Self-organisation and convection of confined magnetotactic bacteria

Collective motion is found at all scales in biological and artificial systems, and extensive research is devoted to describing the interplay between interactions and external cues in collective dynamics. Magnetotactic bacteria constitute a remarkable example of living organisms for which motion can be easily controlled remotely. Here, we report a new type of collective motion where a uniform distribution of magnetotactic bacteria is rendered unstable by a magnetic field. A new state of “bacterial magneto-convection” results, wherein bacterial plumes emerge spontaneously perpendicular to an interface and develop into self-sustained flow convection cells. While there are similarities to gravity driven bioconvection and the Rayleigh–Bénard instability, these rely on a density mismatch between layers of the fluids. Remarkably, here no external forces are applied on the fluid and the magnetic field only exerts an external torque aligning magnetotactic bacteria with the field. Using a theoretical model based on hydrodynamic singularities, we capture quantitatively the instability and the observed long-time growth. Bacterial magneto-convection represents a new class of collective behaviour resulting only from the balance between hydrodynamic interactions and external alignment.

In order to quantify the dynamics of the merging plumes from the sequence of experimental images, we average in the z direction the intensity of the z − part of the image where the plumes originate. This average is taken over a region of size comparable to the growing plume height, approximately z = 50 µm. This allows us to detect peaks in the intensity profile corresponding to plumes. In order to adjust for each experiment the parameters for peak finding (namely the minimal height and width of a peak as well as the minimal distance between neighbouring peaks) we check manually that the obtained peaks match the position of the plumes in the data (Fig. S 1a). The wavelength at a given time is then defined as the mean distance between neighbouring intensity peaks. To avoid fluctuations in the wavelength due to peak detection, we use the moving median of the computed wavelength over 20 images (2 s) (Fig. S 1b).
In Fig. S 2 we present the raw data for the wavelength as a function of time for different channel sizes. This data is then rescaled in Fig. 2 of the main article to show the two distinct regimes of plumes time-evolution, growth and plateau.

ADDITIONAL EXPERIMENTAL PARAMETERS
We identified several key parameters that govern the dynamics of bacterial magnetoconvection. While elucidating quantitatively their role would require a different experimental set-up, we present here some experimental findings that shed some light on their influence over the system dynamics.

Magnetic field
To understand the influence of the magnetic field strength in the plume formation and growth, we experimentally vary the field strength in a single capillary tube sample. We switch on the field, let the plume evolve for 200 s, then switch the field off. Once the plumes disappear, the experiment is re-started using a different field strength with the same population. We check the validity of this procedure by re-using the same field after a few such cycles and ensuring that wavelength time-evolution is unchanged (Fig. S 3a).
We find that there exist a critical field below which magneto-convection does not occur. This critical field is strongly dependent on the bacterial population and concentration, and we measured values ranging from 0.25 to 1 mT.
While we were not able to investigate fully the critical behaviour, we observe that below this threshold value, no plumes are formed. Slightly above this value, we observe smaller , no plumes appear (no wavelength is measured, not shown), close to the critical field (0.3 mT), small plumes appear but do not grow, and above the critical field, plume time-evolution becomes independent of the magnetic field (b) Above the critical magnetic field, the wavelength is independent of the field strength plumes that do not grow, as presented in Fig. S 3a. Above this value, the coarsening is independent of the magnetic field. Thus, beyond some critical magnetic field strength the speed of plume growth is independent of the field (Fig. S 3b). We attribute this to a field that is strong enough to align the bacteria.
Except for the data presented in Fig. S 3, all the experiments were performed above the critical field (2 mT) to rule out any influence of the field strength on the plume growth.

Orientation of the bacteria
To further validate the assumption that the bacteria remain aligned with the magnetic field in the high field regime, we derive here the theoretical orientation distributions of individual bacteria characterized by a magnetic moment µ placed in a magnetic field B at room temperature (T = 293 K). Magnetotactic bacteria have orientation statistics that can be described by a paramagnetic model, and their orientation distributions therefore follow Boltzmann statistics [1]. Considering a magnetic field B along the z axis, α the angle between B and µ, and φ the azimuthal angle, the probability of a particular orientation is given by where is the partition function. In the following, we write A = µB k B T . This model allows us to calculate the probablility for the cells to be at an angle a ≤ α ≤ b from the field B We independently measured the value of µ on bacteria grown in the exact same conditions. We obtained µ = 10 −15 ± 4 × 10 −16 A m 2 (mean ± standard deviation) and reported the full results elsewhere [2]. From this, we compute and report in Fig Furthermore, the average value of cos α can be derived from Eq. 1 and is given by the Langevin function, as expected in the paramagnetic model framework: The graph of cos α as a function of B is presented in Fig. S 4 for µ = 10 −15 A m 2 and µ = 5 × 10 −16 A m 2 , corresponding respectively to the mean and lower bound values of µ expected for our bacteria population. The value of cos α is a quantity of particular interest, as u 0 cos α is in first approximation the average magnitude of the bacteria speed (magnitude u 0 ) along the magnetic field axis. In the high field regime used in our experiments (B = 2 mT), cos α 1, which validates the assumption that the bacteria can only swim along the applied magnetic field. We also note that cos α starts to become visibly lower than 1 for fields lower than 0.4 mT and drops significantly for fields lower than 0.2 mT, which is consistent with our experimental observation that there is a critical field around these values, below which no plumes can form and grow, due to an insufficient alignment with the field.

MODEL Simulation details
In our simulations, we consider N swimmers each located at x k = (x k , y k , z k ), 1 ≤ k ≤ N . Each swimmer consists of two Stokeslets of strength +se z and −se z located respectively at positions (x k , y k , z k + l 0 /2) and (x k , y k , z k − l 0 /2). To implement the no-slip boundary conditions on both walls (z + = a and z − = 0), we use the image system of each Stokeslet, a set of singularities placed in a symmetric position with respect to the wall that cancel out the flow at the wall. For a singular Stokeslet of strength s orthogonal to a single wall, the image system as described in Ref. [3] is a Stokeslet of strength −s, a Stokeslet dipole of strength −2hs and an irrotational source dipole of strength 2h 2 s where h is the distance between the bacterium and the wall, (Fig. S 5). We only include the first image of the swimmers on each wall, and ignore the additional reflections of the image systems.
To avoid singularities in the flow, we use regularised Stokeslets with a spatially distributed force rather than a Dirac distribution, as presented in Ref. [4]. We use a blob of size δ = 0.3l 0 comparable to the size of a bacterium and the regularisation function of equation (18) in Ref. [4], namely The regularised image system is composed of the same singularities but with different regularisation functions, and the corresponding flow is given in equation (21) of Ref. [4]. We consider the flow created at x e by a Stokeslet at se z x s = (x k , y k , z k + l 0 /2). The image points of the Stokeslet on the lower and upper walls are x im,L = (x k , y k , −z k − l 0 /2) and x im,U = (x k , y k , 2a − z k − l 0 /2) respectively. We write x = x e − x s , x L = x e − x im,L and x U = x e − x im,U . The velocity is then given by with H 1 , H 2 , D 1 , D 2 regularisation functions given explicitly in Ref. [4]. The flow created by a swimmer is then obtained by adding the flow of the second Stokeslet of strength −se z at x s = (x k , y k , z k − l 0 /2), which has the same expression as above but with s − → −s and z k + l 0 /2 − → z k − l 0 /2. The value of s is 6πµl 0 u 0 , with non dimensional value s = 6π. Simulations with heterogeneous bacteria population are made using a normal distribution for the downwards velocities, with mean u 0 , standard deviation u 0 /3. We cut negative values so that bacteria all swim in the −z direction. Corresponding strength of hydrodynamic singularities s = 6πµl 0 u 0i is then used to compute bacteria-bacteria interactions.
The time evolution of the system is obtained by computing the flow created on each swimmer by both its image and the other swimmers and their images. We then update the position of each swimmer using a forward Euler method with time step dt = 0.5. The coordinates of each swimmers are stored in a 3 × N matrix updated at each iteration. To obtain the flow and streamlines generated by the swimmers (see Fig. 3 of the main article), we use a 100 × 1000 grid in the channel and compute the velocity vector field.

Characterisation of plume structure
Plume-plume hydrodynamic interactions, and therefore the dynamics of the whole system, depend on the position of individual swimmers relative to the wall and hence on the density profile within each plume. Although we are not able to analyse the experimental structure of the plume, we can investigate in the model the density profile and local flow around plumes with different numbers of swimmers, as shown in Fig S. 6. We find that plumes of all sizes are denser near the wall, with some bacteria sticking to it. For the bacteria detached from the wall, the density decreases linearly from the bottom to the tip of the plume. This stratified structure corresponds to a balance between downward swimming and repulsive interactions between bacteria in the vertical z-direction.

Short time dynamics
We investigate the short time dynamics of the system and we show that the convexity of the wavelength vs. time relationship observed in experiments and simulations, especially at the beginning of plume formation, originates from the time that it takes for bacteria to accumulate at the bottom of the channel.
At short times, the bacteria start at random in the channel. They then swim downwards and accumulate at the lower z − wall. In this first regime, the local concentration in the plume region (close to the z − wall) is increasing with time. As shown in the main text, higher concentrations lead to faster plume dynamics. We therefore expect wavelength growth to accelerate while the bacteria accumulate at the wall. This explanation is borne out by the observation that the wavelength is linear if all the bacteria start instead at the bottom (Fig. S 7). We also evaluate the time for all the bacteria to reach the lower region of the channel and show that it is the same as the characteristic time in which the wavelength growth is accelerating. This time for bacteria to accumulate is made longer by repulsive hydrodynamic interactions between swimmers along the z direction, as shown in Due to the existence of a short-time regime, we expect the timescale of the early accelerating growth of the wavelength to have a different dependence on the system parameters (channel size a and number of swimmers N ) than the time to reach the wavelength plateau, τ p . For an identical number of swimmers, the bacteria will take longer to swim down for larger channels, and the first regime of accelerated growth will be longer relatively to τ p . For identical channel sizes and a larger number of swimmers, repulsive hydrodynamic interactions in the z direction will make the absolute time for all bacteria to reach the z − wall longer. Because of this variation of the early dynamics, there is some scatter when rescaling the wavelength as a function of time as in Fig. 2 of the main article: Fig. S 8 indeed shows that while identical parameters lead to similar wavelength dynamics, varying a and N modifies the variation for λ(t) and leads to scatter.

The role of hydrodynamic images
To gain insight into the role of hydrodynamic image singularities in the plume dynamics, we also performed simulations ignoring the image system on the upper z + wall (Fig. S  9). We observe that, without the hydrodynamic images, the dynamics is faster. At high concentration, we find that at long-time the final state is a single plume, taller than the channel size. On the contrary, when the plumes interact with their images on the z + wall, the dynamics are slowed down. We see then in both simulations and experiments, that the channel width a sets the long term wavelength.

Comparison between experimental and theoretical plume flow structures
We extend here the comparison between our model and the experiments by comparing the flow obtained through PIV to the flow created in the channel by the hydrodynamic images. We plot in Fig. S10 the streamlines as well as the vertical velocity before plume formation and at (i) early, (ii) intermediate and (iii) late times of plume dynamics, for both experiments (a) and model (b). The plumes are seen to generate similar flow structures in our model and experiments, showing good qualitative agreement at different stages of the system dynamics. While the streamlines observed, or those averaged over a large field depth, are less revealing at earlier times, the plumes are characterised at all times by a strong local upward velocity.