### Abstract

Experiments and simulations on single α-actin filaments in the Poiseuille flow through a microchannel show that the center-of-mass probability density across the channel assumes a bimodal shape as a result of pronounced cross-streamline migration. We reexamine the problem and perform Brownian dynamics simulations for a bead-spring chain with bending elasticity. Hydrodynamic interactions between the pointlike beads are taken into account by the two-wall Green tensor of the Stokes equations. Our simulations reproduce the bimodal distribution only when hydrodynamic interactions are taken into account. Numerical results on the orientational order of the end-to-end vector of the modelpolymer are also presented together with analytical hard-needle expressions at zero flow velocity. We derive a Smoluchowski equation for the center-of-mass distribution and carefully analyze the different contributions to the probability current that causes the bimodal distribution. As for flexible polymers,hydrodynamic repulsion explains the depletion at the wall. However, in contrast to flexible polymers, the deterministic drift current mainly determines migration away from the centerline and thereby depletion at the center. Diffusional currents due to a position-dependent diffusivity become less important with increasing polymer stiffness.

We would like to thank T. Pfohl, R. Winkler, and G. Gompper for helpful discussions. This work was supported by the Deutsche Forschungsgemeinschaft (DFG) through the research training group GRK 1558.

I. INTRODUCTION

II. MODELING

A. Semiflexible polymer

B. Hydrodynamic interactions

C. Brownian dynamics simulation

D. Parameters

III. RESULTS

A. Center-of-mass probability distribution

B. Polymer orientation within the channel

IV. KINETIC THEORY FOR A SEMIFLEXIBLE POLYMER

A. Smoluchowski equation for center-of-mass current

B. Analysis of the lateral center-of-mass current

V. SUMMARY AND CONCLUSION

### Key Topics

- Poiseuille flow
- 56.0
- Polymers
- 56.0
- Tensor methods
- 31.0
- Hydrodynamics
- 29.0
- Probability theory
- 18.0

## Figures

A semiflexible polymer modeled by a bead-spring chain with bending rigidity is confined between two parallel planar plates with distance 2*W*. The space vector points from the centerline at *z* = 0 to bead *i*.

A semiflexible polymer modeled by a bead-spring chain with bending rigidity is confined between two parallel planar plates with distance 2*W*. The space vector points from the centerline at *z* = 0 to bead *i*.

Sketch of the bead-spring chain. *N* beads with position vectors are connected by springs. The vector connects bead *i* to *i* + 1 and denotes the end-to-end vector.

Sketch of the bead-spring chain. *N* beads with position vectors are connected by springs. The vector connects bead *i* to *i* + 1 and denotes the end-to-end vector.

Field lines of the flow fields induced by a point force acting close to a wall either (a) along or (b) perpendicular to a wall.^{35}

Field lines of the flow fields induced by a point force acting close to a wall either (a) along or (b) perpendicular to a wall.^{35}

Field lines of the flow fields induced by a point force at the centerline; calculated by the two-wall Green tensor (a) and by the approximate Green tensor (b) based on the Blake tensor.^{35}

Field lines of the flow fields induced by a point force at the centerline; calculated by the two-wall Green tensor (a) and by the approximate Green tensor (b) based on the Blake tensor.^{35}

Snapshots of the bead-spring chain for different persistence lengths at the centerline (top) and near the wall (bottom) at *v* _{0} = 2.5 mm/s. (a) *L* _{ p }/*L* = 1, (b) *L* _{ p }/*L* = 4, and (c) *L* _{ p }/*L* = 16.

Snapshots of the bead-spring chain for different persistence lengths at the centerline (top) and near the wall (bottom) at *v* _{0} = 2.5 mm/s. (a) *L* _{ p }/*L* = 1, (b) *L* _{ p }/*L* = 4, and (c) *L* _{ p }/*L* = 16.

Center-of-mass probability distribution plotted from the centerline (*z* _{ C } = 0) to the wall (*z* _{ C } = *W*) for different flow velocities *v* _{0} and a persistence length of *L* _{ P }/*L* = 1. The inset compares the simulation results for zero flow velocity *v* _{0} to the hard-needle distribution of Eq. (16).

Center-of-mass probability distribution plotted from the centerline (*z* _{ C } = 0) to the wall (*z* _{ C } = *W*) for different flow velocities *v* _{0} and a persistence length of *L* _{ P }/*L* = 1. The inset compares the simulation results for zero flow velocity *v* _{0} to the hard-needle distribution of Eq. (16).

Center-of-mass probability distributions for different persistence lengths *L* _{ P }/*L* at a fixed center flow velocity *v* _{0} = 2.5 mm/s.

Center-of-mass probability distributions for different persistence lengths *L* _{ P }/*L* at a fixed center flow velocity *v* _{0} = 2.5 mm/s.

Center-of-mass probability density for different ratios *l*/*a* at a fixed persistence length *L* _{ P }/*L* = 2 and a fixed flow velocity *v* _{0} = 2.5 mm/s. In order to change the ratio *l*/*a*, we keep the length *L* and the number of beads *N* constant and vary the radius *a* of the beads. This corresponds to increasing the thickness of our model polymer.

Center-of-mass probability density for different ratios *l*/*a* at a fixed persistence length *L* _{ P }/*L* = 2 and a fixed flow velocity *v* _{0} = 2.5 mm/s. In order to change the ratio *l*/*a*, we keep the length *L* and the number of beads *N* constant and vary the radius *a* of the beads. This corresponds to increasing the thickness of our model polymer.

Center-of-mass probability distribution simulated without hydrodynamic interactions for different flow velocities *v* _{0} and persistence length *L* _{ P }/*L* = 1.

Center-of-mass probability distribution simulated without hydrodynamic interactions for different flow velocities *v* _{0} and persistence length *L* _{ P }/*L* = 1.

Order parameter *S* and average end-to-end distance 〈*R*〉 plotted versus the lateral center-of-mass position *z* _{ C }/*W*: (a) and (b) for a fixed stiffness *L* _{ p }/*L* = 1 and different flow velocities; (c) and (d) for a fixed flow velocity *v* _{0} = 2.5 mm/s and different stiffnesses *L* _{ p }/*L*. Inset in (a): For *v* _{0} = 0 mm/s, the simulated order parameter *S* is compared to the analytic result of Eq. (19).

Order parameter *S* and average end-to-end distance 〈*R*〉 plotted versus the lateral center-of-mass position *z* _{ C }/*W*: (a) and (b) for a fixed stiffness *L* _{ p }/*L* = 1 and different flow velocities; (c) and (d) for a fixed flow velocity *v* _{0} = 2.5 mm/s and different stiffnesses *L* _{ p }/*L*. Inset in (a): For *v* _{0} = 0 mm/s, the simulated order parameter *S* is compared to the analytic result of Eq. (19).

Plots of the different contributions of the lateral center-of-mass current across the channel at flow velocity *v* _{0} = 2.5 mm/s for different bending rigidities: (a) *L* _{ P }/*L* = 1, (b) *L* _{ P }/*L* = 2, (c) *L* _{ P }/*L* = 4, and (d) *L* _{ P }/*L* = 16.

Plots of the different contributions of the lateral center-of-mass current across the channel at flow velocity *v* _{0} = 2.5 mm/s for different bending rigidities: (a) *L* _{ P }/*L* = 1, (b) *L* _{ P }/*L* = 2, (c) *L* _{ P }/*L* = 4, and (d) *L* _{ P }/*L* = 16.

(a) At the centerline, the relaxing U-shaped filament initiates flow fields relative to the applied Poiseuille flow. The resulting hydrodynamic interactions drive the filament away from the centerline (see also the second video of the supplemental material^{43}). The strength of the flow is given by the color code in arbitrary units. (b) Close to the wall the filament is under tension. This initiates flow fields illustrated in Fig. 3 on the left that drive the filament away from the wall (see also the third video of the supplemental material^{43}).

(a) At the centerline, the relaxing U-shaped filament initiates flow fields relative to the applied Poiseuille flow. The resulting hydrodynamic interactions drive the filament away from the centerline (see also the second video of the supplemental material^{43}). The strength of the flow is given by the color code in arbitrary units. (b) Close to the wall the filament is under tension. This initiates flow fields illustrated in Fig. 3 on the left that drive the filament away from the wall (see also the third video of the supplemental material^{43}).

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