A drag force interpolation model for capsule-shaped cells in fluid flows near a surface.

We report an interpolation model to calculate the hydrodynamic force on tethered capsule-shaped cells in micro-fluidic flows near a surface. Our model is based on numerical solutions of the full Navier-Stokes equations for capsule-shaped objects considering their geometry, aspect ratio and orientation with respect to fluid flow. The model reproduced the results from computational fluid dynamic simulations, with an average error of <0.15 % for objects with an aspect ratio up to 5, and the model exactly reproduced the Goldman approximation of spherical objects close to a surface. We estimated the hydrodynamic force imposed on tethered Escherichia coli cells using the interpolation model and approximate models found in the literature, for example, one that assumes that E. coli is ellipsoid shaped. We fitted the 2D-projected area of a capsule and ellipsoid to segmented E. coli cells. We found that even though an ellipsoidal shape is a reasonable approximation of the cell shape, the capsule gives 4.4 % better agreement, a small difference that corresponds to 15 % difference in hydrodynamic force. In addition, we showed that the new interpolation model provides a significantly better agreement compared to estimates from commonly used models and that it can be used as a fast and accurate substitute for complex and computationally heavy fluid dynamic simulations. This is useful when performing bacterial adhesion experiments in parallel-plate flow channels. We include a MATLAB script that can track cells in a video time-series and estimate the hydrodynamic force using our interpolation formula.