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Simulating the swimming motion of a flagellated bacterium in a microstructured bio-fluid

Arjun Sharma, Sabarish V. Narayanan, Sarah Hormozi, Donald L. Koch

Abstract

We develop a numerical framework to simulate the locomotion of a flagellated bacterium with a spheroidal head (such as Escherichia coli) in biological fluids like mucus, which are entangled polymer solutions exhibiting elasto-viscoplastic (EVP) rheology and porous microstructure. To account for the scale disparity between the large bacterial head and the slender flagellar bundle, whose thickness is comparable to the pore size, we employ a two-fluid model in which the bundle directly drives the solvent and exchanges momentum with the polymer phase via drag proportional to their relative velocity. The numerical implementation combines a finite-difference discretization of the two-fluid equations with a slender-body theory (SBT) to model flagellar forcing. A key observation is that the coupled mass and momentum equations for these inertialess flows, together with SBT, are linear in the pressure and velocity fields and in the force distribution along the flagellar bundle. By treating the polymer stress as a body force, we decompose the flow field and hydrodynamic moments into three additive contributions: kinematic (motion), flagellar forcing, and polymer stress. This decomposition allows several components of the flow to be precomputed and enables the determination of swimming velocity via a resistivity formulation driven by polymer-induced forces, which greatly improves computational efficiency during transient calculations of the polymer stress and the resulting flow. We validate the method and use it to analyze how polymer microstructure and its interactions with the bacterial head and tail affect motility in complex biofluids.

Simulating the swimming motion of a flagellated bacterium in a microstructured bio-fluid

Abstract

We develop a numerical framework to simulate the locomotion of a flagellated bacterium with a spheroidal head (such as Escherichia coli) in biological fluids like mucus, which are entangled polymer solutions exhibiting elasto-viscoplastic (EVP) rheology and porous microstructure. To account for the scale disparity between the large bacterial head and the slender flagellar bundle, whose thickness is comparable to the pore size, we employ a two-fluid model in which the bundle directly drives the solvent and exchanges momentum with the polymer phase via drag proportional to their relative velocity. The numerical implementation combines a finite-difference discretization of the two-fluid equations with a slender-body theory (SBT) to model flagellar forcing. A key observation is that the coupled mass and momentum equations for these inertialess flows, together with SBT, are linear in the pressure and velocity fields and in the force distribution along the flagellar bundle. By treating the polymer stress as a body force, we decompose the flow field and hydrodynamic moments into three additive contributions: kinematic (motion), flagellar forcing, and polymer stress. This decomposition allows several components of the flow to be precomputed and enables the determination of swimming velocity via a resistivity formulation driven by polymer-induced forces, which greatly improves computational efficiency during transient calculations of the polymer stress and the resulting flow. We validate the method and use it to analyze how polymer microstructure and its interactions with the bacterial head and tail affect motility in complex biofluids.

Paper Structure

This paper contains 20 sections, 61 equations, 18 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Polymer stress and constitutive modeling
  4. Slender-body theory and hydrodynamic coupling
  5. Force and torque free constraints
  6. Operator formulation and model overview
  7. Exact linear decomposition of flow fields and hydrodynamic moments
  8. Precomputations for flow due to kinematic motion ($\hat{\mathcal{P}}_\text{MFP}^{(M)}$)
  9. Pre-computations for flows due to flagellar forcing ($\hat{\mathcal{P}}_\text{MFP}^{(\text{Flag})}$ and $\hat{\mathcal{P}}_\text{Flag}$)
  10. Reformulating the SBT problem ($\hat{\mathcal{P}}_\text{SBT}$)
  11. Time-stepping algorithm for coupled swimmer–polymer dynamics
  12. Validation
  13. Handling two fluids
  14. EVP constitutive law
  15. Flagellar bundle forcing: Stokeslet next to a sphere
  16. ...and 5 more sections

Figures (18)

  • Figure 1: Schematic of a bacterium with a spheroidal head and helical flagellar bundle, showing helix geometry and key variables.
  • Figure 2: Plot of the motor torque $g$ as a function of motor angular velocity $\omega_M$. See table \ref{['tab:EcoliBergPoon']} for typical values of the constants.
  • Figure 3: Schematic showing the coupling between the polymer stress, $\bm{\Pi}$ and three subproblems involving unknowns in equation \ref{['eq:Unknowns']}. The three subproblems are the main fluid–polymer system, the flagellum-only flow, and the slender-body theory solver.
  • Figure 4: Discretization of the helical line of singularities into $N_S$ points. Each point on the helix (blue stars) carries a two-fluidlet, a fundamental singularity of the two-fluid medium.
  • Figure 5: Plot of the dimensionless drag force on a translating sphere due to (a) the solvent and (b) polymer as a function of $L_B/R_H$, for different viscosity ratios, simulated using the two-fluid Newtonian solver and compared with analytical solutions from Sabarish. (c) shows the solvent torque on a rotating sphere. The polymer torque is zero due to the slip condition.
  • ...and 13 more figures