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  1. Author Response:

    Evaluation Summary:

    This paper will be of interest to researchers who perform single-molecule fluorescence imaging experiments as well as those who want to include machine learning in their data analyses. The authors have developed a machine learning algorithm that addresses some of the data analysis challenges in the field of single-molecule fluorescence imaging. The methods are rigorously benchmarked using simulated data and tested using real data. There are some concerns whether Tapqir is general enough for use by the broader community of single-molecule fluorescence researchers.

    We thank the reviewers for their thorough review of the manuscript. In response to the reviewer comments, we posted to bioRxiv a revised manuscript with new data and edits to text. Concerns about generality are addressed in the revised manuscript and in the responses to specific reviewer comments below.

    Reviewer #1 (Public Review):

    "Bayesian machine learning analysis of single-molecule fluorescence colocalization images" by Ordabayev, et al. reports the development, benchmarking, and testing of a Bayesian machine learning-based method, which the authors name Tapqir, for analyzing single-molecule fluorescence colocalization data. Unlike currently available, more conventional analysis methods, Tapqir attempts to holistically model the microscopy images that are recorded during a colocalization experiment. Tapir uses a physics-based, global model with parameters describing all of the features of the experiment that are expected to contribute to the recorded microscopy images, including shot noise of the spots and background, camera noise, size and shape of the spots, and specific- and non-specific binders. Based on benchmarking on simulated data with widely varying properties (e.g., signal-to-noise; amounts, rates, and locations of specific and non-specific binders; etc.), Tapqir generally does as well and, in some cases, better than currently existing methods. The authors also test Tapqir on real microscopy images with similarly varying properties from studies that have been previously published by their research group and demonstrate that their Tapqir-based analysis is able to faithfully reproduce the previously published results, which were obtained using the more conventional analysis methods available at the time the data were originally published. This is a well-designed and executed study, Tapqir represents a conceptual and practical advance in the analysis of single-molecule fluorescence colocalization experiments, and its performance has been comprehensively and rigorously benchmarked on simulated data and tested on real data. The conclusions of this study are well supported by the data, but some of the limitations of the method need to be clarified and discussed in more depth, as outlined below.

    1. Given that the AOI is centered at the target molecule and there is a strong prior for the binder also being located at the center of the AOI, the performance of Tapqir is dependent on several variables of the microscopy/optical system (e.g., the microscope point-spread function, magnification, accurate alignment of target and binder imaging channels, accurate drift correction, etc.). Although this caveat is mentioned and some of these factors are listed in the main text of the manuscript, the authors could have expanded this discussion in order to clarify the extent to which the performance of Tapqir depends on these factors.

    We added relevant new data to the revised manuscript in Table 5. The question about alignment accuracy is now discussed in the Materials and Methods:

    “Tests on data simulated with increasing proximity parameter values σxy (true) (i.e., with decreasing precision of spatial mapping between the binder and target image channels) confirm that the cosmos model accurately learns σxy (fit) from the data (Figure3–Figure Supplement 3D; Table 5). This was the case even if we substituted a less-informative σxy prior (Uniform vs. Exponential; Table 5).

    The CoSMoS technique is premised on colocalization of the binder spots with the known location of the target molecule. Consequently, for any CoSMoS analysis method, classification accuracy will in general decline when the images in the target and binder channels are less accurately mapped. However, for the Tapqir cosmos model, low mapping precision has little effect on classification accuracy at typical non-specific binding densities (λ = 0.15; see MCC values in Table 5).”

    The more general point about priors is now addressed in the Materials and Methods as follows:

    “All simulated and experimental data sets in this work were analyzed using the prior distributions and hyperparameter values given above, which are compatible with a broad range of experimental conditions (Table 1). Many of the priors are uninformative and we anticipate that these will work well with images taken on variety of microscope hardware. However, it is possible that highly atypical microscope designs (e.g., those with effective magnifications that are sub-optimal for CoSMoS) might require adjustment of some fixed hyperparameters and distributions (those in Eqs. 6a, 6b, 11, 12, 13, 15, and 16). For example, if the microscope point spread function is more than 2 pixels wide, it may be necessary to increase the range of the w prior in Eq. 13. The Tapqir documentation (https://tapqir.readthedocs.io/en/stable/) gives instructions for changing the hyperparameters.”

    1. The Tapqir model has many parameters, each with its own prior. The majority of these priors are designed to be uninformative and/or weak and the only very strong prior is the probability that a specific binder is located at or very near the center of the AOI. The authors could have tested and commented on how the strength of the prior on the location of a specific binder affects the performance of Tapqir.

    The revised manuscript includes new data on and expanded discussion of this point. In our model, the position of a target-specific spot relative to the target position has a prior distribution illustrated as the green curve in Figure 2-Figure supplement 2. Importantly, the peak in this distribution does not have an a priori set width. Instead, the width of the peak is a model hyperparameter, σxy, that is learned from the image data set without user intervention. To make sure that this point is understood, we expanded and clarified the relevant Methods section and modified the legend of Figure 2-Figure supplement 2.

    To address the reviewers’ specific question, we constructed simulated data sets with different mapping precision values and analyzed them; the results are presented in the (new) Table 5 and discussed:

    “The CoSMoS technique is premised on colocalization of the binder spots with the known location of the target molecule. Consequently, for any analysis method, classification accuracy declines when the images in the target and binder channels are less accurately mapped. For the Tapqir cosmos model, low mapping precision has little effect on classification accuracy at typical non-specific binding densities (λ = 0.15; see MCC values in Table 5).”

    1. Given the priors and variational parameters they report, the authors show that Tapqir performs robustly and seems to require no experiment-to-experiment optimization. This is expected to be the case for the simulated data, since they were simulated using the same model that Tapqir uses to perform the analysis. With regard to the real data, however, it is quite likely that this is due to the fact that the analyzed data all come from the same laboratory and, therefore, likely the same microscope(s). It would have therefore been very useful if the authors would have listed and discussed which microscope settings, experimental conditions, and/or other considerations, beyond those described in point 1 above, would result in a need for re-optimization of the priors and/or variational parameters.

    As noted above, we now address this point in the Materials and Methods as follows:

    “All simulated and experimental data sets in this work were analyzed using the prior distributions and hyperparameter values given above, which are compatible with a broad range of experimental conditions (Table 1). Many of the priors are uninformative and we anticipate that these will work well with images taken on variety of microscope hardware. However, it is possible that highly atypical microscope designs (e.g., those with effective magnifications that are sub-optimal for CoSMoS) might require adjustment of some fixed hyperparameters and distributions (those in Eqs. 6a, 6b, 11, 12, 13, 15, and 16). For example, if the microscope point spread function is more than 2 pixels wide, it may be necessary to increase the range of the w prior in Eq. 13. The Tapqir documentation (https://tapqir.readthedocs.io/en/stable/) gives instructions for changing the hyperparameters.”

    1. Based on analysis of the simulated data shown in Figure 5, where the ground truth is known, the use of Tapqir to infer kinetics is less accurate that the use of Tapqir to infer equilibrium binding constants. The authors do a great job of discussing possible reasons for this. In the case of the real data analyzed in Figure 6 and in Figure 6 - Figure Supplements 1 and 2, the kinetic results obtained using Tapqir have different means and generally larger error bars than those obtained using Spot-Picker. To more comprehensively assess the performance of Tapqir versus Spot-Picker, the authors could have used the association and dissociation rates to calculate the corresponding equilibrium binding constants and then compared these kinetically calculated equilibrium binding constants to the population-calculated equilibrium binding constants that the authors calculate and report in the bottom plot in Panel D of Figure 6 and Figure 6 - Figure Supplements 1 and 2. This would provide some information on the accuracy of the kinetics in that the closer the kinetically and population-calculated equilibrium binding constants are to each other, the more accurately the kinetics have been estimated. Performing this type of analysis for the kinetics obtained using Tapqir and Spot-Picker would have allowed a more comprehensive comparison of the two methods.

    This comment seems to reflect a misunderstanding. Fig. 6 and its figure supplements do not report any dissociation kinetics or binding equilibrium constants. Instead, they report ka (pseudo first-order target-specific association rate constant), kns (pseudo first-order target non-specific association rate constant), and Af (the active faction, i.e., the fraction of target molecules capable of association with binder). ka and Af values from the two methods agree within experimental uncertainty for all four data sets analyzed. kns values differ, but as we point out:

    “We noted some differences between the two methods in the non-specific association rate constants kns. Differences are expected because these parameters are defined differently in the different non-specific binding models used in Tapqir and spot-picker (see Materials and Methods).”

    (There is additional discussion of this point in Materials and Methods). The reviewer is correct that the estimated uncertainties (i.e., error bars in panels D) in ka and Af are generally larger for Tapqir than for spot-picker. This is expected, for the reasons that we explain:

    “In general, previous approaches in essence assume that spot classifications are correct, and thus the uncertainties in the derived molecular properties (e.g., equilibrium constants) are systematically underestimated because the errors in spot classification, which can be large, are not accounted for. By performing a probabilistic spot classification, Tapqir enables reliable inference of molecular properties, such as thermodynamic and kinetic parameters, and allows statistically well-justified estimation of parameter uncertainties. This more inclusive error estimation likely accounts for the generally larger kinetic parameter error bars obtained from Tapqir compared to those from the existing spot-picker analysis method (Figure 6, Figure 6–Figure Supplement 1, Figure 6–Figure Supplement 2, and Figure 6–Figure Supplement 3). ”

    Reviewer #2 (Public Review):

    The work by Ordabayev et al. details a Bayesian inference-based data analysis method for colocalization single molecule spectroscopy (CoSMoS) experiments used to investigate biochemical and biophysical mechanisms. By using this probabilistic framework, their method is able to quantify the colocalization probabilities for individual molecules while accounting for the uncertainty in individual binding events, and accounting for camera and optical noise and even non-specific binding. The software implementation of this method, called Tapqir, uses a Python-based probabilistic programming language (PPL) called pyro to automate and speed-up the optimization of a variational Bayes approximation to the posterior probability distribution. Overall, Tapqir is a powerful new way to analyze CoSMoS data.

    Tapqir works by analyzing small regions (14x14 pixels) of fluorescence microscopy images surrounding previously identified areas of interest (AOI). The collection of images of these AOIs through time are then analyzed collectively using a probabilistic model that accounts for each time frame of each AOI and is able to determine whether up to K "binders" (K=2 here) are present and which of them is specifically bound. This approach of directly modeling the contents of the image data is relatively novel, and few other examples exist. The details of the probabilistic model used incorporate an impressive amount of physical insight (e.g., camera gain) without overparameterization.

    We thank the reviewer for these positive comments.

    The gamma-distributed noise model used in Tapqir captures quite a lot of physics and, given the analyses in Figs. 3-6, clearly works, but might be limited to certain types of cameras used in the fluorescence microscopy (e.g., EMCCDs). For instance, sCMOS cameras have pixel-dependent amplification and noise profiles, rather than a single gain parameter, and are sometimes approximately modeled as normal distributions with both mean and variance having an intensity-dependent and independent contribution that is different for each pixel on the camera. It is unclear how Tapqir performs on different cameras.

    In the revised manuscript, we expanded the discussion of the Image likelihood component of our model to emphasize that 1) all data sets we analyze are experimental or simulated EMCCD images, 2) sCMOS images have the different noise characteristics alluded to by the reviewer, and 3) optimal sCMOS image analysis might require a modified model, possibly including the ability to use per-pixel calibration data as a prior as was done in super-resolution work (now cited) that uses sCMOS data.

    sCMOS cameras have in recent years become very popular for some kinds of single-molecule imaging (e.g., PALM/STORM or live-cell single-particle tracking). However, for the low-background/low-signal in vitro single-molecule TIRF that is our target application for the approach described in the manuscript, EMCCD is still preferable over sCMOS for many, but not all, imaging conditions (see https://andor.oxinst.com/learning/view/article/what-is-the-best-detector-for-single-molecule-studies). Thus, we think there will be plenty of interest in the approach we describe in the manuscript even if (which is not certain) the program functions better with EMCCD than with sCMOS images.

    Going forward to develop and test an sCMOS-targeted version of the model, as we have done for EMCCD, will require revised model and code, but will also necessitate accurately simulating sCMOS CoSMoS images, obtaining experimental sCMOS CoSMoS images reflecting a broad range of realistic experimental conditions, and using the simulated and experimental images to test the new model. These may well be useful things to do in the future but would be a considerable step beyond the scope of the present manuscript.

    The variational Bayes solution used by Tapqir provides many computational benefits, such as numerical tractability using pyro and speed. It is possible that the exact posterior, e.g., as obtained using a Markov chain Monte Carlo method, would be insignificantly different with the amount of data typical for CoSMoS experiments; however, this difference is not explored in the current work.

    We agree. However, since we have not done any analyses using MCMC, there is nothing in particular that we can say about it in the context of CoSMoS data analysis. Implementation of an MCMC approach using our model will be easier in the future because the Pyro developers are currently working to optimize the implementations of MCMC methods in their software.

    The intrinsic use of prior probability distributions in any Bayesian inference algorithm is extremely powerful, and in Tapqir offers the opportunity to "chain together" subsequent analyses by using the marginalized posteriors from one experiment as the basis for the priors for subsequent experiments (e.g., in \sigma^{xy}) for extremely high accuracy inference. While the manuscript discusses setting and leveraging the power of priors, it does not explore the power of such "chaining" and the positive effects upon accuracy.

    Chaining is beneficial in principle. However, in practice it will help significantly only if the uncertainty in the posterior parameter values from the non-chained analysis is larger than the experiment-to-experiment variability in the “true” parameter values. For σxy we obtain very narrow credence intervals without chaining (Table 1). In our judgement, these are unlikely to be made more accurate by using prior information from another experiment where such factors as microscope focus adjustment may be slightly different.

    A significant number of CoSMoS experiments use multiple, distinct color fluorophores to probe the colocalization of different species to the target. The current work focuses only upon analyzing data with a single color-channel. Extensions to multiple independent wavelengths are computationally trivial, given the automated variational inference ability of PPLs such as pyro, and would increase the impact of the work in the field.

    Our current approach can be used to analyze multi-channel data simply by analyzing each channel independently. However, we agree that there would be advantages to joint analysis of multiple wavelength channels (especially if there is crosstalk between channels) and that implementing multi-channel analysis is a logical extension of our study. It is straightforward (though not trivial, in our experience) to implement such multi-wavelength models. However, testing the functioning of candidate models and validating them using simulation and experimental data would require extensive work that in our view goes beyond what is reasonable to include in the present manuscript.

    Tapqir analysis provides time series of the probability of a specific binding event, p(specific), for each target analyzed (c.f., Fig. 5B), and kinetic parameters are extracted from these time series using secondary analyses that are distinct from Tapqir itself.

    The method reported here is well designed, sound, and its utility is well supported by the analyses of simulated and experimental data sets reported here. Tapqir is a cutting-edge image analysis approach, and its proper treatment of the uncertainty inherent to CoSMoS experiments will certainly make an impact upon the analysis of CoSMoS data. However, many of the (necessary) assumptions about the data (e.g., fluorescence microscopy) and desired information (e.g., off-target vs on-target binding) are quite specific to CoSMoS experiments and therefore limit the direct applicability of Tapqir for the analysis of other single-molecule microscopy techniques. With that in mind, the direct Bayesian inference-based analysis of image data, as opposed to integrated time series, as demonstrated here is very powerful, and may encourage and inspire related methods to be developed.

    Our approach is a powerful way to analyze CoSMoS data in part because it is specific to CoSMoS – it is premised on a physics-based model that incorporates known features of CoSMoS experiments. We agree that the general approach could be adapted to other image analysis applications.

    Reviewer #3 (Public Review):

    In this manuscript, the authors seek to improve the reproducibility and eliminate sources of bias in the analysis of single molecule colocalization fluorescence data. These types of data (i.e., CoSMoS data) have been obtained from a number of diverse biological systems and represent unique challenges for data analysis in comparison with smFRET. A key source of bias is what constitutes a binding event and if those events are colocalized or not with a surface-tethered molecule of interest. To solve these issues, the authors propose a Bayesian-based method in which each image is analyzed individually and locally around areas of interest (AOIs) identified from the surface tethered molecules. A strength of the research is that the approach eliminates many sources of bias (i.e., thresholding) in analysis, models realistic image features (noise), can be automated and carried out by novice users "hands-free", and returns a probability score for each event. The performance of the method is superb under a number of conditions and with varying levels of signal-to-noise. The analysis on a GPU is fairly quick-overnight-in comparison with by-hand analysis of the traces which can take days or longer. Tapqir has the potential to be the go-to software package for analysis of single molecule colocalization data.

    The weaknesses of this work involve concerns about the approach and its usefulness to the single-molecule community at large as wells as a lack of information about how users implement and use the Tapqir software. For the first item, there are a number of common scenarios encountered in colocalization analysis that may exclude use of Tapqir including use of CMOS rather than EM-CCD cameras, significant numbers of tethered molecules on the surface that are dark/non-fluorescent, a high density/overlapping of AOIs, and cases where event intensity information is critical (i.e., FRET detection or sequential binding and simultaneous occupancy of multiple fluorescent molecules at the same AOI). In its current form, the use of Tapqir may be limited to only certain scenarios with data acquired by certain types of instruments.

    In the following paragraphs, we address 1) concerns about application to CMOS, 2) dark target molecules, 3) overlapping AOIs, and 4) application to methods (e.g., smFRET) that require extraction of both colocalization and intensity data.

    1. Application to CMOS images.

    In the revised manuscript, we expanded the discussion of the Image likelihood component of our model to emphasize that 1) all data sets we analyze are experimental or simulated EMCCD images, 2) sCMOS images have the different noise characteristics alluded to by the reviewer, and 3) optimal sCMOS image analysis might require a modified model, possibly including the ability to use per-pixel calibration data as a prior as was done in super-resolution work (now cited) that uses sCMOS data.

    sCMOS cameras have in recent years become very popular for some kinds of single-molecule imaging (e.g., PALM/STORM or live-cell single-particle tracking). However, for the low-background/low-signal in vitro single-molecule TIRF that is our target application for the approach described in the manuscript, EMCCD is still preferable over sCMOS for many, but not all, imaging conditions (see https://andor.oxinst.com/learning/view/article/what-is-the-best-detector-for-single-molecule-studies). Thus, we think there will be plenty of interest in the approach we describe in the manuscript even if (which is not certain) the program functions better with EMCCD than with sCMOS images.

    Going forward to develop and test an sCMOS-targeted version of the model, as we have done for EMCCD, will require revised model and code, but will also necessitate accurately simulating sCMOS CoSMoS images, obtaining experimental sCMOS CoSMoS images reflecting a broad range of realistic experimental conditions, and using the simulated and experimental images to test the new model. These may well be useful things to do in the future but would be a considerable step beyond the scope of the present manuscript.

    1. Dark target molecules.

    In their detailed comments, the reviewers suggested a “no target molecules in sample” (NTIS) control instead of the “no fluorescent target molecules in control AOIs” (NFTICA) design that we illustrate in Fig. 1. Both types can be used as a Tapqir control dataset without any modification of the program or model. We have edited the Fig. 1 caption to explain that either type is acceptable. The reviewers are correct that, all else being equal, NTIS may be better if the target molecules are incompletely labeled. However, in practice experimenters usually know the fraction of molecules that are labeled and reduce the fluorescent target molecule surface density to hold the fraction of spots with two or more coincident target molecules (fluorescent or not) below a chosen threshold (typically 1 % or less), negating the possible advantage of NTIS (but at the expense of collecting less data per sample). On the other hand, NFTICA has the practical advantage that it is a control internal to the sample and is thus immune to problems caused by temporal or sample-to-sample variability (e.g., of surface properties).

    1. Overlapping AOIs.

    The method does not require non-overlapping AOIs – we used partially overlapping AOIs in the experimental data analyzed in the manuscript. Even though our analysis used larger AOI sizes (and hence, more overlap) than the spot-picker method, there was good agreement in the results, indicating that overlap does not cause any undue problems.

    In the revised manuscript Results section we added the following discussion of the effect of AOI size:

    “Since target-nonspecific spots are built into the cosmos model, there is no need to choose excessively small AOIs in an attempt to exclude non-specific spots from analysis. We found that reducing AOI size (from 14 x 14 to 6 x 6 pixels) did not appreciably affect analysis accuracy on simulated data (Table 2). In analysis of experimental data, smaller AOI sizes caused occasional changes in calculated p(specific) values reflecting apparent missed detection of a few spots (Figure 3–Figure supplement 4). Out of caution, we therefore used 14 x 14 pixel AOIs routinely, even though the larger AOIs somewhat reduced computation speed (Table 2 and Figure 3–Figure Supplement 4).”

    1. Methods requiring extraction of intensity data.

    The cosmos model we describe in the manuscript does not incorporate phenomena where the spot intensity at a single target changes, such as when there is FRET or multiple binders. As we point out in the final paragraph of the Discussion, more elaborate versions of the cosmos model that incorporate these phenomena could be developed. This would entail implementation, optimization, and validation with simulations and real data of the new model, which is beyond the scope of the present manuscript.

    Second, for adoption by non-expert users information is missing in the main text about practical aspects of using the Tapqir software including a description of inputs/outputs, the GUI (I believe Taqpir runs at the command line but the output is in a GUI), and if Tapqir integrates the kinetic modeling or not.

    This information is given in the online Tapqir documentation. The kinetic analysis (as in Fig. 6) is a simple Python script that is run after Tapqir; the instructions for using it are included in the documentation. Tapqir runs can be initiated using either a CLI or GUI. Output can be viewed in Tensorboard, in a Tapqir GUI, and/or passed to a Jupyter notebook or Python script for further analysis, plotting, etc.

    Given that a competing approach has already been published by the Grunwald lab, it would be useful to compare these methods directly in both their accuracy, usefulness of the outputs, and calculation times.

    The reviewer does not explain why comparing with the Grunwald method would be preferable to the comparison with spot-picker that is included in the manuscript. To be sure there is no misunderstanding, the following are the same for the two methods and therefore are not reasons to prefer one or the other of these methods for the comparison in Fig. 6 (see also Discussion):

    1. Like Tapqir, both spot-picker and Grunwald methods analyze 2-D images, not integrated intensities.

    2. Unlike Tapqir, neither spot-picker nor Grunwald is fully objective; both require subjective selection of classification thresholds by the analyst in order to tune the algorithm performance for analysis of a particular dataset.

    3. Neither spot-picker nor Grunwald is a Bayesian method. “Bayesian” in the Grunwald paper title refers to their excellent work on a separate analytical method (described in the same paper) for evaluating the number of binder molecules colocalized with a target spot; this method is not relevant to a comparison with the model presented in our manuscript.

    4. Unlike Tapqir, neither spot-picker nor Grunwald estimate classification probabilities. Instead, they simply assign binary spot/no-spot classifications that do not convey to downstream analyses the extent of uncertainty in each classification.

    5. Neither spot-picker nor Grunwald has been validated previously using simulated image data. Consequently, the validity of image classification has not been established for either.

    The comparison of Fig. 6 and supplements does not claim to and is not intended to show that Tapqir is better than spot-picker for real experimental data; we cannot make such a claim for these or any other methods because we do not know the true kinetic process and rate constants that generated the experimental data. Instead, our comparison uses experimental data sets with a broad range of characteristics (Table 1) to show that Tapqir yields similar association rate constants to those produced by spot-picker even though the former is objective and automatic while the latter requires subjective tuning by an analyst. Our choice to use spot-picker over Grunwald for this comparison was dictated by the fact that among the co-authors we have such an expert in the use of spot-picker, whereas we lack comparable expertise with Grunwald. We have little doubt that Grunwald would also produce results similar to the other methods in the hands of an expert user who is able to subjectively adjust classification parameters.

    Along these lines, the utility of calculating event probability statistics (Fig. 6A) is not well fleshed-out. This is a key distinguishing feature between Tapqir and methods previously published by Grunwald et al. In the case of Tapqir, the probability outputs are not used to their fullest in the determination of kinetic parameters. Rather a subjective probability threshold is chosen for what events to include. This may introduce bias and degrade the objective Tapqir pipeline used to identify these same events.

    This comment reflects a misunderstanding. No probability threshold is used in the kinetic analyses (Figs. 5 and 6). Instead, we make full use of the p(specific) probability output using the posterior sampling strategy that is illustrated in Fig. 5B and is described in the Results and in Materials and Methods. In the revised manuscript we modified the Results section to further emphasize this point.

    Finally, the manuscript could be improved by clearly distinguishing between the fundamental approach of Bayesian image analysis from the Tapqir software that would be used to carry this out.

    We have revised the manuscript to adopt this recommendation. We now call the mathematical model “the cosmos model” and use “Tapqir” to refer to the software.

    A section devoted to describing the Tapqir interface and the inputs/outputs would be valuable. In the manuscript's current form, the lack of information on the interface along with the potential requirement for a GPU and need for the use of a relatively new programming language (Pyro) may hamper adoption and interest in colocalization methods by general audiences.

    Description of the interface and inputs/outputs is given in the online Tapqir documentation.

    Users do not need to own a GPU; they can instead run the program on a readily available cloud computing service. We have now added to Table 1 data showing that computation time on the Google Colab Pro cloud service is actually faster than that on our local GPU system. Colab Pro is inexpensive, readily accessible, and user friendly. We have added to the user manual a tutorial that shows how to run a sample data set using Tapqir on Colab.

    Users do not need any knowledge of Pyro to use Tapqir; Pyro is merely used internally in the coding of Tapqir.

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  2. Evaluation Summary:

    This paper will be of interest to researchers who perform single-molecule fluorescence imaging experiments as well as those who want to include machine learning in their data analyses. The authors have developed a machine learning algorithm that addresses some of the data analysis challenges in the field of single-molecule fluorescence imaging. The methods are rigorously benchmarked using simulated data and tested using real data. There are some concerns whether Tapqir is general enough for use by the broader community of single-molecule fluorescence researchers.

    (This preprint has been reviewed by eLife. We include the public reviews from the reviewers here; the authors also receive private feedback with suggested changes to the manuscript. Reviewer #2 agreed to share their name with the authors.)

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  3. Reviewer #1 (Public Review):

    "Bayesian machine learning analysis of single-molecule fluorescence colocalization images" by Ordabayev, et al. reports the development, benchmarking, and testing of a Bayesian machine learning-based method, which the authors name Tapqir, for analyzing single-molecule fluorescence colocalization data. Unlike currently available, more conventional analysis methods, Tapqir attempts to holistically model the microscopy images that are recorded during a colocalization experiment. Tapir uses a physics-based, global model with parameters describing all of the features of the experiment that are expected to contribute to the recorded microscopy images, including shot noise of the spots and background, camera noise, size and shape of the spots, and specific- and non-specific binders. Based on benchmarking on simulated data with widely varying properties (e.g., signal-to-noise; amounts, rates, and locations of specific and non-specific binders; etc.), Tapqir generally does as well and, in some cases, better than currently existing methods. The authors also test Tapqir on real microscopy images with similarly varying properties from studies that have been previously published by their research group and demonstrate that their Tapqir-based analysis is able to faithfully reproduce the previously published results, which were obtained using the more conventional analysis methods available at the time the data were originally published. This is a well-designed and executed study, Tapqir represents a conceptual and practical advance in the analysis of single-molecule fluorescence colocalization experiments, and its performance has been comprehensively and rigorously benchmarked on simulated data and tested on real data. The conclusions of this study are well supported by the data, but some of the limitations of the method need to be clarified and discussed in more depth, as outlined below.

    1. Given that the AOI is centered at the target molecule and there is a strong prior for the binder also being located at the center of the AOI, the performance of Tapqir is dependent on several variables of the microscopy/optical system (e.g., the microscope point-spread function, magnification, accurate alignment of target and binder imaging channels, accurate drift correction, etc.). Although this caveat is mentioned and some of these factors are listed in the main text of the manuscript, the authors could have expanded this discussion in order to clarify the extent to which the performance of Tapqir depends on these factors.

    2. The Tapqir model has many parameters, each with its own prior. The majority of these priors are designed to be uninformative and/or weak and the only very strong prior is the probability that a specific binder is located at or very near the center of the AOI. The authors could have tested and commented on how the strength of the prior on the location of a specific binder affects the performance of Tapqir.

    3. Given the priors and variational parameters they report, the authors show that Tapqir performs robustly and seems to require no experiment-to-experiment optimization. This is expected to be the case for the simulated data, since they were simulated using the same model that Tapqir uses to perform the analysis. With regard to the real data, however, it is quite likely that this is due to the fact that the analyzed data all come from the same laboratory and, therefore, likely the same microscope(s). It would have therefore been very useful if the authors would have listed and discussed which microscope settings, experimental conditions, and/or other considerations, beyond those described in point 1 above, would result in a need for re-optimization of the priors and/or variational parameters.

    4. Based on analysis of the simulated data shown in Figure 5, where the ground truth is known, the use of Tapqir to infer kinetics is less accurate that the use of Tapqir to infer equilibrium binding constants. The authors do a great job of discussing possible reasons for this. In the case of the real data analyzed in Figure 6 and in Figure 6 - Figure Supplements 1 and 2, the kinetic results obtained using Tapqir have different means and generally larger error bars than those obtained using Spot-Picker. To more comprehensively assess the performance of Tapqir versus Spot-Picker, the authors could have used the association and dissociation rates to calculate the corresponding equilibrium binding constants and then compared these kinetically calculated equilibrium binding constants to the population-calculated equilibrium binding constants that the authors calculate and report in the bottom plot in Panel D of Figure 6 and Figure 6 - Figure Supplements 1 and 2. This would provide some information on the accuracy of the kinetics in that the closer the kinetically and population-calculated equilibrium binding constants are to each other, the more accurately the kinetics have been estimated. Performing this type of analysis for the kinetics obtained using Tapqir and Spot-Picker would have allowed a more comprehensive comparison of the two methods.

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  4. Reviewer #2 (Public Review):

    The work by Ordabayev et al. details a Bayesian inference-based data analysis method for colocalization single molecule spectroscopy (CoSMoS) experiments used to investigate biochemical and biophysical mechanisms. By using this probabilistic framework, their method is able to quantify the colocalization probabilities for individual molecules while accounting for the uncertainty in individual binding events, and accounting for camera and optical noise and even non-specific binding. The software implementation of this method, called Tapqir, uses a Python-based probabilistic programming language (PPL) called pyro to automate and speed-up the optimization of a variational Bayes approximation to the posterior probability distribution. Overall, Tapqir is a powerful new way to analyze CoSMoS data.

    Tapqir works by analyzing small regions (14x14 pixels) of fluorescence microscopy images surrounding previously identified areas of interest (AOI). The collection of images of these AOIs through time are then analyzed collectively using a probabilistic model that accounts for each time frame of each AOI and is able to determine whether up to K "binders" (K=2 here) are present and which of them is specifically bound. This approach of directly modeling the contents of the image data is relatively novel, and few other examples exist. The details of the probabilistic model used incorporate an impressive amount of physical insight (e.g., camera gain) without overparameterization.

    The gamma-distributed noise model used in Tapqir captures quite a lot of physics and, given the analyses in Figs. 3-6, clearly works, but might be limited to certain types of cameras used in the fluorescence microscopy (e.g., EMCCDs). For instance, sCMOS cameras have pixel-dependent amplification and noise profiles, rather than a single gain parameter, and are sometimes approximately modeled as normal distributions with both mean and variance having an intensity-dependent and independent contribution that is different for each pixel on the camera. It is unclear how Tapqir performs on different cameras.

    The variational Bayes solution used by Tapqir provides many computational benefits, such as numerical tractability using pyro and speed. It is possible that the exact posterior, e.g., as obtained using a Markov chain Monte Carlo method, would be insignificantly different with the amount of data typical for CoSMoS experiments; however, this difference is not explored in the current work.

    The intrinsic use of prior probability distributions in any Bayesian inference algorithm is extremely powerful, and in Tapqir offers the opportunity to "chain together" subsequent analyses by using the marginalized posteriors from one experiment as the basis for the priors for subsequent experiments (e.g., in \sigma^{xy}) for extremely high accuracy inference. While the manuscript discusses setting and leveraging the power of priors, it does not explore the power of such "chaining" and the positive effects upon accuracy.

    A significant number of CoSMoS experiments use multiple, distinct color fluorophores to probe the colocalization of different species to the target. The current work focuses only upon analyzing data with a single color-channel. Extensions to multiple independent wavelengths are computationally trivial, given the automated variational inference ability of PPLs such as pyro, and would increase the impact of the work in the field.

    Tapqir analysis provides time series of the probability of a specific binding event, p(specific), for each target analyzed (c.f., Fig. 5B), and kinetic parameters are extracted from these time series using secondary analyses that are distinct from Tapqir itself.

    The method reported here is well designed, sound, and its utility is well supported by the analyses of simulated and experimental data sets reported here. Tapqir is a cutting-edge image analysis approach, and its proper treatment of the uncertainty inherent to CoSMoS experiments will certainly make an impact upon the analysis of CoSMoS data. However, many of the (necessary) assumptions about the data (e.g., fluorescence microscopy) and desired information (e.g., off-target vs on-target binding) are quite specific to CoSMoS experiments and therefore limit the direct applicability of Tapqir for the analysis of other single-molecule microscopy techniques. With that in mind, the direct Bayesian inference-based analysis of image data, as opposed to integrated time series, as demonstrated here is very powerful, and may encourage and inspire related methods to be developed.

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  5. Reviewer #3 (Public Review):

    In this manuscript, the authors seek to improve the reproducibility and eliminate sources of bias in the analysis of single molecule colocalization fluorescence data. These types of data (i.e., CoSMoS data) have been obtained from a number of diverse biological systems and represent unique challenges for data analysis in comparison with smFRET. A key source of bias is what constitutes a binding event and if those events are colocalized or not with a surface-tethered molecule of interest. To solve these issues, the authors propose a Bayesian-based method in which each image is analyzed individually and locally around areas of interest (AOIs) identified from the surface tethered molecules. A strength of the research is that the approach eliminates many sources of bias (i.e., thresholding) in analysis, models realistic image features (noise), can be automated and carried out by novice users "hands-free", and returns a probability score for each event. The performance of the method is superb under a number of conditions and with varying levels of signal-to-noise. The analysis on a GPU is fairly quick-overnight-in comparison with by-hand analysis of the traces which can take days or longer. Tapqir has the potential to be the go-to software package for analysis of single molecule colocalization data.

    The weaknesses of this work involve concerns about the approach and its usefulness to the single-molecule community at large as wells as a lack of information about how users implement and use the Tapqir software. For the first item, there are a number of common scenarios encountered in colocalization analysis that may exclude use of Tapqir including use of CMOS rather than EM-CCD cameras, significant numbers of tethered molecules on the surface that are dark/non-fluorescent, a high density/overlapping of AOIs, and cases where event intensity information is critical (i.e., FRET detection or sequential binding and simultaneous occupancy of multiple fluorescent molecules at the same AOI). In its current form, the use of Tapqir may be limited to only certain scenarios with data acquired by certain types of instruments.

    Second, for adoption by non-expert users information is missing in the main text about practical aspects of using the Tapqir software including a description of inputs/outputs, the GUI (I believe Taqpir runs at the command line but the output is in a GUI), and if Tapqir integrates the kinetic modeling or not. Given that a competing approach has already been published by the Grunwald lab, it would be useful to compare these methods directly in both their accuracy, usefulness of the outputs, and calculation times. Along these lines, the utility of calculating event probability statistics (Fig. 6A) is not well fleshed-out. This is a key distinguishing feature between Tapqir and methods previously published by Grunwald et al. In the case of Tapqir, the probability outputs are not used to their fullest in the determination of kinetic parameters. Rather a subjective probability threshold is chosen for what events to include. This may introduce bias and degrade the objective Tapqir pipeline used to identify these same events.

    Finally, the manuscript could be improved by clearly distinguishing between the fundamental approach of Bayesian image analysis from the Tapqir software that would be used to carry this out. A section devoted to describing the Tapqir interface and the inputs/outputs would be valuable. In the manuscript's current form, the lack of information on the interface along with the potential requirement for a GPU and need for the use of a relatively new programming language (Pyro) may hamper adoption and interest in colocalization methods by general audiences.

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