Longitudinal dispersion of DNA in nanochannels

A theory is presented of the longitudinal dispersion of DNA under equilibrium confined in a nanochannel. Orientational fluctuations of the DNA chain build up to give rise to substantial fluctuations of the coil in the longitudinal direction of the channel. The translational and orientational degrees of freedom of the polymer are described by the Green function satisfying the usual Fokker-Planck equation. It is argued that this is analogous to the transport equation occurring in the theory of convective diffusion of particles in pipe flow. Moreover, Taylor’s method may be used to reduce the Fokker-Planck equation to a diffusion equation for long DNA although subtleties arise connected with the orientational distribution of segments within the channel. The longitudinal “step length” turns out to be proportional to the typical angle of a DNA segment to the sixth power. The dispersion is underestimated compared to experiment, probably because the harmonic approximation is used to describe the polymer confinement.

💡 Research Summary

The paper develops a theoretical framework for the longitudinal dispersion of DNA confined in a nanochannel at equilibrium. Starting from a worm‑like chain description, the authors treat each infinitesimal segment by its position r(s) and unit tangent u(s). The channel walls impose a transverse confinement potential, which the authors approximate as harmonic, (U(r_{\perp})=\frac{1}{2}k r_{\perp}^{2}). Under this approximation the joint probability density (P(\mathbf{r},\mathbf{u},s)) obeys a Fokker‑Planck (or Smoluchowski) equation that couples translational diffusion in the transverse and longitudinal directions with rotational diffusion of the tangent vector.

The central insight is to recognize the formal analogy between this Fokker‑Planck equation and the convective‑diffusion equation that appears in the classic Taylor‑Aris analysis of solute transport in pipe flow. In Taylor’s problem, a fast transverse mixing caused by the parabolic velocity profile leads, after a short transient, to an effective longitudinal diffusion coefficient that is much larger than the molecular diffusivity. By performing a multiscale expansion—separating the rapid transverse equilibration time (\tau_{\perp}) from the slow longitudinal evolution time (\tau_{\parallel})—the authors reduce the original equation to a one‑dimensional diffusion equation for the mean longitudinal coordinate.

Mathematically, the reduction yields an effective longitudinal diffusivity
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