We address the problem of inferring the number of independently blinking fluorescent light emitters, when only their combined intensity contributions can be observed at each timepoint. This problem occurs regularly in light microscopy of objects that are smaller than the diffraction limit, where one wishes to count the number of fluorescently labelled subunits. Our proposed solution directly models the photo-physics of the system, as well as the blinking kinetics of the fluorescent emitters as a fully differentiable hidden Markov model. Given a trace of intensity over time, our model jointly estimates the parameters of the intensity distribution per emitter, their blinking rates, as well as a posterior distribution of the total number of fluorescent emitters. We show that our model is consistently more accurate and increases the range of countable subunits by a factor of two compared to current state-of-the-art methods, which count based on autocorrelation and blinking frequency, Further-more, we demonstrate that our model can be used to investigate the effect of blinking kinetics on counting ability, and therefore can inform experimental conditions that will maximize counting accuracy.

The inherent limitations of fluorescence microscopy, notably the restricted number of color channels, have long constrained comprehensive spatial analysis in biological specimens. Here, we introduce cycleHCR technology that leverages multicycle DNA barcoding and Hybridization Chain Reaction (HCR) to surpass the conventional color barrier. cycleHCR facilitates high-specificity, single-shot imaging per target for RNA and protein species within thick specimens, mitigating the molecular crowding issues encountered with other imaging-based spatial omics techniques. We demonstrate whole-mount transcriptomics imaging of 254 genes within an E6.5\~7.0 mouse embryo, achieving precise three-dimensional gene expression and cell fate mapping across a specimen depth of \~ 310 µm. Utilizing expansion microscopy alongside protein cycleHCR, we unveil the complex network of 10 subcellular structures in primary mouse embryonic fibroblasts. Furthermore, in mouse hippocampal slice, we image 8 protein targets and profile the transcriptome of 120 genes, uncovering complex gene expression gradients and cell-type specific nuclear structural variances. cycleHCR provides a unifying framework for multiplex RNA and protein imaging, offering a quantitative solution for elucidating spatial regulations in deep tissue contexts for research and potentially diagnostic applications.