Robust model-based analysis of single-particle tracking experiments with Spot-On
Advances in imaging technologies , genetically encoded tags and fluorophore development have made single-particle 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 model-dependent ( 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 quasi-immobile chromatin-bound fraction and a freely diffusing fraction inside the nucleus . However , while SPT of slow-diffusing membrane proteins is an established technology ( Weimann et al . , 2013 ) , 2D-SPT 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 , fast-diffusing molecules move and spread out their emitted photons over multiple pixels causing a ‘motion-blur’ artifact ( Berglund , 2010; Deschout et al . , 2012; Frost et al . , 2012; Goulian and Simon , 2000; Izeddin et al . , 2014 ) , whereas immobile or slow-diffusing molecules resemble point spread functions ( PSFs; Figure 1A ) . This results in under-counting of the fast-diffusing 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 slow-diffusing molecules will generally remain in-focus until they photobleach and therefore exhibit long trajectories , whereas fast-diffusing molecules in 3D rapidly move out-of-focus , thus resulting in short trajectories ( we refer to this as ‘defocalization’; Figure 1C ) . This results in a time-dependent under-counting of fast-diffusing 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 Spot-On . Spot-On is available as a web-interface ( https://SpotOn . berkeley . edu ) as well as Python and Matlab packages .
Spot-On is a user-friendly web-interface that pedagogically guides the user through a series of quality-checks of uploaded datasets consisting of pooled single-molecule trajectories . It then performs kinetic model-based analysis that leverages the histogram of molecular displacements over time to infer the fraction and diffusion constant of each subpopulation ( Figure 2 ) . Spot-On does not directly analyze raw microscopy images , since a large number of localization and tracking algorithms exist that convert microscopy images into single-molecule trajectories ( for a comparison of particle tracking methods , see ( Chenouard et al . , 2014 ) ; moreover , Spot-On can be one-click interfaced with TrackMate ( Tinevez et al . , 2017 ) , which allows inspection of trajectories before uploading to Spot-On ) . To use Spot-On , a user uploads their SPT trajectory data in one of several formats ( Figure 2 ) . Spot-On then generates useful meta-data for assessing the quality of the experiment ( e . g . localization density , number of trajectories etc . ) . Spot-On also allows a user to upload multiple datasets ( e . g . different replicates ) and merge them . Spot-On then calculates and displays histograms of displacements over multiple time delays . The next step is model fitting . Spot-On models the distribution of displacements for each subpopulation using Brownian motion under steady-state conditions without state transitions ( full model description in Materials and Methods ) . Spot-On also accounts for localization errors ( either user-defined or inferred from the SPT data ) . Crucially , Spot-On corrects for defocalization bias ( Figure 1C ) by explicitly calculating the probability that molecules move out-of-focus as a function of time and their diffusion constant ( Video 1 ) . In fact , Spot-On uses the gradual loss of freely diffusing molecules over time as additional information to infer the diffusion constant and size of each subpopulation . Spot-On considers either 2 or 3 subpopulations . For instance , TFs in nuclei can generally exist in both a chromatin-bound state characterized by slow diffusion and a freely diffusing state associated with rapid diffusion . In this case , a 2-state model is generally appropriate ( ‘bound’ vs . ‘free’ ) . Spot-On 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 , Spot-On allows easy download of publication-quality figures and relevant data ( Figure 2; Full tutorial in Supplementary file 1 ) . We first evaluated whether Spot-On could accurately infer subpopulations ( Figure 1D ) and successfully account for known biases ( Figure 1C ) using simulated data . We compared Spot-On 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 ‘MSDi’ ) as well as a sophisticated Hidden-Markov Model-based 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 ( 2-state 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 ) . Spot-On 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: Spot-On 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 MSDi method failed under most conditions regardless of whether all trajectories were used ( MSDi ( all ) ) or a fitting filter applied ( MSDi ( R2 >0 . 8 ) ; Figure 3A–B; Table 1 ) . vbSPT performed almost as well as Spot-On for slow-diffusing proteins , but showed larger deviations for fast-diffusing 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 ( DFREE: 2 . 0 µm²/s; FBOUND: 70%; Δτ: 7 ms; Figure 3C ) . Spot-On and vbSPT accurately inferred both DFREE and FBOUND . In contrast , MSDi ( R2 > 0 . 8 ) greatly underestimated FBOUND ( 13 . 6% vs . 70% ) , whereas MSDi ( all ) slightly overestimated FBOUND . Since MSDi-based methods apply two thresholds ( first , minimum trajectory length: here five frames; second , filtering based on R2 ) in many cases less than 5% of all trajectories passed these thresholds and this example illustrate how sensitive MSDi-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 out-of-focus ( DFREE: 14 . 0 µm²/s; FBOUND: 50%; Δτ: 20 ms; Figure 3D ) . Spot-On again accurately inferred FBOUND , and slightly underestimated DFREE 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 FBOUND in this case ( Figure 3D ) . Consistent with this , Spot-On without defocalization-bias correction also strongly overestimates the bound fraction ( Figure 3—figure supplement 5 ) . We conclude that correcting for defocalization bias is critical . The MSDi-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 DFREE and FBOUND 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 Spot-On and vbSPT . Spot-On 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 Spot-On is accurate , we next tested whether it was also robust . Spot-On’s ability to infer DFREE and FBOUND 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 time-scale of state changes is significantly longer than the frame rate ( Figure 3—figure supplement 10 ) . Moreover , Spot-On inferred the localization error with nanometer precision provided that a significant bound fraction is present ( Figure 3—figure supplement 11 ) . Finally , we sub-sampled the data sets and found that just ~3000 short trajectories ( mean length ~3–4 frames ) were sufficient for Spot-On to reliably infer the underlying dynamics ( Figure 3—figure supplement 12 ) . We conclude that Spot-On is robust . Taken together , this analysis of simulated SPT data suggests that Spot-On 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 . Having validated Spot-On on simulated data , which is not subject to experimental biases ( Figure 1A–B ) , we next sought to evaluate Spot-On on experimental data . To generate SPT data with minimal acquisition bias we performed stroboscopic photo-activation SPT ( spaSPT; Figure 4A ) , which integrates previously and separately published ideas to minimize experimental biases . First , spaSPT minimizes motion-blurring , 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 photo-stable dyes PA-JF549 and PA-JF646 ( Grimm et al . , 2016a ) in combination with the HaloTag ( ‘Halo’ ) labeling strategy made it possible to achieve a signal-to-background ratio greater than 5 with just 1 ms excitation pulses , thus providing a good compromise between minimal motion-blurring and high signal ( Figure 4B ) . Second , spaSPT minimizes tracking errors ( Figure 1B ) by using photo-activation ( Figure 4A ) ( Grimm et al . , 2016a; Manley et al . , 2008 ) . Tracking errors are generally caused by high particles densities . Photo-activation 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 H2B-Halo ( overwhelmingly bound; fast diffusion; Figure 4C ) , Halo-CTCF ( Hansen et al . , 2017 ) ( largely bound; slow diffusion; Figure 4D ) and Halo-NLS ( overwhelmingly free; very fast diffusion; Figure 4F ) in human U2OS cells and Halo-Sox2 ( Teves et al . , 2016 ) ( largely free; intermediate diffusion; Figure 4E ) in mouse embryonic stem cells ( mESCs ) . We labeled Halo-tagged proteins in live cells with the HaloTag ligands PA-JF549 or PA-JF646 ( Grimm et al . , 2016a ) and performed spaSPT using HiLo illumination ( Video 2 ) . To generate a large dataset to comprehensively test Spot-On , we performed 1064 spaSPT experiments across 60 different conditions . First , we studied whether Spot-On 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 . Spot-On 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 Spot-On is also robust to anomalous diffusion despite modeling Brownian motion . While the ground-truth is unknown when considering experiments , Spot-On gave biologically reasonable results: histone H2B was overwhelmingly bound and free Halo-3xNLS was overwhelmingly unbound ( comparison with vbSPT: Figure 4—figure supplement 3 ) . These results provide additional validation for the bias corrections implemented in Spot-On . We also note that although Spot-On was validated on spaSPT data , SPT data with non-photoactivatable dyes is also suitable for Spot-On analysis provided that the density is sufficiently low to minimize tracking errors ( see also Appendix 3: "Which datasets are appropriate for Spot-On ? ” ) . Finally , we demonstrated above that just ~3000 short trajectories ( mean length ~3–4 frames ) were sufficient for Spot-On to accurately infer DFREE and FBOUND ( 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 cell-to-cell variability in TF dynamics . Having validated Spot-On on experimental SPT data , we next applied Spot-On to estimate the effect of motion-blurring on the estimation of subpopulations . As mentioned , since most localization algorithms ( Chenouard et al . , 2014; Sergé et al . , 2008 ) achieve super-resolution through PSF-fitting , this may cause motion-blurred molecules to be undersampled , resulting in a bias towards slow-moving 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 PA-JF549 and PA-JF646 dyes ( Grimm et al . , 2016a ) . We used Spot-On 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 fast-diffusing proteins like Halo-3xNLS and H2B-Halo , motion-blurring 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 , motion-blurring caused a ~2 fold overestimate for rapidly diffusing Halo-3xNLS ( 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 JF549 appears to better capture the dynamics of proteins with a minimal bound fraction such as Halo-3xNLS ( Figure 4J–K ) . Finally , we note that the extent of the bias due to motion-blurring will likely be very sensitive to the localization algorithm . Here , using the MTT-algorithm ( Sergé et al . , 2008 ) , motion-blurring caused up to a 2-fold error in both the DFREE and FBOUND estimates . Taken together , these results suggest that Spot-On 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 ) .
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 fast-diffusing 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 ) . Spot-On now complements these tools by providing a bias-corrected , comprehensive open-source framework for inferring subpopulations and diffusion constants from pooled SPT data and makes this platform available through a convenient web-interface . 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 fast-diffusing 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 ‘gold-standard’ acquisition and analysis procedures for SPT .
Spot-On 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 ( 1Δτ , 2Δτ , . . . ) . This is performed by fitting a semi-analytical model to the empirical histogram of displacements using non-linear 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 , Δτ , is given by:P ( r , Δτ ) =Nr2DΔτe−r24DΔτ Here N is a normalization constant with units of length . Spot-On 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 , σ , Spot-On also accounts for this ( Matsuoka et al . , 2009 ) :P ( r , Δτ ) =r2 ( DΔτ+σ2 ) e−r24 ( DΔτ+σ2 ) Many proteins studied by SPT can generally exist in a quasi-immobile state ( e . g . a chromatin-bound state in the case of transcription factors ) and one or more mobile states . We will first consider the 2-state model . Under most conditions , state transitions can be ignored ( ( Hansen et al . , 2017 ) and Figure 3—figure supplement 10 ) . Thus , the steady-state 2-state model considered by Spot-On becomes:P ( r , Δτ ) =FBOUNDr2 ( DBOUNDΔτ+σ2 ) e−r24 ( DBOUNDΔτ+σ2 ) + ( 1−FBOUND ) r2 ( DFREEΔτ+σ2 ) e−r24 ( DFREEΔτ+σ2 ) Here , the quasi-immobile subpopulation has diffusion constant , DBOUND , and makes up a fraction , FBOUND , whereas the freely diffusing subpopulation has diffusion constant , DFREE , and makes up a fraction , FFREE=1-FBOUND . To account for defocalization bias ( Figure 1C ) , Spot-On explicitly considers the probability of the freely diffusing subpopulation moving out of the axial detection range , Δz , during each time delay , Δτ . This is important . For example , only ~25% of freely-diffusing molecules will remain in focus for at least five frames ( assuming Δτ = 10 ms; Δz=700 nm; one gap allowed; D = 5 µm²/s ) , resulting in a 4-fold undercounting if uncorrected for . If we assume absorbing boundaries such that any molecule that contacts the edges of the axial detection range located at zMAX=Δz/2 and zMIN=−Δz/2 is permanently lost , the fraction of freely diffusing molecules with diffusion constant , DFREE , that remain at time delay , Δτ , is given by ( Carslow and Jaeger , 1959; Kues and Kubitscheck , 2002 ) :Premaining ( Δτ , Δz , DFREE ) =1Δz∫−Δz/2Δz/2{1−∑n=0∞ ( −1 ) n[erfc ( ( 2n+1 ) Δz2−z4DFREEΔτ ) +erfc ( ( 2n+1 ) Δz2+z4DFREEΔτ ) ]}dz However , this analytical expression overestimates the fraction lost since there is a significant probability that a molecule that briefly contacted or exceeded the boundary re-enters the axial detection range . The re-entry probability depends on the number of gaps allowed in the tracking ( g ) , Δτ , and Δz and can be approximately accounted for by considering a corrected axial detection range , Δzcorr , larger than Δz: Δzcorr>Δz:Δzcorr ( Δz , Δτ , DFREE , g ) =Δz+a ( Δz , Δτ , g ) DFREE+b ( Δz , Δτ , g ) Although Δzcorr 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 one-dimensionally along the z-axis drawn from a uniform distribution from zMIN=−Δz/2 to zMAX=Δz/2 . Next , using a time-step Δτ , one-dimensional Brownian diffusion was simulated along the z-axis using the Euler-Maruyama scheme . For time delays from 1Δτ to 15Δτ , the fraction of molecules that were lost was calculated in the range of D=[1;12] μm2/s . a ( Δz , Δτ , g ) and b ( Δz , Δτ , g ) were then estimated through least-squares fitting of Premaining ( Δτ , Δzcorr , D ) to the simulated fraction remaining . The process was repeated over a grid of plausible values of ( Δz , Δτ , g ) to derive a grid of 134 , 865 ( a , b ) parameter pairs . This pre-calculated library of ( a , b ) parameters enables Spot-On to perform model fitting on nearly any SPT dataset with minimal overhead . Thus , the 2-state model Spot-On uses for kinetic modeling of SPT data is given by:P2 ( r , Δτ ) =FBOUNDr2 ( DBOUNDΔτ+σ2 ) e−r24 ( DBOUNDΔτ+σ2 ) +ZCORR ( Δτ , Δzcorr , DFREE ) ( 1−FBOUND ) r2 ( DFREEΔτ+σ2 ) e−r24 ( DFREEΔτ+σ2 ) where:ZCORR ( Δτ , Δzcorr , DFREE ) =1Δzcorr∫−Δzcorr2Δzcorr2{1−∑n=0∞ ( −1 ) n[erfc ( ( 2n+1 ) Δzcorr2−z4DFREEΔτ ) +erfc ( ( 2n+1 ) Δzcorr2+z4DFREEΔτ ) ]}dz Having derived the 2-state model , generalization to a 3-state model with 1 bound and 2 diffusive states is straightforward . If the three subpopulations have diffusion constants DBOUND , DSLOW , DFAST , and fractions FBOUND , FSLOW , FFAST , such that FBOUND+FSLOW+FFAST=1 , then the 3-state model considered by Spot-On becomes:P3 ( r , Δτ ) =FBOUNDr2 ( DBOUNDΔτ+σ2 ) e−r24 ( DBOUNDΔτ+σ2 ) +ZCORR ( Δτ , Δzcorr , DSLOW ) FSLOWr2 ( DSLOWΔτ+σ2 ) e−r24 ( DSLOWΔτ+σ2 ) +ZCORR ( Δτ , Δzcorr , DFAST ) ( 1−FBOUND−FSLOW ) r2 ( DFASTΔτ+σ2 ) e−r24 ( DFASTΔτ+σ2 ) Where ZCORR ( Δτ , Δzcorr , D ) is as described above . Spot-On calculates the empirical histogram of displacements based on a user-defined bin width . Spot-On allows the user to choose between PDF- and CDF-fitting of the kinetic model to the empirical displacement distributions; CDF-fitting is generally most accurate for smaller datasets and the two are similar for large datasets ( Figure 3—figure supplement 9 ) . The integral in ZCORR ( Δτ , Δzcorr ) 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 non-linear least squares algorithm ( Levenberg-Marquardt ) . Random initial parameter guesses are drawn uniformly from the user-specified parameter range . The optimization is then repeated several times with different initialization parameters to avoid local minima . Spot-On constrains each fraction to be between 0 and 1 and for the sum of the fractions to equal 1 . Although Spot-On performs well on both experimental and simulated SPT data , the model implemented by Spot-On 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 , Spot-On 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 signal-to-background 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/slow-moving particles . In that case , one would expect Spot-On 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 DfreeΔτ>> σ2 ( 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 σ 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 Spot-On 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 ) , Spot-On 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 time-scale of tens to a few hundreds of milliseconds , Spot-On may not be appropriate . Fifth and finally , Spot-On ignores correlations between adjacent displacements , although taking such information into account can potentially improve the parameter inference ( Vestergaard et al . , 2014 ) . Halo-Sox2 ( Teves et al . , 2016 ) knock-in 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 pre-coated with a 0 . 1% autoclaved gelatin solution ( Sigma-Aldrich , St . Louis , MO , G9391 ) under feeder free conditions in knock-out 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 Penicillin-streptomycin ( ThermoFisher #15140122 ) , 4 . 6 μL 2-mercapoethanol ( Sigma-Aldrich 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 . Halo-3xNLS , H2B-Halo-SNAP and knock-in C32 Halo-CTCF ( 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 Penicillin-streptomycin ( ThermoFisher #15140122 ) ) and were passaged every 2–4 days before reaching confluency . For live-cell 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 ( MCO-18AIC ( UV ) ) at 37°C/5 . 5% CO2 . 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 . The indicated cell line was grown overnight on plasma-cleaned 25 mm circular no 1 . 5H cover glasses ( Marienfeld , Germany , High-Precision 0117650 ) either directly ( U2OS ) or MatriGel coated ( mESCs; Fisher Scientific , Hampton , NH , #08-774-552 according to manufacturer’s instructions just prior to cell plating ) . After overnight growth , cells were labeled with 5–50 nM PA-JF549 or PA-JF646 ( 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 red-free medium keeping all other aspects of the medium the same . Single-molecule imaging was performed on a custom-built Nikon TI microscope ( Nikon Instruments Inc . , Melville , NY ) equipped with a 100x/NA 1 . 49 oil-immersion TIRF objective ( Nikon apochromat CFI Apo TIRF 100x Oil ) , EM-CCD camera ( Andor , Concord , MA , iXon Ultra 897; frame-transfer mode; vertical shift speed: 0 . 9 μs; −70°C ) , a perfect focusing system to correct for axial drift and motorized laser illumination ( Ti-TIRF , 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% CO2 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 PA-JF549; 633 nm ( 1 W , Genesis Coherent , Pala Alto , CA ) for PA-JF646; 405 nm ( 140 mW , OBIS , Coherent ) for all photo-activation experiments . The excitation lasers were modulated by an acousto-optic Tunable Filter ( AA Opto-Electronic , France , AOTFnC-VIS-TN ) 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 multi-band dichroic ( 405 nm/488 nm/561 nm/633 nm quad-band , Semrock , Rochester , NY ) and then focused in the back focal plane of the objective . Fluorescence emission light was filtered using a single band-pass filter placed in front of the camera using the following filters: PA-JF549: Semrock 593/40 nm bandpass filter; PA-JF646: Semrock 676/37 nm bandpass filter . The microscope , cameras , and hardware were controlled through NIS-Elements software ( Nikon ) . The spaSPT experimental settings for Figure 4G–H were as follows: 1 ms 633 nm excitation ( 100% AOTF ) of PA-JF646 was delivered at the beginning of the frame; 405 nm photo-activation 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 motion-blur spaSPT experiments ( Figure 4I–K ) , the camera exposure was fixed to 9 . 5 ms and photo-activation performed as above . To keep the total number of delivered photons constant , we generated an AOTF-laser 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 custom-written Matlab implementation of the MTT-algorithm ( Sergé et al . , 2008 ) and the following settings: Localization error: 10-6 . 25; deflation loops: 0; Blinking ( frames ) : 1; max competitors: 3; max D ( μm2/s ) : 20 . The spaSPT trajectory data was then analyzed using the Matlab version of Spot-On ( 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]; We 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 β . Then , the particle diffuses in 3D until it bleaches . Diffusion is simulated by drawing jumps following a normal law of parameters N ( 0 , 2DΔτ ) , where D is the diffusion coefficient and Δτ the exposure time . Finally , a localization error ( N0 , σ ) is added to each ( x , y , z ) localization in the simulated trajectories . For comparisons of Spot-On , MSDi and vbSPT using a 2-state 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 in-focus 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 Spot-On , vbSPT and MSDi are given in Appendix 1 . We also considered a 3-state 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 Spot-On and vbSPT ( Figure 3—figure supplement 6 ) , since the MSDi methods did not perform well . As in the 2-state 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 in-focus trajectories were recorded . We considered three different subpopulation conditions: ( 1 ) FBOUND = 25%; FSLOW = 25%; FFAST = 50%; ( 2 ) FBOUND = 25%; FSLOW = 50%; FFAST = 25%; ( 3 ) FBOUND = 50%; FSLOW = 25%; FFAST = 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 Spot-On ( all ) were constrained to three subpopulations . Full details on how the simulations were analyzed by Spot-On and vbSPT are given in Appendix 1 . All raw 1064 spaSPT experiments ( Figure 4 ) as well as the 3480 simulations ( Figure 3 ) are freely available in Spot-On 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 MSDi and vbSPT analysis as well as for generating the simulated data at: https://zenodo . org/record/835171 Spot-On is fully open-source . The web-interface can be found at: https://SpotOn . berkeley . edu . All raw code is available at GitLab: https://gitlab . com/tjian-darzacq-lab . The web-interface code can be found at https://gitlab . com/tjian-darzacq-lab/Spot-On; the Matlab command-line version of Spot-On can be found at: https://gitlab . com/tjian-darzacq-lab/spot-on-matlab; the Python command-line version of Spot-On can be found at https://gitlab . com/tjian-darzacq-lab/Spot-On-cli; the SPT simulation code ( simSPT ) can be found at: https://gitlab . com/tjian-darzacq-lab/simSPT; finally , the ‘TrackMate to Spot-On connector’ plugin , which adds an extra menu to TrackMate which allows one-click upload of datasets to Spot-On can be found at: https://gitlab . com/tjian-darzacq-lab/Spot-On-TrackMate
Single-particle 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 ‘Spot-On’ , an intuitive web-interface . Spot-On implements a kinetic modeling framework that accounts for known biases , including molecules moving out-of-focus , and robustly infers diffusion constants and subpopulations from pooled single-molecule trajectories . To minimize inherent experimental biases , we implement and validate stroboscopic photo-activation SPT ( spaSPT ) , which minimizes motion-blur bias and tracking errors . We validate Spot-On using experimentally realistic simulations and show that Spot-On outperforms other methods . We then apply Spot-On to spaSPT data from live mammalian cells spanning a wide range of nuclear dynamics and demonstrate that Spot-On consistently and robustly infers subpopulation fractions and diffusion constants .
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 . Single-particle 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 single-particle 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 pre-existing methods to improve the experimental set-up . 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 fast-moving 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 user-friendly algorithm package called Spot-On to correct for this issue . Spot-On was thoroughly benchmarked against other solutions , demonstrating both its accuracy and robustness . Single-particle 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 .