Author response: Recovering mixtures of fast-diffusing states from short single-particle trajectories
Year: 2022
Type: peer-review
Abstract: Article Figures and data Abstract Editor's evaluation Introduction Results Discussion Materials and methods Data availability References Decision letter Author response Article and author information Metrics Abstract Single-particle tracking (SPT) directly measures the dynamics of proteins in living cells and is a powerful tool to dissect molecular mechanisms of cellular regulation. Interpretation of SPT with fast-diffusing proteins in mammalian cells, however, is complicated by technical limitations imposed by fast image acquisition. These limitations include short trajectory length due to photobleaching and shallow depth of field, high localization error due to the low photon budget imposed by short integration times, and cell-to-cell variability. To address these issues, we investigated methods inspired by Bayesian nonparametrics to infer distributions of state parameters from SPT data with short trajectories, variable localization precision, and absence of prior knowledge about the number of underlying states. We discuss the advantages and disadvantages of these approaches relative to other frameworks for SPT analysis. Editor's evaluation This paper will be of interest to the cellular biologists who perform single-particle tracking experiments and develop new tracking methodologies. The authors investigate a new way of estimating an unknown number of diffusion states from short single-molecule trajectories. Ideas developed in the paper are likely to be used for further algorithm development. The authors give the users access to a repository on GitHub that contains comprehensive code that supports the paper. https://doi.org/10.7554/eLife.70169.sa0 Decision letter Reviews on Sciety eLife's review process Introduction Biological processes are driven by interactions between molecules. To understand the role of a molecular species in a process, a central challenge is to measure subpopulations of the molecule engaged in distinct interactions without perturbing the living system. Some interactions – such as complex formation – cause changes in a molecule’s mobility. As a result, live-cell single-particle tracking (SPT), by separately observing the motion of individual molecules, is a promising tool to meet this challenge (Shen et al., 2017). While SPT originally targeted proteins on cellular membranes, advances in the past two decades led to intracellular applications (Barak and Webb, 1982, Ghosh and Webb, 1994, Kubitscheck et al., 2000, Goulian and Simon, 2000). These include the use of stochastic labeling to isolate a single emitter’s path (Manley et al., 2008), a principle that can be extended into intracellular settings with genetically encoded photoconvertible proteins (Ando et al., 2002, Wiedenmann et al., 2004) or cell-permeable dyes (Grimm et al., 2015, Grimm et al., 2016). Another advance is pulsed or ‘stroboscopic’ excitation, which reduces blur associated with fast-diffusing emitters (Elf et al., 2007). Together with modifications of TIRF microscopes (Tokunaga et al., 2008), these techniques have facilitated the application of SPT to intracellular settings with fast-moving subpopulations (English et al., 2011, Persson et al., 2013, Izeddin et al., 2014, Normanno et al., 2015, Hansen et al., 2017). Following Manley et al., 2008, we refer to this experiment as ‘sptPALM’ (Figure 1A, Video 1). Figure 1 with 1 supplement see all Download asset Open asset Overview of sptPALM. (A) Schematic of experimental setup. An inclined illumination source is used in combination with a high-numerical aperture (NA) objective to resolve molecules in a thin slice in a cell. The excitation laser is pulsed to limit motion blur. Tracking yields a set of short trajectories (mean track length 3–5 frames). Trajectories shown are from a 7.48 ms tracking movie with retinoic acid receptor α-HaloTag (RARA-HaloTag) labeled with photoactivatable JF549 in U2OS nuclei. Asterisks in the movie frames mark particles at the edge of the focus. (B) Schematic of our inference problem. Each trajectory’s state is assumed to be a random draw from a distribution of state parameters. The goal is to recover this distribution from the observed trajectories. (C) Effects of particle mobility on trajectory length. RARA-HaloTag trajectories from U2OS nuclei were binned into five groups based on their mean squared displacement (MSD). Individual data points are the mean trajectory length of each group for three distinct knock-in clones of RARA-HaloTag (c156: 36961 trajectories, c239: 27543 trajectories, c258: 60347 trajectories); bar heights are the means across clones. Video 1 Download asset This video cannot be played in place because your browser does support HTML5 video. You may still download the video for offline viewing. Download as MPEG-4 Download as WebM Download as Ogg Example of sptPALM data. NPM1-HaloTag in U2OS osteosarcoma nuclei was labeled with 100 nM PA-JFX549-HTL for 5 min followed by washes (‘Materials and methods’), then imaged with a HiLo setup at 7.48 ms frame intervals with 1.5 ms excitation pulses. The pixel size after accounting for magnification is 160 nm. Dots and lines indicate the output of the detection and tracking algorithm; each trajectory has been given a distinct color. sptPALM experiments on fast-moving emitters in 3D settings pose several challenges for analysis (Hansen et al., 2018). First, apparent motion in sptPALM reflects both the true motion of the emitter and error associated with the estimate for its position (‘localization error’) (Martin et al., 2002, Matsuoka et al., 2009). Like fixed cell PALM and STORM microscopies (Betzig et al., 2006, Rust et al., 2006), the magnitude of localization error in sptPALM depends on the number of photons collected from each emitter (Thompson et al., 2002). But unlike fixed cell microscopies, sptPALM has another component of error due to motion blur, the convolution of the microscope’s point spread function with the path of the emitter. This component of error is not trivial: the mean 2D displacement of a Brownian particle with diffusion coefficient 10 μm2 s-1 during a 1 ms integration is ~180 nm, substantially larger than typical localization error in fixed cell PALM/STORM (Figure 1—figure supplement 1B). Consequently, localization error in sptPALM depends on both the emitter’s mobility and its distance from the focus and is not simple to measure (Kubitscheck et al., 2000, Berglund, 2010, Michalet and Berglund, 2012). Pulsed excitation can be used to reduce motion blur (Elf et al., 2007), but because the laser pulse still has nonzero duration (usually ≥1 ms), motion blur remains an important part of the measurement (Deschout et al., 2012, Lindén et al., 2017). Second, the high numerical aperture (NA) objectives required to resolve single emitters induce short depths of field, typically less than a micron. Whereas bacteria such as Escherichia coli are often small enough to fit into the resulting focal volume, mammalian cells – with depths ≥5–10 μm – cannot. As a result, intracellular SPT experiments only capture short transits of emitters through the focal volume, a behavior termed defocalization (Figure 1C, Video 2, Video 3; Kues and Kubitscheck, 2002, Mazza et al., 2012, Hansen et al., 2018). The duration of each transit depends on the emitter’s mobility. This creates a sampling problem: slow particles with long residences inside the focal volume contribute a few long trajectories, while fast particles with short residences contribute many short trajectories. Mean trajectory length is often as little as 3–4 frames, severely limiting the ability to infer dynamic parameters (such as diffusion coefficient) from any single trajectory. Fast multifocal imaging may mitigate this problem (Abrahamsson et al., 2013), but such methods currently require higher photon budgets and are not yet applicable to fast-diffusing targets with high motion blur. Meanwhile, the use of cylindrical optics to encode axial position in PSF astigmatism (Kao and Verkman, 1994), while popular in fixed cell PALM/STORM, is complicated in sptPALM by its resemblance to motion blur. Video 2 Download asset This video cannot be played in place because your browser does support HTML5 video. You may still download the video for offline viewing. Download as MPEG-4 Download as WebM Download as Ogg Illustration of defocalization for a single regular Brownian state. Trajectories were simulated in a 5 × 5 × 10 μm ellipsoid μm using the Euler–Maruyama scheme for regular Brownian motions with specular reflections at the ellipsoid boundaries. The diffusion coefficient for all trajectories was held constant at 2.0 μm2 s-1, while trajectories were randomly photoactivated at any point in the sphere and were subject to Poisson bleaching at 14 Hz. The left panel shows the 3D context of the trajectories, with dotted lines indicating the boundaries of the focal volume. The depth of the focal volume was 700 nm, which is roughly equivalent to the measured depth of field for our oil immersion objectives. The right panel shows the projection of the trajectories that coincide with the focal volume onto a hypothetical idealized camera. Notice that particles may make multiple transits through the focal volume that manifest as distinct trajectories. Video 3 Download asset This video cannot be played in place because your browser does support HTML5 video. You may still download the video for offline viewing. Download as MPEG-4 Download as WebM Download as Ogg Illustration of defocalization for multistate regular Brownian motion. Trajectories were drawn from two states – a fast state with diffusion coefficient 5.0 μm2 and a slow state with diffusion coefficient 0.05 μm2 s-1 – and simulated with a spherical nucleus with 5 μm radius. The left panel shows the trajectories in their native three dimensions while the right panel shows trajectories projected through the focal volume. Third, the true number of dynamic subpopulations or ‘states’ for a protein of interest is usually unknown a priori. Proteins often participate in many complexes with distinct dynamics. Model-dependent analyses that assume a fixed number of states (Mazza et al., 2012, Hansen et al., 2017, Hansen et al., 2018), while powerful when combined with complementary measurements (Izeddin et al., 2014, Hansen et al., 2020), are limited to measuring coefficients of known models. To compound model complexity, a protein may behave differently in distinct subcellular environments. Indeed, although sptPALM directly observes the spatial context for each trajectory (Xiang et al., 2020), analyses such as jump distribution modeling often discard this information by aggregating jumps across all subcellular locations. The central problem for sptPALM analysis is to recover the underlying dynamic states for a protein of interest given a set of observed trajectories in the presence of these three challenges. A common approach to recover subpopulations from sptPALM is to construct histograms of the mean squared displacement (MSD), the maximum likelihood estimator for the diffusion coefficient in the absence of localization error. The MSD is highly variable for short trajectories and, when used to estimate diffusion coefficient, becomes especially error-prone when the variance of localization error is unknown (Michalet and Berglund, 2012). More problematically, MSD histograms assume that sampling from slow and fast states with equal occupation produces the same number of trajectories, which leads to severe state biases in the presence of defocalization (Mazza et al., 2012, Hansen et al., 2018). Common preprocessing steps to select for long trajectories compound the problem by introducing biases for slow emitters that remain in focus. Methods based on least-squares fitting of the jump length cumulative distribution function (CDF) have interpreted sptPALM data with two- and three-state models while accounting for defocalization (Mazza et al., 2012, Hansen et al., 2018), but extend poorly to more complex models due to overfitting and do not provide a way to select between competing models. A different approach to model selection is represented by vbSPT, a variational Bayesian framework for reaction-diffusion models (Persson et al., 2013). vbSPT relies on the evidence lower bound to identify the number of states, and it excels at recovering occupations and transition rates for a small number of diffusing states from short trajectories. However, it is not appropriate to apply in situations where the target’s dynamic profile is not discrete and does not consider defocalization or localization error, although it can be complemented with a separate estimate of localization error (Lindén et al., 2017). As such, there is a need for methods that combine the advantages of Bayesian methods like vbSPT with a model that can accommodate nondiscrete dynamic profiles, while accounting for biases induced by sptPALM imaging geometry. Here, we examine two alternative methods for recovering an sptPALM target’s dynamic profile. The first is based on a Dirichlet process mixture model (DPMM) and the second on a finite state approximation to the DPMM that we refer to as a state array (SA). Exploring these techniques on simulated and real datasets, we find that although both DPMMs and SAs recover complex mixtures of states and can be applied to nondiscrete distributions of diffusion coefficients, SAs far outperform DPMMs due to their robustness to variable localization error variance. Both methods share the limitation that they do not deal with transitions between states. We investigate how this limitation affects apparent state occupations recovered with these methods. The SA method is publicly available as the pip-installable Python package saspt (source: https://github.com/alecheckert/saspt; Heckert, 2022c documentation: https://saspt.readthedocs.io/en/latest/). Results Two approaches to infer subpopulations in sptPALM datasets We considered how to infer dynamic subpopulations from the short, fragmented trajectories produced by sptPALM in a manner robust to the effects of localization error and defocalization (Figure 1). A simple and popular approach to this problem is to make a separate estimate for the parameters of each trajectory, then compile a histogram of the results. In the case of Brownian motion, we refer to this method as the ‘MSD histogram’ approach since the MSD is the maximum likelihood estimator for the diffusion coefficient of a Brownian motion with no localization error. Real estimates of a particle’s position, however, are invariably associated with localization error. In sptPALM, this problem is more significant due to motion blur, which increases the magnitude of the error (Figure 1—figure supplement 1). To incorporate these effects, we refer to the combination of regular Brownian motion with normally distributed, mean-zero localization error as ‘RBME’ (‘Materials and methods’). Each RBME is characterized by two parameters: the diffusion coefficient and the localization error variance. (For brevity, we refer to the latter simply as ‘localization error.’) Importantly, the increments of RBME are only Markovian when the localization error is zero (Martin et al., 2002; Figure 1—figure supplement 1). Because individual trajectories produced by sptPALM are usually too short to estimate localization error, and because it does not take into account other effects like defocalization, the MSD histogram approach is prone to large systematic biases (Michalet and Berglund, 2012, Hansen et al., 2018). While techniques exist to mitigate some biases of MSD fitting (Kepten et al., 2015), most are difficult to apply at the single trajectory level due to the small number of points per trajectory. A distinct approach is represented by Bayesian finite state mixture models (Marin et al., 2005, McLachlan et al., 2019; Figure 2A, Figure 2—figure supplement 1A). Such models are comprised of a collection of states labeled k=1,…,K. Each state is associated with an occupation τk (describing the probability to observe trajectories from that state) and a vector of state parameters θk (describing the kind of trajectories produced by that state). Importantly, θk can also incorporate measurement parameters like the localization error. The probability to observe a particular trajectory x is then ∑k=1KτkpX(x|θk), where pX(x|θk) is a distribution over trajectories produced by state k and depends on the type of motion being considered. The goal is to infer τk and θk for each state given some observed set of trajectories F. A challenge with such methods is choosing the number of states K as well as the high computational cost when pX(x|θ) is nonconjugate to the prior over θ. Figure 2 with 1 supplement see all Download asset Open asset Schematic comparison of finite state mixtures, Dirichlet process mixtures, and state arrays (SAs). (A) Finite state mixture models use a discrete set of K states. Challenges include estimating K and producing intelligible output when the underlying dynamic profile is not discrete. (B) Dirichlet process mixture models (DPMMs) address the problem of nondiscrete dynamic profiles by using a continuous distribution over state parameters. Inference routines are slow, so in this work we use approximative motion models. (C) SAs, a special case of the finite state mixture. SAs approximate DPMMs by using a discrete grid of state parameters and have a faster inference routine. Challenges with SAs include the choice of the parameter grid. Potential solutions can be found in the Bayesian nonparametric class of methods. These approaches begin with a single model comprising a very large or infinite collection of states. A Bayesian inference algorithm is then used to prune away superfluous complexity, leaving a sparse subset of states sufficient to explain the observed trajectories. The foundational example is the DPMM (Ferguson, 1973), which has the distinct advantage of being able to approximate essentially any mixture of states, discrete or continuous (Neal, 1992, Teh, 2010; Figure 2B). Its disadvantage is the high computational cost associated with inference, which becomes especially severe when considering types of motion with multiple parameters (such as RBME) (Neal, 2000, Andrieu et al., 2003). We considered two responses to this challenge. First, we constructed a DPMM that uses a cheap approximation to RBME by treating the RBME as a Markov process (Matsuda et al., 2018; Figure 3C). This assumption is strictly true only when the localization error is zero and is the same assumption used to estimate diffusion coefficient via the MSD (Michalet and Berglund, 2012). Because localization error is never actually zero, we were curious to see when and how this method breaks down. Figure 3 with 2 supplements see all Download asset Open asset Application of state arrays and Dirichlet process mixture models (DPMMs) to mixtures of Brownian (A) Brownian motion with localization error is a motion model that two parameters: diffusion coefficient and localization error variance. (For brevity, we refer to the latter simply as ‘localization error.’) Brownian motion, RBME has between jumps due to the of localization error. (B) array inference for The occupation estimate is the estimate for the which is through variational the of inference, we localization error to distributions over the diffusion (C) DPMM inference for mixtures of Brownian Because the sampling for a DPMM is slow, we use an approximative motion model that the of the in Example of state arrays on simulated sptPALM. Tracking was simulated in a spherical nucleus with 700 focal 14 bleaching 7.48 ms frame and variable localization error. The lines the state array mean occupations for of the same The second approach we is a model we refer to as a (SA). This model is a special case of the finite state by a large number of states K and the state parameters to the of an that some parameter (Figure Figure 2—figure supplement 1). the array for RBME a of diffusion coefficients and localization error An array for an diffusion model may also incorporate or more parameters. The occupation of each in this array is through a variational Bayesian the occupation of most states to zero to a set sufficient to explain the (‘Materials and methods’). Importantly, SAs infer a distribution over the state parameters with distributions for each trajectory. The of the variational inference algorithm means that the distribution is a mean of these We focus our on the distribution in this with some of the individual distributions for each trajectory at the Because the parameters for each state in an SA are the most can be and As a result, SAs can more complex models than In this we use a 2D SA for RBME a of diffusion coefficients and localization error inference, we the localization error part to of the diffusion coefficient (Figure This about localization error SAs more robust to in localization error than DPMMs (Figure supplement 1). DPMMs and SAs work with to of of trajectories. This often aggregating trajectories across multiple cells, which can cell-to-cell variability. To cell-to-cell we also found it to have a and estimate of state occupation that with a number of trajectories. This is from the SA and is simply the of the RBME likelihood function across all of the trajectories observed in a cell. We refer to this as the occupation it like a less of the SA method (‘Materials and methods’). to account for defocalization we developed a method applicable to the distributions of both DPMMs and SAs (Figure supplement 2, and methods’). DPMMs and SAs on simulated sptPALM data As the for inference, we considered a mixture of in a spherical with a thin focal volume the with dimensions to a mammalian cell and the sphere and are only observed when their coincide with the focal volume. Because no are during the is a highly fragmented set of trajectories with mean length 3–5 We settings to approximate real sptPALM with bleaching rates diffusion coefficients in the μm2 s-1, and localization error between and We the ability of SAs, and MSD histograms to recover the underlying distribution of diffusion coefficients from this data. We these into with In class localization error for all states was as a known constant to the (Figure Figure supplement 1A). In class 2, localization error was held constant for all states but was unknown to the (Figure Figure supplement 1B). In class localization error was to between states and was also unknown to the (Figure Figure supplement for class we simulated that incorporate localization error, motion blur, tracking and (Figure supplement Figure supplement In these the localization error is for each emitter and depends on the emitter’s axial position, the stochastic number of photons it during each and its of motion blur Video Figure with supplements see all Download asset Open asset of the mean squared displacement Dirichlet process mixture model and state array methods to recover dynamic profiles from trajectory of diffusing states were simulated in a 700 focal volume with 7.48 ms frame were into three of based on the of localization error as in the each trajectories were occupations for five are on each of state occupation estimates for each method as a function of Each method was on trajectory from an underlying three-state dynamic model μm2 s-1 μm2 s-1 5.0 μm2 s-1 then occupations were by the distribution produced by each of integration were set to μm2 s-1 μm2 s-1 or μm2 s-1 were per Mean error in state occupation estimates for the in Each is the across all mixtures of diffusing states with diffusion coefficients using each a of trajectories were simulated with the underlying state of state transitions on the and SA We the transition constant between two diffusing states, trajectories per Video Download asset This video cannot be played in place because your browser does support HTML5 video. You may still download the video for offline viewing. Download as MPEG-4 Download as WebM Download as Ogg Example of a simulated SPT Two Brownian states with diffusion coefficients and 5.0 μm2 s-1 were imaging was simulated with settings to our experimental SPT an objective with numerical aperture immersion with image pixel size frame 7.48 and 2 ms excitation pulses. were using the Video 5 Download asset This video cannot be played in place because your browser does support HTML5 video. You may still download the video for offline viewing. Download as MPEG-4 Download as WebM Download as Ogg SPT at variable excitation pulse of Brownian motions were simulated as in Video we used or ms excitation and a frame of The mixture diffusing states with the diffusion μm2 s-1 μm2 s-1 μm2 s-1 and μm2 s-1 DPMMs and SAs both recovered the dynamic profile for in class 1 with a that the MSD histogram large of trajectories, DPMMs and SAs nondiscrete distributions of states (Figure Figure supplement 1A). knowledge of the localization error was 2 and the SA approach both the MSD and DPMM The was especially when the of diffusion and error to jump variance were likely due to its of localization error. Meanwhile, the dynamic profile by SAs was by in the localization error (Figure and Figure supplement and the from in class 3 we found that the mean squared of the from the true was for SAs, while it was for both the MSD histogram and DPMM approaches (Figure supplement The dynamic profiles produced by the and SA approaches can be to occupation estimates over particular diffusion coefficient We the and of these estimates with discrete or models (Figure Figure supplement Figure supplement As the number of trajectories occupations by DPMMs and SAs to of the true In the MSD approach was associated with large systematic an (Mazza et al., 2012, Hansen et al., 2018). and dynamic in class SAs also the DPMM approach (Figure supplement Figure supplement the was for small diffusion coefficients, for which the DPMM state occupation estimates were severely Both methods difficult
Source: eLife
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $2,500
OpenAlex ID: W4294732812