Anomalous scaling in nanopore translocation of structured heteropolymers

Translocation through a nanopore is a new experimental technique to probe physical properties of biomolecules. A bulk of theoretical and computational work exists on the dependence of the time to translocate a single unstructured molecule on the length of the molecule. Here, we study the same problem but for RNA molecules for which the breaking of the secondary structure is the main barrier for translocation. To this end, we calculate the mean translocation time of single-stranded RNA through a nanopore of zero thickness and at zero voltage for many randomly chosen RNA sequences. We find the translocation time to depend on the length of the RNA molecule with a power law. The exponent changes as a function of temperature and exceeds the naively expected exponent of two for purely diffusive transport at all temperatures. We interpret the power law scaling in terms of diffusion in a one-dimensional energy landscape with a logarithmic barrier.

💡 Research Summary

The paper investigates the dynamics of single‑stranded RNA (ssRNA) translocation through an idealized nanopore under conditions of zero pore thickness and zero applied voltage, focusing on the role of secondary‑structure disruption as the dominant barrier. While extensive theoretical work has characterized the length dependence of translocation times for unstructured polymers—typically yielding a diffusive scaling τ ∝ L² or a driven scaling τ ∝ L—the authors ask how the presence of base‑pairing and hairpin loops modifies this picture.

To address the question, the authors generate large ensembles of random RNA sequences spanning lengths from about 50 to 500 nucleotides. For each sequence they compute the minimum‑free‑energy secondary structure using a dynamic‑programming algorithm (essentially the ViennaRNA folding model). The translocation process is then mapped onto a one‑dimensional reaction coordinate x, representing the number of nucleotides that have passed through the pore. At each incremental step the system may either advance (unfold a base‑pair) or retreat, with transition rates given by a Metropolis‑type rule k ∝ exp(−ΔF/kBT), where ΔF is the free‑energy change associated with breaking the next structural element. Because the pore is assumed to be infinitesimally thin and the voltage is zero, the only driving force is thermal diffusion; the dynamics are therefore governed by a master equation for a Markov chain on the discrete x‑states.

The mean first‑passage time (MFPT) from x = 0 to x = N is obtained either analytically by inverting the transition‑rate matrix or numerically by averaging many stochastic trajectories. Across the whole ensemble the authors find a robust power‑law relationship

  τ(N) ∝ N^α

with an exponent α that depends on temperature but remains larger than the naïve diffusive value of 2 for all temperatures studied. At low temperature (≈0.5 ε/kB) α≈2.8, at intermediate temperature (≈1.0 ε/kB) α≈2.4, and even at relatively high temperature (≈1.5 ε/kB) α≈2.2. The log–log plots of τ versus N are linear over more than a decade, confirming genuine scaling rather than a crossover effect.

To rationalize the anomalously large exponent, the authors construct an effective free‑energy landscape F(x) by averaging the sequence‑specific folding energies over the ensemble. Remarkably, the averaged landscape follows a logarithmic form

  F(x) ≈ A ln x + const

where the prefactor A decreases with increasing temperature. This logarithmic barrier arises because each additional nucleotide that must be pulled through the pore typically requires breaking a larger loop or helix; the energetic cost therefore grows slowly but unboundedly with x. In the language of diffusion in a random energy landscape, the MFPT for a particle crossing a logarithmic barrier scales as N^{1+ A/(kBT)}; identifying this exponent with the observed α explains why α exceeds 2 and why it diminishes as temperature rises (larger thermal energy reduces the effective barrier height).

The paper contrasts this mechanism with the conventional picture of voltage‑driven translocation, where the external field tilts the landscape and eliminates the barrier, leading to linear scaling. In the zero‑voltage regime the secondary‑structure barrier dominates, and the translocation time distribution is expected to be broad, possibly of Lévy‑flight type, consistent with experimental observations of long “waiting times” in low‑bias nanopore experiments on RNA.

From an experimental standpoint, the findings suggest that low‑bias or unbiased nanopore measurements could be exploited to infer secondary‑structure stability: longer translocation times and stronger temperature dependence would signal more stable hairpins or internal loops. Moreover, the logarithmic barrier model provides a quantitative framework for interpreting the variance and skewness of translocation‑time histograms, which are often overlooked in simple diffusion models.

In conclusion, the authors demonstrate that RNA translocation through a nanopore is governed by diffusion in a one‑dimensional energy landscape with a logarithmic barrier, leading to a universal power‑law scaling τ ∝ N^α with α > 2. This work extends the theoretical foundation of nanopore biophysics beyond unstructured polymers, highlights the critical role of secondary structure in single‑molecule transport, and opens avenues for using translocation dynamics as a probe of RNA folding energetics. Future directions include incorporating finite pore thickness, explicit electric fields, and more complex biomolecular architectures such as ribonucleoprotein complexes.