3D particle tracking on the two-photon microscope

We have developed a 3D single-particle-tracking (SPT) system based around the two-photon laserscanning fluorescence microscope that can track particles in all three dimensions and at a high frequency response. We have implemented two different techniques to achieve this goal. The techniques employ feedback control in order to track the particle but differ in the approach they use to ascertain the particle's 3D position. The first technique scans a small volume around a particle to build up a volumetric image that is then used to determine the particle position. The second technique scans only a single plane but utilizes optical aberrations which have been introduced into the optical system that break the axial symmetry of the point spread function and serves as an indicator ofthe particle's axial position. The current system has a frequency response at the video rate and an axial range of 100 tim. We tested the system by tracking welldefined test trajectories, and by tracking particles in model systems. We identified several different modes of motion in sucrose solutions and agarose gels, including the transient trapping ofparticles in the microdomains ofagarose gels.

Single particle tracking (SPT) is an optical microscopy technique where the motion of a single particle is followed.The first paper describing SPT was by Barak and Webb', which used SPT to trace the motion of fluorescently tagged lowdensity lipoprotein receptors in the plasma membrane ofhuman fibroblasts.The technique was extended to both bright field and differential interference contrast microscopy and used to follow the motion ofgold particles2'3.
SPT has several strengths that makes it an attractive technique for use in biological research.First, SPT possesses a spatial resolution as high as five nanometers.In contrast, a technique such as FRAP (fluorescence recovery after photobleaching) is limited to the dimensions ofthe point-spread function (PSF) ('-O.5 tm).Second, SPT has a fast temporal resolution.Many ofthe experiments have been done at the video rate (33 Hz) and some have achieved kinetics as fast as 200 Hz4.Another advantage of SPT is that experiments can be done in-vivo.Besides being nondestructive, in-vivo studies also allow the biochemical microenvironment around the particle to be probed.In addition, by its very nature, SPT allows the motion of individual particles to be studied rather than the ensemble averaged behavior of a population ofparticles.This can provide much more detailed information than other techniques can, particularly when dealing with heterogeneous populations.For example, SPT has helped reveal the different sub-populations of receptor molecules in the plasma membrane5.
SPT has some technical challenges: there can be artifacts due to labeling the particle with a relatively large latex or gold sphere; ambiguities may arise in the interpretation of the trajectories of particles; photobleaching or low signal to noise ratios can be a factor with fluorescent probes; and for high spatial resolution studies (-10 nm), it may be necessary to mechanically isolate the instrument from vibrations.
The early SPT systems were all two-dimensional systems.These systems were not able to effectively address fundamentally 3D phenomena such as endocytosis, intracellular transport and diffusion within the cytoplasm.To address this limitation, research groups have developed some 3D SPT tracking systems.Kao and Verkman6 developed a 3D SPT system that employed cylindrical optics in order to break the axial symmetry of the PSF and thus enable 3D single particle tracking.They obtained a frequency response of3-4 Hz and resolution of 12 nm in the axial direction and 5nm in the radial direction.Peters et al.7 who monitored the position of a laser beam focused onto a polystyrene sphere in order to track a particle in 3D.They were able to obtain a 1 kHz frequency response with an accuracy of 1 nm.Pralle et al.8 used a quadrant photodiode to detect the scattered light from a trapped sphere for 3D tracking.This technique has a high spatial and temporal resolution but a fairly limited range and calibrations are required for different axial depths.
Twophoton excitation is an excited state transition caused by the simultaneous absorption of two photons.Its application to microscopy was first implemented by Denk et aL.Since the excitation goes as the second power ofthe laser intensity in twophoton microscopy, appreciable excitation occurs only near the focal plane where the photon density is the highest.For high numerical aperture objectives the excitation region is typically 0.3 microns in the radial direction and 0.9 microns in the axial direction.Excitation localization not only reduces cell damage and photobleaching, but also creates a 3 D sectioning effect.A further advantage of twophoton microscopy is excellent background rejection due to the excitation wavelength being widely separated from the emission light and thus it can be easily eliminated with filters.Furthermore two-photon microscopy is particularly well suited for high-resolution studies ofthick tissues and cells.The longer wavelength used in twophoton microscopy has a deep penetration depth and low phototoxicity compared to the conventional confocal microscopy.For many highresolution tissue studies it is often the ideal technique to employ.
With this in mind, we have built a 3D SPT that utilizes the inherent 3D localization oftwo-photon excitation.The 3D SPT system we have developed has a frequency response at the video rate, an effective spatial resolution of less than I 5 nm, and an axial range that is limited by the working depth ofthe objective.20Materials and Methods

Instrument
Figure 1 shows the experimental setup for the two-photon SPT experiments.The system is built around a Zeiss Axiovert 1 10 epifluorescent microscope (Carl Zeiss, Thomwood, NY).The excitation source is a Mira 900 Ti-Sapphire mode-locked laser that is pumped by an Innova 3 10 argon-ion laser (Coherent Inc., Palo Alto, CA).The excitation pulses from the Ti-Sapphire laser are 150 fs in duration at full-width-half-maximum and have a repetition rate ofSO MHz.A galvanometer x-y scanner (Cambridge Technology, Watertown, MA directs the light into the microscope.A short pass dichroic mirror (Chronia Technology, Inc., Brattleboro, VT) reflects the laser light entering the microscope to the objective.To perform axial scanning we move the objective using a piezo 721, 1 PIFOC microscope focusing drive and a E612.C0 amplifier to drive the piezo Polytec P1, Costa Mesa, CA).The dichroic mirror in the microscope passes the fluorescence from the sample to the detection system that consists ofa 1(7400 photomultiplier tube Haniamatsu, Bridgewater, NI) operating in photon-counting mode.A home-built PCI card in a PC controls both the x-y scanner and the z-stage, and in addition counts the pulses from the discriminator.The software used to control the PCI card and to collect and display the data is custom-written in C++, ftm piz.
Figure 1: The standardsetupforthe Iwo-photon microscope.Most ofthemodificationsforparticle tracking are software rather than hardware based with the exception ofa signal-outfrom the scanner wh/ch servers tosynchronfre the start of the ta acquisitionwith the software.Thisallows us to operate the scanner near it bandwidth-limitedrate with greater accuracy.
We have implemented two different techmques to accomplish 3D tracking on the two-photon microscope The first method is use 3D volumetric scanning to localize the particle.The second technique is similar to Kao and Verkman's approach in that it utilizes aberrations in the PSF to determine the axial position ofthe particle There are three main ideas behind both of the techniques.The first is that the 3D sectioning effect of twophoton microscopy innately helps localize the axial position ofthe particle The second idea is that by scanning only a small region in the vicinity of the particle we can achieve a higher frequency response than we could if we were scanning a larger area And the third major idea is active feedback control in the trackmg routine As explained below, ifthe computer can calculate the position ofparticle and then quickly reposition the scan region, we can dynamically follow the motion ofthe particle over extended distances even though we are only scanmng a small region at any given time In the volumetric tracking approach an xy plane is built up pixel by pixel in the focal plane of the objective by scanning the laser beam with the galvanometer xy scannmg mirrors The counts in each pixel are recorded by detecting the fluorescence with a photomultiplier tube operating in photon counting mode.
Immediately after scanning a plane, the axial piezo scanner repositions the objective and another xy plane is scanned The process repeats until we have built up sufficient volumetric information to determine the position of the particle For this study we typically chose either three to six different axial positions This was few enough to insure a sufficient frequency response and many enough to allow an accurate determination ofthe axial position ofthe particle After acquiring a 3D volume the computer then calculates the location of the particle For reasons of ease and computational efficiency, we have the computer perform a simple center of mass calculation to determine the particle's position within the volume The computer then repositions the scan region by offsetting the origm ofthe new scan and place the particle in the center ofthe new scan region As long as the procedure is fast enough such that the particle does not move outside the scan region while the volume is being scanned it will be possible to dynamically follow the motion of the particle The other method we have implemented to track particles in three dimensions involves introducing aberrations that break the axial symmetry of the PSF into the optical system of the microscope This symmetry breaking allows us to determine the full 3D position ofthc particle by merely scanmng a single xy plane in the vicinity ofthe particle We find that applying an uneven torque to the dichroic mirror in the microscope conveniently produces the needed aberrations by bending the mirror into a slightly cylindrical shape Figure 2a is a plot of the PSF at several axial positions The axial position ofthe particle is then determined from the shape ofthe PSF by calculating the generalized moment of the image, which is given by: (M-M) where and M are the second moments ofthe images Ex,2 I, -x, i, and x1 andy7 are the pixel position at the ith pixel and /is the intensity at that position Figure 2b is a plot ofthe GM for the PSF pictured in 2a.When tracking the particle the radial position is determined by a center of mass ca1cu1atiot To determine the axial position the computer calculates the GM ofthe image and then relates this back to the axial position ofthe particle by using the inverse ofthe GM versus distance curve show in 2b.

3 Computer Simulations
In order to test the theory for 3D SPT, we performed 3D random walks on both a lattice and continuum model.For the continuum model, we specify the distance the particle moves by generating a Gaussian random deviate ofunit variance Two uniform random deviates determined the azimuthal and radial angle ofthe particle motion For the lattice model calculations we chose a cubic lattice and generated a umform random deviate to specify one ofthe six possible directions the particle could move at each step.All the random deviates were generated with the C routines gasdev and ran2 (Press et al., 1992).
For both ofthe models we scaled the calculations using the relation 12 6Dt (3)

4 Data Analysis
The aim ofdata analysis in SPT is to extract from a particle trajectory the relevant parameters that characterize the motion (such as the diffusion coefficient velocity and escape time) and then use this information to assign various modes of motion to the trajectory or sub4rajectory10.Some ofthe modes ofmotion investigated are free diffusion, diffusion with flow, and diffusion in environments where a particle can be trapped for a period oftime before again diffusing away.
The difficulty in classifying trajectories is that a pure random walk can mimic a wide variety ofmotions In many cases, it is not a trivial matter to make a conclusive categorization ofeven a simulated trajectory.Matters are made worse for (a) 0.0 0.5 1.0 1.5 nicrcns The difficulty in classifying trajectories is that a pure random walk can mimic a wide varietyofmotions.In many cases it is not a tnvial matter to make a conclusive categorization ofeven a simulated trajectory Matters are made worse for expenmental data due to the spatial and temporal limitations ofthe instrument and the finite length ofthe trajectories To help classify the trajectories we will employ both MSD plots and statistical assays that quantify the probability that a particle with a known diffusion coefficient D stays within a certain radius R forall times t.

MSD versus Time Plots
TheMSD plot for a trajectory ofN points is given mathematically as MSD= MSD(n&) + MSD(,tht) + MS'D(nzt) and n is the lag time being considered.Thus for example, MSD3wouldbe the average square distance a particle moved in three time steps.
There are two main types ofmotion with which we will be concerned with in this paper when dealing with MSD plots.They are represented mathematically by: (r2 Diffusion with flow (7) To determine the diffusion coefficient ofa particle, we have fit a straight line to the first three points ofthe mean square distance (MSD) versus time plot for the trajectory.Ifthe entire MSD versus time curve is fit to find the diffusion coefficient, the resulting distribution ofdiffusion coefficients is so broad as to make the calculation useless".The reason being is that there are fewer sub4rajectories oflonger length in a trajectory, and thus there will be a greater variance in the value ofthe MSD for the longer lag times This leads to a very broad distribution in the slopes ofthe fits to the MSD versus time plots.An alternative and frequently equivalent approach is to use a weighted fit to the MSD plot that gives greater weight to the shorter time lag points.

Transient Trapping of a Particle
Next we turn to methods to detect confinement of diffusing particles.The discussion will follow the work done by Saxton12 and Simson et al .who both dealt with the two-dimensional case, In many instances in SPT it is necessary to distinguish between a trajectory that is a pure random walk and one that is constrained within a certain region For example, one may wish to ascertain whether a receptor molecule is trapped within a microdomain in the plasma membrane or whether a vesicle is trapped within a region m the cytoplasm As noted in the last section, we can m pnnciple solve the problem by plotting the MSD versus time For a sufficiently long trajectoiy or ensemble oftrajectories, the character ofthe motion will be evident by the shape of the curve.However, in cases where the particle undergoes transient trapping, or where there is only short trajectory available to examine, it is more complicated to establish whether the motion was the result ofactually being trapped within a region or was merely due to Brownian motion.To test the instrument we tracked immobilized 200 rim latex spheres that were dried on a coverslip (Figure 4).We obtained a standard deviation of approximately 30 nm in both radial directions and 40 nm in the axial direction.This is higher than what we would expect giventhephoton count rate for the particles, but is consistent with whatwe obtained with standard wide field video microscopy obtained on the same microscope The optical table we performed these particular measurements on was not vibration isolated This is what we thrnk is causing the relatively high standard deviations In figure 4b we measured the interparticle distance oftwo particles as a fbnction oftime (whichshould be more independent of vibrations than the position measurement of a single particle) and obtained a standard deviation of 14 nm which is more consistent with what we expect gwen the number ofdetected photons

Test Trajectories
To further test the instrument, we mounted a xypiezo stage on the microscope (P730 Piezo xyscanner with E6I2CO controller) in order to be able to move the sample in welldefined trajectories.The particular dataset shown in Figure 5was obtained with the aberration method and with a frequency response of 14 Hz We obtained similar results with other parameters The piezo was mounted vertically such that the sample could be moved along the axial axis of the microscope system and along one ofthe radial axis.First we compared the trajectory we obtained by moving the sample along the same trajectory first along the yaxis and then along the z-axis A 5 im sine wave with a frequency of 0 1 Hz was input into the xypiezo.This same input signal was then used to drive the other axis for comparison.The recovered trajectories were with 5% of the accepted axial calibration for the microscope system and had identical frequencies given the precision of the instrument as described in Figure 3.In Figure 5b we tested whether the amplitude of the motion would scale appropriately by changing the amplitude of the input function.A 5 jim and 2 pm signal were separately input into the piezo, and the recovered motions were within 3 8% of one another This is higher than what we would expect and is most likely due to the fact that the zpiezo used to drive the objective was operating in open-loop mode and the hysteresis of the piezo was not taken into account.

Model Systems
We examined a number of other model systems in order to see if we could identify different modes of motiox To do this we tracked 200 nm latex spheres in sucrose solutions The two types of motion we expect for trajectories free in solution are pure diffusion and diffusion with flow Diffusion with flow will occur if the system contains convection currents otherwise we will have pure diffusion Both modes of motion are mathematically welIcharacterized phenomena, so they make suitable systems to test the instmment Sucrose solutions were used to increase the viscosIty of the solutions relative to that of water.This was done so as to make the diffusion coefficients physiological and to further test the instrument by tracking particles over a range ofdiffusion coefficients Figure 6a shows a typical trajectory of particle tracked in 60% sucrose.The motion appears to be Brownian in all three dimensions The data consisted ofa sequence of400 coordinates that were recorded 250 ms apart and were taken with the volumetric tracking approach.Figure 6b contains the MSD versus time plots ofthe trajectory offigure 6a.The resulting curves are very nearly the straight line we would expect from pure Browman motion The slope of the MSD for the individual axis is approximately one4hird the slope of the radial MSD which we expect for isotropic diffusion The deviations from the straight line are due to the stochastic nature of the Brownian motion and are within the anticipated statistical fluctuations.Figure 6c shows the MSD plots for lag times up to 0.5 s.The recovered parameters from the linear fits are 4. 13E-1O cm2/s, 4,33E-1O cm2/s, and 4.53E40 cm2/s, for the individual x, y, and z axis, and 4.30E40cm2/s for the overall diffusion coefficient.The average recovered diffusion coefficient for 10 different trajectories of 200 nm spheres in 60% sucrose is 4. lB 10 cmZ/s.This is within 10% ofthe value of38E-1O crn2/s we expect from the StokesEinstein relationship given by: kT D=-- (10)

6,z-iR
where k is Boltzmann's constant, T is the temperature (295 K), I? is the radius of the particle, and i is the viscosity of the solution.
Figure 7a shows an example of diffusion with flow.The sample consisted of 200 nm spheres in a 50% sucrose solution.This sample was less viscous than the 60% sucrose solution and had greater convection currents.The data consisted of a sequence of 1000 coordinates that were recorded 125 ms apart and was obtained using the volumetric tracking approach.To obtain the diffusion coefficient we fit a straight line to the first three points ofthe trajectory as shown in Figure 7c.The recovered value of 14.4E10 crn2/s was within 2% of the expected value as determined from the Stokes-Einstein relation.To calculate the velocities, we fit the trajectory to equation 5, with D and V as parameters.The fit is shown in Figure 7b.The recovered speeds for the flow were 0.35 pm/s, 0.24 .tm/s,and 0. 19 run/s for the x, y. and z-axis respectively, and 0.467 tim/s for the overall speed.Caution must be exercised in the interpretation of the velocity parameters.We assumed uniform flow over the entire length of the trajectory.which may not be a vaid assumption since tne velocity could have been changing in this interval.This is a general concern when trying to apply the model of pure diffusion with flow to experimental trajectories.One way to address the problem is to break the trajectory into shorter segments and analyze each segment separately.The velocity in each segment would then more likely be constant.At any rate, the velocities that were recovered by the fit are reasonable given the dynamics ofthe system.
: : :: r Li:: ., Figure 8 is a plot ofrecovered diffusion coefficients for particles tracked in 6O%, 56%, 5O%, and 40% sucrose using the volumetric scanmng method The diffusion coefficients were plotted versus inverse viscosity The resulting fit is linear as expected from the Stokes-Einstein relationship and has a slope of2 09 N which is wiThin 3 3% ofthe value of2 16 N we expect for the conditions ofthe experiment.lNiscoslty (1O2cP) We also tracked the motion oflatex spheres in 0.3% agarose gels.Agarose gels consist ofa network of cross4inked polymers that on the microscopic scale, form an mhomogeneous media with pores ofdifferent sizes that can transiently trap a diffusing particle.
Figure 9 shows the some ofthe results obtained from tracking a sphere in agarose.The data for the agarose experIments consisted ofa sequence of400 points each 256 ms apart obtained with the volumetric approach The profile plot for the trajectory ofthe particle in the agarose gel was plotted m Figures 9b The profile plots for the sphere in the agarose gels display significant trapping The value of32 in Figure 9b corresponds to only a 1E33 probability that the motion ofthe particle in this region was due to pure Brownian motion In comparison, the c-profile plots m Figures 9c for a particle Ifl 60% sucrose did not display any sigmficant trapping with the highest value being only 1 2 This is consistent with our earlier classification ofthe motion of200 nm spheres in 60% sucrose as pure Browman motion Figure 9: () 3D lrajecto,y ofa 200 nm spheres in a 03% agarose solution. (#b) Profi1ep1otsfor ).Thepartick started at the origin andwas trappedrn a microdomainfor approximately 20 seconds and then was trapped twice more between 30 and50 seconds into the trajectory andagarn at approximately 90 seconds (c) For comparison theprofileplotsfor 200 nm spheres in 60% sucrosefrom Figure 6 No sigrnJIcant tracking was observed as is expected We have constructed a novel SPT instrument that simultaneously tracks particles in all three dimensions using two differentbut related techniques The volumetric scanning approach has a frequency response of up to approximately 68 Hz and is very straightforward to implement.The aberration method allows us to track particle in all three dimensions at a currently maximum rate ofapproximately 28 Hz.We tested the instrument by moving the sample in well-defined trajectories with a xy piezo scanner attached to the microscope and tracking the particle with the techniques.We also investigated different methods that can help classify the 3D trajectories we obtain from various modes of motion The diffusion coefficients recovered for 200 nm latex spheres in solution were in agreement with the StokesEmstein relation, and the standard deviation in the distribution of diffusion coefficients were withm what we expected given the statistical and instrumental uncertainties of the method.None of the trajectories in the free solution displayed any trappingas determined by the plots oftheir -profiles In contrast the motion ofthe spheres in the agarose gel did display sigmficant trapping The data we have obtained so far are all selfconsistent and establishes that the basic methodology we havedeveloped is sound The technique can be improved in a number ofrelatively easy ways.First, the galvanometer scannerwe are using is an older model and is bandwidth limited to 500 Hz Current scanners can go as fast 1 kHz, which will gives us a factor of two mcrease in performance and allow us to track particles at 60 Hz Second, the dichroic we used to generate the aberrations in the aberration tracking method can be fabncated to a specific curvature This may allow us to extend the range overwhich we can ascertain the 3D position ofthe particle without having to reposition the objective.Also, specifically engineering a PSF may allow us to increase the precision oftbe axial position measurement.Thirdly, we can incorporate simultaneous widefleld imaging, which will allow us to momtor the position ofthe particle relative to any larger structures ofinterest.This would be ofparticular interest where the global structure ofthe sample is also changing, such as in cell migration studies.

Conclusions
We have developed a SPT system on the twophoton microscope that is capable of tracking a particle in all three dimensions over an extended axial range at a fast frequency response.The system is built around a twophoton fluorescence microscope and utilizes the inherent 3D localization of two-photon excitation.We have studied the motion of particles in model systems consisting of sucrose solutions and agarose gels.We have been able to extract the relevant parameters that characterize the trajectories and classify the mode of motion as either pure diffusion diffusion with flow, or transient trapping. 6Acknowledgemen Wewould like to acknowledge NIH grants RRO3 155 to E 0 andR290M56486-O1 to P T S

Fig 2 :
Fig 2: (a) Successive axial images taken 750 nm apart ofa 200nmlatex sphere embedded in an agarose gel taken with a C Apochromat 'lOx water immersion objective.(b) The resulting GMcurvefor the central axial layers.

Figure 4
Figure 4 Immobilized200 nm spheres dried on a microscope coverslip The objective was a Zeiss 63x oil immersion objective (a) The time tracefor each ofthe axisfor a smgleparticle The standarddeviation were 29 6 nm 29 7 nm and 39 7 nmfor xy andzaxis Note the traces were slightly offsetfrom one another (b) The interparticle chstance between two particles dried on a coversl4.The standarddeviatIon ofthe measurement was 14. 1 nm.

Figure 5 :
Figure 5: Recovered trajectories obtainedfrom tracking aparticle dried on a coversl4.The entire samplewas mounted to a xypiezo scanner that couldmove the particle along both the z andyaxis independently.The input signals were sine waves ft om afunction generator.