Stretching weakly bending filaments with spontaneous curvature in two dimensions

Some important biomolecules (for instance, bacterial FtsZ and eukaryotic DNA) are known to posses spontaneous (intrinsic) curvature. Using a simple extension of the wormlike chain model, we study the response of a weakly bending filament in two dimensions to a pulling force applied at its ends (a configuration common in classical in-vitro experiments and relevant to several in-vivo cell cases). The spontaneous curvature of such a chain or filament can in general be arc-length dependent and we study a case of sinusoidal variation, from which an arbitrary case can be reconstructed via Fourier transformation. We obtain analytic results for the force-extension relationship and the width of transverse fluctuations. We show that spontaneous-curvature undulations can affect the force-extension behavior even in relatively flexible filaments with a persistence length smaller than the contour length.

💡 Research Summary

The authors extend the classic worm‑like chain (WLC) model to incorporate a spontaneous (intrinsic) curvature that may vary along the contour of a filament. Their focus is on a weakly bending filament confined to two dimensions and subjected to a tensile force applied at its ends—a geometry that mirrors many in‑vitro single‑molecule pulling experiments and also occurs in vivo (e.g., during bacterial cytokinesis or DNA packaging).

Starting from the standard bending energy (A/2\int (\partial^{2}\mathbf r/\partial s^{2})^{2} ds) and adding a term that penalizes deviations from a prescribed curvature (\kappa_{0}(s)), the Hamiltonian becomes
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