Irreversibility and symmetry breaking in the creation and annihilation of defects in active living matter

TL;DR

This work finds that defect creation and annihilation undergoes spatial symmetry breaking, and proposes that these results stem from a fundamental dualism between nematic structural organization and generated polar forces, which are intrinsic to living systems.

Abstract

Active living matter continuously creates and annihilates topological defects in a process that remains poorly understood. Here, we investigate these dynamics in two distinct active living systems: swarming bacteria and human bronchial epithelial cells. Despite their entirely different evolutionary origins, biological functions, and physical scales, both systems exhibit half-integer defects, consistent with the nematic phase. However, in contrast to active nematic theory, we find that defect creation and annihilation undergoes spatial symmetry breaking. We propose that these results stem from a fundamental dualism between nematic structural organization and generated polar forces, which are intrinsic to living systems. Furthermore, estimation of entropy production reveals that creation and annihilation are not reversed processes. Our findings challenge conventional nematic models and emphasize the role of defect-mediated dynamics in non-equilibrium biological systems as a major source of entropy production.

Figures (13)

• Figure 1: Director field and nematic defects in swarming Bacillus subtilis and human bronchial epithelial cells (HBECs) monolayers. The nematic director field, colored according to the orientations (ranging from $-\pi/2$ to $\pi/2$), and half-integer defects in (a) swarming B. subtilis and (b) HBECs. $+^1/_2$ types are represented as pink comet shapes and $-^1/_2$ as violet tripods. (c, d) Rates of pair-creation (blue) and annihilation (red) of oppositely charged $\pm^1/_2$ defects; (c) swarming B. subtilis and (d) HBECs.
• Figure 2: Configuration of $+^1/_2$ and $-^1/_2$ defect-pairs during creation and annihilation. Center sketch: Definitions of orientational angles $\varphi^+$(pink) and $\varphi^-$(violet) for paired $+^1/_2$ and $-^1/_2$ defects. These angles are defined relative to the vector pointing from the $+^1/_2$ to the $-^1/_2$ defect, shown by the solid black arrow.(a, b) Distribution of the pair orientation offset angle, $\delta=\varphi^+ -3\varphi^- -\pi$, as a function of time for creation (blue) and annihilation (red) events. $\delta=0$ implies that the nematic director fields of the paired-defects are in phase. (c, d) Distribution of $\varphi^+$ as a function of time for the creation (blue) and annihilation (red) events. The creation/annihilation instant is indicated by $t=0$, with negative time indicating time to annihilation, and positive time is from creation. Red and blue color intensities indicate probability density. The bright line close to $t=0$ indicates a region in which the distance between the defects is up to half cell-body length ($3-4~\mu m$ for bacteria and $7-8 ~\mu m$ for cells). At these distances the continuous nematic field is not well defined and the discrete nature of particles is of concern.(e-g)Angular distribution of $\varphi^+$ and $\varphi^-$ for defect pairs at small separation ($d<15 ~\mu m$ for bacteria and $d<35 ~\mu m$ for cells). Pairs are classified as "up" ($\langle \varphi^+ \rangle >0$, green) or "down" ($\langle \varphi^+ \rangle <0$, yellow).
• Figure 3: Spiraling trajectories of creation and annihilation.Center sketch: Co-rotation angle $\gamma$—the line connecting between the location of defect centers with respect to the angle close to the creation/annihilation instant ($\gamma_{d \rightarrow 0}$).(a, b) Distance between $\pm^1/_2$ defect-pairs as a function of time after creation (blue) and before annihilation (red), for bacteria and cells. The creation/annihilation instant is indicated by $t({d \rightarrow 0})$, with negative values indicating time to annihilation, and positive ones indicating time from creation. Solid lines represent the average trajectories, while red and blue color intensities indicate probability density. (c, d) Time evolution of $\gamma$ (the angle of the line connecting the defect-pairs, aligned to $\gamma_{d \rightarrow 0}=0$). Solid lines represent averages for "up" (green) and "down" (yellow) configurations. Shaded areas indicate the standard deviation. (e, f) Average trajectories of $+^1/_2$ (pink) and $-^1/_2$ (violet) defects during creation and annihilation, measured in the lab frame of reference. Trajectories are centered around the creation/annihilation position (black point), rotated to $\gamma_{d \rightarrow 0}=0$ and mirrored to the "down" configuration.
• Figure 4: Symmetry breaking in the flow around $+^1/_2$.(a–b) The distribution of $\phi_V$ in bacteria, the angle between the $+^1/_2$ velocity and defect orientation (inset in panel a). Average flow fields around $+^1/_2$ defects are obtained by separately averaging trajectories according to their defect-pair configurations: "up" ($\langle \varphi^+ \rangle > 0$) or "down" ($\langle \varphi^+ \rangle < 0$). Flows are shown for defects with overall displacement to the right (blue--gray) and to the left (pink). The velocity scale indicated by the black arrow is $10~\mu\mathrm{m}/\mathrm{s}$. Scale bar: $10~\mu\mathrm{m}$.
• Figure 5: The interaction between nematic and polar symmetries.(a) Defect creation via bend instability with the overlaid nematic (black lines) and polar (red arrows) fields. A grain boundary in the polar field is introduced (blue dashed line). The polar field leads to a torque on the defects (green arrow), which will reorient the $+^1/_2$ defect, causing it to self-propel and spiral around the $-^1/_2$ defect (inset). Shown here is the "down" configuration. $\zeta_p\neq0$. (b) Nematic and polar fields around a pair of defects before annihilation. Here, the grain boundaries do not directly connect. A similar torque, acting on the $+^1/_2$ defect, leads to a spiraling trajectory (inset). (c) Total co-rotation ($\Delta\gamma$) and (d) final positive defect orientation ($\Delta\varphi^+$) of the defect pair as a function of the applied polar body force ($\zeta_p$). Dashed lines indicate creation and solid lines for annihilation, green "up" and yellow "down" configuration. (e) Predicted Stokes flow around a $+^1/_2$ defect in the annihilation "up" (or creation "down") configuration. Shown with $|\zeta_p/\zeta_Q| = 0.5$ and $\phi_g=0$. (f) Angular offset of defect self-propulsion ($\phi_V$) and (g) net vorticity around a $+^1/_2$ defect as a function of the relative polar body force ($\zeta_p/\zeta_Q$) and position of the grain boundary ($\phi_g$). $\phi_g=0$ implies that the grain boundary alignment with the tail of the $+^1/_2$ defect is as in (a) and (b).