Statistical and Biological Physics

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Coupling of transverse and longitudinal response in stiff polymers

Benedikt Obermayer, and Oskar Hallatschek

For a comprehensive picture of the viscoelasticity of cellular material a thorough understanding of single molecule properties is necessary, in particular of the response of single filaments under force. Biological examples include cytoskeletal polymers such as actin or microtubules, but also stretched DNA, which by now is almost ubiquitious in single-molecule experiments. The large bending stiffness of these polymers makes their static and dynamic features highly anisotropic: fluctuations and response are larger in the transverse direction (perpendicular to the local tangent) than in the longitudinal one. However, these filaments are also nearly inextensible, and motion in the two directions is therefore not independent. We show that beyond the linear level, where the coupling between transverse and longitudinal dynamics is not yet relevant, the dynamic response to transverse forces becomes nonlinearly coupled to the longitudinal one, even in the weakly-bending limit of an almost straight contour. The underlying physical mechanism can easily be understood (see figure): a transverse point force leads to the growth of a bulge in the contour. Due to the backbone inextensibility, this bulge can keep growing only by pulling in contour length from the filament's tails. Slowed down by viscous solvent friction, the resulting effective longitudinal pulling force straightens the tails by reducing the thermal roughness of the contour. This feedback mechanism leads to a significant weakening of the transverse response by the coupling to the friction-constrained longitudinal one. We analyze this effect via scaling arguments and by a systematic theory that also contains the experimentally and biologically relevant case of prestretched filaments initially under tension. Our results apply not only to the single filament response, but have implications also for the collective dynamics in crosslinked networks and tensegrity structures.