Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2018 / Paper 1 / Section A(iii) / 5 / iv

Interpret the stipulated model as a freely jointed chain with $N$ independent, isotropically oriented segments of length $b$. With end-to-end vector $\mathbf{R}={\sum }_{j=1}^N{\mathbf{b}}_j$, cross terms vanish on averaging:Using only the model's supplied segment size and count givesThe mean vector is zero, not the mean magnitude. In the three-dimensional large-$N$ Gaussian chain limit,Thus the common root-mean-square interpretation of an average chain size is $10.1\mathrm{nm}$, while the mean scalar distance is about $9.32\mathrm{nm}$. The contour length $Nb=320\mathrm{nm}$ is a different quantity. A count of $999$ links between $1000$ positions changes these estimates by less than $0.1\%$.

These are results of the explicitly idealized statistical chain, not a prediction about the conformation or persistence length of an actual biological molecule. TODO (user-deferred biology): material-specific biological interpretation of that model.