Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Curvature dependent dynamics of a bacterium confined in a giant unilamellar vesicle

Olivia Vincent, Aparna Sreekumari, Manoj Gopalakrishnan, Vishwas V Vasisht, Bibhu Ranjan Sarangi

TL;DR

This work shows that a single bacterium confined within a curved GUV boundary accumulates in a bi-exponential manner, characterized by two distinct length scales that depend on the vesicle radius. By combining experiments, ABP simulations, and an analytical Fokker-Planck treatment, the authors reveal that the short length scale is governed by translational diffusion while the long length scale is controlled by rotational diffusion and self-propulsion, with curvature entering via two dimensionless parameters. The study provides a cohesive framework linking boundary-induced orientation to spatial density through a linearized SFPE and a semi-mean-field closure, supported by simulations and data collapse. These insights advance understanding of active matter in curved confinement and have implications for designing systems that sort or route active particles using geometry.

Abstract

We investigate the positional behavior of a single bacterium confined within a vesicle by measuring the probability of locating the bacterium at a certain distance from the vesicle boundary. We observe that the distribution is bi-exponential in nature. Near the boundary, the distribution exhibits rapid exponential decay, transitioning to a slower exponential decay, and eventually becoming uniform further away from the boundary. The length scales associated with the decay are found to depend on the confinement radius. We interpret these observations using molecular simulations and analytical calculations based on the Fokker-Planck equation for an Active Brownian Particle model. Our findings reveal that the small length scale is strongly influenced by the translational diffusion coefficient, while the larger length scale is governed by rotational diffusivity and self-propulsion. These results are explained in terms of two dimensionless parameters that explicitly include the confinement radius. The scaling behavior predicted analytically for the observed length scales is confirmed through simulations.

Curvature dependent dynamics of a bacterium confined in a giant unilamellar vesicle

TL;DR

This work shows that a single bacterium confined within a curved GUV boundary accumulates in a bi-exponential manner, characterized by two distinct length scales that depend on the vesicle radius. By combining experiments, ABP simulations, and an analytical Fokker-Planck treatment, the authors reveal that the short length scale is governed by translational diffusion while the long length scale is controlled by rotational diffusion and self-propulsion, with curvature entering via two dimensionless parameters. The study provides a cohesive framework linking boundary-induced orientation to spatial density through a linearized SFPE and a semi-mean-field closure, supported by simulations and data collapse. These insights advance understanding of active matter in curved confinement and have implications for designing systems that sort or route active particles using geometry.

Abstract

We investigate the positional behavior of a single bacterium confined within a vesicle by measuring the probability of locating the bacterium at a certain distance from the vesicle boundary. We observe that the distribution is bi-exponential in nature. Near the boundary, the distribution exhibits rapid exponential decay, transitioning to a slower exponential decay, and eventually becoming uniform further away from the boundary. The length scales associated with the decay are found to depend on the confinement radius. We interpret these observations using molecular simulations and analytical calculations based on the Fokker-Planck equation for an Active Brownian Particle model. Our findings reveal that the small length scale is strongly influenced by the translational diffusion coefficient, while the larger length scale is governed by rotational diffusivity and self-propulsion. These results are explained in terms of two dimensionless parameters that explicitly include the confinement radius. The scaling behavior predicted analytically for the observed length scales is confirmed through simulations.
Paper Structure (14 sections, 38 equations, 10 figures, 1 table)

This paper contains 14 sections, 38 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Bacterium inside a Vesicle
  3. Active Brownian Particle Model
  4. Theory: Fokker-Planck equation
  5. Laplace transform-based solution of Eq. \ref{['eq4']}
  6. A semi mean-field approximation for the orientation variable
  7. Solution for $P(z)$
  8. Discussion and Conclusions
  9. E.coli Confinement in GUVs
  10. Image Analysis
  11. Derivation of the SFPE
  12. On the roots of the equation $\Gamma(s)=0$
  13. Inverse Laplace transform of Eq. \ref{['eq7']}
  14. The reflecting boundary condition and the fast exponential decay of $P(z)$

Figures (10)

  • Figure 1: Experiments (top panel) Phase contrast microscopy image of a bacterium (highlighted by the cyan border) encapsulated inside a vesicle of different radius (a) 29$\mu$m, b) 18$\mu$m, c) 8$\mu$m, d) 3$\mu$m). (middle panel) Trajectory of the encapsulated bacterium for the corresponding vesicle radius. The yellow dashed line shows the boundary of the vesicle. The scale bar is 10$\mu$m. (bottom panel) Positional distribution of the bacterium for the corresponding radius. The scaled distance from the circular boundary, z, is defined as $1- r/R$, such that z = 0 is on the boundary and z=1 is at the center.
  • Figure 2: Experiments (a) (main panel) Scaled positional distribution function for different vesicle radius. The dashed lines correspond to bi-exponential fits for selected data. (inset) Full positional distribution. (b) The length scales $\zeta_1$ (main panel) and $\zeta_2$ (inset) extracted from a bi-exponential fitting function. Error bars indicate 95% confidence intervals, calculated from the upper and lower confidence bounds of the fit
  • Figure 3: Simulations (a) Scaled positional distribution for fixed $D_r=0.0025$ and varying confinement radius ($R=$1, 2, 5, 10, 20 and 50). (b) Scaled positional distribution for fixed $R=5$ and varying rotational diffusion ($D_r=$0.1, 0.02, 0.01 and 0.0025). (c) Scaled positional distribution for fixed $D_r=0.01$ and $R=1$ and varying $D_t$ ($10^{-3}, 10^{-4}$ and $10^{-5}$). To highlight that the distribution becomes independent of $D_t$ at larger $z$ values, the curves have been shifted relative to one another, using one curve as a reference, and shown in the inset
  • Figure 4: Length scales from simulation (symbols) and SFPE (lines) (a) The small length scale $\zeta_1(R)$ for fixed $D_t=10^{-4}$ and varying $D_r=0.1, 0.02, 0.01, 0.0025$. Inset (i): For a fixed $D_r=0.0025$, $\zeta_1(R)$ is shown for $D_t=10^{-3}$ (filled circle), $10^{-4}$ (filled square) and $10^{-5}$ (filled diamond). In the inset (ii) we show the same data, with $R$ scaled by $D_t$. (b) Simulation data for the larger length scale $\zeta_2$ is shown, in comparison with the theoretical predictions. In the inset we show the same data, with $R$ scaled by $u_0/D_r$, which is the persistence length $l_P$ of the ABP (see text for explanation).
  • Figure 5: (a) Angular distribution $g(\chi)$ (Eq. \ref{['eq6']}) for varying $D_r$ at fixed $R=1$. (b) The conditional average of $\cos \chi$ (Eq. \ref{['eq19+']}) as a function of $z$ for varying $D_r$ and fixed $R=1$. Both (a) and (b) are obtained from simulations. (c) Average of $\cos \chi$, from three different approaches, i.e., (i) the global mean computed using $g(\chi)$ in (a) (open circles, black) (ii) the $z\to 0$ limit of the conditional mean in (b) (open squares, red) (iii) the semi-mean-field expression for the global mean in Eq. \ref{['eq29']}, for varying $\eta$. Here the data is shown only for $R=1$, with $D_r$ varying.
  • ...and 5 more figures