## Tracking-FCS: Fluorescence correlation spectroscopy of individual particles

Optics Express, Vol. 13, Issue 20, pp. 8069-8082 (2005)

http://dx.doi.org/10.1364/OPEX.13.008069

Acrobat PDF (485 KB)

### Abstract

We exploit recent advances in single-particle tracking to perform fluorescence correlation spectroscopy on *individual* fluorescent particles, in contrast to traditional methods that build up statistics over a sequence of many measurements. By rapidly scanning the focus of an excitation laser in a circular pattern, demodulating the measured fluorescence, and feeding these results back to a piezoelectric translation stage, we track the Brownian motion of fluorescent polymer microspheres in aqueous solution in the plane transverse to the laser axis. We discuss the estimation of particle diffusion statistics from closed-loop position measurements, and we present a generalized theory of fluorescence correlation spectroscopy for the case that the motion of a single fluorescent particle is actively tracked by a time-dependent laser intensity. We model the motion of a tracked particle using Ornstein-Uhlenbeck statistics, using a general theory that contains a number of existing results as specific cases. We find good agreement between our theory and experimental results, and discuss possible future applications of these techniques to passive, single-shot, single-molecule fluorescence measurements with many orders of magnitude in time resolution.

© 2005 Optical Society of America

## 1. Introduction

1. D. Magde, E. L. Elson, and W. W. Webb, “Thermodynamic fluctuations in a reacting system - measurement by fluorescence correlation spectroscopy,” Phys. Rev. Lett. **29**, 705–708 (1972). [CrossRef]

2. E. L. Elson and D. Magde, “Fluorescence correlation spectroscopy. 1. Conceptual basis and theory,” Biopolymers **13**, 1–27 (1974). [CrossRef]

3. D. Magde, E. L. Elson, and W. W. Webb, “Fluorescence correlation spectroscopy. 2. Experimental realization,” Biopolymers **13**, 29–61 (1974). [CrossRef] [PubMed]

4. O. Krichevsky and G. Bonnett, “Fluorescence correlation spectroscopy: the technique and its applications,” Rep. Prog. Phys. **65**, 251–297 (2002). [CrossRef]

5. S. T. Hess, S. Huang, A. A. Heikal, and W. W. Webb, “Biological and Chemical Applications of Fluorescence Correlation Spectroscopy: A Review,” Biochemistry **41**, 697–705 (2002). [CrossRef] [PubMed]

6. A. J. Berglund, A. C. Doherty, and H. Mabuchi, “Photon statistics and dynamics of Fluorescence Resonance Energy Transfer,” Phys. Rev. Lett. **89**, 068101 (2002). [CrossRef] [PubMed]

7. H. D. Kim, G. U. Nienhaus, T. Ha, J. W. Orr, J. R. Williamson, and S. Chu, “Mg^{2+}-dependent conformational changes of RNA studied by fluorescence correlation and FRET on immobilized single molecules,” Proc. Natl. Acad. Sci. U.S.A. **99**, 4284–4289 (2002). [CrossRef] [PubMed]

8. K. C. Neuman and S. M. Block, “Optical Trapping,” Rev. Sci. Instrum . **75**, 2787–2809 (2004). [CrossRef]

9. M. J. Saxton and K. Jacobson, “Single-particle tracking: applications to membrane dynamics,” Annu. Rev. Bio-phys. Biomolec. Struct. **26**, 373–399 (1997). [CrossRef]

10. A. E. Cohen and W. E. Moerner, “Method for trapping and manipulating nanoscale objects in solution,” Appl. Phys. Lett. **86**, 093109 (2005). [CrossRef]

11. A. E. Cohen, “Control of Nanoparticles with Arbitrary Two-Dimensional Force Fields,” Phys. Rev. Lett. **94**, 118102 (2005). [CrossRef] [PubMed]

12. T. Meyer and H. Schindler, “Simultaneous Measurement of Aggregation and Diffusion of Molecules in Solutions and in Membranes,” Biophys. J. **54**, 983–993 (1988). [CrossRef] [PubMed]

13. T. Ha, D. S. Chemla, T. Enderle, and S. Weiss, “Single molecule spectroscopy with automated positioning,” Appl. Phys. Lett. **70**, 782–784 (1997). [CrossRef]

14. J. Enderlein, “Tracking of fluorescent molecules diffusing within membranes,” Appl. Phys. B **71**, 773–777 (2000). [CrossRef]

15. J. Enderlein, “Positional and Temporal Accuracy of Single Molecule Tracking,” Sing. Mol. **1**, 225–230 (2000). [CrossRef]

16. R. S. Decca, C.-W. Lee, and S. R. Wassall, “Single molecule tracking scheme using a near-field scanning optical microscope,” Rev. Sci. Instr. **73**, 2675–2679 (2002). [CrossRef]

17. V. Levi, Q. Ruan, K. Kis-Petikova, and E. Gratton, “Scanning FCS, an novel method for three-dimensional particle tracking,” Biochem. Soc. Trans. **31**, 997–1000 (2003). [CrossRef] [PubMed]

18. A. J. Berglund and H. Mabuchi, “Feedback controller design for tracking a single fluorescent molecule,” Appl. Phys. B **78**, 653–659 (2004). [CrossRef]

19. K. Kis-Petikova and E. Gratton, “Distance measurement by circular scanning of the excitation beam in a two-photon microscope,” Microsc. Res. Tech. **63**, 34–49 (2004). [CrossRef]

20. V. Levi, Q. Ruan, and E. Gratton, “3-D particle tracking in a two-photon microscope. Application to the study of molecular dynamics in cells,” Biophys. J. **88**, 2919–2928 (2005). [CrossRef] [PubMed]

21. M. A. Digman, P. Sengupta, P. W. Wiseman, C. M. Brown, A. R. Horwitz, and E. Gratton, “Fluctuation Correlation Spectroscopy with a Laser-Scanning Microscope: Exploiting the Hidden Time Structure,” Biophys. J. **88**, L33–L36 (2005). [CrossRef] [PubMed]

22. M. A. Digman, C. M. Brown, P. Sengupta, P. W. Wiseman, A. R. Horwitz, and E. Gratton, “Measuring fast dynamics in solutions and cells with a laser scanning microscope,” Biophys. J. **89**, 1317–1327 (2005). [CrossRef] [PubMed]

*et. al*. [17

17. V. Levi, Q. Ruan, K. Kis-Petikova, and E. Gratton, “Scanning FCS, an novel method for three-dimensional particle tracking,” Biochem. Soc. Trans. **31**, 997–1000 (2003). [CrossRef] [PubMed]

19. K. Kis-Petikova and E. Gratton, “Distance measurement by circular scanning of the excitation beam in a two-photon microscope,” Microsc. Res. Tech. **63**, 34–49 (2004). [CrossRef]

20. V. Levi, Q. Ruan, and E. Gratton, “3-D particle tracking in a two-photon microscope. Application to the study of molecular dynamics in cells,” Biophys. J. **88**, 2919–2928 (2005). [CrossRef] [PubMed]

## 2. Experimental methods and results

*ω*

_{0}= 2

*π*× 8 kHz in order to translate the excitation laser in a circular path about its axis. This circular motion modulates the laser intensity at off-axis positions within the sample volume (on-axis, a particle sees constant laser intensity). We estimate a particle’s position by demodulating the fluorescence signal at the laser rotation frequency using a lock-in amplifier. The resulting

*xy*position estimates are filtered by a programmable microcontroller (Analog Devices, Norwood, MA; Keil

*μ*Vision software, Plano, TX) and fed back to a two-axis piezoelectric sample stage (Polytec PI, Inc., Auburn, MA) that translates the entire sample volume. A stochastic state-space model of this system along with optimal feedback controller design methods that incorporate realistic experimental limitations, such as noisy fluorescence measurements and piezo-actuator bandwidths, can be found in [18

18. A. J. Berglund and H. Mabuchi, “Feedback controller design for tracking a single fluorescent molecule,” Appl. Phys. B **78**, 653–659 (2004). [CrossRef]

18. A. J. Berglund and H. Mabuchi, “Feedback controller design for tracking a single fluorescent molecule,” Appl. Phys. B **78**, 653–659 (2004). [CrossRef]

*w*

_{xy}rotating at a radius

*r*

_{0}and angular frequency

*ω*

_{0}about its direction of propagation. In any fixed plane perpendicular to the axis of rotation, we may approximate the rate of photon detection from a particle at position (

*x*,

*y*) ↦ (

*ρ*,

*θ*) by a two dimensional Gaussian “detectivity” function, given in cylindrical coordinates by

*ω*

_{0}, the quadrature outputs are given by the cosine and sine transforms of Φ

_{t}(

*ρ*,

*θ*) at frequency

*ω*

_{0}. Denoting these quadratures by

*f*

_{c}(

*ρ*,

*θ*) and

*f*

_{s}(

*ρ*,

*θ*) respectively, we find (neglecting finite-bandwidth filtering at the lock-in amplifier’s output)

*I*

_{1}is the modified Bessel function of order 1. This pair of transformations maps curves in the (

*x*,

*y*) plane to curves in the (

*f*

_{c},

*f*

_{s}) plane, with the convenient property that the angular coordinate

*θ*is unchanged. Lock-in detection may therefore incorrectly estimate the

*distance*of a particle from the origin, but never the

*direction*. In the limit

*ρ*/

*w*

_{xy}≪ 1, with

*r*

_{0}~

*w*

_{xy}, we have

*h*(

*ρ*)∝

*ρ*for small

*ρ*, shows that the lock-in output is a

*linear*function of the particle’s position near the origin and provides an error signal suitable for feedback control. While a particle is locked to the laser axis by the feedback controller, the non-linear character of

*h*(

*ρ*) is not apparent. This is a general strategy for feedback control in mildly nonlinear systems, in which a suitable control law linearizes the system dynamics about a fixed point. Note that when a rotating laser is focused through a microscope objective, the beam converges and then diverges, with the proportionality constant between the particle position and the linearized lock-in quadrature outputs becoming a function of the axial coordinate

*z*, changing signs upon passing through the focal plane (See Fig. 1). In this way, we see that our two-dimensional controller may track transverse motion over a range of depths, so long as the control law provides enough gain margin to absorb the variation in position estimation and provided that the sample volume does not cross the focal plane.

*μ*l, 0.1 nM solution of microspheres was placed between glass cover slips. We used excitation light with 532 nm wavelength and only ~ 1 – 10

*μ*W optical power for exciting the fluorescently-labelled beads. Our overall photon count rates of ~ 10

^{5}/s are comparable to the count rates from individual dye molecules excited by a correspondingly higher laser power. In this configuration, the fluorescent beads provide a good test-case toward the eventual goal of tracking individual molecules labelled with a single fluorescent marker. The

*z*position of the objective was adjusted such that the focus lay just outside the sample boundary, so that the particle cannot diffuse across the focal plane of the microscope objective. In calibration experiments at high fluorophore concentration, we have estimated the depth of our sample in this configuration to be ~ 1

*μ*m (see Fig. 1).

*μ*m along both the

*x*and

*y*axes. The residual fluctuations in the fluorescence will be discussed in detail in the next section. Here, we simply remark that they arise from the feedback control bandwidth and also from the uncontrolled motion of the particle in the axial (

*z*) direction.

*D*for this microsphere from the position of the sample stage during tracking. While the particle is (approximately) locked on the laser axis by the feedback controller, the

*xy*position of the sample stage provides a bandwidth-limited filtration of the particle’s position. Let the change in a particle’s

*x*position during a time interval Δ

*t*be given by Δ

*x*, and similarly for

*y*. Then we may construct a simple estimator

*D̂*for a particle’s diffusion coefficient

*D*based on the sample stage trajectory:

*t*. If Δ

*t*is smaller than the inverse closing bandwidth of the controller, then the sample stage will exhibit reduced variations on this timescale, and we will tend to underestimate the resulting diffusion coefficient with Eq. (5). It is easy to see this by considering an extreme case: imagine binning the particle’s position over extremely small intervals, much smaller than the response time of our piezoelectric sample stage (say, 1

*μ*s). Then in each time interval Δ

*t*, the sample stage cannot track the diffusing particle (although the particle does not move far enough to escape the tracking controller) and only follows an “average” trajectory. The sample stage will move a distance proportional to Δ

*t*(

*i*.

*e*. at a fixed velocity along the averaged particle trajectory) for these very small bin times, while the particle itself moves a distance proportional to √Δ

*t*(the characteristic property of Brownian motion). For small enough Δ

*t*, then, the estimator given by Eq. (5) will dramatically underestimate the particle’s diffusion v coefficient (by a factor proportional to √Δ

*t*). As Δ

*t*becomes much larger than the inverse closing bandwidth, however, the sample stage can move sufficiently fast to track the detailed motion of the particle within a single bin interval. In this case, we expect a good statistical estimate of

*D*from Eq. (5), but with a concomitant increase in the estimator variance due to the smaller number of bin intervals per fixed-length trajectory. Estimates constructed in this way are shown in Fig. 3, along with error bars calculated for the estimator Eq. (5) assuming underlying Brownian motion statistics (see [23] for details about statistical estimation and error estimates). For large bin times, the estimate of

*D*converges to a value 6.2

*μ*m

^{2}/s, close to the value 7.2

*μ*m

^{2}/s predicted by the Stokes-Einstein relation for 60 nm-diameter beads in water at room temperature. It is possible that surface adhesion slightly reduced the effective diffusion coefficient, or that this particular microsphere was closer to 70 nm in diameter, than the batch average of 60 nm. We do not suspect induced optical dipole forces to give any significant trapping effect, since only a moderate trapping effect was observed for 1 mW of near-IR laser power on microspheres with 276 nm diameter in [20

20. V. Levi, Q. Ruan, and E. Gratton, “3-D particle tracking in a two-photon microscope. Application to the study of molecular dynamics in cells,” Biophys. J. **88**, 2919–2928 (2005). [CrossRef] [PubMed]

*μ*W of 532 nm light. A detailed analysis of optical trapping effects in FCS experiments can be found in [24

24. G. Chirico, C. Fumagalli, and G. Baldini, “Trapped Brownian Motion in Single- and Two-Photon Excitation Fluorescence Correlation Experiments,” J. Phys. Chem. B **106**, 2508–2519 (2002). [CrossRef]

## 3. Fluorescence fluctuations and Tracking-FCS

*z*direction. In this section, we develop a full theoretical model for the residual fluorescence fluctuations that arise during a typical tracking period under Gaussian illumination. With this model, we may study the both the dynamics of the feedback controller and also the intrinsic statistics of the particle’s diffusive motion solely by examining fluorescence fluctuation statistics. In this way, our theory is analogous to the theory of FCS. The main ingredient of our model is the use of Ornstein-Uhlenbeck statistics (see below) for modelling the motion of a tracked (or trapped) particle.

*trapped*in the laboratory reference frame. We will no longer model the particle as a free Brownian particle in the frame of the sample stage; rather, we characterize the action of the tracking controller as a

*damping force*centered at the coordinate origin (the lock-point of the tracking controller). Since diffusion along the Cartesian axes is independent, consider the particle’s motion along one axis only. Let the particl’s position be given by

*X*

_{t}and its diffusion coefficient be

*D*. The damping force due to tracking may be modelled as a rate

*γ*

_{x}, the step-response of the closed-loop servo system, such that the particle’s position follows an approximate Ornstein-Uhlenbeck process:

*dW*

_{t}is the differential of a Wiener process [25, 26]. Equation (6) is a stochastic differential equation whose solutions

*X*

_{t}satisfy Ornstein-Uhlenbeck statistics for the stationary distribution

*P*

_{0}(

*x*) = lim

_{t→∞}

*P*(

*X*

_{t}=

*x*) and transition probability

*P*

_{τ}(

*x*

_{2}|

*x*

_{1}) = lim

_{t→∞}

*P*(

*X*

_{t+π}=

*x*

_{2}|

*X*

_{t}=

*x*

_{1}):

*a*

_{τ}= exp(-

*γ*

_{x}

*τ*),

*x*¯

^{2}[1 - exp(-2

*γ*

_{x}

*τ*)] and the length-scale is given

*x*¯

^{2}=

*D*/

*γ*

_{x}. We consider only stationary distributions, since we are interested in the steady-state of the tracking controller, not initial transients.

*arbitrary*time-dependent path

*w*

_{x}

*χ*

_{t}(in one dimension only), where

*χ*

_{t}is the dimensionless laser path written in units of the beam waist

*w*

_{x}. We assume the laser intensity Φ

_{x}(

*x*) is Gaussian with beam waist wx, so that the rate of photon detections is also Gaussian with the same waist. The fluorescence autocorrelation function is given in this case by

*t*. See Appendix A for a derivation of Eq. (9). It is convenient to define a dimensionless “confinement parameter”

*ζ*

_{x}by

*τ*

_{x}=

*D*is the usual diffusion time.

*ζ*

_{x}is a measure of the localization of the trapped particle with respect to the excitation beam waist. Free Brownian motion results are recovered in the limit

*γ*

_{x}→ 0 (

*ζ*

_{x}→ 0), while

*ζ*

_{x}→ ∞(

*ζ*

_{x}→ 1) represents the perfect tracking limit.

_{x}(

*τ*;

*χ*

_{t}), we may explicitly evaluate for Gaussian Φ

_{x}and probabilities given by Eqs. (7) and (8):

*G*(

*τ*) = 〈

_{x}(

*τ*;

*χ*

_{t})〉

_{t}. If we now consider a full three-dimensional case, in which the particle is trapped with rate

*γ*

_{k}along axis

*k*∊ {

*x*,

*y*,

*z*} in a Gaussian laser with waist

*w*

_{k}in the

*k*direction, and the laser centroid follows the three-dimensional path

**r**

_{t}, we have an explicit expression for the full fluorescence autocorrelation function

**r**

_{t}. It reproduces the usual results for free Brownian motion (

*γ*

_{k}= 0) and time-independent laser intensity (

**r**

_{t}=

**r**

_{0}) in the appropriate limits.

*D*symmetrically “trapped” along the

*x*and

*y*axes with rate

*γ*

_{xy}ina rotating radially symmetric Gaussian beam with transverse waist

*w*

_{xy}. for our experiment,

*γ*

_{xy}is exactly the feedback tracking bandwidth;

*i*.

*e*. it is the reciprocal of the (exponential) response time for a unit-step input to the tracking controller. The laser rotates at radius

*w*

_{xy}

*ρ*

_{0}and angular frequency

*ω*

_{0}. In our experiment, microspheres are confined in the

*z*direction by the boundaries of the sample volume, which we may take to be reflecting boundaries for the relevant case of low surface adhesion. Green’s functions for free particle motion and corresponding fluorescence autocorrelation functions have been calculated as an infinite series for a Gaussian beam focused

*symmetrically*between two reflecting planes [2

2. E. L. Elson and D. Magde, “Fluorescence correlation spectroscopy. 1. Conceptual basis and theory,” Biopolymers **13**, 1–27 (1974). [CrossRef]

27. A. Gennerich and D. Schild, “Fluorescence correlation spectroscopy in small cytosolic compartments depends critically on the diffusion model used,” Biophys. J. **79**, 3294–3306 (2000). [CrossRef] [PubMed]

*asymmetric*configuration. Instead of tackling this difficult analytical problem, we instead approximate the reflecting boundaries in a way that naturally fits the formalism developed here: we introduce a third Ornstein-Uhlenbeck particle trap in the

*z*dimension with corresponding rate

*γ*

_{z}. This trap is

*not*due to the closed-loop particle tracking in our case; it is simply a tractable approximation for two reflecting planes that are placed asymmetrically about the laser focus. For this case, we have

**r**

_{t}/

*w*

_{x}= (

*ρ*

_{0}cos

*ω*

_{0}

*t*,

*ρ*

_{0}sin

*ω*

_{0}

*t*,

*z*

_{0}). Defining

*ω*

_{0}in Eq. (15) are an artifact of the laser rotation, and they occur at a much larger frequency than the servo response frequency

*γ*

_{xy}and the

*z*relaxation time

*γ*

_{z}. We are more interested in the contribution of these latter relaxation rates, and their interaction with a particle’s diffusion dynamics. In order to decouple these two effects, we may average the fluorescence correlation function over “coarse-grained” time bins, which are large compared to the rotation frequency but small compared to all other rates. Let us average

*G*(

*τ*) over bins of size

*T͂*≈

*NT*, where

*T*= 2

*π*/

*ω*

_{0}is the rotation period. (There exists an integer

*N*for which the approximation

*T͂*≈

*NT*is valid whenever

*T͂*≫

*T*, and of course, the approximation is exact when

*T͂*is equal to an integer number of rotation periods.) Denoting the coarse-grained autocorrelation function by

*G͂*(

*τ*), we have

*g*(

*τ*),

*λ*

_{τ,xy}, and

*λ*

_{τ,z}are all approximately constant over the coarse grained time bin

*T͂*. The remaining integral can be reduced to the form

*I*

_{0}is the modified Bessel function of order 0. After performing the integration in Eq. (16), we find

*average*of the shape of the laser over a rotation period. In our experiment, we expect that the laser is not particularly well-approximated by a rotating Gaussian excitation profile because we focus the laser into an aqueous solution using an oil-immersion objective. Even though the full expression for

*G*(

*τ*) is not particularly good in our case, the approximate expression matches the experimental data quite well. See Fig. 4. The fit parameters are consistent with the estimate of

*D*from Fig. 3 for a transverse beam waist

*w*

_{xy}= 1

*μ*m. Furthermore, the tracking bandwidth of 134 Hz matches well with the electronic design parameters and the 200 Hz designed bandwidth in [18

**78**, 653–659 (2004). [CrossRef]

*z*direction of

*ω*

_{0}to be much faster than whatever dynamics he or she wishes to measure. By increasing the number of periods over which the lock-in signal is integrated, the choice of

*ω*

_{0}will not affect the signal-to-noise ratio of the position estimation. Thus, the utility of the coarse-grained approximation is not restricted to the timescales presented here, nor does it in principle reduce the time-resolution of the tracking-FCS method. To resolve faster oscillations in the fluorescence autocorrelation curve, the experimenter simply needs to choose a sufficiently large

*ω*

_{0}, then average over these oscillation in the data analysis. On the other hand, fluorescence fluctuations on timescales

*faster*than the laser rotation frequency are also decoupled in the autocorrelation curve. It therefore simply remains for the experimenter to choose a laser rotation frequency outside of the dynamical timescale of interest, which may be done in principle with no degradation of tracking performance.

## 4. Conclusions

*individual*particle is actively tracked by real-time feedback control methods. The results of this paper are twofold: first, we presented a simplified method of particle tracking using fluorescence modulation/demodulation techniques. Second, we presented a full theory of fluorescence fluctuations in the case that a particle’s motion is not so slow that the controller may be considered “perfect.” The key concept that underlies the tracking-FCS theory is the separation of

*controlled*motion, represented by Ornstein-Uhlenbeck trapping statistics, uncontrolled but deterministic motion of the laser centroid, and uncontrolled Brownian motion. The tracking-FCS theory naturally unites closed-loop control parameters, such as the feedback controller’s step response time

*γ*

_{xy}, with dynamical parameters such as the particle’s diffusion coefficient

*D*. Furthermore, the coarse-grained approximation suppresses the large amplitude (deterministic) oscillations in a measured fluorescence autocorrelation curve, thus reducing the apparent experimental artifacts in measured results. The theory presented here should be immediately applicable to three-dimensional tracking of the type in [20

**88**, 2919–2928 (2005). [CrossRef] [PubMed]

*μ*m

_{2}/s and detected photon count rates as low as 10

^{4}s

^{-1}should be feasible with the present techniques. A complementary question arises in the single-fluorophore case: since typical fluorophores may only emit a total of ~ 10

^{6}photons before photobleaching, the absolute length of time over which a single fluorophore may be tracked is limited. However, beautiful recent experiments have revealed folding transitions in single ribozymes [29

29. X. Zhuang, L. E. Bartley, H. P. Babcock, R. Russell, T. Ha, D. Hershlag, and S. Chu, “A Single-Molecule Study of RNA Catalysis and Folding,” Science **288**, 2048–2051 (2000). [CrossRef] [PubMed]

30. B. Okumus, T. J. Wilson, D. M. J. Lilley, and T. Ha, “Vesicle Encapsulation Studies Reveal that Single Molecule Ribozyme Heterogeneities Are Intrinsinc,” Biophys. J **87**, 2798–2806 (2004). [CrossRef] [PubMed]

31. E. Rhoades, E. Gussakovsky, and G. Haran, “Watching proteins fold one molecule at a time,” Proc. Natl. Acad. Sci. U.S.A. **100**, 3197–3202 (2003). [CrossRef] [PubMed]

^{4}s

^{-1}, we may expect to use the techniques described here to track (and observe) a single dye-labelled biomolecule for a time ~ 4s, which is sufficient for revealing these (and presumably other) interesting conformational transitions. The tracking-FCS methods presented here may then be useful for further resolving faster conformational and kinetic dynamics, such as those accessible by traditional FCS, on

*individual*molecules with no ensemble averaging.

## A. Derivation of Equation (9)

9. M. J. Saxton and K. Jacobson, “Single-particle tracking: applications to membrane dynamics,” Annu. Rev. Bio-phys. Biomolec. Struct. **26**, 373–399 (1997). [CrossRef]

*X*

_{t}be a real-valued stochastic process, and let Φ

_{t}(

*x*) be some deterministic (sure) function of

*x*, indexed by the time

*t*. Here, we do not make any restriction on the form of Φ

_{t}or

*X*

_{t}, except that

*X*

_{t}is at least ergodic and homogenous (that is, stationary in the long-time limit [25]). Homogeneity ensures that the steady-state distributions

_{t}(

*x*) to be the rate of photon arrivals from a particle at position

*x*at time

*t*. The stochastic, time-dependent photon arrival rate

*σ*

_{t}is then a sure function of the stochastic particle position

*X*

_{t}:

*σ*= Φ

_{t}_{t}(

*X*

_{t}). Its autocorrelation function is given by

_{t}(

*q*), the Fourier transform of Φ

_{t}(

*x*), we have

*X*

_{t},

*X*

_{t+τ}), which exists for all suitably bounded real-valued processes

*X*

_{t}[26]. Inserting Eq. (23) into Eq. (22), we note that Φ͂

_{t}and Φ͂

_{t+τ}are the only remaining functions of

*t*. We may therefore pull the time averaging brackets outside the integrals and perform the integrals over

*q*

_{1}and

*q*

_{2}to find

## Acknowledgments

## References and links

1. | D. Magde, E. L. Elson, and W. W. Webb, “Thermodynamic fluctuations in a reacting system - measurement by fluorescence correlation spectroscopy,” Phys. Rev. Lett. |

2. | E. L. Elson and D. Magde, “Fluorescence correlation spectroscopy. 1. Conceptual basis and theory,” Biopolymers |

3. | D. Magde, E. L. Elson, and W. W. Webb, “Fluorescence correlation spectroscopy. 2. Experimental realization,” Biopolymers |

4. | O. Krichevsky and G. Bonnett, “Fluorescence correlation spectroscopy: the technique and its applications,” Rep. Prog. Phys. |

5. | S. T. Hess, S. Huang, A. A. Heikal, and W. W. Webb, “Biological and Chemical Applications of Fluorescence Correlation Spectroscopy: A Review,” Biochemistry |

6. | A. J. Berglund, A. C. Doherty, and H. Mabuchi, “Photon statistics and dynamics of Fluorescence Resonance Energy Transfer,” Phys. Rev. Lett. |

7. | H. D. Kim, G. U. Nienhaus, T. Ha, J. W. Orr, J. R. Williamson, and S. Chu, “Mg |

8. | K. C. Neuman and S. M. Block, “Optical Trapping,” Rev. Sci. Instrum . |

9. | M. J. Saxton and K. Jacobson, “Single-particle tracking: applications to membrane dynamics,” Annu. Rev. Bio-phys. Biomolec. Struct. |

10. | A. E. Cohen and W. E. Moerner, “Method for trapping and manipulating nanoscale objects in solution,” Appl. Phys. Lett. |

11. | A. E. Cohen, “Control of Nanoparticles with Arbitrary Two-Dimensional Force Fields,” Phys. Rev. Lett. |

12. | T. Meyer and H. Schindler, “Simultaneous Measurement of Aggregation and Diffusion of Molecules in Solutions and in Membranes,” Biophys. J. |

13. | T. Ha, D. S. Chemla, T. Enderle, and S. Weiss, “Single molecule spectroscopy with automated positioning,” Appl. Phys. Lett. |

14. | J. Enderlein, “Tracking of fluorescent molecules diffusing within membranes,” Appl. Phys. B |

15. | J. Enderlein, “Positional and Temporal Accuracy of Single Molecule Tracking,” Sing. Mol. |

16. | R. S. Decca, C.-W. Lee, and S. R. Wassall, “Single molecule tracking scheme using a near-field scanning optical microscope,” Rev. Sci. Instr. |

17. | V. Levi, Q. Ruan, K. Kis-Petikova, and E. Gratton, “Scanning FCS, an novel method for three-dimensional particle tracking,” Biochem. Soc. Trans. |

18. | A. J. Berglund and H. Mabuchi, “Feedback controller design for tracking a single fluorescent molecule,” Appl. Phys. B |

19. | K. Kis-Petikova and E. Gratton, “Distance measurement by circular scanning of the excitation beam in a two-photon microscope,” Microsc. Res. Tech. |

20. | V. Levi, Q. Ruan, and E. Gratton, “3-D particle tracking in a two-photon microscope. Application to the study of molecular dynamics in cells,” Biophys. J. |

21. | M. A. Digman, P. Sengupta, P. W. Wiseman, C. M. Brown, A. R. Horwitz, and E. Gratton, “Fluctuation Correlation Spectroscopy with a Laser-Scanning Microscope: Exploiting the Hidden Time Structure,” Biophys. J. |

22. | M. A. Digman, C. M. Brown, P. Sengupta, P. W. Wiseman, A. R. Horwitz, and E. Gratton, “Measuring fast dynamics in solutions and cells with a laser scanning microscope,” Biophys. J. |

23. | M. H. DeGroot, |

24. | G. Chirico, C. Fumagalli, and G. Baldini, “Trapped Brownian Motion in Single- and Two-Photon Excitation Fluorescence Correlation Experiments,” J. Phys. Chem. B |

25. | C. W. Gardiner, |

26. | N. G. Van Kampen, |

27. | A. Gennerich and D. Schild, “Fluorescence correlation spectroscopy in small cytosolic compartments depends critically on the diffusion model used,” Biophys. J. |

28. | A. J. Berglund and H. Mabuchi, “Performance bounds on single-particle tracking by fluorescence modulation,” in preparation (2005). |

29. | X. Zhuang, L. E. Bartley, H. P. Babcock, R. Russell, T. Ha, D. Hershlag, and S. Chu, “A Single-Molecule Study of RNA Catalysis and Folding,” Science |

30. | B. Okumus, T. J. Wilson, D. M. J. Lilley, and T. Ha, “Vesicle Encapsulation Studies Reveal that Single Molecule Ribozyme Heterogeneities Are Intrinsinc,” Biophys. J |

31. | E. Rhoades, E. Gussakovsky, and G. Haran, “Watching proteins fold one molecule at a time,” Proc. Natl. Acad. Sci. U.S.A. |

**OCIS Codes**

(180.2520) Microscopy : Fluorescence microscopy

(180.5810) Microscopy : Scanning microscopy

(300.6280) Spectroscopy : Spectroscopy, fluorescence and luminescence

**ToC Category:**

Research Papers

**History**

Original Manuscript: August 9, 2005

Revised Manuscript: September 20, 2005

Published: October 3, 2005

**Citation**

Andrew Berglund and Hideo Mabuchi, "Tracking-FCS: Fluorescence correlation spectroscopy of individual particles," Opt. Express **13**, 8069-8082 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-20-8069

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### References

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