Allowing the two ends of a modeled polymer to move changes a basic prediction about translocation: the probability of ending on one side is nonlinear in the starting gap, according to an exact analysis. With fixed boundaries, the same probability is linear.
That result comes from a mathematical model that represents translocation as three ordered stochastic walkers: two diffusing outer boundaries and a pore walker between them. The calculation follows which side is reached first, along with the time taken and the separation that remains on the other side.
The shift comes from letting the boundaries move
Rather than treating the polymer’s effective length as fixed, the setup lets its boundaries fluctuate. Three diffusion constants, D1, D2 and D3, describe the mobility built into the Brownian walkers, while the fixed-boundary case provides a reference for comparison.
To calculate the chance of a left-versus-right outcome, the authors use a backward Fokker–Planck approach. In practical terms, this is an analytical method for working from the possible end states back to the model’s starting configuration. The resulting moving-boundary probability is exact within the stated Brownian-walker model.
The contrast with fixed boundaries is the main conceptual change. In the fixed case, the starting gap and splitting probability follow a straight-line relationship. Once the boundaries diffuse, that relationship bends, so the starting geometry no longer leads to a simple linear rule for the outcome.
Long waits are part of the model’s picture
The paper goes beyond the chance of a left or right outcome. It derives the joint distribution of the time when the left side is hit and the separation that remains to the right. At large times and large separations, the Brownian distribution has an explicit asymptotic scaling form.
The separate completion-time distribution decays as a power law at long times. The distribution of the final chain length—or, in the walker picture, the final right-boundary separation—also has a power-law decay at large values.
These tails describe how the model assigns probability to unusually late completion and unusually large final sizes. They do not share one universal exponent: the reported tail exponents vary continuously with the three diffusion constants, D1, D2 and D3.
Within the Brownian model, changing the relative mobility of the two boundaries and the pore therefore changes the predicted shape of the long-event distributions. The statistics of translocation time and final length are tied to the motion specified in the model, rather than to a single fixed tail law.
The equations were checked with simulated walks
The analytical Brownian predictions were compared with simulated walker trajectories. The simulations generated Brownian steps from independent, identically distributed Gaussian random numbers with zero mean and unit variance.
The authors report excellent agreement between the analytical predictions and the numerical simulations, including the predicted curves and asymptotic behavior for the settings they tested.
For numerical time resolution, the simulations mostly used a step of 0.00001 time units and also tested 0.0001. The text says that this change did not significantly alter the results.
That agreement is an internal check on the mathematical model and its numerical implementation, not a measurement from a physical translocation system. The study uses simulated stochastic trajectories rather than a physical or biological sample.
A fractional version changes the numbers
The study also tests a different kind of motion for the pore: fractional Brownian motion. Its increments are correlated, and the authors generated them numerically with the Davies–Harte fast-Fourier-transform method.
The fractional version retains the broad qualitative features of the ordinary Brownian case, but its splitting probabilities and asymptotic exponents vary with the Hurst exponent used to define the process.
This extension is numerical rather than an exact analytical solution. Its exponents were fitted over simulated ranges, so the preprint does not establish that the same power-law behavior holds outside those ranges.
The distinction is therefore quantitative as well as technical: the Brownian treatment supplies exact results for the stated walker model, while the fractional extension shows how changing the process’s temporal correlations can shift the predicted probabilities and exponents.
A model result, not a measurement
The authors present the work as a unified framework for first-passage problems with fluctuating boundaries. Its conclusions apply to the three-walker representation itself; they do not show that the model quantitatively predicts translocation in a particular real polymer or nanopore.
The analysis points to several next steps. The fractional-Brownian case would need an analytical treatment, while models that include interactions, external driving or additional polymer degrees of freedom could test how robust the reported patterns are under more realistic dynamics.
The paper’s status
The document is an arXiv version 1 preprint dated 20 Aug 2026. The generated data are stated to be publicly available in the DARE repository of the University of Oldenburg.
The acknowledgments report support from ANR, the Alexander von Humboldt Foundation, the DFG and the Ministry of Science and Culture of Lower Saxony.
Paper data and sources
Original title: Splitting probabilities for Brownian motion with diffusing boundaries: Application to polymer translocation
Authors: Alexander K. Hartmann, Satya N. Majumdar, Alberto Rosso
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text