# The counterbend dynamics of cross-linked filament bundles and flagella

## TL;DR

This paper investigates how cross-linking proteins in flagella influence their bending dynamics, revealing complex length-dependent behaviors and counter-waves that challenge simpler mechanical models.

## Contribution

It demonstrates that cross-linking mechanics induce a bimodal length-dependent response, altering traditional Euler-Bernoulli predictions and affecting flagellar wave regulation.

## Key findings

- Counterbend induces counter-waves opposing oscillations.
- Cross-linking mechanics cause a bimodal length-dependent response.
- Simpler theories may lead to paradoxical interpretations.

## Abstract

Cross-linked filament bundles, such as in cilia and flagella, are ubiquitous in biology. They are considered in textbooks as simple filaments with larger stiffness. Recent observations of flagellar counterbend, however, show that induction of curvature in one section of a passive flagellum instigates a compensatory counter-curvature elsewhere, exposing the intricate role of the diminutive cross-linking proteins at large-scales. We show that this effect, a material property of the cross-linking mechanics, modifies the bundle dynamics non-trivially, and induces a bimodal $L^2-L^3$ length-dependent material response that departs from the Euler-Bernoulli theory. Hence, the use of simpler theories to analyse experiments can result in paradoxical interpretations. Remarkably, the counterbend dynamics instigates counter-waves in opposition to driven oscillations in distant parts of the bundle, with potential impact on the regulation of flagellar bending waves. These results have a range of physical and biological applications, including the empirical disentanglement of material quantities via counterbend dynamics.