Detecting brief changes in time-series data remains a major challenge in fields where short-lived states carry meaning. In single-molecule localisation microscopy, this problem is particularly acute as fluorescent molecules used to tag protein oligomers display heterogenous photophysical behaviour that can complicate photobleach step analysis; a key step in resolving nanoscale protein organisation. Existing methods often require extensive filtering or prior calibration, and can fail to accurately account for blinking or reversible dark states that may contaminate downstream analysis. In this paper, an extension to RJMCMC is proposed for change point detection with heterogeneous temporal dynamics. This approach is applied to the problem of estimating per-frame active fluorophore counts from one-dimensional integrated intensity traces derived from Fluorescence Localisation Imaging with Photobleaching (FLImP), where compound change point pair moves are introduced to better account for short-lived events known as blinking and dark states. The approach is validated using simulated and experimental data, demonstrating improved accuracy and robustness when compared with current photobleach step analysis methods and with the existing analysis approach for FLImP data. This Compound RJMCMC (CRJMCMC) algorithm performs reliably across a wide range of fluorophore counts and signal-to-noise conditions, with signal-to-noise ratio (SNR) down to 0.001 and counts as high as seventeen fluorophores, while also effectively estimating low counts observed when studying EGFR oligomerisation. Beyond single molecule imaging, this work has applications for a variety of time series change point detection problems with heterogeneous state persistence. For example, electrocorticography brain-state segmentation, fault detection in industrial process monitoring and realised volatility in financial time series.
Identifying change points in time-series data is fundamental to understanding dynamic systems in a variety of applications from financial markets to healthcare monitoring [1]. Reversible jump Markov chain Monte Carlo (RJMCMC) offers a principled solution to such problems, enabling trans-dimensional estimation of the number and location of change points, and so is capable of modelling discrete events embedded in continuous, noisy data. However, when temporal dynamics are heterogeneous within the model, short-lived level changes are often missed or mistakenly considered as noise or outliers, despite the potential to mark biologically, physically, or operationally meaningful events [2][3][4][5][6].
Standard RJMCMC approaches often fail to account for short-lived events accurately for two main reasons. 1) Short-lived events are weakly penalised in reversible jump Markov chain Monte Carlo, as their brief duration contributes little to the overall likelihood, highlighting a broader challenge in capturing heterogeneous temporal dynamics. 2) Modelling these short-lived events typically requires the addition of two closely spaced change points; one to enter and one to exit the state. The first proposed change point often introduces a temporary mismatch with the observed data, reducing the model likelihood and, in the case of RJMCMC, resulting in a low acceptance probability. This makes addition of both change points unlikely unless an exhaustive search of possible configurations is performed. As a result, chains can exhibit poor convergence, with short-lived states inconsistently identified [2]. In cases where multiple change points can be added simultaneously [7], the aforementioned problems persist as acceptance is unlikely unless both change points are simultaneously correctly placed, and so samplers remain inefficient in the presence of short-lived events, as provisions are not made to explore closely-spaced changepoints.
One field where this issue is particularly pronounced is photobleach step analysis in single molecule localisation microscopy. In photobleach step analysis, fluorescently tagged protein subunits are imaged over time. The small separations between these sub-units (as low as 5nm to 50nm) lie well below the diffraction limit of conventional light microscopy (∼ 250nm) and so the fluorophore point spread functions overlap appearing as a single point spread function, from which the individual fluorophores cannot be resolved. Discrete changes in one-dimensional fluorescence intensity traces are therefore used to count the number of active fluorophores per-frame. In this field, data naturally takes the form of step functions, corresponding to discrete fluorophore counts. These counts can be used to estimate the oligomeric distribution of protein populations in a sample, or, when considering the frame-wise active fluorophore count, to determine nanoscale protein separations and study protein oligomerisation. However, the reliability of each application depends directly on the accuracy of the initial fluorophore counting [8][9][10][11][12][13][14][15][16][17].
In photobleach step analysis, fluorophores most often reside in an active, fluorescent state and eventually move into an irreversible, inactive, photobleached state. However, fluorophores can also display complex photophysical behaviour, including short-lived ‘off’ states known as blink states and longer-lived dark states [18], as visualised in Fig. 1. These temporary changes in fluorescence can lead to incorrect estimation of per-frame active fluorophore counts, or may force partial or complete exclusion of traces, particularly when such deviations are misclassified as noise or considered analytically intractable. This issue is especially pronounced during separation localisation, where accurate per-frame fluorophore counts are critical, and the misattribution of short lived states can compromise subsequent positional measurements [9,[19][20][21].
Several approaches to fluorophore counting in photobleach step analysis have been proposed. However, many existing methods assume intensity is monotonically decaying and so cannot accommodate blink or dark states [19,22]. Those that do support reversibility often suffer from high computational demands [23] or limited scalability [24]. Certain approaches address the unknown number of fluorophores, and consequently the unknown number of changepoints, by overestimating this number and representing the presence or absence of each with a binary indicator [23,25,26]. This formulation allows for changepoints to be identified but introduces inefficiencies in the estimation process. Correctly identifying related intensity parameters, namely single-fluorophore and background intensity, is also essential, yet existing methods often require prior user knowledge [23], fix intensity parameters at the beginning of analysis [24], or rely on calibration from labelled traces [22], which are often not available in this
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