Denoise Stepwise Signals by Diffusion Model Based Approach

Stepwise signals are ubiquitous in single-molecule detections, where abrupt changes in signal levels typically correspond to molecular conformational changes or state transitions. However, these features are inevitably obscured by noise, leading to uncertainty in estimating both signal levels and transition points. Traditional frequency-domain filtering is ineffective for denoising stepwise signals, as edge-related high-frequency components strongly overlap with noise. Although Hidden Markov Model-based approaches are widely used, they rely on stationarity assumptions and are not specifically designed for signal denoising. Here, we propose a diffusion model-based algorithm for stepwise signal denoising, named the Stepwise Signal Diffusion Model (SSDM). During training, SSDM learns the statistical structure of stepwise signals via a forward diffusion process that progressively adds noise. In the following reverse process, the model reconstructs clean signals from noisy observations, integrating a multi-scale convolutional network with an attention mechanism. Training data are generated by simulating stepwise signals through a Markov process with additive Gaussian noise. Across a broad range of signal-to-noise ratios, SSDM consistently outperforms traditional methods in both signal level reconstruction and transition point detection. Its effectiveness is further demonstrated on experimental data from single-molecule Forster Resonance Energy Transfer and nanopore DNA translocation measurements. Overall, SSDM provides a general and robust framework for recovering stepwise signals in various single-molecule detections and other physical systems exhibiting discrete state transitions.

💡 Research Summary

The manuscript introduces a novel denoising framework for stepwise signals commonly encountered in single‑molecule experiments, called the Stepwise Signal Diffusion Model (SSDM). Traditional approaches—low‑pass filtering and Hidden Markov Model (HMM) based methods—struggle with stepwise data because high‑frequency components of abrupt transitions overlap with noise, and HMMs require a predefined number of states and assume stationarity. To overcome these limitations, the authors adopt the Denoising Diffusion Probabilistic Model (DDPM) paradigm, which treats denoising as a reverse stochastic process that progressively removes Gaussian noise injected during a forward diffusion phase.

The core of SSDM is a one‑dimensional U‑Net architecture enhanced with residual blocks and attention modules. The encoder‑decoder structure captures multi‑scale features, while attention layers emphasize transition edges and long‑range dependencies. At each diffusion timestep t (out of T = 1000), the network receives a noisy signal x_t and a sinusoidal time embedding, and predicts the noise component ε̂(x_t, t). The predicted noise is used to compute the mean of the reverse Gaussian transition, following a cosine‑scheduled β_t noise schedule. The variance is fixed to β_t I, simplifying sampling.

Training data are synthetically generated stepwise traces using a continuous‑time Markov chain. The authors simulate balanced 2‑, 3‑, and 4‑state signals, each 1000 points long, under five signal‑to‑noise ratios (SNR = 0.25, 0.5, 1, 3, 5). For each combination of transition matrix, state count, and SNR, 100 independent trajectories are created, yielding 10 800 noise‑free signals. Gaussian noise is added to obtain the noisy observations. An independent test set of 3 600 traces with faster transitions is used for evaluation. The authors also train separate models on single‑state‑count datasets to assess the impact of prior knowledge of the number of states.

A custom loss function combines a Smooth L1 base term with two dynamic weights: an amplitude weight that penalizes large residuals and an edge weight that focuses on first‑ and second‑order differences of the underlying clean signal, smoothed by a 1 × 3 kernel to create a localized penalty band around transitions. Importance sampling over diffusion steps (p(t) ∝ exp(−3t/T)) concentrates training on early timesteps where the signal is less corrupted, improving efficiency.

Performance is quantified by mean squared error (MSE) for amplitude recovery and F1‑score for transition‑point detection (a tolerance of ±2 samples). A composite score merges both metrics. On the synthetic test set, SSDM achieves MSE = 0.0041, F1 = 0.96, and a composite score of 8.31, substantially outperforming low‑pass filters and HMMs across all SNRs. Notably, transition detection remains above 95 % accuracy even at SNR = 1, where conventional methods drop below 70 %.

The authors validate SSDM on two experimental datasets. First, sm‑FRET optical traces (100 Hz sampling) containing 19 trajectories (≈226 k points) are denoised, revealing clear step levels and precise transition times despite substantial photon‑shot noise. Second, λ‑DNA nanopore current recordings (10 kHz sampling) exhibit rapid, low‑amplitude translocations; SSDM restores the underlying stepwise conductance changes, facilitating accurate dwell‑time and amplitude analysis. In both cases, the denoised signals align closely with manually curated ground truth, demonstrating practical utility.

Limitations are acknowledged. SSDM is presently limited to one‑dimensional time series; extending to multi‑dimensional modalities (e.g., imaging‑based single‑molecule data) would require architectural adaptations. The training assumes additive Gaussian noise; real‑world non‑Gaussian artifacts (e.g., flicker, spikes) may demand more sophisticated noise models or robust diffusion schedules. Finally, the need for predefined state counts for thresholding could be mitigated by integrating Bayesian model selection or an adaptive clustering layer.

Future directions suggested include: (i) incorporating non‑Gaussian noise distributions into the diffusion process; (ii) hybridizing the U‑Net‑attention backbone with transformer‑style self‑attention for even longer temporal contexts; (iii) embedding a Bayesian inference module to infer the number of hidden states automatically; and (iv) optimizing the reverse sampling for real‑time streaming applications. Such extensions would broaden SSDM’s applicability across diverse single‑molecule platforms and potentially other domains featuring discrete state transitions.

In summary, the paper presents a well‑designed diffusion‑model‑based denoising pipeline that outperforms established methods in both synthetic benchmarks and real experimental data, offering a promising new tool for the quantitative analysis of stepwise signals in single‑molecule biophysics.