Resolving Cytosolic Diffusive States in Bacteria by Single-Molecule Tracking

Abstract

The trajectory of a single protein in the cytosol of a living cell contains information about its molecular interactions in its native environment. However, it has remained challenging to accurately resolve and characterize the diffusive states that can manifest in the cytosol using analytical approaches based on simplifying assumptions. Here, we show that multiple intracellular diffusive states can be successfully resolved if sufficient single-molecule trajectory information is available to generate well-sampled distributions of experimental measurements and if experimental biases are taken into account during data analysis. To address the inherent experimental biases in camera-based and MINFLUX-based single-molecule tracking, we use an empirical data analysis framework based on Monte Carlo simulations of confined Brownian motion. This framework is general and adaptable to arbitrary cell geometries and data acquisition parameters employed in two-dimensional or three-dimensional single-molecule tracking. We show that, in addition to determining the diffusion coefficients and populations of prevalent diffusive states, the timescales of diffusive state switching can be determined by stepwise increasing the time window of averaging over subsequent single-molecule displacements. Time-averaged diffusion analysis of single-molecule tracking data may thus provide quantitative insights into binding and unbinding reactions among rapidly diffusing molecules that are integral for cellular functions.

Introduction

The ability to probe the positions and motions of single molecules in living cells has made single-molecule localization and tracking microscopy a powerful experimental tool to study the molecular basis of cellular functions (1, 2, 3). Single-molecule trajectories, if sampled in sufficient numbers, provide the distribution of molecular motion behavior in cells, and the statistical analyses of localization and trajectory data have been used to resolve the prevalent diffusive states as well as their population fractions. A key benefit of tracking single molecules is that individual trajectories can be sorted according to predefined (quality) metrics (for example, to include only nonblinking molecules (4) or molecules localized in specific subcellular regions of interest (5)). These advantages are not shared by ensemble-averaged measurements, such as fluorescence recovery after photobleaching and fluorescence correlation spectroscopy (6).

Bacteria are ideally suited specimens for single-molecule localization and tracking microscopy. Unlike eukaryotic cells, the small size of bacteria (∼1 μm in diameter) guarantees that all molecules remain in focus during imaging (7), in particular when the microscope uses an engineered three-dimensional (3D) point-spread function (PSF), such as an astigmatic (8) or a double-helix PSF (DHPSF) (9, 10). Early applications of single-molecule localization microscopy in bacteria focused on differentiating stationary versus freely diffusing molecules and quantifying the relative population fractions and lifetimes of these diffusive states. For example, DNA-bound lac repressors in search of their promoter region appear stationary at 10-ms frame rates and can thus be clearly distinguished from unbound lac repressors that explore the entire Escherichia coli cell volume on the same timescale (11). Similarly, the E. coli chromosome-partitioning protein MukB forms stationary clusters only when incorporated into the quasistatic DNA-bound structural maintenance of chromosomes complex (12). In both of these cases, the stationary, DNA-bound states represent the biologically active form of the protein, whereas the unbound diffusive state represents the inactive protein. However, other proteins, in particular those involved in delocalized regulatory and signaling networks, may not exhibit such stationary states. These proteins may instead form oligomeric complexes that diffuse at measurably different rates (13, 14, 15, 16, 17). A major objective for single-molecule tracking microscopy is therefore to resolve the different diffusive states that manifest in the cytosol of living cells.

Assigning a single molecule to a specific diffusive state is challenging, especially for fast-diffusing cytosolic species. The molecular displacements measured in single-molecule tracking can be used to compute apparent diffusion coefficients for each detected single molecule, but such estimates are prone to large errors, particularly when the trajectories are short and the number of available molecular displacements is low (15, 18). Short trajectories (<20 displacements) are the norm in camera-based single-molecule tracking with genetically encodable fluorescent protein labels. However, genetically encoded fluorescent proteins offer unmatched labeling specificity and efficiency and therefore remain preferable when off-target labeling with chemical dyes may lead to artifacts (19). For slowly diffusing molecules in bacteria, it is possible to resolve multiple diffusive states by fitting the experimentally measured distributions of molecular displacements, r, or apparent diffusion coefficients, D^∗, using analytical equations describing Brownian (i.e., normal) diffusion (15, 18, 20, 21, 22). Such analytical approaches produce acceptable results only if biomolecular motion is slow enough that confinement effects can be ignored. However, a typical cytosolic protein undergoing Brownian diffusion at a rate D = 10 μm²/s can traverse the entire width of a rod-shaped bacterial cell in as little as 10–25 ms. As a result, the observed motion of cytosolic proteins in bacteria is strongly confined by the cell boundaries, and molecular displacements will, on average, be smaller than those expected for unconfined diffusion. Approaches assuming unconfined Brownian motion are therefore not suitable when tracking fast diffusing molecules in the cytosol of bacterial cells.

Several approaches have been developed in recent years to extract the diffusion rates and population fractions of different diffusive states that manifest for unbound molecules in confined cellular environments. These approaches account for confinement effects by the cell boundaries either (semi-) analytically (23, 24, 25, 26) or numerically through a Monte Carlo simulation of Brownian diffusion trajectories (7, 13, 17, 27, 28). To date, there is no clear consensus in the field as to what the most effective imaging and analysis methods for resolving different diffusive states are. For example, should tracking data be acquired in two or three dimensions? Should the distributions of displacements or apparent diffusion coefficients be analyzed? How should localization precision and localization accuracy be taken into account? In addition, determining the transition rates between diffusive states has proven to be a challenging problem (15, 26). To address these questions and challenges, we test and experimentally validate a numerical analysis framework based on Monte Carlo simulations for both two-dimensional (2D) and 3D single-molecule tracking in bacterial cells (Fig. 1). By the explicit modeling of confinement and the motion blur of diffusing molecules inside small bacterial cells, we directly account for single-molecule localization errors that themselves depend on the molecule’s diffusion rates and are therefore difficult to treat analytically. We validate the method by extracting the unconfined diffusion coefficients for two genetically encoded fluorescence proteins, enhanced yellow fluorescent protein (eYFP) and mEos3.2, in living Yersinia enterocolitica cells. Using simulated 2D or 3D single-molecule tracking data of a known diffusive state composition, we quantify to what extent two or more simultaneously present diffusive states can be resolved by the numerical fitting of the displacement or apparent diffusion coefficient distributions. Finally, we consider the influence of dynamic transitions between different diffusive states that may manifest upon the association and dissociation of freely diffusing molecules. We propose a new approach, termed time-averaged diffusion (TAD) analysis, to determine the timescales of association and dissociation dynamics between diffusing molecules. We apply TAD analysis to both camera-based single-molecule tracking and MINFLUX tracking. MINFLUX microscopy is a recently developed method for single-molecule localization and tracking, which offers 10 times more localizations per track at 10–100 times higher temporal resolution by minimizing the number of photons necessary for each localization (29). We conclude that quantitative numerical analysis of 2D and 3D single-molecule trajectories, as described here, can provide accurate estimations of diffusion rates, population fractions, and interconversion rates of prevalent intracellular diffusive states. Such information is crucial for investigating the dynamic molecular-level events that regulate the functional outputs of signaling and control networks in living cells.

Figure 1.

Diagram of numerical diffusion fitting analysis workflow. Experimental and simulated data are analyzed using the same data processing routines so that experimentally determined apparent diffusion coefficient (or displacement) distributions can be analyzed using linear combinations of simulated distributions.

Materials and Methods

Super-resolution fluorescence imaging setup

Experiments were performed on a custom-built dual-color inverted fluorescence microscope based on the RM21 platform (Mad City Labs, Madison, WI). Immersion oil was placed between the objective lens (UPLSAPO 100× 1.4 numerical aperture) and the glass coverslip (number 1.5, 22 mm × 22 mm; VWR, Radnor, PA). A 514 nm laser (Genesis MX514 MTM; Coherent, Santa Clara, CA) was used for the excitation of eYFP (∼350 W/cm²), and a 561 nm laser (Genesis MX561 MTM; Coherent) was used for the excitation of mEos3.2 (∼350 W/cm²). A 405 nm laser (OBIS 405 nm LX; Coherent) was used to activate mEos3.2 (∼20 W/cm²) simultaneously with 561 nm excitation. Single-molecule images were obtained by utilizing eYFP photoblinking (30) and mEos3.2 photoswitching. Under these imaging conditions, more than 50% of the imaged cells underwent cell division on the coverslip (Fig. S1). Zero-order quarter-wave plates (WPQ05M-405, WPQ05M-514, WPQ05M-561; Thorlabs, Newton, NJ) were used to circularly polarize all excitation lasers. The spectral profile of the 514 nm laser was filtered using a bandpass filter (ET510/10 bp; Chroma Technology, Bellows Falls, VT). Fluorescence emission was passed through a shared filter set (LP02-514RU-25, Semrock NF03-561E-25, and Chroma ET700SP-2P8; Semrock, Rochester, NY). A dichroic beam splitter (Chroma T560lpxr-uf3) was then used to split the emission pathway into “green” and “red” channels to image eYFP and mEos3.2, respectively. An additional 561 nm notch filter (Chroma ZET561NF) was inserted into the “red” channel to block scattered laser light. Each emission path contains a wavelength-specific dielectric phase mask (Double Helix, Boulder, CO) that is placed in the Fourier plane of the microscope to generate a DHPSF (10, 31). The fluorescence signals in both channels were detected on two separate sCMOS cameras (ORCA-Flash 4.0 V2; Hamamatsu, Bridgewater, NJ). Up to 20,000 frames were collected per field of view with an exposure time of 25 ms. Exposure times of 25 ms were used for all experiments to maximize the fluorescent signal to background ratio (32). A flip mirror in the emission pathway enables toggling the microscope between fluorescence imaging and phase contrast imaging modes without having to change the objective lens of the microscope.

Raw data processing

Raw single-molecule PSF images were processed and analyzed using MATLAB (The MathWorks, Natick, MA). Standard PSF images were analyzed using centroid estimation (33). DHPSF images were analyzed using a modified version of the easyDHPSF code (34). Specifically, the maximal likelihood estimation based on a double-Gaussian PSF model was used to extract the 3D localizations of single-molecule emitters (35). For experimental data, the background was estimated using a median filter with a time window of 10 frames (36).

To assign localizations to individual cells, cell outlines were generated based on the phase contrast images using the open-source software OUFTI (37). The cell outlines were transformed to overlay on the fluorescence data by a two-step 2D affine transformation using the “cp2tform” function in MATLAB. First, five control point pairs were manually selected by estimating the position of the cell poles of the same five cells in both the single-molecule localization data and cell outlines. A rough transformation was generated, and cell outlines containing less than 10 localizations within their boundaries were removed. In addition, cells positioned partly outside the field of view were manually removed so that they do not skew the final transformation. The center of mass for all remaining cell outlines and single-molecule localizations within them then served as a larger set of control point pairs to compute the final transformation function. Only localizations that lay within the cell outlines after transformation were considered for further analysis.

Single-molecule tracking analysis

Molecular displacements were computed as the Euclidean distance between subsequent localizations of the same molecule using a distance threshold of 2.5 μm. Displacements were linked into trajectories and considered for further analysis only if at least three subsequent (i.e., localizations in adjacent frames) displacements were available. In addition, if two or more localizations were present in the cell simultaneously during the length of the trajectory, the trajectory was discarded. These steps minimized the misassignment of two or more molecules to the same trajectory (38).

To obtain apparent diffusion coefficients for a given trajectory, its mean-squared displacement (MSD) was calculated using

$$MS{D}_N=\frac{1}{N-1}\sum _{n=2}^N{\left(x_n-x_{n-1}\right)}^2,$$ (1)

where N is the total number of localizations in the trajectory, and x_n is the position of the molecule at time point n. The apparent diffusion coefficient, D^∗, was then computed by

$$D^{\ast }=\frac{MSD}{2\cdot m\cdot \mathit{\Delta }t},$$ (2)

where m = 2 or 3 is the dimensionality, and Δt = 25 ms is the camera exposure time used in all our experiments and simulations. We note that the estimated single-step apparent diffusion coefficients and displacements do not directly take into account static and dynamic localization errors (18) or the effect of confinement within the bacterial cells. We instead accounted for these effects through the explicit simulation of experimental data, as described in the following section.

Monte Carlo simulations for camera-based tracking

The calculation of the apparent diffusion coefficients for a large number of tracked molecules will result in a distribution of values, even if the molecular diffusion is governed by a single diffusive state. In addition, for confined diffusion within small bacterial cell volumes, the movement of molecules is restricted in space. Such confinement results in an overall left shift of the apparent diffusion coefficient distributions for a given diffusive state (Fig. 2 a, dashed lines). The shape of the confined distribution is dependent on the size and shape of the confining volume.

Figure 2.

Monte Carlo simulations of expected experimental distribution. (a) Probability density functions show the effect of spatial confinement. The apparent diffusion coefficients are computed based on the time-integrated (25 ms) center-of-mass coordinates of simulated particles undergoing Brownian diffusion in a cylindrical volume (radius = 0.4 μm, length = 5 μm). The confined distributions are left shifted (dashed lines) compared to the unconfined distributions. (b) The fraction of successfully localized single molecules. Time-integrated (25 ms) single-molecule fluorescence signals produce images that resemble PSFs that are blurred to different extents (insets). Faster moving molecules are localized less efficiently because of motion blurring. (c) Shown are the expected distributions of apparent diffusion coefficients when confinement and motion blur is taken into account. The similarity of the distributions increase for faster diffusion coefficients. (a) and (c) are adapted from (4). To see this figure in color, go online.

To generate libraries of simulated distributions for arbitrary diffusion coefficients, we performed Monte Carlo simulations of confined Brownian motion inside the volume of a cylinder using a set of 64 diffusion coefficients ranging from 0.05 to 20 μm²/s as input parameters. The size of the confining cylinder was chosen to match the average size of a typical rod-shaped bacterial cell (radius = 0.4 μm, length = 5 μm). The apparent diffusion coefficient distribution for the average cell length and width are not different from the distribution arising from a population-weighted distribution of cell sizes (Document S1. Figs. S1–S8 and Table S1, Document S2. Article plus Supporting Material). The starting position of the trajectory was randomly set within the volume of the cylinder, and Brownian motion was simulated using short time intervals of 100 ns. If a molecule was displaced outside of the volume of the cylinder within a time step, it was redirected back toward the inside of the cylinder at a random angle. Choosing a short time step ensured that the entire volume of the cylinder, including the interfacial region near the cell boundary, could be sampled by the diffusing molecule. Although a rod-shaped bacterium more closely resembles a spherocylindrical shape, the apparent diffusion coefficient distributions in a spherocylinder were indistinguishable from those in a cylinder of the same length and radius (Fig. S2 b).

To simulate the raw experimental observable, we generated noisy, motion-blurred single-molecule images. For 2D simulations, we summed 50 standard PSFs (approximated as 2D Gaussians with full width at half maximum [FWHM] ∼325 nm) corresponding to 50 periodically sampled positions of a fluorescent emitter during the camera exposure time (25 ms). Similarly, for 3D simulations, we summed 50 DHPSFs. Because the DHPSF has a larger cross section than the standard PSF, fewer photons are necessary for localizing emitters in 2D. To match photon counts measured experimentally, we scaled the photon count of each simulated image to 500 photons per localization for the standard PSF and 1000 photons per localization for the DHPSF. To normalize to the total photon budget, we simulated 3D trajectories with five displacements (3D) and 2D trajectories with 11 displacements. To each simulated frame, we added a laser background of ∼13 photons/pixel and introduced Poisson noise based on the final photon count in each pixel. A dark offset (50 photons/pixel on average) with Gaussian-read noise (σ ∼1.5 photons) was added as well to produce the final image. The resulting image was then multiplied by the experimentally measured pixel-dependent gain of our sCMOS camera to obtain an image in units of detector counts.

By explicitly simulating spatially blurred emission profiles with realistic signal-to-noise ratios, we can account for both static and dynamic localization error (Fig. S3). Static localization error is the result of finite numbers of fluorescence signal photons that provide an imprecise measure of the PSF shape and thus result in single-molecule localizations of limited precision (1). Dynamic localization errors manifest for moving emitters that generate motion-blurred images on the detector (Fig. 2 b, inset). When analyzed using common fitting algorithms (which are based on data fitting to well-defined PSF shapes), motion-blurred images provide 2D or 3D position estimates with limited accuracy and precision (39). If the motion blur is too severe, then the PSF of the molecule may become too distorted to result in a successful fit. Motion blur therefore limits the detection efficiency of fast-diffusing molecules (Fig. 2 b).

We simulated n = 5000 single-molecule trajectories for each of the 64 input diffusion coefficients to obtain 5000 apparent diffusion coefficient estimates and 5 × 5000 = 25,000 molecular displacements (3D data) or 11 × 5000 = 55,000 molecular displacements (2D data). The corresponding probability density functions PDF(D^∗) and the empirical cumulative distribution functions CDF(D^∗), or alternatively PDF(r) and CDF(r), were then smoothed by B-spline interpolation of order 25 and normalized individually (Figs. 2 c, S4, and S5). The interpolated distributions were then interpolated again along the D-axis (D is the unconfined (input) diffusion coefficient) using the “natural” interpolation method in the “scatteredInterpolant” MATLAB function. This two-step interpolation provides a continuous function that provides the experimentally expected distribution for any species whose Brownian motion is governed by a diffusion coefficient value in the range of 0.05 and 20 μm²/s. The simulated distributions account for the effects of molecular confinement due to the cell boundaries, signal integration over the camera exposure time, and the experimentally calibrated signal-to-noise levels.

Data fitting

To estimate the number of diffusive states, their diffusion coefficients, and their population fractions, we fit the experimentally measured cumulative distribution functions using linear combinations of simulated CDF(r) or CDF(D^∗). Using the CDF for fitting instead of a PDF histogram eliminates bin-size ambiguities that can bias the fitting results. To determine the number of diffusive states, we performed a constrained linear least-squares fit (using the “lsqlin” function in MATLAB) and a periodically sampled array of simulated CDFs. We combined diffusive states that had diffusion coefficient values within 20% of each other into a single diffusive state by a weighted average based on their population fractions. The resulting vector of fitting parameters, consisting of diffusion coefficients of individual diffusive states and their respective population fractions, was used as a starting point to create arrays of trial fitting parameter vectors with different numbers of diffusive states, ranging from a single diffusive state to a user-defined maximal number of states (five in all cases considered here). We generated the trial parameter vectors as follows: We either combined adjacent diffusive states through weighted averaging, or we split diffusive states into two states with equal population fractions and diffusion coefficient 20% above and below the original value. We considered all state combination and splitting possibilities. We used each trial vector as a starting point for nonlinear least-squares fitting of five separate subsets of the data (using the “fmincon” function in MATLAB). In each case, the quality of the fit (quantified as the residual sum of squares) was found by comparing the quality of the fit with respect to the remaining subsets (data cross-validation). The average residual sum of squares was used to quantify the quality of the fit corresponding to a given trial vector. This method yielded multiple trial vectors given the number of diffusive states.

For each number of diffusive states, only the trial vector with the best quality of fit was retained. The optimal number of states was then determined by identifying the last trial vector for which adding an additional state resulted in at least a 5% improvement in the quality of the fit. Finally, this trial vector was then used as the starting point to fit the full data set using nonlinear least-squares fitting. To estimate the error in each of the fitted parameters, we resampled the data set 100 times by bootstrapping and then fit them individually, initializing the fit with the same starting parameter vector. To limit the number of parameters for nonlinear least-squares fitting, the population fractions of diffusive states below 0.5 μm²/s were not refined but instead assigned to slowly diffusing or stationary molecules. This choice was made because we focus here on fast cytosolic diffusion. Depending on the biological system, however, this threshold may need to be lowered further. For simplicity, all data and fits are displayed as PDFs instead of CDFs throughout this manuscript.

Simulation of MINFLUX trajectories

To simulate experimental tracking data obtained by MINFLUX microscopy, we first computed 3D isotropic Brownian motion trajectories, sampled at a high time resolution and confined within a cylinder of length l = 5 μm and radius r = 0.4 μm (same as for camera-based tracking). The short time step for each displacement was 1 μs, and the total trajectory length was 20 ms. We assumed exponentially distributed fluorescence blinking on- and off-times with t_on = 2 ms and t_off = 0.6 ms, in agreement with experimental measurements of the fluorescent protein mEos2 (29). As before, we simulated 5000 trajectories for 64 diffusion coefficients in the range of D ∈ [0.05, 15] μm²/s to create libraries of distributions used for the fitting of simulated experimental data. We then projected the 3D motion trajectories onto the xy plane and tracked the blinking emitters using a doughnut intensity profile scanned over the emitter using a four-step multiplex cycle, as described previously (29). The doughnut size parameter was set to FWHM = 800 nm, and the field-of-view scanning parameter was set to L = 400 nm. Choosing larger values for FWHM and L minimizes the probability of fast-moving emitters (D > 5 μm²/s) escaping from the MINFLUX observation region during tracking. The multiplex cycle time was Δt = 200 μs. To account for motion blurring during a multiplex cycle, we considered the excitation and emission probabilities from each of the computed emitter positions (sampled at 1 μs time steps). The detected photon counts were assumed to follow Poisson statistics. Emitter localization was performed with the previously described modified least mean-squared estimator (29), with k = 2, β₀ = 0.96, and β₁ = 5.75. The resulting trajectories each had 100 localizations, which were sampled every 200 μs.

Modeling state transition simulation

To address the effect of a dynamic equilibrium between two diffusive states, we simulated trajectories for which one or more state transitions take place during a single-molecule trajectory. 3D state-switching trajectories were simulated with the track lengths of five displacements. 2D MINFLUX state-switching trajectories were simulated with the track lengths of 99 displacements. We considered a two-state system in which molecules spend equal amounts of time in each state, resulting in a populations fraction of 50% for each state. The average time, T, that a molecule takes to switch from one state to the other and back again is

$$T=t_1+t_2,$$ (3)

where t₁ and t₂ are the average time spent in states 1 and 2, respectively. The state-switching kinetics were modeled as follows: Each individual molecule trajectory randomly started in one of the two states. The time t spent in a given state before transitioning to the other was modeled as the exponential decay

$$p\left(t\right)=e^{-\frac{t}{t_1}}.$$ (4)

Thus, the time spent in a given state is given by

$$t=-\ln \left(p\left(t\right)\right)\cdot t_1,$$ (5)

where the value of p(t) was a value between 0 and 1 randomly chosen from a uniform distribution.

This process was repeated, allowing the molecule to switch back and forth between the two states, until the total amount of time reached the total length of the trajectory. State-switching trajectories were then simulated for camera-based or MINFLUX-based tracking as described above.

TAD analysis of MINFLUX simulations

We quantified two observables based on the mean percentage error and mean percentage deviation curves of MINFLUX state-switching simulations (Fig. 7). We first quantified the averaging time point, N_i,T, after which the mean percentage error/deviation increases linearly as a function of averaging time, N_i. This point was found by smoothing the respective curves by interpolation and then taking the derivative of the result. The value of N_i,T was defined as the shortest averaging time that had a positive derivative, a positive derivative for each of the following 10 averaging times, and a negative derivative for the previous averaging time. In addition, if the first five time points of the curve had slopes above one, the first time point was chosen. If these steps did not produce a satisfactory N_i,T (because of a noisy curve at short time points), a manual time point was input as an initial guess. The N_i,T was then found by increasing the initial guess until an averaging time was found that had 10 consecutive averaging times with a greater mean percentage error/deviation. The linear section of the curve was found by fitting a linear function to the curve between N_i,T and the longest averaging time point. If the fitting resulted in a root mean-squared error greater than a threshold of 0.5, the section of the curve being fit was decreased by one time point. This step was repeated until the linear fit produced a root mean-squared error below the threshold value (i.e., only the linear portion of the curve was fit). This process was used to avoid fitting the nonlinear asymptotic portion of the curve.

Figure 7.

Resolving diffusive states in the presence of dynamic state transitions for MINFLUX data. (a) Shown is the mean percentage error in the parameter estimates compared to the ground truth for various switching times (D₁ = 1 μm²/s, D₂ = 10 μm²/s, k₁ = k₂). Initial slope determinations (dashed black lines) are shown for the T = 10, 20, and 40 ms data sets. The averaging time is the value of N_i multiplied by the multiplex cycle time Δt = 200 μs. (b) Shown is the averaging time at which the mean percentage error begins to linearly increase. (c) Slope of the initial linear increase of the mean percentage error is shown. Switching times of 0.2 and 2 ms are not included here because the linear section of their curves in the panel are not sufficiently resolved. (d) Shown is the mean percentage deviation in the parameter estimates relative to the parameter estimates at N_i = 3. Again, initial slope determinations (dashed black lines) are shown for the T = 10, 20, and 40 ms data sets. (e) Shown is the averaging time at which the mean percentage deviation in (d) begins to linearly increase. (f) Slope of the initial linear increase of the mean percentage deviation in (d) is shown. Again, switching times of 0.2 and 2 ms are not included. To see this figure in color, go online.

Bacterial strains and plasmids

Plasmids for the inducible exogenous expression of fluorescent and fluorescently tagged proteins were derived from isopropyl β-D-1-thiogalactopyranoside -inducible pAH12 and arabinose-inducible pBAD vectors. The coding sequences of eYFP were PCR amplified using Q5 DNA polymerase (New England BioLabs, Ipswich, ME) from pXYFPN-2 (40). The PCR product was isolated using a gel purification kit (Invitrogen, Carlsbad, CA) and used as a megaprimer for amplification and introduction into a pAH12 derivative containing a kanamycin-resistance cassette, LacI, and a lac promoter to generate pAH12-eYFP. The pAH12 backbone was a gift from Carrie Wilmot.

For the pBAD-mEos3.2, the protein coding sequence was amplified from a mEos3.2-N1 plasmid, gifted to us by Michael Davidson (plasmid number 54525; Addgene, Watertown, MA). The PCR products were gel purified, and both the PCR products and the pBAD backbone were digested with EcoRI and XhoI restriction enzymes (New England BioLabs). Digested vector and inserts were ligated using T4 DNA ligase and transformed into E. coli TOP10 cells. Colonies were PCR screened for the presence of the correct insert using GoTaq DNA Polymerase (Thermo Fisher Scientific, Hampton, NH), and the plasmid was isolated from positive clones (Omega Bio-tek, Norcross, GA).

All plasmids were sequenced by GENEWIZ (South Plainfield, NJ) before electroporation into Y. enterocolitica for analysis. Transformed cells were plated on Luria broth agar (10 g/L peptone, 5 g/L yeast extract, 10 g/L NaCl, and 1.5% agar) (Thermo Fisher Scientific) containing kanamycin (50 μg/mL) or ampicillin (200 μg/mL). For electroporation of Y. enterocolitica pIML421asd cells, recovery media and plates also contained diaminopimelic acid (dap). A list of all strains and plasmids can be found in Table S1.

Cell culture

Y. enterocolitica cultures were inoculated from a freezer stock in brain heart infusion (BHI) media (Sigma-Aldrich, St. Louis, MO) with nalidixic acid (Sigma-Aldrich) (35 μg/mL) and 2,6-dap (Chem-Impex International, Wood Dale, IL) (80 μg/mL) one day before an experiment and grown at 28°C with shaking. After 24 h, 300 μL of overnight culture was diluted in 5 mL fresh BHI, nalidixic acid, and dap and grown at 28°C for another 60–90 min. In addition, the inoculation media also contained kanamycin or ampicillin for pAH12- or pBAD-based plasmids, respectively. Cultures of cells containing pAH12- or pBAD-based plasmids were induced with isopropyl β-D-1-thiogalactopyranoside (Sigma-Aldrich) (0.2 mM, final) or arabinose (Chem-Impex) (0.2%), respectively, for the final 2 h of incubation. Cells were pelleted by centrifugation at 5000 × g for 3 min and washed three times with M2G (4.9 mM Na₂HPO₄, 3.1 mM KH₂PO₄, 7.5 mM NH₄Cl, 0.5 mM MgSO₄, 10 μM FeSO₄ [EDTA chelate; Sigma-Aldrich], 0.5 mM CaCl₂, and 0.2% glucose). The remaining pellet was then resuspended in M2G and dap_. Cells were plated on 1.5–2% agarose pads in M2G containing dap.

Results and Discussion

eYFP and mEos3.2 undergo confined Brownian diffusion in Y. enterocolitica

To experimentally validate the numerical analysis framework based on Monte Carlo simulations of confined diffusion (Fig. 1), we tracked the 3D motion of individual eYFP and mEos3.2 fluorescent proteins in living Y. enterocolitica cells. Previous studies in E. coli (28, 41) and Caulobacter crescentus (42) have established that small cytosolic proteins undergo Brownian motion. Nonspecific interactions due to macromolecular crowding reduce the diffusion coefficient for small cytosolic proteins but do not by themselves lead to measurable deviations from normal Brownian diffusion (43). In contrast, the motion of large macromolecular complexes (>30 nm in diameter) is best described by anomalous diffusion due to glass-like properties of the bacterial cytoplasm (44).

The experimentally measured distributions of apparent diffusion coefficients were fit well using a single diffusive state with D = 11.3 μm²/s (for eYFP, Fig. 3 a) and D = 15.0 μm²/s (for mEos3.2, Fig. 3 b). The close agreement between simulations and experiment confirms that the assumption of spatially confined Brownian diffusion is valid for both eYFP and mEos3.2 in Y. enterocolitica under our experimental conditions. These diffusion coefficient values are in agreement with previously measured values of GFP in bacteria (28, 45, 46, 47, 48, 49, 50). The structure and molecular weights of eYFP (27 kDa) and mEos3.2 (26 kDa) are very similar. The differences in their diffusion coefficients may thus be due to differences in nonspecific transient interactions with other cellular components. We also note that there is a small (6% or less) stationary (<0.5 μm²/s) population for both fluorescence proteins. We find small numbers of stationary trajectories in all of our single-molecule tracking data sets, which indicates that even freely diffusing cytosolic proteins may become immobilized. However, we did not find that these stationary molecules exhibit any subcellular preference.

Figure 3.

The 3D diffusion of cytosolic fluorescent proteins eYFP and mEos3.2 in Y. enterocolitica can be explained using a single diffusive state. (a) eYFP diffuses at 11.5 μm²/s (red). (b) mEos3.2 diffuses at 15.0 μm²/s (red). A small fraction (<6%) of stationary trajectories is present in both data sets (blue). The total fit is shown as a dashed black line. To see this figure in color, go online.

2D versus 3D single-molecule tracking to estimate diffusion coefficients

Most single-molecule tracking results reported to date utilize the standard PSF for 2D single-molecule tracking. Acquiring 3D trajectories requires engineered PSFs, such as astigmatic, DHPSF, or tetrapod PSFs (10, 51, 52, 53, 54, 55). A common feature of engineered PSFs is their increased footprint on the detector compared to the standard PSF. Because of their increased size, engineered PSFs require higher photon counts to achieve lateral localization precisions equivalent to those obtained with the standard PSF. Given the finite photon budgets of fluorescent labels, 2D tracking can thus yield longer single-molecule trajectories that contain roughly twice the number of displacements than 3D trajectories acquired with engineered PSFs.

To determine whether diffusion coefficients are more accurately estimated by 2D or by 3D tracking, we repeated the 3D DHPSF simulations using the standard PSF. We generated simulated distributions of apparent 2D diffusion coefficients in the same way as for the 3D data (Materials and Methods). However, the simulated 2D trajectories had twice as many displacements as the 3D trajectories to provide an equivalent total photon count over the course of a trajectory. We found that the resulting 2D apparent diffusion coefficient distributions are broader, and their peaks are systematically right-shifted compared to their 3D equivalents (Fig. 4 a). The increased left shift of the 3D distribution is due to the additional confinement of the molecule’s motion in the z dimension that is not measured in 2D tracking.

Figure 4.

Comparison of 2D and 3D tracking. (a) Shown is a comparison of 2D and 3D apparent diffusion coefficient distributions corresponding to 1 and 10 μm²/s. The distributions for 3D tracking are left shifted to a larger extent because of the additional confinement in the third dimension. (b and c) Shown are the relative errors in determining the diffusion coefficient of a single diffusive state using 2D (b) and 3D (c) single-molecule tracking. Shown are the averages and SD of four independent simulations containing n = 5000 trajectories, each resampled 10 times by bootstrapping. To see this figure in color, go online.

We then performed numerical fitting of simulated 2D tracking data to estimate the diffusion coefficient. We found that there is a slight increase in accuracy when fitting 2D data compared to 3D data for a single diffusive state, particularly for fast diffusion (Fig. 4, b and c). The improved accuracy of 2D tracking may be due to the decreased similarity of the 2D distributions for fast diffusion coefficients (Fig. S4), which enables more accurate parameter estimation.

Single-molecule tracking can be used to resolve different diffusive states

The free fluorescent proteins examined in the previous section each exhibited a single predominant diffusive state, which means that these two proteins do not exhibit stable interactions with other cellular components. This property is important for their use as nonperturbative labels that do not alter the diffusive behaviors of the target proteins beyond an overall reduction in their native diffusion rate. An overall reduction in diffusion rate is expected because of the increased molecular weight and hydrodynamic radius of the fusion protein. If the target protein stably interacts with cognate binding partners to form homo- or hetero-oligomeric complexes of different sizes, then single-molecule tracking of nonperturbatively labeled target proteins may be used to resolve the corresponding diffusive states. Examples of different diffusive states reported in the recent literature include the cytosolic preassembly of the bacterial type 3 secretion system proteins SctQ and SctL (4); ternary complex formation of the elongation factor Tu, which can bind to aminoacyl-transfer RNA, GTP, and translating ribosomes (17); the nucleotide excision repair initiation molecule UvrB (14); and short-lived ribosome binding of EF-P (13).

To test the resolving capability of single-molecule tracking, we simulated mixed distributions of 3D displacements or apparent diffusion coefficients that contain two different diffusive states. We then fit these distributions to obtain the unconfined diffusion coefficients and relative population fractions of each diffusive state. By systematically varying the diffusion coefficients, we assessed the error in the optimized fitting parameters for various combinations. We examined both equal (50:50) and unequal population fractions (80:20). In all cases, the distributions were based on 5000 trajectories with five displacements each. We found that the errors in the optimized fitting parameters increased when the diffusion coefficients were similar as evidenced by the wedge-shaped diagonal (Fig. 5 a). Slight differences in diffusion rate are thus more readily resolved for slowly diffusing molecules than for faster moving ones. We reason that the ability to resolve fast diffusive states is further compromised by the confinement effect, which causes the distributions of apparent diffusion coefficients to become more similar in the high diffusion coefficient limit (Fig. 2 c).

Figure 5.

Multiple diffusive states can be resolved by the numerical fitting of single-molecule tracking data using 2D and 3D tracking. (a and b) Shown are the relative errors for determining the diffusion coefficients and population fractions of binary mixtures of diffusive states using 3D (a) and 2D (b) tracking. The relative population fractions in the two-state mixtures were either 50–50% (left) or 20–80% (right). The relative error for each fitting parameter (diffusion coefficients D₁ and D₂ and their corresponding population fractions f₁ and f₂) is represented as a matrix for different diffusion coefficient combinations. Each pixel represents the mean (relative) error of the parameter’s fit value after analyzing 10 data sets (resampled by bootstrapping), each containing 5000 tracks. To see this figure in color, go online.

Current detector technologies, in particular, large field-of-view sCMOS detectors, have made it possible to readily acquire single-molecule trajectories in thousands of cells in a single imaging session. Thus, 5000 trajectories can be obtained even for proteins expressed at low levels. For highly expressed proteins, up to 100,000 trajectories can be obtained. We therefore repeated our analysis using distributions based on 100,000 trajectories. As expected, the errors in the parameter estimates decreased (∼7% on average) when fitting the now more thoroughly sampled distributions (Fig. S6). Therefore, the resolving capability improves when additional measurements are available to sample the shape of experimental distributions. However, larger errors persist along the diagonal of the error matrices, highlighting the difficulty in resolving states with similar diffusion coefficients. When the population fractions are split 80:20, larger errors manifest because of the smaller number of proteins in the diffusive state with a 20% population fraction. In those cases, the relative error in the smaller fraction can approach 100% (i.e., the smaller fraction is completely eliminated when the fitting routine converges on a one-state solution) (Materials and Methods).

To test whether the above results may be extrapolated to more complex state distributions, we simulated a few selected examples of mixed distributions containing three and four diffusive states, maintaining n = 5000 total trajectories in each case. We found that three states can be simultaneously resolved as long as their diffusion coefficients are sufficiently different, and their population fractions are similar (Document S1. Figs. S1–S8 and Table S1, Document S2. Article plus Supporting Material). Again, the errors in the fitting parameters increase for faster (i.e., more similar) diffusion coefficients (Fig. S7 b). In the case of a four-state population, the distribution is best fit with three-state results, even when the values of the diffusion coefficients are well separated (Document S1. Figs. S1–S8 and Table S1, Document S2. Article plus Supporting Material). Specifically, the two fastest states are combined into a single state with a correspondingly larger population fraction. The three- and four-state simulations thus recapitulate the trends observed for binary diffusive state mixtures.

To test whether 2D tracking is also more discriminating when multiple diffusive states are present, we constructed simulated two-state distributions of apparent diffusion coefficients based on 2D data. Again, we observed only a slight increase in the accuracy of the fitting (∼3%) for the 2D fitting compared to 3D for a two-state fitting (Fig. 5 b). We therefore conclude that 2D and 3D single-molecule tracking are roughly equivalent in their ability to resolve different diffusive states. We note, however, that 3D single-molecule localization microscopy has the additional advantage of providing more detailed spatial information on the subcellular locations of diffusing molecules, which may provide important additional information in select cases. We also note that the above analysis only pertains to the diffusion of cytosolic proteins. The diffusion of membrane proteins is subject to different confinement effects that may make it more appropriate to track in 3D (5).

Transitions between diffusive states

Thus far, we have only considered diffusive states that do not interconvert on the timescale of a single-molecule trajectory (∼100–300 ms on average). Under physiological conditions, however, molecules may frequently bind to or dissociate from cognate interaction partners and thereby transition between different diffusive states. The time resolution for making single-step displacement measurements (∼25 ms) is shorter than the time resolution for determining apparent diffusion coefficients (∼5 × 25 ms = ∼125 ms). We therefore hypothesized that, in the presence of diffusive state switching, more accurate parameter estimates may be obtained by fitting single-step displacement distributions. To test this hypothesis, we simulated distributions for two states, D₁ = 1 μm²/s and D₂ = 10 μm²/s, that can interconvert on timescales comparable to a single-molecule trajectory. We then gradually decreased the average diffusive state-switching time T = (k₁)⁻¹ + (k₂) ⁻¹ = t₁ + t₂ and imposed k₁ = k₂ to keep the population fractions equal (Materials and Methods). To fit the single-step displacement distributions, we generated a library of simulated single-step displacement distributions as described before for apparent diffusion coefficients (Fig. S5). Both the apparent diffusion coefficient distributions and single-step displacement distributions were then fit with their respective library. To quantify the overall accuracy of the fit, we averaged the relative errors of all fitting parameters (in this case, the diffusion coefficients D₁ and D₂ and the population fractions f₁ and f₂ = 1 − f₁). We found that, in the limit of infinitely long switching times (no state transitions), both approaches produce parameter estimates with similar accuracy (Figs. 6, a and b and S8). As the average switching time is decreased, the mean relative errors start to increase for both methods. Importantly, fitting the distributions of apparent diffusion coefficients produced parameter estimates that deviated sooner from the ground truth (as a function of decreasing average switching time) than those obtained by fitting single-step displacement distributions. In the limit of short switching times, the fitting of both the apparent diffusion coefficient and single-step displacement distributions produced large errors because a single molecule can sample both diffusive states repeatedly during the timescale of the measurement. When using 25 ms exposure times, accurate parameter estimates can be made for this two-state system, if T > 75 ms and T > 500 ms for displacement and apparent diffusion coefficient fitting, respectively. For accurate extraction of the parameters, the time resolution of the measurement should be about three times shorter than the average switching time T.

Figure 6.

Resolving diffusive states in the presence of dynamic state transitions. (a) Shown are the mean relative errors of the fitting parameters for a two-state mixture (D₁ = 1 μm²/s, D₂ = 10 μm²/s, 50:50 population fraction) as a function of different switching times between two diffusive states. The mean percentage error obtained by fitting the single-step displacement distributions diverges for T < 75 ms, whereas the mean percentage error obtained by apparent diffusion coefficient fitting diverges for T < 500 ms. (b) Shown are individual parameter estimates as a function of state-switching time for the same simulations as in (a). Population fraction f₂ = 1 − f₁ is not shown for clarity. The shaded areas represent 10% error limits for each parameter. (c) The mean relative errors of the fitting parameters as a function of the number of averaged displacements are shown. (d) Parameter estimates as a function of averaged displacements for the same simulations as in (c) are shown. The color scheme is the same as the legend in (c). Gray lines represent the ground truth. The fitted individual parameter value produces horizontal curves for both the very short (2 ms) and very long (10⁴ ms) switching times. For intermediate switching times (50 ms), the fitted values trend away from the true value as the number of averaged displacements increases. (e) Shown is the mean deviation relative to the single displacement parameter estimates (N_i = 2) for different switching times. To see this figure in color, go online.

The above observations suggest that it should be possible to estimate the timescale of diffusive state switching by TAD analysis (i.e., by varying the number of averaged displacements). We therefore evaluated the apparent diffusion coefficients for overlapping subtrajectories having different numbers of displacements/localizations. Specifically, within each single-molecule trajectory, we defined overlapping subtrajectories with N_i localizations and N_i − 1 displacements. The number of subtrajectories for a given N_i is S = N − N_i + 1, where N is the number of localizations in the full-length trajectory. Defining the first localization in the subtrajectories as P, we modified Eq. 1 to

$$MS{D}_{N_i,P}=\frac{1}{N_i-1}\sum _{n=P+1}^{N_i+P-1}{\left(x_n-x_{n-1}\right)}^2,$$ (7)

to obtain mean-squared displacement values for different subtrajectory lengths and starting points, namely N_i = 2, 3, …, 6 and P=1…S.

Based on these sets of observables, we generated five new apparent diffusion coefficient libraries corresponding to the five different values of N_i (on average, our experimental 3D trajectories are five displacements long). The state-switching trajectories were then reanalyzed using Eq. 7 and fit with the corresponding library. Again, we used the mean relative error over all fitting parameters to quantify the overall accuracy of the fit for each value of N_i (Fig. 6 c). Consistent with the results above, the accuracy of the fitting parameters was poor for short switching times and good for long switching times. Importantly, the mean relative errors are constant for all N_i in both of these limiting cases. Thus, if the state-switching time is substantially shorter or longer than the time resolution of the measurement, then the mean error does not change. In contrast, the mean errors increase for increasing N_i, if switching times are comparable to the timescale of a single-molecule trajectory (0.05–0.5 s). The same trends were also observed when plotting the individual parameter fitting results (Fig. 6 d). Based on these results, we conclude that the timescale of diffusive state switching can be estimated by determining the rate of change of individual fitting parameters as a function of the number of averaged displacements. For example, based on the results in Fig. 6, c and d, observing a consistent increase or decrease of individual fitting parameters as a function of N_i would indicate a diffusive state-switching time between 20 and 500 ms. We note that the ground truth is unknowable in experimental work. We therefore computed an error relative to the parameter values obtained when fitting single displacement distributions (i.e., N_i = 2). Single displacement distributions offer the best time resolution and thus should be least affected by diffusive state averaging. The parameter deviations relative to the parameter estimates at N_i = 2 displayed similar trends as those referenced to the ground truth (Fig. 6 e).

It is clear that the dynamic range of TAD analysis improves if trajectories contain a large number of displacements. However, in camera-based tracking of fluorescent fusion proteins, only n = 5 or n = 11 displacements can be observed on average for 3D and 2D tracking, respectively. Longer trajectories can be acquired using chemical dyes (24, 56, 57) or multiple fluorophores as labels (58), but the potential of nonspecific labeling or the size of multivalent fluorescent tags have to be weighed against this benefit. An important advantage of camera-based tracking is that the temporal dynamic range is tunable to access slow switching timescales (>500 ms) by adjusting the exposure time and/or by acquiring single-molecule trajectories in time-lapse mode (17, 27, 59). On the other hand, exposure times shorter than a few milliseconds come at the expense of data acquisition throughput because the full chip of current sCMOS cameras cannot be read out faster than 100 Hz (17). Thus, faster timescales are difficult to assess by camera-based tracking.

A solution to access faster timescales is MINFLUX microscopy (29). The time resolution of MINFLUX-based single-molecule tracking is two orders of magnitude better than camera-based tracking (0.2 ms vs 25 ms) and the number of localizations N is larger by one order of magnitude (n = ∼100 vs n = ∼10). MINFLUX microscopy may thus be able to provide access to state-switching dynamics on 0.2–20 ms timescales, whereas camera-based tracking can cover state-switching dynamics on millisecond to minute timescales. To test the capability of MINFLUX microscopy to quantify fast state-switching times, we applied TAD analysis to simulated MINFLUX data. MINFLUX trajectories were generated in the same way as the camera-based trajectories (i.e., through Monte Carlo simulations of confined Brownian diffusion), but the MINFLUX localization algorithm was used instead of PSF fitting (Materials and Methods). We then used libraries of N_i-fold averaged MINFLUX displacement distributions to fit state-switching trajectories for different switching times T (D₁ = 1 μm²/s, D₂ = 10 μm²/s, k₁ = k₂). We found that the mean percentage error versus N_i curves (Fig. 7 a) displayed two key characteristics that correlate linearly with switching time T or with switching rate 1/T. First, for each switching time T, there exists a threshold value N_i,T, after which the mean percentage error increases linearly as a function of N_i. N_i,T and T are linearly correlated (Fig. 7, a and b). Second, the slope of the initial linear increase and the switching rate 1/T are linearly correlated as well (Fig. 7, a and c). Based on these linear relationships, we conclude that the timescale of state transitions can be determined from the position of N_i,T and from the slope of the following linear increase.

Because the ground truth is not accessible by experiment, we repeated the above analysis by referencing all parameter estimates to the parameters obtained at N_i = 3 (Fig. 7 d). N_i = 3 corresponds to a time resolution of 600 μs. The curves obtained by plotting the mean percentage deviation from the N_i = 3 parameter estimates versus N_i displayed the same characteristic linear increases as a function of N_i. The onset of the linear increase N_i,T and the slope of the linear increase still correlated linearly with T and 1/T, respectively (Fig. 7, d–f). These results show that the switching rate between two diffusive states can be reliably determined by TAD analysis of 2D and 3D single-molecule tracking data.

Conclusions

In this work, we present and test a robust analysis method for estimating the diffusive state parameters of fluorescently labeled biomolecules in confined bacterial cell volumes based on single-molecule tracking. We show that it is possible to resolve the unconfined diffusion coefficients and the population fractions of multiple diffusive states based on a few thousand short single-molecule trajectories obtained by camera-based tracking. The numerical analysis framework presented is generally applicable to both 2D and 3D tracking and any confinement geometry. We show that 2D and 3D single-molecule tracking are roughly equivalent in their ability to resolve multiple diffusive states. To address the issue of diffusive state switching during the timescale of the measurement, we propose TAD analysis. By averaging over different numbers of subsequent displacements, the timescale of state switching can be determined, if that timescale is comparable to the duration of the recorded trajectories. For example, MINFLUX microscopy can provide access to state-switching dynamics occurring on 2–200 ms timescales using data acquisition parameters relevant for fluorescent protein localization in living cells. On the other hand, camera-based tracking can be used to detect state-switching dynamics on 20 ms to seconds timescales either by using longer exposure times or by acquiring data in time-lapse mode. TAD analysis of experimental single-molecule trajectories thus provides a general and robust approach to quantify the diffusive states and diffusive state transitions that manifest in living cells.

Author Contributions

J.R. and A.G. designed research. J.R. and T.Y. performed research. J.R., J.C., C.R., and A.G. contributed new reagents/analytic tools. J.R., J.C., and A.G. analyzed data. J.R. and A.G. wrote the article.

Editor: Julie Biteen.