Topological Defects in a Living Nematic Ensnare Swimming Bacteria

Active matter exemplified by suspensions of motile bacteria or synthetic self-propelled particles exhibits a remarkable propensity to self-organization and collective motion. The local input of energy and simple particle interactions often lead to complex emergent behavior manifested by the formation of macroscopic vortices and coherent structures with long-range order. A realization of an active system has been conceived by combining swimming bacteria and a lyotropic liquid crystal. Here, by coupling the well-established and validated model of nematic liquid crystals with the bacterial dynamics, we develop a computational model describing intricate properties of such a living nematic. In faithful agreement with the experiment, the model reproduces the onset of periodic undulation of the director and consequent proliferation of topological defects with the increase in bacterial concentration. It yields a testable prediction on the accumulation of bacteria in the cores of $+1/2$ topological defects and depletion of bacteria in the cores of $-1/2$ defects. Our dedicated experiment on motile bacteria suspended in a freestanding liquid crystalline film fully confirms this prediction. Our findings suggest novel approaches for trapping and transport of bacteria and synthetic swimmers in anisotropic liquids and extend a scope of tools to control and manipulate microscopic objects in active matter.

Popular Summary

Liquid crystals exhibit great diversity in mechanical and optical properties that make them invaluable materials for a wide range of applications such as electronic displays, optical filters, and some lasers. Their unique behavior comes from an internal structure that blurs the line between liquids and solids, allowing for many possible molecular arrangements. Living liquid crystals are a new class of these materials that combine living mobile bacteria and water-based nontoxic liquid crystals. These substances exhibit optical and mechanical properties not present in their inanimate counterparts. We present a computational model, verified by experiments, that describes the behavior of living liquid crystals and suggests new ways of controlling bacteria in such compounds.

Unlike previous models, we incorporate two populations of bacteria that travel in opposite directions within the liquid crystal—a behavior that has been observed in previous experiments. We also include the effects of a preferred molecular orientation in liquid crystals due to a specific treatment of the surfaces. Computational modeling points to several new phenomena, including the accumulation of bacteria in the cores of positive topological defects and a corresponding depletion in the negative ones. We confirmed these predictions by observing a fluorescent strain of the bacteria Bacillus subtilis swimming in a film of the liquid crystal disodium cromoglycate.

Our findings show how, by introducing and guiding defects in a liquid crystal, one can manipulate bacteria either to study the bacteria populations themselves or even to develop bacteria-powered micromachines.

Figure 1

Schematics of experimental cell. (a) A liquid crystal suspension of motile bacteria is sandwiched between two glass plates with the separation $h$ [32]. A specific treatment of the glass plate inner surface anchors the LC director $\mathbf{n}$ along the $x$ axis. Bacteria exhibit apolar interaction: bacteria swimming in the opposite direction can pass without collision. (b) Illustration of bacterial alignment along the LC director $\mathbf{n}$. Swimming bacteria impose a pair of forces $\pm \mathbf{F}$ (force dipole) on the LC.

Figure 2

Results of computational modeling. Nematic orientations and magnitudes of the order parameter (a)–(c), concentration fields (d)–(f), and flow velocity magnitudes with streamlines (g)–(i). Panels (a), (d), and (g) depict a state with defects for zero anchoring and average concentration $c_0=0.2$. Some $\pm 1/2$ topological defects are shown. Panels (b), (e), and (h) display stable periodic configuration with surface anchoring (${\xi }_{\mathrm{an}}=0.05$) and $c_0=0.54$, i.e., near the instability threshold. The thick black arrow indicates the prescribed director orientation set by the surface anchoring. Panels (c), (f), and (i) illustrate hysteretic behavior: the configuration with defects exists for the same parameter values (${\xi }_{\mathrm{an}}=0.05$, $c_0=0.54$) as in (b), (e), and (h). Other model parameters are $\tau =66.6$, $D_c=4$, $\mathrm{Er}=3.75$, $\mathrm{\Lambda }=1$, $\zeta =0.2$, $\mathrm{\Gamma }=0.5$. See also Videos 1–4 in Supplemental Material [39].

Figure 3

Characterization of defect dynamics. (a) Phase diagram. Symbols circle, times depict the simulations results. The black solid line indicates the linear stability limit, Eq. (9). Stable director undulations exist between defect and homogeneous aligned states (yellow band). The transition boundaries are obtained by gradually increasing the average concentration $c_0$ from a slightly perturbed homogeneous initial condition. Hatching shows the hysteresis band where nonvanishing defects coexist with the undulated or homogeneous aligned states, $\tau =66.6$ (corresponding, approximately 20–30 sec for the conditions of our experiment). (b) Mean velocity $\overline{V}$ of $+1/2$ defects (circle, red) and $-1/2$ defects (square, blue) vs $c_0$. Error bars are the velocity standard deviations. The velocity distributions for $-1/2$ (c) and $1/2$ defects (d) for ${\xi }_{\mathrm{an}}=0$ and for $c_0=1.8$ (blue) and $c_0=0.3$ (brown). Red solid lines are Gaussian fits and solid black lines are the stretched exponential fits.

Figure 4

Concentration distributions. Relative concentration difference $\mathrm{\Delta }c/c_0$ vs $c_0$ for $\tau =66.6$ (a) and vs $\tau$ for $c_0=0.3$ (b) for $1/2$ (circle) and $-1/2$ (times) defects. Inset: $\mathrm{\Delta }c/c_0$ for $1/2$ defects vs defect velocity $V$. Blue symbols in (b) indicate $\tau =66.6$ used in most of the simulations. Error bars are the standard deviations. (c) Close-up view of the concentration field $c$ for $c_0=0.3$. Some defects are shown in red circles, white arrow indicates the direction of motion. Domain size is $40\times 40$ units of length. Steady-state concentration distributions near $1/2$ defect (d) and $-1/2$ defect (e) obtained from Eq. (11) for $V=0.5$. Size of the integration domain $40\times 20$ units; due to reflection symmetry, only the upper half of the field $c$ is shown.

Figure 5

Experimental verification. (a) Fluorescent image of swimming bacteria suspended in a freestanding LC film in the regime of chaotic motion. Average bacterial concentration $c\approx 2\times 10^9\mathrm{cell}/{\mathrm{cm}}^3$. Scale bar is $50\mu \mathrm{m}$. See also Videos 5 and 8 in Supplemental Material [39]. (b) Digitally processed image of (a). Green lines show the LC nematic director orientation reconstructed from the bacterial orientation. Several defects are present. (c) Bacterial concentration distribution in the area indicated by the blue dashed box in (b). Colors represent the concentration of bacteria extracted from the averaged fluorescence intensity. Two topological defects ($1/2$ and $-1/2$) are highlighted. White lines show defect orientations. Scale bar is $50\mu \mathrm{m}$. (d) Bacterial concentration profiles around the defects. Blue line represents a concentration profile in the vicinity of $1/2$ defect along the white dashed line in (d). $x=0$ corresponds to the center of $1/2$ defect. The red line represents a concentration profile in the vicinity of $-1/2$ defect. Concentration profiles along three white dashed lines are averaged. Inset: Bacterial concentration profiles along the lines passing through the cores of defects extracted from the computational model for $c_0=0.3$, $\tau =66.6$ and corresponding to Fig. 4.

Figure 6

Illustration of accumulation and depletion of bacteria in topological defects. (a) Accumulation of bacteria in $+1/2$ defect. Bacteria accumulate in the yellow region due to the convergence of bacterial trajectories. (b) Bacteria escape from the dark blue region of $-1/2$ defect independently of their initial orientation.

Figure 7

Linear stability analysis of the aligned state. (a) Density plot of the growth rate $\mathrm{Re}[\sigma (\mathbf{k})]$ vs $k_x$, $k_y$ for ${\xi }_{\mathrm{an}}=0$. Maximum growth rate occurs along the nematic direction ($x$ axis). (b) $\mathrm{Re}[\sigma (\mathbf{k})]$ vs $k_x$, $k_y=0$, for different ${\xi }_{\mathrm{an}}$ values, $c_0=0.2$. (c) Critical wavelength $L_{\mathrm{cr}}=2\pi /k_{\mathrm{cr}}$ vs $c$, from Eq. (8). The dashed line is the fit $L_{\mathrm{cr}}\sim 1/\sqrt{c-c_{\mathrm{cr}}}$. Inset: $k_{\mathrm{cr}}$ vs ${\xi }_{\mathrm{an}}$, from Eq. (10).