Hidden Markov modeling of single-particle diffusion with stochastic tethering.

The statistics of the diffusive motion of particles often serve as an experimental proxy for their interaction with the environment. However, inferring the physical properties from the observed trajectories is challenging. Inspired by a recent experiment, here we analyze the problem of particles undergoing two-dimensional Brownian motion with transient tethering to the surface. We model the problem as a hidden Markov model where the physical position is observed and the tethering state is hidden. We develop an alternating maximization algorithm to infer the hidden state of the particle and estimate the physical parameters of the system. The crux of our method is a saddle-point-like approximation, which involves finding the most likely sequence of hidden states and estimating the physical parameters from it. Extensive numerical tests demonstrate that our algorithm reliably finds the model parameters and is insensitive to the initial guess. We discuss the different regimes of physical parameters and the algorithm's performance in these regimes. We also provide a free software implementation of our algorithm.

Machine-Learning-Based Single-Molecule Quantification of Circulating MicroRNA Mixtures.

MicroRNAs (miRs) are small noncoding RNAs that regulate gene expression and are emerging as powerful indicators of diseases. MiRs are secreted in blood plasma and thus may report on systemic aberrations at an early stage via liquid biopsy analysis. We present a method for multiplexed single-molecule detection and quantification of a selected panel of miRs. The proposed assay does not depend on sequencing, requires less than 1 mL of blood, and provides fast results by direct analysis of native, unamplified miRs. This is enabled by a novel combination of compact spectral imaging and a machine learning-based detection scheme that allows simultaneous multiplexed classification of multiple miR targets per sample. The proposed end-to-end pipeline is extremely time efficient and cost-effective. We benchmark our method with synthetic mixtures of three target miRs, showcasing the ability to quantify and distinguish subtle ratio changes between miR targets.

Percolation perspective on sites not visited by a random walk in two dimensions.

We consider the percolation problem of sites on an L×L square lattice with periodic boundary conditions which were unvisited by a random walk of N=uL^{2} steps, i.e., are vacant. Most of the results are obtained from numerical simulations. Unlike its higher-dimensional counterparts, this problem has no sharp percolation threshold and the spanning (percolation) probability is a smooth function monotonically decreasing with u. The clusters of vacant sites are not fractal but have fractal boundaries of dimension 4/3. The lattice size L is the only large length scale in this problem. The typical mass (number of sites s) in the largest cluster is proportional to L^{2}, and the mean mass of the remaining (smaller) clusters is also proportional to L^{2}. The normalized (per site) density n_{s} of clusters of size (mass) s is proportional to s^{-τ}, while the volume fraction P_{k} occupied by the kth largest cluster scales as k^{-q}. We put forward a heuristic argument that τ=2 and q=1. However, the numerically measured values are τ≈1.83 and q≈1.20. We suggest that these are effective exponents that drift towards their asymptotic values with increasing L as slowly as 1/lnL approaches zero.