Phases of active matter composed of multicellular magnetotactic bacteria near a hard surface

We investigate the motion of multicellular magnetotactic bacteria (MMB) near a hard surface. Directing MMB towards a flat surface causes them to accumulate about the wall as an active gas. MMB are exponentially distributed about the surface with a penetration depth determined by the typical distance an MMB swims before it aligns with the ambient field. Increasing the magnetic field past a critical value, at which the penetration depth is of order the size of an MMB, causes the active gas to condense into a two-dimensional active fluid. Measurements of MMB motion in this phase reveal that contact with the surface enhances rotational diffusion by a factor of 8 and that velocity fluctuations are Laplace distributed. The fluid undergoes a percolation transition at a critical magnetic field. Supercritical active fluids exhibit dynamic voids, which limit the relaxation of density fluctuations. We describe the possible ecological significance of these results.

Figure 1

Multicellular magnetotactic bacteria live in spherical colonies composed of about 50 bacteria that grow and move like a single organism [22]. (a)–(f) As the cells in a colony divide, the colony grows, elongates, and divides symmetrically. The scale bar is 4 $\mu \mathrm{m}$. Panels (a)–(f) are modified slightly from Ref. [11] for legibility. Panel (g) shows the angle $\theta$ between the colony velocity $\mathbf{U}$ and the ambient magnetic field $\mathbf{B}$ averaged over several hundred colonies as they turn to align with the field. The average alignment rate $\gamma$ is fit for each magnetic field. The solid black line is the solution of Eq. (1).

Figure 2

MMB scatter from a surface. (a) A snapshot of an experiment $(B=0.29\mathrm{mT})$ is shown. Red circles show positions of colonies near the chamber wall (black line). The blue curve shows the trajectory of a colony that collided with the wall. (b) Colonies are distributed exponentially about the wall. The distance $X$ from the wall is normalized by $\ell$. The black line shows the best-fit exponential.

Figure 3

MMB are directed to collide normal to a coverslip by an applied magnetic field. (a) A schematic of the experiment is shown. The microscope is focused one colony radius below the top surface. (b) At the highest magnetic field examined (4 mT), colonies localize on the glass surface, where they diffuse laterally and collide with one another. The contact network is shown to highlight the local hexagonal order of the active fluid and the many vacancies, dislocations, and voids. The scale bar is 10 $\mu \mathrm{m}$.

Figure 4

At high magnetic field, MMB diffuse laterally over the chamber wall. (a) The measured distance $\mathbf{r}$ between isolated colonies at a time $-t$ before they collide is consistent with the motion of hard Brownian disks; we fit the two-dimensional diffusion coefficient $D_{\mathrm{eff}}$ at each magnetic field. The solid line is quadratic for $-t<{\tau }_{\mathrm{B}}$ and then increases linearly with slope $16D_{\mathrm{eff}}$ [32]. (b) $D_{\mathrm{eff}}$ decreases with increasing magnetic field. The red line shows the fit to $D_{\mathrm{eff}}=\alpha B^{-2}$.

Figure 5

The probability density functions of tangential velocities of colonies are shown for a range of magnetic fields. The number of instantaneous measurements from which the distributions is calculated range from $6.5\times {10}^5$ ($B=0.9\mathrm{mT}$) to $1.6\times {10}^6$ ($B=4.0\mathrm{mT}$). Normal and Laplace distributions are shown as guides to the eye.

Figure 6

Colonies self-organize into active fluids. Increasing the magnetic field increases the concentration of colonies but decreases their effective temperature. Representative snapshots show the geometry of the active fluids at magnetic fields (a) 0.9 mT, (b) 2.0 mT, (c) 2.8 mT, and (d) 4.0 mT. Blue curves and red curves show the boundaries of finite clusters and spanning clusters, respectively. Scale bars are 10 $\mu \mathrm{m}$. (e) The fluids undergo a percolation transition at a critical magnetic field $B_c\approx 2\mathrm{mT}$. Near $B_c$, the spanning cluster rapidly appears and disappears as colonies reorganize. The distribution of cluster sizes $N$ and the weight of the spanning cluster are averaged in time. The pair-correlation function $g\left(r\right)$ [panel (f)] and the structure factor $S\left(q\right)$ [panel (g)] are consistent with a compressible fluid. (h) The intermediate scattering function is shown for $B=4.0\mathrm{mT}$. All distances $r$ and wave numbers $q$ are measured relative to the average colony diameter.

Figure 7

The two-dimensional active fluid formed by colonies at the highest magnetic field examined (4 mT) shows a prevalence of large voids. The typical size of a void $\approx 200 \mu {\mathrm{m}}^2$ is much larger than the cross-sectional area of a colony 40 $\mu {\mathrm{m}}^2$. Voids form as colonies escape the surface. The voids diffuse through the fluid and merge (see Supplemental Video SV3.avi [23]). To characterize these dynamics we measure the average area $\langle A_{\mathrm{void}}\rangle$ of voids that are entirely surrounded by colonies. (a) Like the velocity fluctuations of individual colonies, instantaneous fluctuations of void areas are Laplace distributed. The average change is zero, consistent with the persistence of voids. (b) The power spectrum of the time series shown in panel (a) shows a $1/f$ dependence (red curve).