 Solution

For a polymer curve parameterized by contour distance $s$, let $\mathbf{t}(s)$ be its unit tangent. The persistence length is the characteristic contour distance over which tangent orientations lose correlation. In the usual three-dimensional worm-like chain convention,where ${\kappa }_b$ is the bending stiffness. Segments much shorter than $L_p$ look approximately straight; a much longer chain can contain many statistically distinct orientations.

A surface image can be traced to estimate tangent correlations as a function of contour separation, after accounting for tracing resolution and substrate effects. A strictly equilibrated two-dimensional chain has a different relation between bending stiffness and correlation decay length from the three-dimensional convention. The definition must therefore state whether the surface preserves a projected conformation or allows in-plane equilibration. This is a statement about statistical filament mechanics; biological interpretation is deferred.