (Fig. 3d,e), consistent with Fig. 2d. The quantity *b* which indicates the width of the distribution of *l* increases in the direction parallel to the magnetic field but decreases in the perpendicular direction, indicating that the horizontal nematic domain becomes more prominent along with the magnetic field strength. Together, the above observations suggest that the nematic order over the entire region is enhanced by the magnetic field, but the size of the local nematic domain in which the bacteria align is constant with respect to the direction parallel to the magnetic field, regardless of the magnetic field strength.

The constant *l*_{n,∥} reflects the presence of a characteristic undulating orientational structure, as can be seen prominently in Fig. 2b (*B* = 28.2 mT). To extract the wavelength of the bending deformation of the bacterial orientations, we define a characteristic length *λ* as the first local minimum value of the correlation functions, as depicted in Fig. 3f. A slight decrease in *λ*_{n,∥} with increasing field strength is observed at weak magnetic fields, after which the length *λ*_{n,∥} becomes nearly constant around 45 *μ*m at high field strengths (Fig. 3c). The half wavelength defined here allows for quantitative comparison with theoretical predictions, as discussed later.

To quantify the characteristics of the flow fields, we analyzed the normalized velocity correlation in space defined as

where the ensemble average is carried out as above. Fig. 3g and h show the velocity correlation in the directions parallel and perpendicular to the magnetic field, respectively. Since the flow field includes vortical structures with clockwise and anti-clockwise handedness, the correlation function takes a local minimum, and hence we can obtain characteristic lengths *λ*_{v,∣∣} and *λ*_{v,⊥} in both directions, as with *λ*_{n,∣∣}. The insets in Fig. 3g and h indicate the *B*-dependence of the characteristic lengths, showing that as the magnetic field increases, *λ*_{v,∣∣} becomes constant, while *λ*_{v,⊥} increases. Notably, the length *λ*_{v,∣∣} is comparable to *λ*_{n,∣∣}, suggesting that the periodic longitudinal flow is attributed to the periodic undulating orientation field of bacteria.

Furthermore, we analyzed the temporal normalized correlation function of velocity defined as follows: \({C}_{v}(\Delta t)={\left\langle {\langle {{{{{{{\bf{v}}}}}}}}({{{{{{{\bf{r}}}}}}}},t+\Delta t)\cdot {{{{{{{\bf{v}}}}}}}}({{{{{{{\bf{r}}}}}}}},t)\rangle }_{t}/{\langle {{{{{{{\bf{v}}}}}}}}({{{{{{{\bf{r}}}}}}}},t)\cdot {{{{{{{\bf{v}}}}}}}}({{{{{{{\bf{r}}}}}}}},t)\rangle }_{t}\right\rangle }_{{{{{{{{\bf{r}}}}}}}}}\), where the ensemble average is taken as above. By fitting it with an exponential function, \(\exp [-\Delta t/{\tau }_{v}]\), we can obtain a typical correlation time (Fig. 3i). In a weak or intermediate magnetic field, the time *τ*_{v} is approximately 1 s and corresponds to a typical lifetime of turbulent vortices^{24}, but in the strong magnetic field, it increases by a factor of two due to the long persistence of the nematic ordered phase.

### Turning the active turbulence on and off magnetically: intrinsic hydrodynamic instability

Since the external magnetic field and torque can be switched on and off quickly, our magnetic approach allows probing transitions between different collective states. Such control is impossible using, e.g., geometric boundaries that are stationary^{35,36,37,38,39,40,41,42}. In the case of the nematically aligned dense *B. subtilis* population, turning off the magnetic field leads to the rapid growth of the underlying minute undulation (Fig. 4a,b and Supplementary Movie 5). The transition from a nematic-aligned state with large correlations over the entire region to an active turbulent state with decaying correlations took place, accompanied by instantaneous stripe-like structures in the correlations (Fig. 4c). The half wavelength of the bending deformation is obtained from the local minimum of the orientation correlation in the direction parallel to the magnetic field 0.5 s after turning the magnetic field off (Fig. 4d), giving *λ*_{n,∣∣} = 44.2 *μ*m. We argue that this length scale indeed corresponds to the intrinsic wavelength associated with the onset of the active turbulence, which is different from the wavelength associated with the chaotic turbulent steady-states at low magnetic fields (Fig. 3c). From this direct measurement of the wavelength of undulation, we conclude that the characteristic wavelengths are nearly independent of the magnetic field strength. The emergence of transverse flow is characterized by vorticity maps with velocity streamlines and velocity correlations with stripe-like structures, which are prominent on short time scales (≤0.5 s) after the magnetic field is turned off (Fig. 4e–g). At longer time scales (≥1 s), the full-fledged active turbulence is recovered.

As shown in refs. ^{11,12,17,19,20}, swimming bacteria like *B. subtilis* are pusher-type microswimmers that exert force dipoles on their surrounding fluid^{16}, and thus the active stress is known to induce bending deformation of aligned structures, mediated by fluid flows. Such a self-amplifying bending deformation is an important feature of extensile active nematic systems^{21,22}. Taking our cue from the activity-induced hydrodynamic instability demonstrated in those earlier works, we investigate whether the transverse flows we observed stem from the active stress by pushers by considering the 2D Stokes equation governing the fluid flow velocity **u**(**r**, *t*)^{26,35}:

where *μ* is the viscosity coefficient, the pressure *p* is the Lagrange multiplier for the incompressibility condition ( ∇ ⋅ **u** = 0), *α* term is the effective friction with the substrate, and the right-hand side of the equation represents the active force with the coefficient *f*_{0} (*f*_{0} > 0 for pushers) determined by the bacterial orientation field **n**(**r**, *t*). We assume the uniformity of density distribution of bacteria over space even when a magnetic field is applied due to the sufficiently high concentration. Accordingly, the strength of active force *f*_{0} = *q**c*_{0}, where *q* and *c*_{0} are the strength of dipoles and the number density of bacteria, respectively, is assumed to be constant. As shown in Fig. 5a and b, we calculated the fluid velocity field **u**(**r**, *t*) from the instantaneous orientation field **n**(**r**, *t*) by following equation (5) with appropriate values of *μ*, *α*, and *f*_{0}, which is similar to the velocity **v**(**r**, *t*) from PIV analysis, not only in the velocity orientations but also in the distributions of velocity magnitude (Fig. 5c). Although the velocity field obtained from PIV may differ from that of the fluid, the coincidence of the measured velocity field and the calculated fluid flow suggests that the bacteria are nematically aligned, leading to the apparent cancellation of the swimming velocities of the individual bacteria in the PIV analysis, so that the velocity field from PIV represents essentially the fluid flow field. This result indicates that the emergent transverse flow originates from active stresses exerted on the fluid toward the convex direction of the locally bent-oriented structure, as shown in Fig. 4, leading to velocity vortices aligned in the direction perpendicular to the magnetic field. The optimal parameters in equation (5) are searched in a manner proposed in ref. ^{26}, with some modifications. We introduce a parameter *Q* defined as \(Q(\mu /\alpha ,f_{0}/\alpha )={\langle | {{{{{{{\bf{u}}}}}}}}({{{{{{{\bf{r}}}}}}}},t)-{{{{{{{\bf{v}}}}}}}}({{{{{{{\bf{r}}}}}}}},t){| }^{2}/| {{{{{{{\bf{u}}}}}}}}({{{{{{{\bf{r}}}}}}}},t)| | {{{{{{{\bf{v}}}}}}}}({{{{{{{\bf{r}}}}}}}},t)| \rangle }_{{{{{{{{\bf{r}}}}}}}},t}\), where 〈⋅〉_{r,t} denotes the spatial and time averages, in order to quantify the extent to which the calculated flow field **u**(**r**, *t*) coincides with the experimentally obtained **v**(**r**, *t*). The quantity *Q* was averaged over 0.5 s after the magnetic field (*B* = 28.2 mT) was turned off, and the map of *Q* was obtained by varying the parameters of *μ*/*α* and *f*_{0}/*α* (Fig. 5d). The minimum value of *Q* yields the optimal parameter set (*μ*/*α*, *f*_{0}/*α*) = (605 *μ*m^{2}, 3820 *μ*m^{2}s^{−1}).

Bacterial turbulence is, in general, known to be well reproduced by single-field models which focus exclusively on the mean bacterial velocity field^{13,23,25}. However, in the case of our experimental system, since the collective state that emerges under the application of a strong magnetic field is not polar but nematic-aligned, and since both the fluid velocity and bacterial orientation fields can be measured experimentally separately, it is more reasonable to consider a two-field model consisting of the fluid velocity (Eq. (5)) and bacterial orientation fields in order to illustrate the magnetically controlled collective states. Since we focus on the nematic ordered state and its hydrodynamic instability, we consider the bacterial orientation field **n**(**r**, *t*) that is symmetric under a reversal of orientation, **n** = − **n**. With reference to earlier works (refs. ^{11,12,13,14,35,50}), the evolution equation of the bacterial orientation can be given in the following:

On the left-hand side, the bacterial alignment is advected by **u**. On the right-hand side, the first term denotes the diffusion with the coefficient *D*_{n} that penalizes orientation distortions. The second term reorients the bacterial alignment via fluid flow velocity by solvent strain tensor **E** = (∇^{T}**u** + ∇ **u**^{T})/2 and vorticity one **W** = (∇^{T}**u** − ∇ **u**^{T})/2, with a shape parameter − 1≤ *γ* ≤1. As discussed in the case of dilute suspensions (Fig. 1), bacteria are aligned with respect to the magnetic field, and hence we incorporate the nematic magnetic torque with the strength *J*_{B} and the unit vector of the field \(\hat{{{{{{{{\bf{B}}}}}}}}}\) in the last term.

To understand the undulation instability and its magnetic controllability, we here analyzed the linear stability of a nearly aligned state along the magnetic field direction. The presence of the magnetic field that forces bacteria to align in parallel to the *x* − *y* plane allows us to simply consider a two-dimensional system, where the magnetic field direction is set to \(\hat{{{{{{{{\bf{B}}}}}}}}}=(1,0)\), and only perturbations in the *y* direction perpendicular to the magnetic field: **n**(**r**, *t*) = **e**_{x} + **n**_{⊥} (∣**n**_{⊥}∣ ≪ 1), where **e**_{x} = (1, 0) and **e**_{x} ⋅ **n**_{⊥} = 0 (Fig. 5e). Fourier transformation of the orientation field, \({{{{{{{{\bf{n}}}}}}}}}_{\perp }({{{{{{{\bf{r}}}}}}}})=\tilde{{{{{{{{\bf{n}}}}}}}}}({{{{{{{\bf{k}}}}}}}}){e}^{i{{{{{{{\bf{k}}}}}}}}\cdot {{{{{{{\bf{r}}}}}}}}+\sigma t}\), yields the growth rate along the direction (*x*) parallel to the magnetic field (see Methods for details):

As for the characteristic lengths of undulation, we can obtain a typical wavenumber that maximizes the obtained growth rate (7) by differentiating it with respect to *k*:

where *F* = *f*_{0}(1 + *γ*)/(2*α*), *M* = *μ*/*α*, and we set *γ* to 0.9 due to the rod shape of bacteria^{35}. It is noteworthy that the characteristic wavelength (Eq. (8)) is independent of the magnetic coefficient *J*_{B}, which is consistent with our experimental observations. This independence allows for a direct comparison between the theoretically predicted characteristic wavelength, 2*π*/*k*_{c}, and 2*λ*_{n,∣∣} = 88.4 *μ*m at *B* = 0 obtained in the experiments, thus giving the estimation *D*_{n} = 221 *μ*m^{2}s^{−1}.

Here, let us discuss the validity of the estimated physical values. Given that *μ* is about 1 mPa s even in the presence of magnetic nanoparticles (see Methods for details), the optimized parameters of *μ*/*α* and *f*_{0}/*α* yield the substrate friction *α* = 2 × 10^{−6} Pa s *μ*m^{−2} and the strength of the force dipole *q* = 0.1 pN *μ*m, which is defined by *f*_{0} = *q**c*_{0} (*c*_{0} = 0.06 *μ*m^{−3}). The force dipole strength *q* is still slightly smaller than observed in previous work, ~ 1 pN *μ*m^{16}, but seems reasonable considering the possible overestimation of the bacterial number density *c*_{0} (e.g., due to vertical heterogeneity) or possible biological reasons such as lack of oxygen due to the high concentration of bacteria. Furthermore, in ref. ^{14,51}, the diffusion coefficient of orientations, *D*_{n}, is expected to be an elastic constant *K* divided by a rotational drag coefficient *ξ*. The constant *K* is assumed to be \(\sim 2{\phi }_{b}^{2}\) [pN] with the volume fraction of bacteria *ϕ*_{b}, and *ξ* is assumed to be *μ**L*^{2}*ϕ*_{b}/6*R*^{2} where *μ* is the fluid viscosity and *L* and *R* denote the length and width of bacterial body. Given that *L* ~ 7 *μ*m, *R* ~ 0.8 *μ*m, and *ϕ*_{b} ~ 0.2 in our experiment, *D*_{n} = *K*/*ξ* ~ 30 *μ*m^{2}s^{−1}. We note that the value is considerably smaller than the one estimated from the characteristic wavelength. The difference may originate, e.g., from the approximative determination of the elastic constant of the dense bacterial suspensions, which would benefit from future rheological investigations.

Using the parameters estimated above, the growth rate is mapped against *k* and *J*_{B} in Fig. 5f and g. At small values of *J*_{B} (below a critical *J*_{B}), the presence of the positive region of *σ* at long wavelengths means that the nematic-aligned state becomes unstable to slight bending deformation, ultimately leading to an active turbulent state. In contrast, at large values of *J*_{B} (above a critical *J*_{B}), the unstable mode of orientation perturbations disappears, as the growth rate is suppressed by *J*_{B}. The transition point above which a nematic-aligned state is stable is given by

*J*_{B} indicates the strength of nematic alignment by the magnetic field. *J*_{B} is related to the magnetic field *B* by considering the Langevin equation for a rod-shaped object in a ferrofluid under an applied magnetic field^{48}, and the specific form of *J*_{B} can be given by *J*_{B} = 2Γ*ϵ*_{1}*B*^{2} where Γ represents the response coefficient of bacterial orientation to magnetic fields and *ϵ*_{1} is a magnetic constant describing the magnetic properties and geometry of bacteria (Eq. (10); see Methods for details). To derive a condition of magnetic field strength under which the nematic-aligned state is stabilized, the prefactor, 2Γ*ϵ*_{1}, needs to be estimated. The distribution of bacterial orientation in the steady state should follow \({e}^{-2\Gamma {\epsilon }_{1}{B}^{2}{\theta }^{2}/{D}_{\theta }}\) (\(={e}^{-{\theta }^{2}/(2\langle \delta {\theta }^{2}\rangle )}\)), where *D*_{θ} is the rotational diffusion coefficient (see Methods for details). Therefore, \(\sqrt{2\Gamma {\epsilon }_{1}/{D}_{\theta }}\) (mT^{−1}) corresponds to the slope of the inverse orientation fluctuation (0.347 mT^{−1}) at a 10% ferrofluid concentration in Fig. 1f, allowing one to estimate the value of 2Γ*ϵ*_{1} using *D*_{θ} = 0.066 rad^{2}s^{−1} obtained from an independent experiment (Supplementary Fig. 4). By substituting this estimated value for equation (9), the transition point *B*_{c} can be estimated to be *B*_{c} ≈ 21 mT, which is approximately a point at which the nematic order parameter measured in the experiment is close to unity (Fig. 2e and Fig. 3d,e), and the velocity magnitude sharply decreases (Fig. 2g).

Our model assumes the instability from a nearly nematic-aligned state, whereas the transition from bacterial turbulence to the nematic state was actually investigated in the experiment (Fig. 2e). This possible discrepancy motivated us to further confirm the transition behavior from a nematic state to bacterial turbulence by implementing the temporal change in the magnetic field (Supplementary Fig. 5). Programming the linear decrease in the magnetic field strength from *B* = 28.2 mT to 0 allowed for a gradual transition from an aligned state to a turbulent state. We found that the order parameter appears to begin to decrease beyond *B* = 21 mT, indicating transition behavior consistent with the transition from active turbulence to the nematic state.

## Conclusion

In summary, we have demonstrated the ability to control the swimming direction and collective states of non-magnetic bacteria by exerting externally controllable torques on them through a biocompatible and magnetizable liquid medium. In dilute bacterial suspensions, the torque can constrain the alignment of the rod-shaped bacteria and their swimming direction to the direction of the magnetic field, leading to externally tunable nematic ordering. In dense suspensions, the magnetic torque sculpts the isotropic active turbulent state to a nematically ordered state – however, flows perpendicular to the magnetic field also appear. As the magnetic field increases, the order of nematic orientation is enhanced, while simultaneously the undulating structure of orientation becomes more pronounced along the magnetic field, and its characteristic length scale is almost independent of the applied magnetic field. We put forward a simple continuum model and show that linear stability analysis yields a characteristic length of the undulation that is independent of the magnetic field and a magnetic condition to determine the emergent collective state, which is in good quantitative agreement with the experimental observation. We highlight that simplifying the emergent collective states through magnetic control allows us to quantitatively compare velocity configurations and intrinsic length scales in active turbulence between experiment and theory, and to reveal physical properties of the system such as active stress, substrate friction, and diffusion constants of the alignment. Therefore, our method would facilitate a quantitative understanding of active matter physics, which is seriously challenged by the large number of unknown physical parameters in various theoretical models.

Our results suggest that externally controllable torque generation on non-magnetic bacteria via a magnetizable medium is a powerful tool to control both individual bacteria as well as their collective states. In contrast to other techniques such as optical^{52} or acoustic tweezers^{53}, our approach leads to the generation of uniform torques everywhere in large samples of up to cm-scales. Moreover, in contrast to controlling active matter via magnetically controlled liquid crystals^{43}, our approach is more direct as it acts in the bulk 3D volume rather than along an interface between the magnetically controlled liquid crystal and the active phase. Another important aspect of our approach is that the functional components enabling the interaction with the external field, the magnetic nanoparticles, are outside the regular non-magnetic bacteria. This likely allows generalization to other bacterial species and even other micro-organisms with different approaches to motility and collective states.

Our findings can open many promising avenues for future work. Given that our technique can fine-tune individual alignment at microscopic scales and can initialize collective states in space and time as desired, it may provide access to more in-depth studies about the phase transition dynamics in active turbulence, such as directed percolation universality^{54}. The utilization of a magnetic field further connects to the much studied field-induced phase transition phenomena in spin systems in the context of condensed matter physics^{55}. Furthermore, dynamic and non-uniform magnetic fields are foreseen to lead to even more advanced spatiotemporal control of both individual bacteria and their collective states via additional translational forces^{56} and torques in a programmable manner^{57}.

## Methods

### Preparation of bacterial suspensions in magnetizable media

*Bacillus subtilis* strain 3610 (Bacillus Genetic Stock Center, Original code: NCIB3610, BGSCID: 3A1) was grown by inoculating a single bacterial colony into 5 mL of Lysogeny broth (LB) medium (created by dissolving a 25 g LB broth base powder (Invitrogen, 12795027) in 1 L MilliQ water and autoclaving) in a 50 mL Falcon tube (Falcon, 352070) and incubating it overnight on an orbital shaker (Grant-bio, PSU-10i) inside an incubator (Memmert, IN55) at 37 °C. Next day, 100 *μ*L of the culture was added to 20 mL of fresh Terrific Broth (TB) medium (Gibco, 11632139) and further incubated on the orbital shaker in the incubator at 37 °C at 160 RPM for ca. 3 hours. After the optical density of the culture, measured at 600 nm in a plastic cuvette with 10 mm path length using a VIS spectrometer (Thermo Scientific, GENESYS 30), reached ca. 0.4, the culture was diluted ca. 10 to 40 times in TB medium to create a dilute bacterial suspension. The dense bacterial suspensions were created by centrifuging (Fisher Scientific, GT R1 Centrifuge) the bacterial suspension (obtained after 3 hours of incubation) at 3000 g at room temperature for 5 minutes to increase the density of the suspension to ~ 20%*v*/*v* at which bacterial turbulence can be observed. The bacterial concentration was estimated from the optical density measured at 600 nm using the relationship that OD = 1.0 at 600 nm corresponds to *c*_{0} ≈ 7 × 10^{8} cells/cm^{3} ^{58}. Finally, the dilute and concentrated bacterial suspensions were mixed with a biocompatible ferrofluid (Ferrotec, PBG300) at a desired proportion: 2%, 5%, or 10%. According to the specifications and physical properties of PBG300 provided by Ferrotec, the viscosity at 27 °C is 3 mPa s, and hence we expect the viscosity of the suspension with ferrofluid to be ca. 1 mPa s even at a 10% ferrofluid concentration in the absence of magnetic fields, which is the almost same as that of the TB medium.

### Sample preparation for microscopy

The dilute bacterial samples were imaged inside rectangular, 50 mm long glass capillaries 4 mm wide and 0.2 mm deep (with wall thickness 0.2 mm, CM scientific, 3524-050). For the dense bacterial samples, a well was constructed on a regular microscopy glass slide using a thermoplastic ionomer film (DuPont Surlyn, 60 *μ*m) that was attached to the glass slide by heating to 130 °C on a hotplate. After filling the well, a 500 *μ*m thick spacer with one 20 mm diameter well (Invitrogen P18174, 10276972) was placed around the well and covered from the top with a glass coverslip to prevent the suspension from evaporating.

### Microscopy under a uniform magnetic field

Imaging of the samples under a uniform horizontal magnetic field was done using a setup described earlier with small modifications^{59}. Briefly, the uniform horizontal field was created using an electromagnet coil pair (GMW 11801523 and 11801524) with a 50 mm gap. The coils were driven with a DC power supply (BK Precision 9205). Microscopy imaging of the sample was done using a custom-made microscope setup consisting of a 20 × objective lens with a numerical aperture of 0.45 (Nikon, TU Plan Fluor), an Epi-Illuminator Module (Thorlabs, CSE2200), and a 0.5 × Camera Tube (Thorlabs, WFA4102) equipped with a CMOS camera (XIMEA, MC050MG-SY-UB). The sample was illuminated from below in transmitted light configuration using a light-emitting diode light source (Thorlabs, MNWHL4) with a collimator which is diffused right before it hits the sample. The images were acquired at 30 frames per second with an exposure time of 33 ms.

### Data processing and analysis

The trajectories of individual swimming bacteria were tracked with Particle Tracking Velocimetry (PTV) analysis by using the TrackMate plugin^{60} in Fiji^{61}. Prior to PTV analysis, raw images from the experiments were post-processed by subtracting the background, enhancing the contrast, smoothing by a Median filter, and inverting the image intensity histogram. Immotile bacteria, such as those adsorbed on the glass capillary surface, were eliminated from the trajectories tracked for over 2 s by accepting only trajectories that showed ballistic motion with mean square displacement in the regime of 1 s proportional to *t*^{c} with *c* greater than 1.5. In addition, to reduce the noise in the velocity determination, the accepted trajectories were smoothed by taking the difference between two data points along the trajectory separated by 10 frames (corresponding to the typical time for a bacterium to move one cell’s body length), instead of considering neighboring data points. These analyses, with the exception of the tracking, were conducted in Matlab (MathWorks) using custom-made scripts.

The velocity field of the bacterial collective motion was obtained using Particle Image Velocimetry (PIV) analysis done using the PIVLab toolbox^{62} in Matlab. The Wiener2 denoise filter was used in the image pre-processing, and the interrogation window size was chosen to be 16 × 16 pixels^{2} (5.52 × 5.52 *μ*m^{2}), corresponding to the typical body length of the studied bacteria. To reduce the noise, the acquired velocity field was further smoothed by averaging over 1 s, which corresponds to the typical lifetime of turbulent vortices (Fig. 3i).

The local orientation of the bacteria was obtained by using OrientationJ plugin^{63} in ImageJ that detects the direction of the largest eigenvector of the structure tensor of the image. We set the local window size of the structure tensor to 15 × 15 pixels (corresponding to 5 *μ*m). The orientation field was further coarse-grained and reduced to the same matrix size as that of the PIV velocity field after the convolution with half of the PIV window size.

The flow velocity field **u** in equation (5) was computed by a spectral method that first solves in Fourier space and then transforms to real space.

### Orientational dynamics of bacteria in a ferrofluid

Bacteria suspended in a ferrofluid under the application of magnetic fields create rod-shaped voids in which magnetic moments anti-parallel to the field are induced^{48}. Their interactions nematically reorient bacteria along the magnetic field. The orientational dynamics of bacteria in a dilute suspension can be described by

The first term is the magnetic torque with the response coefficient Γ of bacterial orientation to the magnetic field and the dimensionless magnetic potential \({U}_{m}={\epsilon }_{1}{B}^{2}{\sin }^{2}\theta +{\epsilon }_{2}{B}^{2}\) where *ϵ*_{1,2} are magnetic constants dependent on the magnetic susceptibility, the permeability of vacuum, and the volume of the bacterium^{48}. The second term is the white Gaussian noise with the rotational diffusion coefficient that satisfies 〈*η*(*t*)〉 = 0 and \(\langle \eta (t)\eta ({t}^{{\prime} })\rangle =\delta (t-{t}^{{\prime} })\) where \(\delta (t-{t}^{{\prime} })\) is a Dirac-delta function. Note that tumbling occurs at most once during the observation time of 10 s, and its effect is considered small enough on the time scale of the observation time to be ignored here for simplicity. According to the conventional procedure upon the assumption that the distribution of bacterial orientation is spatially uniform^{37,64}, one can obtain the Fokker-Planck equation of the probability distribution of bacteria heading *θ*:

The solution in the steady state can be written as

where *A* denotes a normalization factor. This function is fitted to the probability distribution in Fig. 1e. Since this is interpreted as the Gaussian function around *θ* = 0, as shown in Fig. 1f, we analyzed the orientation fluctuation by comparing \({e}^{-{\theta }^{2}/(2\langle \delta {\theta }^{2}\rangle )}\) with equation (12). Thus, 1/〈*δ**θ*^{2}〉 corresponds to 2Γ*ϵ*_{1}*B*^{2}/*D*_{θ}.

To estimate the rotational diffusion coefficient *D*_{θ}, we tracked swimming trajectories in a dilute bacterial suspension with a 10% ferrofluid concentration using PTV analysis. Here, due to the difficulty in tracking bacterial polarity for a long time, let us use the velocity unit vector instead of bacterial orientation, which could be justified by the fact that polarity and velocity are almost identical in the absence of interactions with others. We analyzed mean square angular displacement, defined by 〈(** d**(Δ

*t*)−

**(0))**

*d*^{2}〉 where

**(Δ**

*d**t*) denotes a unit vector of swimming velocity Δ

*t*after an initial condition

**(0) (Supplementary Fig. 4). The ensemble average 〈 ⋅ 〉 was taken for all the bacteria which were tracked for over 10 s and exhibited straight trajectories with mean curvature radii of more than 700**

*d**μ*m to rule out bacteria showing apparent curved trajectories due to their chirality and hydrodynamic interactions with interfaces. Fitting it by \(2(1-\exp (-{D}_{\theta }\Delta t))\) yields the estimate of

*D*

_{θ}= 0.066 rad

^{2}s

^{−1}

^{39}.

### Linear stability analysis

To reveal the undulation instability, we analyzed the linear stability of a uniformly aligned state along the magnetic field factor \({{{{{{{\bf{{J}}}}}}}_{B}}}={J}_{B}\hat{{{{{{{{\bf{B}}}}}}}}}\) with the coefficient *J*_{B} related to the magnetic field strength and the direction \(\hat{{{{{{{{\bf{B}}}}}}}}}=(1,0)\). We consider perturbations for the orientation field in the *y* direction perpendicular to the magnetic field: **n**(**r**, *t*) = **e**_{x} + **n**_{⊥} (∣**n**_{⊥}∣ ≪ 1) where **e**_{x} = (1, 0) and **e**_{x} ⋅ **n**_{⊥} = 0. We also define perturbations for the fluid flow velocity and pressure, \({{{{{{{\bf{u}}}}}}}}({{{{{{{\bf{r}}}}}}}})={{{{{{{\bf{{u}}}}}}}^{{\prime} }}}\) (\(| {{{{{{{\bf{{u}}}}}}}^{{\prime} }}}| \ll 1\)), and *p*(**r**) = *η* (*η* ≪ 1), respectively. Substituting these variables for equations (5) and (6), and retaining first-order terms of the infinitesimal quantities, equations (5) and (6) are readily reduced to the following equations:

Upon the incompressibility condition (\(\nabla \cdot {{{{{{{\bf{{u}}}}}}}^{{\prime} }}}=0\)), Fourier transformation of the fluid flow velocity, \({{{{{{{\bf{{u}}}}}}}^{{\prime} }}}({{{{{{{\bf{r}}}}}}}})=\tilde{{{{{{{{\bf{u}}}}}}}}}({{{{{{{\bf{k}}}}}}}}){e}^{i{{{{{{{\bf{k}}}}}}}}\cdot {{{{{{{\bf{r}}}}}}}}}\), gives the solution of the fluid flow velocity in Fourier space:

where *k* = ∣**k**∣. We assume the direction of **k** to be an elevation angle *φ*, i.e., \({{{{{{{\bf{k}}}}}}}}=(k\cos \varphi ,k\sin \varphi )\). Substitution of the velocity (15) for equation (14) and Fourier transformation of the orientation field, \({{{{{{{{\bf{n}}}}}}}}}_{\perp }({{{{{{{\bf{r}}}}}}}})=\tilde{{{{{{{{\bf{n}}}}}}}}}({{{{{{{\bf{k}}}}}}}}){e}^{i{{{{{{{\bf{k}}}}}}}}\cdot {{{{{{{\bf{r}}}}}}}}+\sigma t}\), gives the growth rate:

The growth rate in the *x* direction parallel to the magnetic field (*φ* = 0) reads equation (7). In addition, the growth rate in the *y* direction perpendicular to the magnetic field (*φ* = *π*/2) is negative throughout the wavenumber. This means that as long as the system is nearly aligned state in the direction parallel to the magnetic field, the alignment is stable in the transverse direction, which is consistent with the increment in the characteristic lengths in the transverse direction around the transition point of the magnetic field (Fig. 3c,h).

### Comparison of magnetic torques between microscopic and continuum descriptions

To derive the specific expression of the coefficient of the magnetic torque in equation (6), we simply write down the evolution equation of the angle *θ*, defined by \({{{{{{{\bf{n}}}}}}}}({{{{{{{\bf{r}}}}}}}},t)=(\cos \theta ,\sin \theta )\), with respect to only the magnetic torque:

Equation (10) readily reads

thus giving *J*_{B} = 2Γ*ϵ*_{1}*B*^{2}.

### Reporting summary

Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.

## Data availability

All the raw data in the main text can be found in Zenodo (DOI: 10.5281/zenodo.11475738).

## Code availability

The custom codes using well-established algorithms used for the experiment analysis and numerical calculation are available from the corresponding author upon reasonable request.

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## Acknowledgements

We thank F. Sohrabi for technical support in bacterial culture, T. Kärki and L. Laiho for establishing protocols for bacterial culture, and C. Rigoni for building the Helmholtz coil setup. J.V.I.T acknowledges funding from ERC (803937). This work was carried out under the Academy of Finland Center of Excellence Program (2022-2029) in Life-Inspired Hybrid Materials (LIBER), project number (346112). K.B. acknowledges support from the Overseas Postdoctoral Fellowship of the Uehara Memorial Foundation and the Overseas Research Fellowship of the Japan Society for the Promotion of Science (JSPS).

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K.B. and J.V.I.T. designed the project and wrote the manuscript. K.B. performed the experiments, the data analysis, and the theoretical analysis.

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Beppu, K., Timonen, J.V.I. Magnetically controlled bacterial turbulence.
*Commun Phys* **7**, 216 (2024). https://doi.org/10.1038/s42005-024-01707-5

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DOI: https://doi.org/10.1038/s42005-024-01707-5

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