Robust modelbased analysis of singleparticle tracking experiments with SpotOn
Abstract
Singleparticle tracking (SPT) has become an important method to bridge biochemistry and cell biology since it allows direct observation of protein binding and diffusion dynamics in live cells. However, accurately inferring information from SPT studies is challenging due to biases in both data analysis and experimental design. To address analysis bias, we introduce ‘SpotOn’, an intuitive webinterface. SpotOn implements a kinetic modeling framework that accounts for known biases, including molecules moving outoffocus, and robustly infers diffusion constants and subpopulations from pooled singlemolecule trajectories. To minimize inherent experimental biases, we implement and validate stroboscopic photoactivation SPT (spaSPT), which minimizes motionblur bias and tracking errors. We validate SpotOn using experimentally realistic simulations and show that SpotOn outperforms other methods. We then apply SpotOn to spaSPT data from live mammalian cells spanning a wide range of nuclear dynamics and demonstrate that SpotOn consistently and robustly infers subpopulation fractions and diffusion constants.
https://doi.org/10.7554/eLife.33125.001eLife digest
Proteins, the molecules that make up the cells’ internal machinery, are responsible for almost every process that keeps cells alive. Watching how proteins move and interact within a living cell can help scientists to better understand these biological mechanisms. Singleparticle tracking is a recent technique that makes these observations possible by taking ‘live’ recordings of individual proteins in a cell. Typically, the goal of a singleparticle tracking experiment is to assign proteins into groups, or subpopulations, based on the way they move in the cell. For example, one subpopulation may be bound to other cellular structures, a second moving freely at a high speed, and a third diffusing slowly. This informs on the biological roles of the proteins.
The method involves an experimental stage and an analysis stage. During the experiment, proteins of interest are labeled with a small dye molecule that produces light when excited by a laser. The laser then illuminates the cell, stimulating all the labels in a thin layer. The position of each molecule is then determined with a microscope and a ‘snapshot’ taken. By repeating this process over multiple images, the movement of each molecule over time can be tracked. However, experimental problems can make the interpretation difficult. Motion blurring takes place when the proteins move so fast they appear as blurs in the images; tracking errors happen when so many proteins are present in the same space their trajectories overlap.
Here, Hansen, Woringer et al. combine two preexisting methods to improve the experimental setup. Using lasers that flash like a strobe light reduces motion blurring by essentially taking snapshots of the proteins at short time intervals. Tracking errors are addressed by a technique whereby only one protein at a time produces light.
Once the images are obtained and analyzed to yield trajectories, the trajectories themselves need to be analyzed to determine the number and properties of the protein subpopulations. Several factors can skew this analysis stage. For example, there is often a bias against fastmoving particles because the laser only lights up a thin layer of the cell. The proteins travelling slowly stay in focus long enough to be detected across many images; the fast ones quickly move out of the layer and are therefore counted less often. Hansen, Woringer et al. designed a free and userfriendly algorithm package called SpotOn to correct for this issue. SpotOn was thoroughly benchmarked against other solutions, demonstrating both its accuracy and robustness.
Singleparticle tracking can lead to misleading results if used incorrectly. It is essential to publically share solutions that help make this technique more rigorous, especially since a growing number of scientists have already started to use the method.
https://doi.org/10.7554/eLife.33125.002Introduction
Advances in imaging technologies, genetically encoded tags and fluorophore development have made singleparticle tracking (SPT) an increasingly popular method for analyzing protein dynamics (Liu et al., 2015). Recent biological applications of SPT have revealed that transcription factors (TFs) bind mitotic chromosomes (Teves et al., 2016), how Polycomb interacts with chromatin (Zhen et al., 2016), that ‘pioneer factor’ TFs bind chromatin dynamically (Swinstead et al., 2016), that TF binding time correlates with transcriptional activity (Loffreda et al., 2017) and that different nuclear proteins adopt distinct target search mechanisms (Izeddin et al., 2014; Rhodes et al., 2017). Compared with indirect and bulk techniques such as Fluorescence Recovery After Photobleaching (FRAP) or Fluorescence Correlation Spectroscopy (FCS), SPT is often seen as less biased and less modeldependent (Goulian and Simon, 2000; Mueller et al., 2013; Shen et al., 2017). In particular, SPT makes it possible to directly follow single molecules over time in live cells and has provided clear evidence that proteins often exist in several subpopulations that can be characterized by their distinct diffusion coefficients (Mueller et al., 2013; Shen et al., 2017). For example, nuclear proteins such as TFs and chromatin binding proteins typically show a quasiimmobile chromatinbound fraction and a freely diffusing fraction inside the nucleus. However, while SPT of slowdiffusing membrane proteins is an established technology (Weimann et al., 2013), 2DSPT of proteins freely diffusing inside a 3D nucleus introduces several biases that must be corrected for in order to obtain accurate estimates of subpopulations. First, while a frame is acquired, fastdiffusing molecules move and spread out their emitted photons over multiple pixels causing a ‘motionblur’ artifact (Berglund, 2010; Deschout et al., 2012; Frost et al., 2012; Goulian and Simon, 2000; Izeddin et al., 2014), whereas immobile or slowdiffusing molecules resemble point spread functions (PSFs; Figure 1A). This results in undercounting of the fastdiffusing subpopulation. Second, high particle densities tend to cause tracking errors when localized molecules are connected into trajectories. This can result in incorrect displacement estimates (Figure 1B). Third, since SPT generally employs 2D imaging of 3D motion, immobile or slowdiffusing molecules will generally remain infocus until they photobleach and therefore exhibit long trajectories, whereas fastdiffusing molecules in 3D rapidly move outoffocus, thus resulting in short trajectories (we refer to this as ‘defocalization’; Figure 1C). This results in a timedependent undercounting of fastdiffusing molecules (Goulian and Simon, 2000; Kues and Kubitscheck, 2002). Fourth, SPT analysis methods themselves may introduce biases; to avoid this, an accurate and validated method is needed (Figure 1D).
Here, we introduce an integrated approach to overcome all four biases. The first two biases must be minimized at the data acquisition stage and we describe an experimental SPT method to do so (spaSPT), whereas the latter two can be overcome using a previously developed kinetic modeling framework (Hansen et al., 2017; Mazza et al., 2012) now extended and implemented in SpotOn. SpotOn is available as a webinterface (https://SpotOn.berkeley.edu) as well as Python and Matlab packages.
Results
Overview of SpotOn
SpotOn is a userfriendly webinterface that pedagogically guides the user through a series of qualitychecks of uploaded datasets consisting of pooled singlemolecule trajectories. It then performs kinetic modelbased analysis that leverages the histogram of molecular displacements over time to infer the fraction and diffusion constant of each subpopulation (Figure 2). SpotOn does not directly analyze raw microscopy images, since a large number of localization and tracking algorithms exist that convert microscopy images into singlemolecule trajectories (for a comparison of particle tracking methods, see (Chenouard et al., 2014); moreover, SpotOn can be oneclick interfaced with TrackMate (Tinevez et al., 2017), which allows inspection of trajectories before uploading to SpotOn).
To use SpotOn, a user uploads their SPT trajectory data in one of several formats (Figure 2). SpotOn then generates useful metadata for assessing the quality of the experiment (e.g. localization density, number of trajectories etc.). SpotOn also allows a user to upload multiple datasets (e.g. different replicates) and merge them. SpotOn then calculates and displays histograms of displacements over multiple time delays. The next step is model fitting. SpotOn models the distribution of displacements for each subpopulation using Brownian motion under steadystate conditions without state transitions (full model description in Materials and Methods). SpotOn also accounts for localization errors (either userdefined or inferred from the SPT data). Crucially, SpotOn corrects for defocalization bias (Figure 1C) by explicitly calculating the probability that molecules move outoffocus as a function of time and their diffusion constant (Video 1). In fact, SpotOn uses the gradual loss of freely diffusing molecules over time as additional information to infer the diffusion constant and size of each subpopulation.
SpotOn considers either 2 or 3 subpopulations. For instance, TFs in nuclei can generally exist in both a chromatinbound state characterized by slow diffusion and a freely diffusing state associated with rapid diffusion. In this case, a 2state model is generally appropriate (‘bound’ vs. ‘free’). SpotOn allows a user to choose their desired model and parameter ranges and then fits the model to the data. Using the previous example of TF dynamics, this allows the user to infer the bound fraction and the diffusion constants. Finally, once a user has finished fitting an appropriate model to their data, SpotOn allows easy download of publicationquality figures and relevant data (Figure 2; Full tutorial in Supplementary file 1).
Validation of SpotOn using simulated SPT data and comparison to other methods
We first evaluated whether SpotOn could accurately infer subpopulations (Figure 1D) and successfully account for known biases (Figure 1C) using simulated data. We compared SpotOn to a popular alternative approach of first fitting the mean square displacement (MSD) of individual trajectories of a minimum length and then fitting the distribution of estimated diffusion constants (we refer to this as ‘MSD_{i}’) as well as a sophisticated HiddenMarkov Modelbased Bayesian inference method (vbSPT) (Persson et al., 2013). Since most SPT data is collected using highly inclined illumination (Tokunaga et al., 2008) (HiLo), we simulated TF binding and diffusion dynamics (2state model: ‘bound vs. free’) confined inside a 4 µm radius mammalian nucleus under realistic HiLo SPT experimental settings subject to a 25 nm localization error (Figure 3—figure supplement 1). We considered the effect of the exposure time (1 ms, 4 ms, 7 ms, 13 ms, 20 ms), the free diffusion constant (from 0.5 µm²/s to 14.5 µm²/s in 0.5 µm²/s increments) and the bound fraction (from 0% to 95% in 5% increments) yielding a total of 3480 different conditions that span the full range of biologically plausible dynamics (Figure 3—figure supplements 2–3; Appendix 1).
SpotOn accurately inferred subpopulation sizes with minimal error (Figure 3A–B, Table 1), but slightly underestimated the diffusion constant (−4.8%; Figure 3B; Table 1). However, this underestimate was due to particle confinement inside the nucleus: SpotOn correctly inferred the diffusion constant when the confinement was relaxed (Figure 3—figure supplement 4; 20 µm nuclear radius instead of 4 µm). This emphasizes that diffusion constants measured by SPT inside cells should be viewed as apparent diffusion constants. In contrast, the MSD_{i} method failed under most conditions regardless of whether all trajectories were used (MSD_{i} (all)) or a fitting filter applied (MSD_{i} (R^{2} >0.8); Figure 3A–B; Table 1). vbSPT performed almost as well as SpotOn for slowdiffusing proteins, but showed larger deviations for fastdiffusing proteins (Figure 3—figure supplements 2–3).
To illustrate how the methods could give such divergent results when run on the same SPT data, we considered two example simulations (Figure 3C–D; more examples in Figure 3—figure supplement 3). First, we considered a mostly bound and relatively slow diffusion case (D_{FREE}: 2.0 µm²/s; F_{BOUND}: 70%; Δτ: 7 ms; Figure 3C). SpotOn and vbSPT accurately inferred both D_{FREE} and F_{BOUND}. In contrast, MSD_{i} (R^{2} > 0.8) greatly underestimated F_{BOUND} (13.6% vs. 70%), whereas MSD_{i} (all) slightly overestimated F_{BOUND}. Since MSD_{i}based methods apply two thresholds (first, minimum trajectory length: here five frames; second, filtering based on R^{2}) in many cases less than 5% of all trajectories passed these thresholds and this example illustrate how sensitive MSD_{i}based methods are to these thresholds. Note that although we show the fits to the probability density function since this is more intuitive (PDF; histogram), we performed the fitting to the cumulative distribution function (CDF). Second, we considered an example with a slow frame rate and fast diffusion, such that the free population rapidly moves outoffocus (D_{FREE}: 14.0 µm²/s; F_{BOUND}: 50%; Δτ: 20 ms; Figure 3D). SpotOn again accurately inferred F_{BOUND}, and slightly underestimated D_{FREE} due to high nuclear confinement (Figure 3—figure supplement 4). Although vbSPT generally performed well, because it does not correct for defocalization bias (vbSPT was developed for bacteria, where defocalization bias is minimal), vbSPT strongly overestimated F_{BOUND} in this case (Figure 3D). Consistent with this, SpotOn without defocalizationbias correction also strongly overestimates the bound fraction (Figure 3—figure supplement 5). We conclude that correcting for defocalization bias is critical. The MSD_{i}based methods again gave divergent results despite seemingly fitting the data well. Thus, a good fit to a histogram of log(D) does not necessarily imply that the inferred D_{FREE} and F_{BOUND} are accurate. A full discussion and comparison of the methods is given in Appendix 1. Finally, we extended this analysis of simulated SPT data to three states (one ‘bound’, two ‘free’ states) and compared SpotOn and vbSPT. SpotOn again accurately inferred both the diffusion constants and subpopulation fractions of each population and slightly outperformed vbSPT (Figure 3—figure supplement 6).
Having established that SpotOn is accurate, we next tested whether it was also robust. SpotOn’s ability to infer D_{FREE} and F_{BOUND} was robust to misestimates of the axial detection range of ~100–200 nm (Figure 3—figure supplement 7), was minimally affected by the number of timepoints considered and fitting parameters (Figure 3—figure supplements 8–9; see also Appendix 2 for parameter considerations) and was not strongly affected by state changes (e.g. binding or unbinding) provided the timescale of state changes is significantly longer than the frame rate (Figure 3—figure supplement 10). Moreover, SpotOn inferred the localization error with nanometer precision provided that a significant bound fraction is present (Figure 3—figure supplement 11). Finally, we subsampled the data sets and found that just ~3000 short trajectories (mean length ~3–4 frames) were sufficient for SpotOn to reliably infer the underlying dynamics (Figure 3—figure supplement 12). We conclude that SpotOn is robust.
Taken together, this analysis of simulated SPT data suggests that SpotOn successfully overcomes defocalization and analysis method biases (Figure 1C–D), accurately and robustly estimates subpopulations and diffusion constants across a wide range of dynamics and, finally, outperforms other methods.
spaSPT minimizes biases in experimental SPT acquisitions
Having validated SpotOn on simulated data, which is not subject to experimental biases (Figure 1A–B), we next sought to evaluate SpotOn on experimental data. To generate SPT data with minimal acquisition bias we performed stroboscopic photoactivation SPT (spaSPT; Figure 4A), which integrates previously and separately published ideas to minimize experimental biases. First, spaSPT minimizes motionblurring, which is caused by particle movement during the camera exposure time (Figure 1A), by using stroboscopic excitation (Elf et al., 2007; Frost et al., 2012). We found that the bright and photostable dyes PAJF_{549} and PAJF_{646} (Grimm et al., 2016a) in combination with the HaloTag (‘Halo’) labeling strategy made it possible to achieve a signaltobackground ratio greater than 5 with just 1 ms excitation pulses, thus providing a good compromise between minimal motionblurring and high signal (Figure 4B). Second, spaSPT minimizes tracking errors (Figure 1B) by using photoactivation (Figure 4A) (Grimm et al., 2016a; Manley et al., 2008). Tracking errors are generally caused by high particles densities. Photoactivation allows tracking at extremely low densities (≤1 molecule per nucleus per frame) and thereby minimizes tracking errors (Izeddin et al., 2014), whilst at the same time generating thousands of trajectories. To consider the full spectrum of nuclear protein dynamics, we studied histone H2BHalo (overwhelmingly bound; fast diffusion; Figure 4C), HaloCTCF (Hansen et al., 2017) (largely bound; slow diffusion; Figure 4D) and HaloNLS (overwhelmingly free; very fast diffusion; Figure 4F) in human U2OS cells and HaloSox2 (Teves et al., 2016) (largely free; intermediate diffusion; Figure 4E) in mouse embryonic stem cells (mESCs). We labeled Halotagged proteins in live cells with the HaloTag ligands PAJF_{549} or PAJF_{646} (Grimm et al., 2016a) and performed spaSPT using HiLo illumination (Video 2). To generate a large dataset to comprehensively test SpotOn, we performed 1064 spaSPT experiments across 60 different conditions.
Validation of SpotOn using spaSPT data at different frame rates
First, we studied whether SpotOn could consistently infer subpopulations over a wide range of frame rates. We experimentally determined the axial detection range to be ~700 nm (Figure 4—figure supplement 1) and performed spaSPT at 200 Hz, 167 Hz, 134 Hz, 100 Hz, 74 Hz and 50 Hz using the four cell lines. SpotOn consistently inferred the diffusion constant (Figure 4G) and total bound fraction across the wide range of frame rates (Figure 4H). This is notable since all four proteins exhibit apparent anomalous diffusion (Figure 4—figure supplement 2) and this demonstrates that SpotOn is also robust to anomalous diffusion despite modeling Brownian motion. While the groundtruth is unknown when considering experiments, SpotOn gave biologically reasonable results: histone H2B was overwhelmingly bound and free Halo3xNLS was overwhelmingly unbound (comparison with vbSPT: Figure 4—figure supplement 3). These results provide additional validation for the bias corrections implemented in SpotOn. We also note that although SpotOn was validated on spaSPT data, SPT data with nonphotoactivatable dyes is also suitable for SpotOn analysis provided that the density is sufficiently low to minimize tracking errors (see also Appendix 3: "Which datasets are appropriate for SpotOn?”). Finally, we demonstrated above that just ~3000 short trajectories (mean length ~3–4 frames) were sufficient for SpotOn to accurately infer D_{FREE} and F_{BOUND} (Figure 3—figure supplement 12). Here we obtain well above 3000 trajectories per cell even at ~1 localization/frame. More generally, with spaSPT this should be generally achievable for all but the most lowly expressed nuclear proteins. Thus, this now makes it possible to study biological celltocell variability in TF dynamics.
Effect of motionblur bias on parameter estimates
Having validated SpotOn on experimental SPT data, we next applied SpotOn to estimate the effect of motionblurring on the estimation of subpopulations. As mentioned, since most localization algorithms (Chenouard et al., 2014; Sergé et al., 2008) achieve superresolution through PSFfitting, this may cause motionblurred molecules to be undersampled, resulting in a bias towards slowmoving molecules (Figure 1A). We estimated the extent of the bias by imaging the four cell lines at 100 Hz and keeping the total number of excitation photons constant, but varying the excitation pulse duration (1 ms, 2 ms, 4 ms, 7 ms, constant; Figure 4I). For generality, we performed these experiments using both PAJF_{549} and PAJF_{646} dyes (Grimm et al., 2016a). We used SpotOn to fit the data and plotted the apparent free diffusion constant (Figure 4J) and apparent total bound fraction (Figure 4K) as a function of the excitation pulse duration. For fastdiffusing proteins like Halo3xNLS and H2BHalo, motionblurring resulted in a large underestimate of the free diffusion constant, whereas the effect on slower proteins like CTCF and Sox2 was minor (Figure 4J). Regarding the total bound fraction, motionblurring caused a ~2 fold overestimate for rapidly diffusing Halo3xNLS (Figure 4K), but had a minor effect on slower proteins like H2B, CTCF and Sox2. Similar results were obtained for both dyes for proteins with a significant bound fraction, but we note that JF_{549} appears to better capture the dynamics of proteins with a minimal bound fraction such as Halo3xNLS (Figure 4J–K). Finally, we note that the extent of the bias due to motionblurring will likely be very sensitive to the localization algorithm. Here, using the MTTalgorithm (Sergé et al., 2008), motionblurring caused up to a 2fold error in both the D_{FREE} and F_{BOUND} estimates.
Taken together, these results suggest that SpotOn can reliably be used even for SPT data collected under constant illumination provided that protein diffusion is sufficiently slow and, moreover, provides a helpful guide for optimizing SPT imaging acquisitions (we include a full discussion of considerations for SPT acquisitions and a proposal for minimum reporting standards in SPT in Appendix 3 and 4).
Discussion
In summary, SPT is an increasingly popular technique and has been revealing important new biological insight. However, a clear consensus on how to perform and analyze SPT experiments is currently lacking. In particular, 2D SPT of fastdiffusing molecules inside 3D cells is subject to a number of inherent experimental (Figure 1A–B) and analysis (Figure 1C–D) biases, which can lead to inaccurate conclusions if not carefully corrected for.
Here, we introduce approaches for accounting for both experimental and analysis biases. Several methods are available for localization/tracking (Chenouard et al., 2014; Sergé et al., 2008) and for classification of individual trajectories (Monnier et al., 2015; Persson et al., 2013). SpotOn now complements these tools by providing a biascorrected, comprehensive opensource framework for inferring subpopulations and diffusion constants from pooled SPT data and makes this platform available through a convenient webinterface. This platform can easily be extended to other diffusion regimes (Metzler et al., 2014) and models (Lee et al., 2017) and, as 3D SPT methods mature, to 3D SPT data. Moreover, spaSPT provides an acquisition protocol for tracking fastdiffusing molecules with minimal bias. We hope that these validated tools will help make SPT more accessible to the community and contribute positively to the emergence of ‘goldstandard’ acquisition and analysis procedures for SPT.
Materials and methods
SpotOn model
Request a detailed protocolSpotOn implements and extends a kinetic modeling framework first described in Mazza et al. (2012) and later extended in Hansen et al. (2017). Briefly, the model infers the diffusion constant and relative fractions of two or three subpopulations from the distribution of displacements (or histogram of displacements) computed at increasing lag time (1$\mathrm{\Delta}\tau$, 2$\mathrm{\Delta}\tau$,. ..). This is performed by fitting a semianalytical model to the empirical histogram of displacements using nonlinear least squares fitting. Defocalization is explicitly accounted for by modeling modeling the fraction of particles that remain in focus over time as a function of their diffusion constant.
Mathematically, the evolution over time of a concentration of particles located at the origin as a Dirac delta function and which follows free diffusion in two dimensions with a diffusion constant D can be described by a propagator (also known as Green’s function). Properly normalized, the probability of a particle starting at the origin ending up at a location r = (x,y) after a time delay, $\mathrm{\Delta}\tau$, is given by:
Here N is a normalization constant with units of length. SpotOn integrates this distribution over a small histogram bin window, Δr, to obtain a normalized distribution, the distribution of displacement lengths to compare to binned experimental data. For simplicity, we will therefore leave out N from subsequent expressions. Since experimental SPT data is subject to a significant mean localization error, $\sigma $, SpotOn also accounts for this (Matsuoka et al., 2009):
Many proteins studied by SPT can generally exist in a quasiimmobile state (e.g. a chromatinbound state in the case of transcription factors) and one or more mobile states. We will first consider the 2state model. Under most conditions, state transitions can be ignored ((Hansen et al., 2017) and Figure 3—figure supplement 10). Thus, the steadystate 2state model considered by SpotOn becomes:
Here, the quasiimmobile subpopulation has diffusion constant, ${D}_{\text{BOUND}}$, and makes up a fraction, ${F}_{\text{BOUND}}$, whereas the freely diffusing subpopulation has diffusion constant, ${D}_{\text{FREE}}$, and makes up a fraction, ${F}_{\text{FREE}}=1{F}_{\text{BOUND}}$. To account for defocalization bias (Figure 1C), SpotOn explicitly considers the probability of the freely diffusing subpopulation moving out of the axial detection range, $\mathrm{\Delta}z$, during each time delay, $\mathrm{\Delta}\tau$. This is important. For example, only ~25% of freelydiffusing molecules will remain in focus for at least five frames (assuming $\mathrm{\Delta}\tau$ = 10 ms; $\mathrm{\Delta}z$=700 nm; one gap allowed; D = 5 µm²/s), resulting in a 4fold undercounting if uncorrected for. If we assume absorbing boundaries such that any molecule that contacts the edges of the axial detection range located at ${z}_{\text{MAX}}=\mathrm{\Delta}z/2$ and ${z}_{\text{MIN}}=\mathrm{\Delta}z/2$ is permanently lost, the fraction of freely diffusing molecules with diffusion constant, ${D}_{\text{FREE}}$, that remain at time delay, $\mathrm{\Delta}\tau$, is given by (Carslow and Jaeger, 1959; Kues and Kubitscheck, 2002):
However, this analytical expression overestimates the fraction lost since there is a significant probability that a molecule that briefly contacted or exceeded the boundary reenters the axial detection range. The reentry probability depends on the number of gaps allowed in the tracking ($g$), $\mathrm{\Delta}\tau$, and $\mathrm{\Delta}z$ and can be approximately accounted for by considering a corrected axial detection range, $\mathrm{\Delta}{z}_{\text{corr}}$, larger than $\mathrm{\Delta}z$: $\mathrm{\Delta}{z}_{\text{corr}}>\phantom{\rule{thinmathspace}{0ex}}\mathrm{\Delta}\phantom{\rule{thinmathspace}{0ex}}z$:
Although $\mathrm{\Delta}{z}_{\text{corr}}$ depend on the number of gaps (g) allowed in the tracking, we will leave it out for simplicity in the following. We determined the coefficients a and b from Monte Carlo simulations. For a given diffusion constant, D, 50,000 molecules were randomly placed onedimensionally along the zaxis drawn from a uniform distribution from ${z}_{\text{MIN}}=\mathrm{\Delta}z/2$ to ${z}_{\text{MAX}}=\mathrm{\Delta}z/2$. Next, using a timestep $\mathrm{\Delta}\tau$, onedimensional Brownian diffusion was simulated along the zaxis using the EulerMaruyama scheme. For time delays from 1$\mathrm{\Delta}\tau$ to 15$\mathrm{\Delta}\tau$, the fraction of molecules that were lost was calculated in the range of D=[1;12] μm^{2}/s. $a\left(\mathrm{\Delta}z,\mathrm{\Delta}\tau ,g\right)$ and $b\left(\mathrm{\Delta}z,\mathrm{\Delta}\tau ,g\right)$ were then estimated through leastsquares fitting of ${P}_{\text{remaining}}\left(\mathrm{\Delta}\tau ,\mathrm{\Delta}{z}_{\text{corr}},D\right)$ to the simulated fraction remaining. The process was repeated over a grid of plausible values of ($\mathrm{\Delta}z,\mathrm{\Delta}\tau ,g$) to derive a grid of 134,865 (a,b) parameter pairs. This precalculated library of (a,b) parameters enables SpotOn to perform model fitting on nearly any SPT dataset with minimal overhead.
Thus, the 2state model SpotOn uses for kinetic modeling of SPT data is given by:
where:
Having derived the 2state model, generalization to a 3state model with 1 bound and 2 diffusive states is straightforward. If the three subpopulations have diffusion constants ${D}_{\text{BOUND}}$, ${D}_{\text{SLOW}}$, ${D}_{\text{FAST}}$, and fractions ${F}_{\text{BOUND}}$, ${F}_{\text{SLOW}}$, ${F}_{\text{FAST}}$, such that ${F}_{\text{BOUND}}+{F}_{\text{SLOW}}+{F}_{\text{FAST}}$=1, then the 3state model considered by SpotOn becomes:
Where ${Z}_{\text{CORR}}\left(\mathrm{\Delta}\tau ,\mathrm{\Delta}{z}_{\text{corr}},D\right)$ is as described above.
Numerical implementation of models in SpotOn
Request a detailed protocolSpotOn calculates the empirical histogram of displacements based on a userdefined bin width. SpotOn allows the user to choose between PDF and CDFfitting of the kinetic model to the empirical displacement distributions; CDFfitting is generally most accurate for smaller datasets and the two are similar for large datasets (Figure 3—figure supplement 9). The integral in ${Z}_{\text{CORR}}\left(\mathrm{\Delta}\tau ,\mathrm{\Delta}{z}_{\text{corr}}\right)$ was numerically evaluated using the midpoint method over 200 points and the terms of the series computed until the term falls below a threshold of 10^{−10}. Model fitting and parameter optimization was performed using a nonlinear least squares algorithm (LevenbergMarquardt). Random initial parameter guesses are drawn uniformly from the userspecified parameter range. The optimization is then repeated several times with different initialization parameters to avoid local minima. SpotOn constrains each fraction to be between 0 and 1 and for the sum of the fractions to equal 1.
Theoretical characteristics and limitations of the model
Request a detailed protocolAlthough SpotOn performs well on both experimental and simulated SPT data, the model implemented by SpotOn has several limitations. First, the kinetic model assumes diffusion to be ideal Brownian motion, even though it is widely acknowledged that the motion of most proteins inside a cell shows some degree of anomalous diffusion. Nevertheless, Figure 4G–H and Figure 4—figure supplement 2 show that the parameter inference for experimental data of proteins presenting various degrees of anomalous diffusion is quite robust.
Second, SpotOn models the localization error as the static mean localization error and this feature can be used to infer the actual localization error from the data. However, the localization error is affected both by the position of the particle with respect to the focal plane (Lindén et al., 2017) and by motion blur (Deschout et al., 2012). Even though a high signaltobackground ratio and fast framerate/stroboscopic illumination help to mitigate these disparities, it is likely that the localization error of fast moving particles will be higher than the bound/slowmoving particles. In that case, one would expect SpotOn to infer a localization error that is the weighted mean of the ‘bound/static’ localization error and the ‘free’ localization error. However, in many situations D_{free}$\mathrm{\Delta}\tau$>> ${\sigma}^{2}$ (even assuming a 2 µm²/s particle imaged at a 5 ms framerate with a ~30 nm localization error, there is still an order of magnitude difference between the two terms). As a consequence, the estimate of $\sigma $ reflects the static localization error (that is, the localization error of the bound fraction), and the localization error estimate becomes less reliable if the bound fraction is very small (Figure 3—figure supplement 11).
Third, following (Kues and Kubitscheck, 2002) the axial detection profile is assumed to be a step function, which is an approximation. However, all simulations here were performed using a detection profile with Gaussian edges (Figure 3—figure supplement 1) and as shown in Figure 3A–B SpotOn still works quite well and moreover is relatively robust to slight mismatches in the axial detection range (Figure 3—figure supplement 7).
Fourth, unlike the original implementation by Mazza et al. (2012), SpotOn ignores state transitions. This reduces the number of fitted parameters and simplifies the generalization to more than two states, but as shown in Figure 3—figure supplement 10 it also causes the parameter inference to fail unless the timescale of state changes is at least 10–50 times longer than the frame rate. Thus, in cases where a molecule is known to exhibit state changes on a timescale of tens to a few hundreds of milliseconds, SpotOn may not be appropriate.
Fifth and finally, SpotOn ignores correlations between adjacent displacements, although taking such information into account can potentially improve the parameter inference (Vestergaard et al., 2014).
Cell culture
Request a detailed protocolHaloSox2 (Teves et al., 2016) knockin JM8.N4 mouse embryonic stem cells ((Pettitt et al., 2009) Research Resource Identifier: RRID:CVCL_J962; obtained from the KOMP Repository at UC Davis) were grown on plates precoated with a 0.1% autoclaved gelatin solution (SigmaAldrich, St. Louis, MO, G9391) under feeder free conditions in knockout DMEM with 15% FBS and LIF (full recipe: 500 mL knockout DMEM (ThermoFisher, Waltham, MA, #10829018), 6 mL MEM NEAA (ThermoFisher #11140050), 6 mL GlutaMax (ThermoFisher #35050061), 5 mL Penicillinstreptomycin (ThermoFisher #15140122), 4.6 μL 2mercapoethanol (SigmaAldrich M3148), 90 mL fetal bovine serum (HyClone Logan, UT, FBS SH30910.03 lot #AXJ47554)) and LIF. mES cells were fed by replacing half the medium with fresh medium daily and passaged every two days by trypsinization. Halo3xNLS, H2BHaloSNAP and knockin C32 HaloCTCF (Hansen et al., 2017) Human U2OS osteosarcoma cells (Research Resource Identifier: RRID:CVCL_0042) were grown in low glucose DMEM with 10% FBS (full recipe: 500 mL DMEM (ThermoFisher #10567014), 50 mL fetal bovine serum (HyClone FBS SH30910.03 lot #AXJ47554) and 5 mL Penicillinstreptomycin (ThermoFisher #15140122)) and were passaged every 2–4 days before reaching confluency. For livecell imaging, the medium was identical except DMEM without phenol red was used (ThermoFisher #31053028). Both mouse ES and human U2OS cells were grown in a Sanyo copper alloy IncuSafe humidified incubator (MCO18AIC(UV)) at 37°C/5.5% CO_{2}. Cell lines were pathogen tested and authenticated through STR profiling (U2OS) as described previously (Hansen et al., 2017; Teves et al., 2016). All cell lines will be provided upon request.
Singlemolecule imaging
Request a detailed protocolThe indicated cell line was grown overnight on plasmacleaned 25 mm circular no 1.5H cover glasses (Marienfeld, Germany, HighPrecision 0117650) either directly (U2OS) or MatriGel coated (mESCs; Fisher Scientific, Hampton, NH, #08774552 according to manufacturer’s instructions just prior to cell plating). After overnight growth, cells were labeled with 5–50 nM PAJF_{549} or PAJF_{646} (Grimm et al., 2016a) for ~15–30 min and washed twice (one wash: medium removed; PBS wash; replenished with fresh medium). At the end of the final wash, the medium was changed to phenol redfree medium keeping all other aspects of the medium the same. Singlemolecule imaging was performed on a custombuilt Nikon TI microscope (Nikon Instruments Inc., Melville, NY) equipped with a 100x/NA 1.49 oilimmersion TIRF objective (Nikon apochromat CFI Apo TIRF 100x Oil), EMCCD camera (Andor, Concord, MA, iXon Ultra 897; frametransfer mode; vertical shift speed: 0.9 μs; −70°C), a perfect focusing system to correct for axial drift and motorized laser illumination (TiTIRF, Nikon), which allows an incident angle adjustment to achieve highly inclined and laminated optical sheet illumination (Tokunaga et al., 2008). The incubation chamber maintained a humidified 37°C atmosphere with 5% CO_{2} and the objective was also heated to 37°C. Excitation was achieved using the following laser lines: 561 nm (1 W, Genesis Coherent, Santa Clara, CA) for PAJF_{549}; 633 nm (1 W, Genesis Coherent, Pala Alto, CA) for PAJF_{646}; 405 nm (140 mW, OBIS, Coherent) for all photoactivation experiments. The excitation lasers were modulated by an acoustooptic Tunable Filter (AA OptoElectronic, France, AOTFnCVISTN) and triggered with the camera TTL exposure output signal. The laser light is coupled into the microscope by an optical fiber and then reflected using a multiband dichroic (405 nm/488 nm/561 nm/633 nm quadband, Semrock, Rochester, NY) and then focused in the back focal plane of the objective. Fluorescence emission light was filtered using a single bandpass filter placed in front of the camera using the following filters: PAJF549: Semrock 593/40 nm bandpass filter; PAJF_{646}: Semrock 676/37 nm bandpass filter. The microscope, cameras, and hardware were controlled through NISElements software (Nikon).
spaSPT experiments and analysis
Request a detailed protocolThe spaSPT experimental settings for Figure 4G–H were as follows: 1 ms 633 nm excitation (100% AOTF) of PAJF_{646} was delivered at the beginning of the frame; 405 nm photoactivation pulses were delivered during the camera integration time (~447 μs) to minimize background and their intensity optimized to achieve a mean density of ≤1 molecule per frame per nucleus. 30,000 frames were recorded per cell per experiment. The camera exposure times were: 4.5 ms, 5.5 ms, 7 ms, 9.5 ms, 13 ms and 19.5 ms.
For the motionblur spaSPT experiments (Figure 4I–K), the camera exposure was fixed to 9.5 ms and photoactivation performed as above. To keep the total number of delivered photons constant, we generated an AOTFlaser intensity calibration curve using a power meter and adjusted the AOTF transmission accordingly for each excitation pulse duration. The excitation settings were as follows: 1 ms, 561 nm 100% AOTF, 633 nm 100% AOTF; 2 ms, 561 nm 43% AOTF, 633 nm 40% AOTF; 4 ms, 561 nm 28% AOTF, 633 nm 27% AOTF; 7 ms, 561 nm 20% AOTF, 633 nm 19% AOTF; constant illumination, 561 nm 17% AOTF, 633 nm 16% AOTF.
spaSPT data was analyzed (localization and tracking) and converted into trajectories using a customwritten Matlab implementation of the MTTalgorithm (Sergé et al., 2008) and the following settings: Localization error: 10^{6.25}; deflation loops: 0; Blinking (frames): 1; max competitors: 3; max D (μm^{2}/s): 20. The spaSPT trajectory data was then analyzed using the Matlab version of SpotOn (v1.0; GitLab tag 1f9f782b) and the following parameters: dZ = 0.7 µm; GapsAllowed = 1; TimePoints: 4 (50 Hz), 6 (74 Hz), 7 (100 Hz), 8 (134 Hz), 9 (167 and 200 Hz); JumpsToConsider = 4; ModelFit = 2; NumberOfStates = 2; FitLocError = 0; LocError = 0.035 µm; D_Free_2State=[0.4;25]; D_Bound_2State=[0.00001;0.08];
SPT simulations
Request a detailed protocolWe developed a utility to simulate diffusing proteins in a confined geometry (simSPT). Briefly, simSPT simulates the diffusion of an arbitrary number of populations of molecules characterized by their diffusion coefficient, under a steady state assumption. Particles are drawn at random between the populations and their location in the 3D nucleus is initialized following a uniform law within the confinement volume. The lifetime of the particle (in frames) is also drawn following an exponential law of mean lifetime $\beta $. Then, the particle diffuses in 3D until it bleaches. Diffusion is simulated by drawing jumps following a normal law of parameters $N\left(0,\sqrt{2D\mathrm{\Delta}\tau}\right)$, where D is the diffusion coefficient and $\mathrm{\Delta}\tau$ the exposure time. Finally, a localization error ($N\left(0,\sigma \right)$) is added to each (x,y,z) localization in the simulated trajectories.
For comparisons of SpotOn, MSD_{i} and vbSPT using a 2state scenario, we parameterized simSPT to consider two subpopulations of particles diffusing in a sphere (the nucleus) of 8 µm diameter illuminated using HiLo illumination (assuming a HiLo beam width of 4 µm), with an axial detection range of ~700 nm, centered at the middle of the HiLo beam with Gaussian edges. Molecules are assumed to have a mean lifetime of 4 frames (when inside the HiLo beam) and of 40 frames when outside the HiLo beam. The localization error was set to 25 nm and the simulation was run until 100,000 infocus trajectories were recorded. More specifically, the effect of the exposure time (1 ms, 4 ms, 7 ms, 13 ms, 20 ms), the free diffusion constant (from 0.5 µm²/s to 14.5 µm²/s in 0.5 µm²/s increments) and the fraction bound (from 0% to 95% in 5% increments) were investigated, yielding a dataset consisting of 3480 simulations. More details on the simulations, including scripts to reproduce the dataset, are available on GitLab as detailed in the ‘Computer code’ section. Full details on how the simulations were analyzed by SpotOn, vbSPT and MSD_{i} are given in Appendix 1.
We also considered a 3state scenario featuring a bound subpopulation (‘bound’), a relatively slow diffusing free subpopulation (‘slow’) and a relatively faster diffusing free subpopulation (‘free’). In this case, we only compared SpotOn and vbSPT (Figure 3—figure supplement 6), since the MSD_{i} methods did not perform well. As in the 2state simulations, we parameterized simSPT to consider that three subpopulations of particles diffusing in a sphere (the nucleus) of 8 µm diameter illuminated using HiLo illumination (assuming a HiLo beam width of 4 µm), with an axial detection range of ~700 nm, centered at the middle of the HiLo beam with Gaussian edges. Molecules are assumed to have a mean lifetime of 4 frames (when inside the HiLo beam) and of 40 frames when outside the HiLo beam. The localization error was set to 40 nm and the simulation was run until 100,000 infocus trajectories were recorded. We considered three different subpopulation conditions: (1) F_{BOUND} = 25%; F_{SLOW} = 25%; F_{FAST} = 50%; (2) F_{BOUND} = 25%; F_{SLOW} = 50%; F_{FAST} = 25%; (3) F_{BOUND} = 50%; F_{SLOW} = 25%; F_{FAST} = 25%. Specifically, for each of these condition, the effect of of the exposure time (1 ms, 4 ms, 7 ms, 10 ms, 13 ms, 20 ms), the slower free diffusion constant (from 0.5 µm²/s to 2.5 µm²/s in 0.5 µm²/s increments) and the faster free diffusion constant (from 4 µm²/s to 11 µm²/s in 1 µm²/s increments) were investigated, yielding a dataset of 720 simulations. Both vbSPT and SpotOn (all) were constrained to three subpopulations. Full details on how the simulations were analyzed by SpotOn and vbSPT are given in Appendix 1.
Data availability
Request a detailed protocolAll raw 1064 spaSPT experiments (Figure 4) as well as the 3480 simulations (Figure 3) are freely available in SpotOn readable Matlab and CSV file formats in the form of SPT trajectories at Zenodo. The experimental data is available at: https://zenodo.org/record/834781; The simulations are available in Matlab format at: https://zenodo.org/record/835541; The simulations are available in CSV format at: https://zenodo.org/record/834787; And supplementary software used for MSD_{i} and vbSPT analysis as well as for generating the simulated data at: https://zenodo.org/record/835171
Computer code
Request a detailed protocolSpotOn is fully opensource. The webinterface can be found at: https://SpotOn.berkeley.edu. All raw code is available at GitLab: https://gitlab.com/tjiandarzacqlab. The webinterface code can be found at https://gitlab.com/tjiandarzacqlab/SpotOn; the Matlab commandline version of SpotOn can be found at: https://gitlab.com/tjiandarzacqlab/spotonmatlab; the Python commandline version of SpotOn can be found at https://gitlab.com/tjiandarzacqlab/SpotOncli; the SPT simulation code (simSPT) can be found at: https://gitlab.com/tjiandarzacqlab/simSPT; finally, the ‘TrackMate to SpotOn connector’ plugin, which adds an extra menu to TrackMate which allows oneclick upload of datasets to SpotOn can be found at: https://gitlab.com/tjiandarzacqlab/SpotOnTrackMate
Appendix 1
Fitting of simulations using SpotOn, vbSPT and MSD_{i}
To systematically evaluate the performance of SpotOn as well as other common analysis tools such as MSD_{i} and vbSPT (Persson et al., 2013), we developed simSPT, a simulation tool to generate a comprehensive set of realistic SPT simulations spanning the range of plausible dynamics (almost a billion trajectories were simulated in total). simSPT is freely available at GitLab: https://gitlab.com/tjiandarzacqlab/simSPT. simSPT simulates 3D SPT trajectories arising from an arbitrary number of subpopulations confined inside a sphere under HiLo illumination and takes into account a limited axial detection range, realistic photobleaching rates and optionally state interconversion. The simulation methods are described in detail at GitLab.
Briefly, we parameterized simSPT to consider that particles diffuse inside a sphere (the nucleus) of 8 µm diameter illuminated using HiLo illumination (assuming a HiLo beam width of 4 µm), with an axial detection range of ~700 nm with Gaussian edges, centered at the middle of the HiLo beam. Molecules are assumed to have a mean lifetime of 4 frames (when inside the HiLo beam) and of 40 frames when outside the HiLo beam.
For the 2state comparisons, the localization error was set to 25 nm and the simulation was run until 100,000 infocus trajectories were recorded. More specifically, the effect of the time between frames (1 ms, 4 ms, 7 ms, 13 ms, 20 ms), the free diffusion constant (from 0.5 µm²/s to 14.5 µm²/s in 0.5 µm²/s increments) and the fraction bound (from 0% to 95% in 5% increments) were investigated, yielding a dataset consisting of 3480 simulations. All 3480 simulated datasets are also available (see Data Availability section). The advantage of simulations is that the ground truth is known.
For the 3state comparisons (Figure 3—figure supplement 6), the localization error was set to 40 nm and the simulation was run until 100,000 infocus trajectories were recorded. We then simulated one bound state (${D}_{\text{BOUND}}$=0.001 µm²/s) and two free states (${D}_{\text{SLOW}}$=0.5 to 2.5 µm²/s in 0.5 µm²/s increments; ${D}_{\text{FAST}}$= 4.0 to 11.0 µm²/s in 1.0 µm²/s increments) and also varying the fractions (${F}_{\text{BOUND}}$=25%, ${F}_{\text{SLOW}}$=25%, ${F}_{\text{FAST}}$= 50%; or ${F}_{\text{BOUND}}$=25%, ${F}_{\text{SLOW}}$=50%, ${F}_{\text{FAST}}$= 25%; or ${F}_{\text{BOUND}}$=50%, ${F}_{\text{SLOW}}$=25%, ${F}_{\text{FAST}}$= 25%;) as was the time between frames (1 ms, 4 ms, 7 ms, 10, 13 ms, 20 ms).
For more specific simulations, extra parameters were varied, such as the width of the axial detection range (Figure 3—figure supplement 7), localization error (Figure 3—figure supplement 11), or the presence/absence of interconversion between states (Figure 3—figure supplement 10).
Comparison of methods for 2state simulations
In the case of the main 3480 simulated SPT datasets for the 2state comparison, we analyzed the data using the Matlab version of SpotOn (either using JumpsToConsider = 4 or all), MSD_{i} (either R^{2} >0.8 or all) or vbSPT. We describe the analysis in details below.
SpotOn (4 jumps)
Rational and parameters
SpotOn allows a user to use the entirety of each trajectory or to use only the first n jumps by adjusting the parameter, JumpsToConsider. For example, consider a trajectory consisting of 6 localizations and without gaps. If JumpsToConsider = 4 and TimePoints = 6, then this trajectory will contribute four displacements to the 1$\mathrm{\Delta}\tau$ histogram, four displacements to the 2$\mathrm{\Delta}\tau$ histogram, three displacements to the 3$\mathrm{\Delta}\tau$ histogram, two displacements to the 4$\mathrm{\Delta}\tau$ histogram and one displacement to the 5$\mathrm{\Delta}\tau$ histogram. Thus, even though the trajectory contains 5 1$\mathrm{\Delta}\tau$ displacements, only the first four will be used for analysis if JumpsToConsider = 4. While on simulated data, using a subset of the trajectories is always slightly less accurate than using the entire trajectory since it slightly underestimates the bound fraction, we previously (Hansen et al., 2017) used this as an empirical way of compensating for all the other experimental biases that cause undercounting of freely diffusing molecules that cannot fully be taken into account in simulations. We therefore also tested this approach in the simulations. To fit the simulations using SpotOn we fed the following parameters to the function SpotOn_core.m (v1.0; GitLab tag 1f9f782b):
dZ = 0.700;
GapsAllowed = 1;
BinWidth = 0.010;
UseAllTraj = 0;
JumpsToConsider = 4;
MaxJump = 6.05;
ModelFit = 2;
DoSingleCellFit = 0;
NumberOfStates = 2;
FitIterations = 2;
FitLocError = 0;
LocError = 0.0247;
D_Free_2State = [0.4 25];
D_Bound_2State = [0.00001 0.08];
TimePoints: 10 if 1 ms; 9 if 4 ms; 8 if 7 ms; 7 if 10 ms; 6 if 13 ms; 5 if 20 ms;
The empirical a,b parameters used to correct for defocalization bias were as follows:
$\mathrm{\Delta}\tau$ = 1 ms; $\mathrm{\Delta}z$ = 0.7 µm; 1 gap: a = 0.0387 s^{12}; b = 0.3189 µm;
$\mathrm{\Delta}\tau$ = 4 ms; $\mathrm{\Delta}z$ = 0.7 µm; 1 gap: a = 0.1472 s^{1/2}; b = 0.2111 µm;
$\mathrm{\Delta}\tau$ = 7 ms; $\mathrm{\Delta}z$ = 0.7 µm; 1 gap: a = 0.1999 s^{1/2}; b = 0.2058 µm;
$\mathrm{\Delta}\tau$ = 10 ms; $\mathrm{\Delta}z$ = 0.7 µm; 1 gap: a = 0.2379 s^{1/2}; b = 0.2017 µm;
$\mathrm{\Delta}\tau$ = 13 ms; $\mathrm{\Delta}z$ = 0.7 µm; 1 gap: a = 0.2656 s^{1/2}; b = 0.2118 µm;
$\mathrm{\Delta}\tau$ = 20 ms; $\mathrm{\Delta}z$ = 0.7 µm; 1 gap: a = 0.3133 s^{1/2}; b = 0.2391 µm;
CDFfitting was then performed in MATLAB R2014b using the Matlab version of SpotOn (v1.0; GitLab tag 1f9f782b) and the estimated free diffusion constant, ${D}_{\text{FREE}}$, and bound fraction, ${F}_{\text{BOUND}}$, recorded for each of the 3480 simulations. The estimated ${D}_{\text{FREE}}$ and ${F}_{\text{BOUND}}$ were then compared to the ground truth known from the simulations. Three parameters were estimated in the fit.
Performance evaluation
SpotOn (4 jumps) performs slightly worse than SpotOn (all) when it comes to estimating ${F}_{\text{BOUND}}$ as expected and essentially identically to SpotOn (all) for estimating ${D}_{\text{FREE}}$. The mean error (bias) for estimating ${F}_{\text{BOUND}}$ was −6.4%, the interquartile range (IQR) was 5.9% and the standard deviation 3.6%. The origin of the error is the undercounting of the bound population due to considering only the first 4 jumps. Since bound molecules remain in focus until they bleach, they always yield only a single trajectory, whereas a single freely diffusing molecule has a probability of yielding multiple trajectories by diffusing infocus for a while, then moving outoffocus for a while and then moving back infocus. For estimating ${D}_{\text{FREE}}$ the bias for SpotOn (4 jumps) was −5.4%, the IQR 3.6% and the standard deviation 3.2%. However, as shown in Figure 3—figure supplements 2 and 4, the slight underestimate of the free diffusion constant is not due to a limitation of SpotOn, but instead due to confinement inside the nucleus (Figure 3—figure supplement 4). For example, a diffusing molecule close to the nuclear boundary moving towards the nuclear boundary will ‘bounce back’ resulting in a large distance travelled, but only a smaller recorded displacement. We validated that this indeed is the origin of the underestimate of ${D}_{\text{FREE}}$ by considering a nucleus with virtually no confinement (20 μm radius) and found that the ${D}_{\text{FREE}}$underestimate was now minimal (Figure 3—figure supplement 4). Finally, SpotOn always estimated the bound diffusion constant, ${D}_{\text{BOUND}}$, with minimal error unlike MSD_{i} or vbSPT, which were not able to accurately estimate ${D}_{\text{BOUND}}$. However, since there is generally less interest in ${D}_{\text{BOUND}}$, we did not use this further for evaluating the performance of the different methods.
SpotOn (all)
Rational and parameters: SpotOn (all) was run on the simulations identically to SpotOn (4 jumps) except the entirety of each trajectory was used for calculating the histograms. To fit the simulations using SpotOn we fed the following parameters to the function SpotOn_core.m (v1.0; GitLab tag 1f9f782b):
dZ = 0.700;
GapsAllowed = 1;
BinWidth = 0.010;
UseAllTraj = 1;
MaxJump = 6.05;
ModelFit = 2;
DoSingleCellFit = 0;
NumberOfStates = 2;
FitIterations = 2;
FitLocError = 0;
LocError = 0.0247;
D_Free_2State = [0.4 25];
D_Bound_2State = [0.00001 0.08];
TimePoints: 10 if 1 ms; 9 if 4 ms; 8 if 7 ms; 7 if 10 ms; 6 if 12 ms; 5 if 20 ms;
The empirical a,b parameters used to correct for defocalization bias were as follows:
o $\mathrm{\Delta}\tau$ = 1 ms; $\mathrm{\Delta}z$ = 0.7 µm; 1 gap: a = 0.0387 s^{1/2}; b = 0.3189 µm;
o $\mathrm{\Delta}\tau$ = 4 ms; $\mathrm{\Delta}z$ = 0.7 µm; 1 gap: a = 0.1472 s^{1/2}; b = 0.2111 µm;
o $\mathrm{\Delta}\tau$ = 7 ms; $\mathrm{\Delta}z$ = 0.7 µm; 1 gap: a = 0.1999 s^{1/2}; b = 0.2058 µm;
o $\mathrm{\Delta}\tau$ = 10 ms; $\mathrm{\Delta}z$ = 0.7 µm; 1 gap: a = 0.2379 s^{1/2}; b = 0.2017 µm;
o $\mathrm{\Delta}\tau$ = 13 ms; $\mathrm{\Delta}z$ = 0.7 µm; 1 gap: a = 0.2656 s^{1/2}; b = 0.2118 µm;
o $\mathrm{\Delta}\tau$ = 20 ms; $\mathrm{\Delta}z$ = 0.7 µm; 1 gap: a = 0.3133 s^{1/2}; b = 0.2391 µm;
As above, CDFfitting was performed and the ${D}_{\text{FREE}}$estimate and ${F}_{\text{BOUND}}$estimate compared to the ground truth for each of the 3480 simulations for which the ground truth is known. Three parameters were estimated in the fit.
Performance evaluation
SpotOn (all) outperformed all other approaches. The mean error (bias) for estimating ${F}_{\text{BOUND}}$ was −1.7%, the interquartile range (IQR) was 1.8% and the standard deviation 1.2%. For estimating ${D}_{\text{FREE}}$ the bias for SpotOn (all) was −4.8%, the IQR 3.5% and the standard deviation 3.3%. But as mentioned above, the slight underestimate of ${D}_{\text{FREE}}$ is simply due to diffusion being confined inside a 4 μm radius nucleus (Figure 3—figure supplement 4). This also helps to emphasize the point that diffusion constants measured inside a nucleus should be interpreted as apparent diffusion constants.
MSD_{i} (R^{2}>0.8)
Rational and parameters
A large number of papers have use different variations of the MSD_{i} approach (Knight et al., 2015; Li et al., 2016; Liu et al., 2014; Schmidt et al., 2016; Zhen et al., 2016). This approach is of course very sensitive to how the MSD is estimated. For example, it is wellknown that accurately estimating diffusion constants from short trajectories (<100 frames) subject to significant localization error is all but impossible as shown by Michalet and Berglund (Michalet and Berglund, 2012). Nevertheless, several papers assign diffusion constants to individual trajectories based on a MSDfit. While the exact method differs somewhat from paper to paper, the most popular approach is to set a threshold of a certain number of localizations per trajectory (most commonly 5; though we note that some reports explicitly attempt to compensate for the bias introduced by setting such a threshold (Zhen et al., 2016)). Each trajectory with at least five localizations are then fit, often using the Matlab library MSDAnalyzer (Tarantino et al., 2014), and thus assigned an apparent diffusion constant. An additional threshold is then applied: only if the fit to the MSD curve is judged sufficiently good, is the diffusion constant then used. Otherwise the trajectory is ignored. This fitting threshold is frequently set based on the coefficient of determination as R^{2}>0.8 in some recent papers (Knight et al., 2015; Li et al., 2016; Schmidt et al., 2016). Next, after analyzing all trajectories in this way, a distribution of diffusion constants is then obtained. The analysis is then performed on the logarithm of these diffusion constants (‘LogD histogram’) (Knight et al., 2015; Li et al., 2016; Schmidt et al., 2016). Both the CDF (Knight et al., 2015) and PDF (Knight et al., 2015; Li et al., 2016; Schmidt et al., 2016; Zhen et al., 2016) can be considered. These are then fitted with a sum of Gaussian distributions: either two (Knight et al., 2015; Schmidt et al., 2016; Zhen et al., 2016) or three (Schmidt et al., 2016; Zhen et al., 2016). We note that it is not immediately clear which distribution fitted diffusion constants should actually follow (e.g. Lognormal, Gamma, Normal, etc.). No justification is given for sums of Gaussians (Knight et al., 2015; Li et al., 2016; Schmidt et al., 2016), though we note that the fit is often quite good both in the previous reports (Knight et al., 2015; Li et al., 2016; Schmidt et al., 2016) and also here as shown in Figure 3—figure supplement 3. Please note that fitting a sum of normal distributions to the LogD histogram is equivalent to fitting a sum of lognormal distributions to the D histogram. We also note here, that in a theoretical study Michalet previously showed that the distribution of diffusion constants is approximately Gaussian, but only under a set of stringent criteria (Michalet, 2010). Since CDFfitting is generally less susceptible to noise from binning and since in this comparison SpotOn also uses CDFfitting, we fit the LogD histogram with a sum of 2 Gaussians using CDFfitting. We refer to this whole procedure as MSD_{i} (R^{2}>0.8). Examples of fits are shown in Figure 3 and Figure 3—figure supplement 3 and the Matlab code to perform the fitting is available together with the data (see “Data availability’). Five parameters were estimated in the fit.
Performance evaluation
Overall, MSD_{i} (R^{2}>0.8) generally performs reasonably well when it comes to estimating ${D}_{\text{FREE}}$, but extremely poorly when it comes to ${F}_{\text{BOUND}}$ and ${D}_{\text{BOUND}}$. The mean error (bias) for estimating ${D}_{\text{FREE}}$ was 8.0%, the interquartile range (IQR) was 4.9% and the standard deviation 28.5%. For estimating ${F}_{\text{BOUND}}$ the bias for MSD_{i} (R^{2}>0.8) was −20.6%, the IQR 32.1% and the standard deviation 26.4%. We note that since ${F}_{\text{BOUND}}$ necessarily has to take a value between 0% and 95% in the simulations and since half the simulations have ${F}_{\text{BOUND}}$<50%, a mean error of −20.6% is actually quite large. Although the bias for ${D}_{\text{FREE}}$ is much smaller, in ~5% of all cases, the error in estimating ${D}_{\text{FREE}}$ is bigger than 2fold. Moreover, in a few very rare cases, not a single trajectory out of the 100,000 simulated trajectories pass both thresholds (R^{2}>0.8; at least five frames). Why is MSD_{i} (R^{2}>0.8) fitting so unreliable? It is instructive to consider an example. In the example dataset provided with the MSD_{i} code (simulation with ${D}_{\text{FREE}}$=2; ${F}_{\text{BOUND}}=$0.75; 1 ms frame rate), the estimated ${D}_{\text{FREE}}$=2.06 is very good, but the estimated ${F}_{\text{BOUND}}=$0.16 is extremely poor. Even though the simulation dataset contains 100,000 simulated trajectories, only 3726 of them actually pass the threshold (R^{2}>0.8; at least five frames). Thus, MSD_{i} (R^{2}>0.8) only uses around 4% of the data. Since the tiny fraction of the dataset that is used for analysis is chosen based on how well it fits an MSDcurve and since displacements of bound molecules are dominated by localization errors and therefore generally poorly fit by MSDanalysis, the procedure enriches for the free population, which is why the estimated bound fraction (16%) is so much lower than the true bound fraction (75%). Additionally, we note that MSD_{i}based analysis is extremely sensitive to the fitting threshold: if instead of R^{2}>0.8, all trajectories had been used the estimated bound fraction would be 87% instead of 16%.
In conclusion, MSD_{i} (R^{2}>0.8) is unreliable for estimating ${F}_{\text{BOUND}}$ when short trajectories are at stake, which is the usual case when performing intracellular SPT of fastdiffusing molecules. MSD_{i} (R^{2}>0.8) most likely fails due to a combination of the following reasons among others. First, it poorly handles localization errors, which dominate the displacements of bound molecules. Second, by only considering trajectories of a certain length (normally at least five frames), it only analyzes a small subsample of the dataset. Third, there is no correction for defocalization bias. Since fastdiffusing molecules move outoffocus and thus have shorter trajectories, the 5frame threshold introduces a large bias against freelydiffusing molecules. Fourth, the fitting threshold (R^{2}>0.8) is relatively arbitrary and the results of the analysis is extremely sensitive to this threshold. Accordingly, in these simulations MSD_{i} (R^{2}>0.8) only analyzes a small fraction (~5%) of all the trajectories; note that this bias against the bound population provides a compensatory bias against the bound population to account for the bias against the free population due to defocalization bias. Fifth, it is difficult to justify the use of Gaussian distributions. Even in cases where the CDFfit to the data is excellent, the fitted ${F}_{\text{BOUND}}$value is often very far off the ground truth. Thus, the goodness of the fit cannot be used to judge how well the parameterestimation went. Finally, we note that several variants of the MSD_{i}based method exist (e.g. the approach used by Zhen et al. (Zhen et al., 2016)) is a bit different than the one used here. However, a full validation test of all MSD_{i}based methods is beyond the scope of this work.
MSD_{i} (all)
Rational and parameters
The MSD_{i} (all) analysis was identical to MSD_{i} (R^{2}>0.8) except for a single difference: instead of only using trajectories of at least five frames where the MSDfit to individual trajectories was judged good (R^{2}>0.8), all trajectories of at least five frames were used, regardless of how good the MSDfit was. five parameters were estimated in the fit.
Performance evaluation
MSD_{i} (all) analysis performed very poorly both when it comes to estimating ${D}_{\text{FREE}}$ and ${F}_{\text{BOUND}}$. The mean relative error (bias) for estimating ${D}_{\text{FREE}}$ was −39.6%, the interquartile range (IQR) was 19.0% and the standard deviation 41.8%. For estimating ${F}_{\text{BOUND}}$ the bias for MSD_{i} (all) was 22.0%, the IQR 17.8% and the standard deviation 15.8%. Thus, in all but a few edge cases, MSD_{i} (all) cannot reliably estimate ${D}_{\text{FREE}}$ or ${F}_{\text{BOUND}}$. As for MSD_{i} (R^{2}>0.8), examples of fits are shown in Figure 3—figure supplement 3 and the Matlab code to perform the fitting is available together with the data (see “Data availability’). In the case of MSD_{i} (all), the main reason for the unreliable estimates is due to defocalization bias. Since fastdiffusing molecules move outoffocus and thus have shorter trajectories, the 5frame threshold introduces a large bias against freelydiffusing molecules. Overall, consistent with previous benchmarking efforts on membrane proteins (Weimann et al., 2013), MSD_{i} (all) performed least well among the tested methods.
vbSPT
Rational and parameters
vbSPT performs singletrajectory classification using HiddenMarkov Modeling (HMM) and Bayesian inference (Persson et al., 2013) and can assign different segments of a single trajectory to different diffusive states, each associated with a particular diffusion constant. vbSPT uses the information from all the estimates on single trajectories to consolidate an estimate of diffusion coefficients and associated fractions in each state.
vbSPT additionally uses a statistical model to infer the most likely number of diffusive states assuming all states to exhibit Brownian motion. Since the simulations used to evaluate vbSPT performed contain only two states, it was not clear how to assign ${D}_{\text{FREE}}$ or ${F}_{\text{BOUND}}$ in cases where e.g. three diffusive states were inferred. Therefore, to optimize the performance of vbSPT and perform the fairest comparison, we restricted vbSPT to two states such that vbSPT would infer the diffusion coefficient of up to two states and provide the associated fractions. This method conceptually differs from the MSD_{i} approach in several ways:
The inferred parameters are not based on the MSD
A specific and rigorous Bayesian statistical model is used to aggregate the parameters estimated on single trajectories to global diffusion states.
vbSPT was initially designed for SPT of diffusing proteins in bacteria (Persson et al., 2013), where defocalization biases are virtually nonexistent since the axial dimension of most bacteria are generally comparable to or smaller than the microscope axial detection range. Furthermore, vbSPT does not explicitly model the localization error. It is then expected that the software performs poorly when the localization error is high, as can be expected when imaging intranuclear factors.
In practice, the following parameters were used to assess vbSPT performance. The software was run on the full set of 3480 simulations. The priors and optimization parameters were left as default and the scripts to perform the analysis are provided together with the experimental data (please see Data Availability section):
dim = 2;
trjLmin = 2;
runs = 3;
maxHidden = 2;
bootstrapNum = 10;
fullBootstrap = 0;
init_D = [0.001, 16];
init_tD = [2, 20]*timestep;
Performance evaluation
Over the 3480 simulations, vbSPT accurately estimated both ${D}_{\text{FREE}}$ and ${F}_{\text{BOUND}}$. The mean relative error (bias) for estimating ${D}_{\text{FREE}}$ was 0.8%, the interquartile range (IQR) was 6.8% and the standard deviation 12.5%. For estimating ${F}_{\text{BOUND}}$ the bias for vbSPT was 5.0%, the IQR 6.1% and the standard deviation 4.6%. Thus, vbSPT estimated values were quite consistent (IQR <7% for both ${D}_{\text{FREE}}$ and ${F}_{\text{BOUND}}$). These values were very close to SpotOn in performance.
When looking at the heatmaps (Figure 3—figure supplement 2) more closely, it appeared that vbSPT performs poorly on the estimation of the free diffusion constant when the mean displacements are small. This case occurs either with small free diffusion constants (0.5–2 µm²/s), or with short frame rates (1 ms) and could be explained by the fact that in such conditions, the displacements of the free population and localization error have comparable magnitudes, and that vbSPT does not account for localization error.
Regarding the estimate of the fraction bound, vbSPT tends to overestimate it more and more as the mean displacement of the free population increases (that is, either the exposure time or ${D}_{\text{FREE}}$). This is most likely because vbSPT does not correct for defocalization bias. Thus, the more free molecules diffuse outoffocus, the more vbSPT will overestimate ${F}_{\text{BOUND}}$. Finally, we note that these two biases somewhat compensate for each other: not considering localization errors causes a small overestimate of the free population, whereas not correcting for defocalization bias causes an underestimate of the free population.
In summary, for conditions where the mean jump length of the free population can be distinguished from the localization error, vbSPT performs reasonably well, while being slightly outperformed by SpotOn.
Comparison of methods for 3state simulations
In the case of the 720 simulated SPT datasets for the 3state comparison, we analyzed the data using the Matlab version of SpotOn (all) and vbSPT. We describe the analysis in details below.
SpotOn (all)
Rational and parameters
SpotOn (all) was run on the simulations identically to the 2state situation above except with one added freely diffusive state. To fit the simulations using SpotOn we fed the following parameters to the function SpotOn_core.m (v1.0; GitLab tag 1f9f782b):
dZ = 0.700;
GapsAllowed = 1;
BinWidth = 0.010;
UseAllTraj = 1;
MaxJump = 6.05;
ModelFit = 2;
DoSingleCellFit = 0;
NumberOfStates = 3;
FitIterations = 8;
FitLocError = 0;
LocError = 0.04;
D_Free1_3State = [0.4 10];
D_Free2_3State = [0.4 25];
D_Bound_3State = [0.00001 0.04];
TimePoints: 10 if 1 ms; 9 if 4 ms; 8 if 7 ms; 7 if 10 ms; 6 if 12 ms; 5 if 20 ms;
The empirical a,b parameters used to correct for defocalization bias were as follows:
$\mathrm{\Delta}\tau$ = 1 ms; $\mathrm{\Delta}z$ = 0.7 µm; 1 gap: a = 0.0387 s^{1/2}; b = 0.3189 µm;
$\mathrm{\Delta}\tau$ = 4 ms; $\mathrm{\Delta}z$ = 0.7 µm; 1 gap: a = 0.1472 s^{1/2}; b = 0.2111 µm;
$\mathrm{\Delta}\tau$ = 7 ms; $\mathrm{\Delta}z$ = 0.7 µm; 1 gap: a = 0.1999 s^{1/2}; b = 0.2058 µm;
$\mathrm{\Delta}\tau$ = 10 ms; $\mathrm{\Delta}z$ = 0.7 µm; 1 gap: a = 0.2379 s^{1/2}; b = 0.2017 µm;
$\mathrm{\Delta}\tau$ = 13 ms; $\mathrm{\Delta}z$ = 0.7 µm; 1 gap: a = 0.2656 s^{1/2}; b = 0.2118 µm;
$\mathrm{\Delta}\tau$ = 20 ms; $\mathrm{\Delta}z$ = 0.7 µm; 1 gap: a = 0.3133 s^{1/2}; b = 0.2391 µm;
As above, CDFfitting was performed and the diffusion constant and subpopulation fraction estimates compared to the ground truth for each of the 720 simulations for which the ground truth is known. Five parameters were estimated in the fit.
Performance evaluation
As in the 2state comparison, SpotOn (all) slightly, but significantly, outperformed vbSPT also in the case of 3 states. The biggest error (bias) in estimating any of the subpopulation fractions was 3% and the biggest standard deviation (3.6% std) was also small (see Figure 3—figure supplement 6 for a full table for statistics). In the case of the diffusion constants, SpotOn also accurately inferred all of these with minimal error. The main limitation of SpotOn 3state fitting, is that it sometimes gets stuck in local minima (we estimate this happens in <1% of cases). Therefore, it was necessary to increase the number of fitting iterations to 8. Nevertheless, SpotOn was very robust and accurately estimated all five parameters with minimal error and outperformed vbSPT.
vbSPT
Rational and parameters
vbSPT analysis was performed exactly as in the 2state case, except with three hidden states instead of 2:
dim = 2;
trjLmin = 2;
runs = 3;
maxHidden = 3;
bootstrapNum = 10;
fullBootstrap = 0;
init_D = [0.001, 16];
init_tD = [2, 20]*timestep;
Although vbSPT was constrained to three states, it occasionally inferred that only 1 or 2 states exist. In case vbSPT inferred less than three states (1 or 2), the inferred diffusion coefficients were matched to the closest diffusion coefficient of the ground truth, and the proportion of the one or two unmatched diffusion coefficients was set to zero.
Performance evaluation
vbSPT generally performed quite well. The maximal error (bias) in estimating any of the subpopulation fractions was 6% and the maximal standard deviation (6.3% std; see Figure 3—figure supplement 6 for a full table for statistics). The main limitation of vbSPT was its inability to infer ${D}_{\text{SLOW}}$: the mean error (bias) for estimating ${D}_{\text{SLOW}}$ was 36.6% and the standard deviation was 64.7%. Therefore, vbSPT performed almost as well as SpotOn for estimating the subpopulation fractions and for estimating ${D}_{\text{FAST}}$, but vbSPT was unable to accurately estimate both ${D}_{\text{BOUND}}$ and ${D}_{\text{SLOW}}$ and thus failed when estimating 2 out of the five parameters. In conclusion, vbSPT performs almost as well as SpotOn when estimating subpopulation fractions, but quite poorly when estimating diffusion constants unless they are very high.
Appendix 2
Considerations for choosing SpotOn parameters
In order to run SpotOn, the user has to set a number of parameters. While some are determined by the acquisition protocol (e.g. time between frames), others will have to be carefully chosen. We provide a discussion of how to choose these here.
JumpsToConsider
Users can either choose to use all displacements from all trajectories (set ‘Use all trajectories’ to ‘Yes’ in the webversion of SpotOn or ‘UseAllTraj = 1’ in the Matlab version of SpotOn) or to use only a subset by controlling the JumpsToConsider variable. For example, consider a trajectory consisting of 6 localizations and without gaps. If JumpsToConsider = 4 and TimePoints = 6, then this trajectory will contribute four displacements to the 1$\mathrm{\Delta}\tau$ histogram, four displacements to the 2$\mathrm{\Delta}\tau$ histogram, three displacements to the 3$\mathrm{\Delta}\tau$ histogram, two displacements to the 4$\mathrm{\Delta}\tau$ histogram and one displacement to the 5$\mathrm{\Delta}\tau$ histogram. Thus, even though the trajectory contains 5 1$\mathrm{\Delta}\tau$ displacements, only the first four will be used for analysis if JumpsToConsider = 4. Why would we want to limit the number of jumps that were used? Since freelydiffusing molecules move outoffocus, almost all very long trajectories will be bound molecules. For example, a single trajectory of 21 localizations will provide 20 displacements to the 1$\mathrm{\Delta}\tau$ histogram, whereas freely diffusing molecules with short trajectories will provide fewer (e.g. 10 trajectories with three localizations would be necessary to also provide 20 displacements to the 1$\mathrm{\Delta}\tau$ histogram). Thus, by limiting JumpsToConsider, one is biasing the displacement histogram against bound molecules. However, as demonstrated in the simulations shown in Figure 3—figure supplement 2, whether all jumps or JumpsToConsider = 4 is used has almost no effect on the ${D}_{\text{FREE}}$estimate, but using JumpsToConsider = 4 causes ${F}_{\text{BOUND}}$ to be underestimated by on average of −5% (percentage points) relative to SpotOn (all). We see a similar ~5–10% difference between SpotOn (four jumps) and SpotOn (all) on the experimental spaSPT data shown in Figure 4. As we have discussed previously (Hansen et al., 2017), restricting JumpsToConsider to four is a way one can compensate for all the many acquisition biases (such as motionblur) that generally cause undercounting for fastdiffusing molecules and which cannot readily be taken into account in simulations. While the optimal value will depend on the trajectory length distribution (JumpsToConsider should not take a value much smaller than the mean trajectory length), we found that JumpsToConsider = 4 provides a good compromise for our experimental data. We strongly recommend including experimental controls (such as histone H2BHalo and Halo3xNLS to ensure that experimental and analysis parameters have been reasonably set).
Number of timepoints
SpotOn considers how the histogram of displacement changes over time for multiple $\mathrm{\Delta}\tau$. The number of $\mathrm{\Delta}\tau$ that will be considered is equal to the number of timepoints – 1. So, if timepoints = 8, the displacements from 1$\mathrm{\Delta}\tau$ to 7$\mathrm{\Delta}\tau$ will be considered. How many timepoints to consider will depend on how much data you have and the framerate. For example, if the mean trajectory length is two frames, setting timepoints to 20 will cause problems since only a tiny fraction of trajectories will be at least 20 frames long and thus contribute to the 19$\mathrm{\Delta}\tau$ histogram. Moreover, the correction for defocalization is approximate, so considering timepoints where more than >95% of free molecules have moved outoffocus is also not recommended; when this happens will further depend on the free diffusion constant. Nevertheless, as long as there is sufficient data to reasonably populate the displacement histograms at all timepoints, SpotOn is highly robust to how this parameter is set (Figure 3—figure supplement 8). As a rule of thumb we generally do not recommend setting timepoints above 10 or considering $\mathrm{\Delta}\tau$ beyond 80 ms.
Iterations for fitting
SpotOn almost always converges optimally in the first iteration, so generally 2 or three is more than sufficient when using the 2state model. For the 3state model, the parameter estimation is more complicated and here we recommend eight iterations as a starting point.
PDF or CDF fitting
Although for large datasets PDF and CDFfitting perform similarly as shown in Figure 3—figure supplement 9, CDFfitting tends to provide more reliable estimates of ${D}_{\text{FREE}}$ and ${F}_{\text{BOUND}}$ when the number of trajectories decreases, likely because PDFfitting is more susceptible to binning noise. Thus, for quantitative analysis we always recommend CDFfitting, though PDFfitting can be convenient for making figures since most people find histograms more intuitive.
Fitting localization error
SpotOn can either use a usersupplied localization error or fit it from the data. As long as there is a significant bound fraction, SpotOn will infer this with nanometer precision (Figure 3—figure supplement 11), though we note that this is an average localization error that mostly reflects the localization error of the bound fraction, and the actual localization error for each individual localization will vary (Deschout et al., 2012; Lindén et al., 2017). In cases, where the bound population is very small, fitting the localization error can be less accurate. Thus, in situations where comparisons are being made between the same protein under different conditions or e.g. between different mutants of the same protein, we recommend fitting to obtain a mean localization error and then keeping it fixed in the comparisons.
Choosing allowed ranges for diffusion constants
SpotOn comes with default allowed ranges. For example, for the 2state model, ${D}_{\text{FREE}}=\left[0.5;25\right]$ and ${D}_{\text{BOUND}}=\left[0.0001;0.08\right]$. These ranges are generally reasonable, but may not be appropriate for all datasets. Whenever SpotOn infers a diffusion constant that is equal to the min or max, caution is needed and it may be necessary to change these limits. In particular, unless a molecule is bound to an unusually dynamic scaffold, ${D}_{\text{BOUND}}$=0.08 µm²/s is almost certainly too high. Thus, we recommend imaging a protein that is overwhelmingly bound, such as histone H2B or H3, fitting the histone data with SpotOn and then use the inferred ${D}_{\text{BOUND}}$ for histone proteins or a slightly larger value as the maximally allowed ${D}_{\text{BOUND}}$ value.
2state or 3state model
SpotOn considers either a 2state or 3state model. Since the 3state model contains two additional fitted parameters, the 3state fit is almost always better. While there are many cases where a 2state model would be inappropriate (e.g. a transcription factor that can exist as either a monomer or tetramer, thus exhibiting two very different diffusive states), generally speaking, we prefer fitting a 2state model for most transcription factors or similar nuclear chromatininteracting proteins. In part, deviations from the 2state model will be due to anomalous diffusion and confinement inside cells, which cause deviation from the ideal Brownian motion model implemented by SpotOn. For this reason, traditional modelselection techniques such as Akaike’s Information Criterion (AIC) or the Bayesian Information Criterion (BIC) can also be misleading.
Appendix 3
SPT acquisition considerations in spaSPT experiments
Considerations for minimizing bias in SPT acquisitions
To obtain a good singlemolecule tracking dataset, a series of requirements have to be met. First of all, it must be possible to image singlemolecules at a high signaltonoise ratio. This is now relatively straightforward thanks to developments in fluorescence labeling strategies and imaging modalities (Lavis, 2017; Liu et al., 2015). The development of the HaloTag proteinlabeling system and bright, photostable organic Halodyes such as TMR and the JF dyes (Grimm et al., 2015) now make it possible to easily visualize single protein molecules inside live cells. Moreover, imaging modalities such as highly inclined and laminated optical sheet illumination (‘HiLo’)(Tokunaga et al., 2008) are relatively straightforward to implement and combined with a highquality EMCCD camera make it possible to image singlemolecules at high signaltonoise suitable for generating highquality 2D SPT data. For details of our imaging setup, which combines HaloTaglabeling with HiLoillumination and which is relatively common and easy to operate, please see the methods section. But we note that many other imaging modalities, e.g. lightsheet or even epifluorescence imaging can generate highquality singlemolecule tracking data.
Thus, in the following we will assume that the above condition is met: namely, that single protein molecules can be tracked inside live cells at high signaltonoise ratio. Nevertheless, even if this condition is met, there are at least four other major sources of bias:
Detection: minimize ‘motionblurring’
Tracking: minimize tracking errors
3D loss: correct for molecules moving outoffocus (defocalization bias)
Analysis methods: infer subpopulations with minimal bias
SpotOn addresses point 3 and 4, as described elsewhere, but point 1 and 2 must be addressed in the experimental design. We discuss strategies to minimize these biases below (spaSPT).
1. Detection – minimizing ‘motionblurring’
Almost all localization algorithms achieve subdiffraction localization accuracy (‘superresolution’) by treating individual fluorophores as pointsource emitters, which generate blurred images that can be described by the PointSpreadFunction (PSF) of the microscope. Modeling of the PSF (typically as a 2dimensional Gaussian) then allows extraction of the particle centroid with a precision of tens of nanometers. But as illustrated in Figure 1A, while this works extremely well for bound molecules, fastdiffusing molecules will spread out their photons over many pixels during the camera exposure and thus appear as ‘motionblurs’. Thus, localization algorithms will reliably detect bound molecules, but may fail to detect fastmoving molecules as has also been observed previously (Berglund, 2010; Deschout et al., 2012; Elf et al., 2007; Izeddin et al., 2014; Lindén et al., 2017). Clearly, the extent of the bias will depend on the exposure time and the diffusion constant: the longer the exposure and higher D, the worse the problem. Assuming Brownian motion, we can calculate the fraction of molecules that will move more than some distance, ${r}_{\text{max}}$, during an exposure time, ${t}_{\text{exp}}$, given a free diffusion constant of ${D}_{\text{FREE}}$ using the following equation:
For example, if we define motionblurring as moving more than two pixels (>320 nm assuming a 160 nm pixel size) during the excitation, an exposure time of 10 ms and a typical free diffusion constant of 3.5 μm^{2}/s (e.g. ~Sox2), we get:
Thus, even for a relatively slowly diffusing protein, with a 10 ms exposure we should expect almost half (48%) of all free molecules to show significant motionblurring, if we assume that molecules move with a constant speed during the exposure. The most straightforward solution, therefore, is to limit the exposure time: in the limit of an infinitely short exposure time, there is no motionblur. In practice, most EMCCD cameras can only image at ~100–200 Hz for reasonably sized ROIs. Moreover, it is generally desirable for the mean jump lengths to be significantly bigger than the localization error, thus for most nuclear factors in mammalian cells it is not desirable to image at above >250 Hz. Accordingly, a reasonable solution is therefore to use stroboscopic illumination. That is, using brief excitation laser pulses that last shorter than the camera frame rate (e.g. 1 ms excitation pulse, 10 ms camera exposure time for a 100 Hz experiment): this achieves minimal motionblurring while maintaining a useful framerate. However, this highlights a key experimental tradeoff: shorter excitation pulses minimize motionblurring, but also minimize the signaltonoise. Therefore, a reasonable compromise has to be determined. Here we use 1 ms excitation pulses: this achieves minimal motion blurring (0.067% > 320 nm using D = 3.5 μm^{2}/s) and still yields very good signal (signaltobackground >5). But users will need to decide this based on their expected D and their experimental setup (signaltonoise). Moreover, different localization algorithms (Chenouard et al., 2014; Deschout et al., 2012) have different sensitivities to motionblurring; thus, the extent of the bias will also depend on the user’s localization algorithm. As we show here, in the case of the MTTalgorithm (Sergé et al., 2008), the estimation of D is quite sensitive to motionblurring, but the estimation of the bound fraction is less sensitive as long as the diffusion constant is <5 μm^{2}/s. But other localization algorithms may be more or less sensitive. Generally speaking, we do not recommend imaging at a signaltobackground <3 and do not recommend using excitation pulses >5 ms, but the optimal conditions will need to be determined on a casebycase basis.
In conclusion, experimentally implementing stroboscopic excitation makes it possible to minimize the bias coming from motionblurring, while still achieving a sufficient signal for reliable localization.
2. Tracking – minimizing tracking errors
It is necessary to minimize tracking errors in order to obtain highquality SPT data. Tracking errors bias the estimation of essentially all parameters we could want to estimate from SPT experiments including diffusion constants, subpopulations, anomalous diffusion etc. While many different tracking algorithms exist, it is fundamentally impossible to perform tracking, that is connecting localized molecules between subsequent frames, at high densities without introducing many tracking errors. Thus, the simplest solution is to image at low densities: in principle, if there is only one labeled molecule per cell, there can be no tracking errors. Yet, because dyes generally bleach quite quickly under most SPT imaging conditions, this has traditionally led to a serious tradeoff between data quality and the number of trajectories which can be obtained. However, with the recent development of bright photoactivatable JFdyes (Grimm et al., 2016a; 2016b) (PAdye), it is now possible to combine the superior brightness of the HaloJF dyes with photoactivation SPT (also called sptPALM (Manley et al., 2008)). That is, a large fraction of Halotagged proteins in a cell can be labeled with HaloPAJF dyes and then photoactivated one at a time: this allows imaging at extremely low densities (<1 fluorescent molecule per cell per frame) and nevertheless tens of thousands of trajectories from a single cell can be obtained. Thus, PAdyes now make it possible to nearly eliminate tracking errors without compromising on signaltonoise or amount of data. In fact, imaging at extremely low densities generally also improves signaltonoise since outoffocus background is reduced and overlapping point emitters are avoided (Izeddin et al., 2014).
Nevertheless, even with paSPT it is still necessary to decide on an optimal density. The key parameters are size of the ROI (ideally the whole nucleus for studies in cells) and D: a large nucleus and a slow D can support a higher density than fastdiffusing molecules in a small nucleus. As a general rule of thumb, we recommend a density of ~1 fluorescent molecule per ROI per frame. This will keep tracking errors at a minimum and still support rapid acquisition of large datasets. All data acquired for this study was acquired at approximately this density.
In practice, keeping an optimal density will require some trialanderror optimization of the 405 nm photoactivation laser intensity. 405 nm excitation does contribute background fluorescence, so we prefer to pulse the 405 nm laser during the camera ‘deadtime’ (~0.5 ms in our case) to avoid this. Moreover, this also makes it easier to keep the photoactivation level constant when changing the frame rate. However, the optimal photoactivation power will depend on the expression level of the protein, protein halflife and the dye concentration and will therefore have to be optimized in each case. We recommend recording initial datasets and then analyzing them using SpotOn which reports the mean number of localizations per frame and then using this information to determine the optimal photoactivation level. However, even then some celltocell variation may be unavoidable: especially in transient transfection experiments where there is large celltocell variation in expression level or when studying proteins expressed from stably integrated transgenes (e.g. Halo3xNLS and H2bHalo in our case). In these cases, some cells will likely exhibit too high a density. To deal with this, SpotOn includes the option to analyze datasets from individual cells first and then excluding a cell with too high a density before analyzing the merged dataset.
Which datasets are appropriate for SpotOn?
In the sections above, we have discussed how to minimize common experimental biases in SPT experiments and proposed spaSPT as a general solution. However, many 2D SPT datasets recorded under different conditions are also appropriate for SpotOn. For example, SPT experiments without photoactivation or with continuous illumination may also be appropriate for analysis with SpotOn. For example, there may be situations where photoactivation SPT is not possible: in such cases, it will be essential to keep the labeling density sufficiently low that tracking errors are minimized and it might thus be necessary to image substantially more cells to get enough statistics. Likewise, as we show in Figure 4JK, motionblurring is a major concern for fastdiffusing molecules, but for a slowly diffusing molecule like HaloCTCF it makes only a small difference. Thus SPT datasets recorded with continuous illumination may also be appropriate provided that the protein of interest is known to diffuse sufficiently slowly.
We also note that since SpotOn uses the loss of fastdiffusing molecules over time to correct for bias and to estimate the free population, it is essential that all trajectories are included in SpotOn for analysis. For example, some tracking and localization algorithms ignore all trajectories below a certain length (e.g. five frames), but this will cause SpotOn to misestimate the loss of molecules moving outoffocus and thus it is imperative that trajectories of all lengths be included when analyzing data using SpotOn. Furthermore, trajectories of only a single localization are required to accurately compute the average number of localizations per frame, which is a key qualitycontrol metric for SPT data.
Moreover, SpotOn does not currently support 3D SPT data. Furthermore, SpotOn assumes diffusion to be Brownian. This is a reasonable approximation even for molecules exhibiting some levels of anomalous diffusion as shown in Figure 4—figure supplement 2, but SpotOn is not appropriate for molecules undergoing directed motion (e.g. a protein moving on microtubules). Additionally, in cases where there are frequent state transitions at a timescale similar to the frame rate (e.g. transcription factor with a 10 ms residence time imaged at 100 Hz), SpotOn may give inaccurate results since it ignores state transitions (Figure 3—figure supplement 10). Finally, the correction for molecules moving outoffocus assumes that molecules are not fully confined within small compartments, that prevent molecules from moving outoffocus.
Appendix 4
Proposed minimal reporting guidelines for SPT data and kinetic modeling analysis
To ensure reproducibility of results and subsequent analyses, datasets, statistics and analysis metrics should be provided. This should allow the reader to quickly assess the quality and statistical significance of the presented results and datasets. So far, to our knowledge, no consensus exists on minimal reporting guidelines for single particle tracking datasets and kinetic modeling analyses. We note, however, that a recent preprint suggests a similar conceptual framework, although less applicable to singlemolecule experiments (Rigano and Strambio De Castillia, 2017),
We propose that published singleparticle datasets be published and reported accompanied with the following metadata. We suggest that these metrics constitute a minimal reporting guideline for singleparticle datasets and subsequent kinetic modeling (though additional information may be appropriate and necessary in some cases).
Dataset description
Criterion  How to obtain it  Example value 

Exposure time  Determined at the acquisition step  5 ms 
Signaltobackground ratio  Mean peak value of detected particle divided by mean background value  5 
Detection algorithm used  MTT (version xxx)  
Tracking algorithm used  MTT (version xxx)  
Number of particles per frame  Provided by SpotOn  Mean: 0.76 
Number of detections  Provided by SpotOn  360000 
Number of trajectories of length >3  Provided by SpotOn  10000 
Mean trajectory length  Provided by SpotOn  4.5 frames 
Localization error  Provided by SpotOn  30 nm 
SpotOn parameters
In addition to these metrics, it is important to report the parameters specified in the detection and tracking algorithms, since this can greatly affect the results. For SpotOn, we recommend reporting the following parameters:
Jump length distribution parameters: BinWidth (µm), Number of timepoints, Jumps to consider or Use all trajectories, MaxJump (µm),
Fitting parameters: Number of states (2 or 3), localization error fitted from data (Yes or No, if no, specify the value, in nm), dZ (µm), a (s^{1/2}), b (µm), PDF or CDF fit (PDF or CDF), number of iterations. Finally, the bounds used for the fitting algorithm should be reported, e.g:
D_{bound}: [0.0005, 0.08] µm²/s
D_{free} [0.15, 25] µm²/s
F_{bound} [0,1]
Obviously, if a 3state model is used, the bounds for the additional subpopulation should also be reported.
In case a custommodified version of SpotOn is used, we recommend that the code be made available and that a summary of the modifications be included in the methods section.
Data availability

Experimental data for "SpotOn: robust modelbased analysis of singleparticle tracking experiments"Publicly available at Zenodo (https://zenodo.org/).

Simulated data for "SpotOn: robust modelbased analysis of singleparticle tracking experiments" (MATLAB format)Publicly available at Zenodo (https://zenodo.org/).

Simulated data for "SpotOn: robust modelbased analysis of singleparticle tracking experiments"Publicly available at Zenodo (https://zenodo.org/).

Software used for "SpotOn: robust modelbased analysis of singleparticle tracking experiments"Publicly available at Zenodo (https://zenodo.org/).
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Article and author information
Author details
Funding
National Institutes of Health (UO1EB021236)
 Xavier Darzacq
National Institutes of Health (U54DK107980)
 Xavier Darzacq
California Institute for Regenerative Medicine (LA108013)
 Xavier Darzacq
Howard Hughes Medical Institute (003061)
 Robert Tjian
Howard Hughes Medical Institute
 Luke D Lavis
Siebel Stem Cell Institute
 Anders S Hansen
The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.
Acknowledgements
ASH and MW contributed equally to this work and are alphabetically listed. We are very grateful to Davide Mazza who inspired this work and provided invaluable comments on SpotOn, to Florian Mueller for suggestions on the webapplication, Christophe Zimmer for insightful discussions, David McSwiggen and Sheila Teves for kindly providing cell lines, Carolyn Elya and Chiahao Tsui for the name ‘SpotOn’, and to members of the Tjian/Darzacq labs and Maxime Dahan for discussions. We also thank Astou Tangara and Anatalia Robles for microscope maintenance. ASH is a postdoctoral fellow of the Siebel Stem Cell Institute. This work was supported by NIH grants UO1EB021236 and U54DK107980 (XD), the California Institute of Regenerative Medicine grant LA108013 (XD), by the Howard Hughes Medical Institute (003061, RT) and used the computational and storage services (TARS cluster) provided by the IT department at Institut Pasteur, Paris.
Version history
 Received: October 27, 2017
 Accepted: January 3, 2018
 Accepted Manuscript published: January 4, 2018 (version 1)
 Version of Record published: February 12, 2018 (version 2)
Copyright
© 2018, Hansen et al.
This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.
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 Structural Biology and Molecular Biophysics
The articles in this special issue highlight how modern cellular, biochemical, biophysical and computational techniques are allowing deeper and more detailed studies of allosteric kinase regulation.

 Developmental Biology
 Structural Biology and Molecular Biophysics
The receptor tyrosine kinase ROR2 mediates noncanonical WNT5A signaling to orchestrate tissue morphogenetic processes, and dysfunction of the pathway causes Robinow syndrome, brachydactyly B, and metastatic diseases. The domain(s) and mechanisms required for ROR2 function, however, remain unclear. We solved the crystal structure of the extracellular cysteinerich (CRD) and Kringle (Kr) domains of ROR2 and found that, unlike other CRDs, the ROR2 CRD lacks the signature hydrophobic pocket that binds lipids/lipidmodified proteins, such as WNTs, suggesting a novel mechanism of ligand reception. Functionally, we showed that the ROR2 CRD, but not other domains, is required and minimally sufficient to promote WNT5A signaling, and Robinow mutations in the CRD and the adjacent Kr impair ROR2 secretion and function. Moreover, using functionactivating and perturbing antibodies against the Frizzled (FZ) family of WNT receptors, we demonstrate the involvement of FZ in WNT5AROR signaling. Thus, ROR2 acts via its CRD to potentiate the function of a receptor supercomplex that includes FZ to transduce WNT5A signals.