Accuracy of the detection of binding events using 3D single particle tracking
© The Author(s) 2017
Received: 21 October 2016
Accepted: 3 March 2017
Published: 23 March 2017
Nanoparticles can be used as markers to track the position of biomolecules, such as single proteins, inside living cells. The activity of a protein can sometimes be inferred from changes in the mobility of the attached particle. Mean Square Displacement analysis is the most common method to obtain mobility information from trajectories of tracked particles, such as the diffusion coefficient D. However, the precision of D sets a limit to discriminate changes in mobility caused by biological events from changes that reflect the stochasticity inherent to diffusion. This issue is of particular importance in an experiment aiming to quantify dynamic processes.
Here, we present simulations and 3D tracking experiments with Gold Nanorods freely diffusing in glycerol solution to establish the best analysis parameters to extract the diffusion coefficient. We applied this knowledge to the detection of a temporary change in diffusion, as it can occur due to the transient binding of a particle to an immobile structure within the cell, and tested its dependence on the magnitude of the change in diffusion and duration of this event.
The simulations show that the spatial accuracy of particle tracking generally does not limit the detection of short binding events. Careful analysis of the magnitude of the change in diffusion and the number of frames per binding event is required for accurate quantification of such events.
KeywordsSingle particle tracking Diffusion Binding event Mean square displacement analysis
Cells present a dynamic environment for the biomolecules that orchestrate life: important processes such as intracellular or intramembrane trafficking [1–3] protein dynamics [4, 5] and gene delivery [6, 7] can be studied in detail by analyzing the mobility of the molecules involved. Single-Molecule Tracking (SMT) is a powerful tool to investigate such dynamic processes. SMT discloses information unobtainable using ensemble techniques, because following molecules individually can reveal variations in behavior that occur during the process, including rare events that are otherwise obscured in the ensemble. The high precision of SMT relies on the possibility to localize a single molecule with higher accuracy than the diffraction limit . Ultimately, the accuracy of localization depends on the optical brightness of the molecule. Because most biomolecules can not be detected using optical microscopy, they need to be labeled with fluorescent markers like organic dyes or fluorescent proteins. Alternatively, metal or semiconductor nanoparticles have been used as labels to track single molecules. Single-Particle Tracking (SPT)  is advantageous over SMT because nanoparticles are generally brighter than fluorophores and can therefore be tracked with better precision. Moreover, as opposed to single fluorophores, nanoparticles don’t bleach, which extends the time span over which a single molecule can be followed. However, nanoparticles are larger than single fluorophores, and will thus affect the mobility of the molecules of interest.
From SPT one can obtain long time traces of single molecules, that are then analyzed to quantify mobility. The Mean Square Displacement (MSD) of the particle reveals characteristic modes of mobility like free diffusion, confined diffusion and active transport, which are characterized by parameters such as diffusion coefficient (D), velocity and confinement size. The ability to track individual molecules, labeled with nanoparticles, with nanometer precision and over long times would make it possible to observe transient changes in the mobility of the molecule that could not be observed using other methods. For example, the binding of a transcription factor to its DNA target has been challenging to detect at the single-molecule level. Though Fluorescence Correlation Spectroscopy (FCS) and SMT approaches have been used to study this process [10, 11], the short length of the traces, due to photobleaching and/or diffusion out of the detection volume, generally directs data analysis to ensemble properties rather than those of single molecules. Therefore, SPT could provide a unique alternative for monitoring the dynamics of an attached molecule.
How reliable are the mobility parameters extracted from such SPT experiment? In the case of active transport, the localization accuracy is the most important factor influencing the precision of the particle velocity. In the case of diffusion an evaluation of the accuracy of D is more complex: diffusion is a stochastic process, and this requires the measurement of many independent localizations to obtain D with high precision. The precision of D is of high relevance for biological experiments, as it sets a threshold to discriminate a biologically meaningful change in diffusion from the intrinsically stochastic variations.
Here we investigate how accurate the diffusion coefficient of a particle can be measured in a SPT experiment, and how well we can detect a transition in its diffusion behavior. The issue of accuracy of diffusion coefficients has been addressed before, with a theoretical approach and simulations [12, 13], but mainly in 2D. 2D SPT can provide higher temporal resolution, but the images are limited in space to single planes and the tracking can be performed only as long as the particle stays in the plane: the use of 2D SPT is therefore limited to tracking in cell compartments that can be approximated to 2D such as the cell membrane [14, 15]. The simulations in this report extend such analysis to 3D tracking experiments.
We used Gold Nanorods (GNRs) as labels for 3D SP using Two Photon excitation. GNRs are cylinder-shaped gold nanoparticles with sizes ranging between few tens to several hundreds of nanometers: they are bigger than organic dyes and fluorescent proteins, and can therefore slow down the tracked molecule. Nevertheless they offer several advantages over fluorophores: their luminescence is up to 100 times higher (yielding a spatial resolution that is 10 times better, ), and can be tracked for hours as they are not affected by bleaching or blinking . Moreover, GNRs are easy to functionalize and are compatible with live cells conditions .
We acquired multiple z sections forming a 3D image stack using a multifocus two photon microscope as described before . Two photon excitation reduces out-of-focus excitation and yields a good spatial resolution both in the longitudinal and axial directions. We scan an array of focal spots using a scanning mirror: this way we obtain a wide-field illumination of the sample.
Some 3D SPT techniques have a higher temporal resolution compared to z sectioning, like for example the use of cylindrical lens to extract 3D positioning . However, the use of astigmatism is not compatible with two-photon excitation, and thus lacks the benefits of higher signal-to-noise; Total Internal Reflection microscopy  gives high spatial and temporal resolution, but within a limited 3D area, not sufficient to cover the entire volume of a cell; orbital tracking  tracks only one particle at the time and cannot benefit from the high throughput of parallel tracking. A good alternative to multi-focus excitation two-photon microscopy is two-photon Light Sheet Microscopy  that provides good penetration depth in the sample and a comparable acquisition speed; for SPT these two techniques present similar challenges.
The outline of this paper is as follows: first, we address the influence of positional accuracy of the 3D tracking scheme on the precision of the extracted MSD with simulations; then we analyze the accuracy of the obtained diffusion coefficient with simulations and experiments; we optimize the parameters that are used to obtain D from the MSD; finally we simulate traces containing a change in diffusion behavior and establish the experimental boundaries for resolving such changes.
The acquisition of 3D movies of single GNRs was performed on a home-built two-photon multifocus scanning microscope as previously reported in , with some small changes. A near IR pulsed laser (Coherent Chameleon Ultra) was used for excitation; the laser beam was split in an array of 625 beams by a diffractive optical element (DOE, custom made by Holoeye). A fast scanning mirror, driven with an Archimedean spiral function, was used to scan the array of beams over the sample: this way we obtained a wide and homogenous excitation on an area of about 60 μm x 60 μm, and collect images of tens of GNRs within this area. A piezo-stage (PIfoc, PI) was used to move the objective in the z-axis to collect 3D images. We acquired images with an EMCCD Camera (Photometrics QuantEM 512SC). The frame size was 400 pixels x 400 pixels, corresponding to 70 μm x 70 μm and the separation between z slices was typically 0.5 μm. We acquired 10 z slices per stack, at a rate of 10 frames/s: the time resolution of our 3D localization was therefore 1 s/stack.
Samples of GNRs of two different sizes were used: 47 ±4 nm x 14 ±2 nm GNRs were synthesized through a seed-mediated method , while 53 ±6 nm x 16 ±3 nm GNRs were purchased from Nanopartz (A12-25-780-CTAB). Both GNRs samples were functionalized with a polyethylene glycol (PEG) layer before use. GNR sizes were obtained from Transmission Electron Microscope (TEM, JEOL JEM 1010) images of both batches. The TEM images also provided a measure for the size dispersion within the two samples. The GNR sizes used for our theoretical calculations were increased by the thickness of a PEG layer. The size of the PEG layer (which cannot be seen in TEM) was measured independently using Fluorescence Correlation Spectroscopy (FCS, ), yielding an effective PEG layer thickness of 8.1 nm (see Additional file 1: Figure S1). GNRs were first suspended in small volumes of demineralized water, then glycerol was added to reach the desired concentration of 95 and 90% glycerol. For SPT in glycerol both GNR samples were excited at a wavelength of 770 nm.
Simulations of movies of diffusing GNRs were performed in LabVIEW using the following procedure: a set of 3D trajectories was created, according to a given diffusion coefficient D (or multiple values of D, in case of changes in behavior); a stack of empty frames was then filled with a 3D Gaussian peak for each time coordinate, and amplitude and standard deviation of the peak were set using typical values obtained experimentally for single GNRs (amplitude=1000 a.u., s xy =300 nm, s z =650 nm); Poissonian noise was added to each pixel in the peak in order to simulate shot-noise; an offset (1000 a.u.) and a background noise (1 a.u.) were added to the entire 3D stack of images, reflecting the camera gain settings and detection noise. As opposed to experimental movies, in simulated movies we introduced only one GNR to prevent incorrect trajectory assignments when GNRs would cross. We simulated videos with a frame rate of 10 frames/s, as typically collected by our setup. The frame size was 300 pixels x 300 pixels (corresponding to about 52 μm x 52 μm, and the separation between z slices was 1 μm.
where s is the width of the PSF, N p is the number of photons, a is the pixel size and b the number of photons in the background noise. As is characteristic for shot-noise, the uncertainty in position decreases with increasing number of photons. The uncertainty in the case of 3D images will depend on the 3D image acquisition scheme. Previously, we reported an experimental increase in x and y accuracy in 3D data that originated from the additional photons recorded in all frames above and below focus that contribute to a 3D peak (). These measurements were made using fixed, immobile GNRs. In the results section we will quantify this effect. However, changes in positions between slices in a stack will affect the positional accuracy.
Accuracy of MSD analysis
Therefore, the optimal number of fitting points to use depends on positional uncertainty, diffusion coefficient and sampling time. In a real experiments the expected D is typically not known, so n is not easy to evaluate. An estimate of the order of magnitude of the D to expect is a first good step. A higher sampling rate or a lower precision increases the value of the optimal n to use. We tested Michalet’s results with 3D simulations using different positional uncertainties and diffusion coefficients. Then we validated these results with experiments using GNRs with known D, compared D to the value measured with SPT and calculated the relative error ρ.
Detection of changes in D
in which Γ is the gamma function.
A p-value is calculated for each point in the D plot, considering two windows of the same size around the point. The minima in the probability plot (Fig. 2 c) correspond to the points in the trace where a diffusion transition is most likely to happen. Rolling windows with different sizes (N 1 and N 2 in Eqs. 10 and 12) didn’t show noticeable differences. We chose a rolling window size of 15 steps, and a Welch test sample size of 15 or 10, when the gap was shorter than 15 steps. Transition points were assigned using a threshold for P and the initial trace was divided in subtraces. We tested different values for the threshold, and we obtained the best compromise between false negative and false positive results with a value of 10-10.
As shown in Fig. 2 c, not all the detected transition points corresponded to real transitions: some were misassigned due to stochastic fluctuations in D. We performed a second Welch test on these subtraces using window sizes corresponding to the entire subtraces length. The transitions confirmed by the second test were accepted as real transition points. Despite this second statistical test, it was not always possible to assign each transition point correctly. For example in Fig. 2 c at t = 25s a change in D was wrongfully detected. A new MSD analysis was finally done on the final subtraces to obtain the mean D, which is plotted in Fig. 2 d.
Results and discussion
Spatial and temporal resolution
In the analysis of dynamic data, the temporal resolution plays an important role: the finite time between acquisitions can obscure fast dynamic processes. Moreover, in real experiments, the movement of the particle occurs also between slices within a 3D stack: we simulated this movement within a stack for a range of diffusion coefficients: as shown in Fig. 3 b, the effect of the movement within stacks can be dramatic for large diffusion coefficients. The positional uncertainty in the x-y plane for the lowest diffusion constant (D = 0.01 μm2/s) is about 9.5 ± 0.6 nm, for the highest (D = 0.5 μm2/s) σ is 143.0 ± 8.4 nm. In the z direction, the uncertainty follows the same trend but is even more pronounced. In the experiments on GNRs performed with our setup, the number of photons collected was very high, due to the high brightness of the two-photon signal of GNRs and low background. From Eq. 1 we calculate a positional accuracy of 4 nm (see Additional file 1: Figure S2) for an average N p of 4000 photons. However, due to GNR movement between slices, the positional uncertainty is increased: considering the diffusion coefficient range expected for our experiments (between 0.02 and 0.07 μm2/s), we expect the effective positional uncertainty in x,y to be around 20 nm, and in z around 40 nm. The uncertainty values obtained from the MSD fit is arund 40 nm: this value includes the x-y and the z components, and is comparable to the σ value in z obtained from simulations.
Factors that determine the uncertainty in the detection of D
In summary, these precautions can reduce the error on the obtained D: using long traces, fixing the positional uncertainty of the MSD fit, and limiting the fit to the first two MSD points. Nevertheless, even with high positional accuracy, one will obtain relatively large errors in D when measuring for finite times due to the stochastic nature of diffusion.
Experimental validation of D accuracy using GNRs in glycerol
In Fig. 6 c, we compare the experimental values (obtained fixing the positional uncertainty) for all the traces, traces longer than 40 points (about 1 min) and 80 points (about 2 min). The first thing to notice is the dramatic decrease in the D error when using longer traces. For the longest traces, the smallest number of MSD fitting points yields the smallest error. Therefore, the minimum error in the calculation of D is obtained using only traces longer than 2 min, fixing the positional uncertainty and using only 2 fitting points: in these conditions we got a relative error as low as 10%.
Detection of changes in diffusion in single particle trajectories
One of the unique possibilities of SPT is to follow a single molecule over a long time, and to directly detect changes in its behavior. The previous discussion on the difficulties to obtain a correct D implies, however, major challenges. In this paragraph we tested how accurately a temporary reduction in diffusion constant of a particle (a ’gap’) can be detected. Following the approach above, MSD analysis was performed with 2 points and fixed positional accuracy calculated from the intensity of the peak. The diffusion coefficient used for the initial and final phases was 0.05 μm2/s, which we typically measure for GNRs inside cells (both in nucleus and cytoplasm). We varied D in the gap from 0.0001 μm2/s to 0.035 μm2/s. The residual mobility of a protein bound to DNA has been reported to be in this range [26, 27]. The initial and final phases were 100 s, while we tested different lengths of the gap phase. We evaluated the effectiveness of the detection in the obtained diffusion coefficient and gap duration (D gap and t gap ) in every set of 100 simulations. We considered D gap correct when it was within ± 20% of the set D, and we considered t gap correct when it was within ± 10% of the set length. An example of a trace simulated using a D gap of 0.0001 μm2/s is shown in Fig. 2 a. In this figure, the gap is not clearly seen in the trajectory (as the time points are very close to each other), but it is easily distinguishable in the D plot. In real experiments the difference in mobility can be smaller.
In the second case, where D gap =0.01μm2/s (Fig. 8 c, d), the transition is also clearly detectable. Both the correct t gap and D gap are detected in about 60% of the cases. In Fig. 8 d, it is clear that most incorrect values originate from a missed transition, which results in a double duration of the gap phase, and an increased D gap .
In the last case (Fig. 8 e, f), D gap = 0.035 μm2/s, only 30% lower than the D outside the gap. Given an uncertainty of at least 10% in the detection of the single diffusion coefficient (see previous paragraph), we expect this difference to be hard to detect. Indeed, looking at D(t) (Fig. 8 e) we can still distinguish a change in D in the common trend, but the fluctuations in each single curve obscure transitions in D. In less than 10% of the traces a gap with the correct length is detected, but the correct D gap is detected in 60% of the cases. This is due to the small difference between D inside and outside the gap: in the cases where the transition is detected at a different point in time, the obtained D will still be good enough, being an average between D and D gap . In about 40% of the simulations no transition is detected (Fig. 8 f). In Additional file 1: Figure S3 more cases with different values of D gap are reported.
Thus, changes in D smaller than 30% can easily be distinguished from averaged data, but in single trajectories a reduction of about 80% is required to detect 60% of such changes.
To improve the precision of the results, a more complex measurement and analysis scheme could be used, that makes use of more parameters to detect subsections; for example, one could simultaneously measure the polarization of the signal. When applicable, other mobility parameters such as direction of the motion, velocity or confinement could be fit to the MSD curves. When such parameters are different during the binding of GNRs to cellular structure, Welch analysis can be performed, as in [28, 29]. The final p-value, obtained by multiplying the p-values of different parameters, will give a more correct assignement of the transition points and consequently more precise estimates of t gap and D gap .
In practice, measuring D and D gap of a protein will depend on the size of the protein and on the local viscosity of the environment. The smaller the molecules, the larger the difference in the diffusion coefficient when it binds to its substrate and the easier to detect the event accurately. The size of the GNR will set an upper limit to the diffusion coefficient that will be measured. The affinity of the protein is directly reflected in the ratio of the time between binding events and the lifetime of the bound complex, the latter being referred to here as t gap . The first may be affected by the presence of a GNR. A wide range of binding times have been reported for example for DNA binding proteins in vivo, ranging from sub-seconds  to several minutes . The ability to track a single protein bound to a GNR will give a more detailed insight in the reaction kinetics and how the complex cellular environment affects this reaction. Here we have shown that using GNRs as labels can, in many conditions, resolve single binding events with nanometer and second accuracy.
use long trajectories: the larger the number of time points in the trace the better the MSD is; in our case, doubling the trace length from 1 min to 2 min yielded a two-fold improvement of the precision in the detected D.
in case of small positional uncertainties, theory and results from simulations suggest to use only the first two points for fitting the MSD.
fixing the positional uncertainty during the MSD fit improves the evaluation of D.
These findings reinforce previous theoretical reports [8, 13]. In our case we tracked GNRs in 3D with an uncertainty of 4 nm based on shot-noise limitations, which increased to about 40 nm due to the movement of particles between each acquisition. With these conditions, the best approximation of D was within 10% of the expected value of D. Such a high precision could not be achieved using fluorophores as GFP or synthetic dyes as quantum dots, because their low signal provides a low spatial resolution, and their bleaching or blinking behavior make it impossible to collect long trajectories.
Given the challenges to extract a precise value of the diffusion coefficient, the analysis of changes in mobility needs extra care. We simulated traces with ‘gaps’ in the diffusion, as it can occur when a particle is temporarily immobilized, for example by specific binding to a cellular structure. The detectability of such gaps depends critically on the difference in the diffusion before and during the binding, determined mainly by the size of the ligands, and the length of the binding event. In our conditions and optimizing the MSD analysis as described, the detection of the gap was possible with a probability equal or higher than 50% only when the gap was longer than 20 s and the D in the gap was less than 5 times smaller than the D in the rest of the trace. These findings are applicable for all types of SPT methods in which individual traces are analyzed without averaging. We expect that many events characterized by a short duration or inducing a limited change in diffusion are overlooked in such experiments because of the stochastic character of diffusion. In any case, using large particles may produce brighter and more stable signals, but reduces the diffusion coefficient, making the difference in D between free and immobile particles smaller.
Note added in Proof
After the manuscript was accepted, we learned that Mortensen et al. (Optimized localization analysis for single-molecule tracking and superresolution microscopy, 2010, Nature Methods) pointed out that Eq. 1 overestimates the 2D positional accuracy by 30%, which agrees very well with the 2D simulation results, as shown in Fig. 3 a. For 3D tracking, this correction is counterweighted by the improved accuracy resulting from fitting multiple slices, also shown in Fig. 3 a. This empirical observation validates using Eq. 1 for estimations of the offset in the MSD plots.
Fluorescence Correlation Spectroscopy
Mean Square Displacement
Single Molecule Tracking
Single Particle Tracking
Transmission electron microscope
We thank Marcel Schaaf, Veer Keizer, Alexander Kros, Aimee Boyle, Michel Orrit and Aquiles Carattino for the critical discussions. We also thank Aimee Boyle for providing us with GNRs samples, and Kirsten Martens for the help in performing FCS measurements.
This work was supported by the Foundation for Fundamental Research on Matter (FOM), which is part of the Netherlands Organisation for Scientific Research (NWO) and by the NWO-VICI research programme, project 680-47-616.
Availability of data and materials
The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.
JC and SC carried out simulations. SC carried out experiments. SC and JvN conceived and coordinated the study. SC and JvN wrote the manuscript. All authors read and approved the final manuscript.
Corresponding author: JvN. Correspondence: email@example.com SC, JC and JvN: Huygens-Kamerlingh Onnes Laboratory, Leiden University, Postbus 9504, 2300RA, Leiden (NL)
The authors declare that they have no competing interests.
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