## Fluctuations in closed-loop fluorescent particle tracking

Optics Express, Vol. 15, Issue 12, pp. 7752-7773 (2007)

http://dx.doi.org/10.1364/OE.15.007752

Acrobat PDF (866 KB)

### Abstract

We present a comprehensive theory of closed-loop particle tracking for calculating the statistics of a diffusing fluorescent particle’s motion relative to the tracking lock point. A detailed comparison is made between the theory and experimental results, with excellent quantitative agreement found in all cases. A generalization of the theory of (open-loop) fluorescence correlation spectroscopy is developed, and the relationship to previous results is discussed. Two applications of the statistical techniques are given: a method for determining a tracked particle’s localization and an algorithm for rapid particle classification based on real-time analysis of the tracking control signal.

© 2007 Optical Society of America

## 1. Introduction

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

02. V. Levi, Q. Ruan, M. Plutz, A. S. Belmont, and E. Gratton, “Chromatin dynamics in interphase cells revealed by tracking in a two-photon excitation microscope,” Biophys. J. **89**, 4275–4285 (2005). [CrossRef] [PubMed]

03. A. J. Berglund and H. Mabuchi, “Tracking-FCS: Fluorescence Correlation Spectroscopy of individual particles,” Opt. Express **13**, 8069–8082 (2005). [CrossRef] [PubMed]

04. A. J. Berglund, K. McHale, and H. Mabuchi, “Feedback localization of freely diffusing fluorescent particles near the optical shot-noise limit,” Opt. Lett. **32**, 145–147 (2007). [CrossRef]

05. 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]

06. H. Cang, C. M. Wong, C. S. Xu, A. H. Rizvi, and H. Yang, “Confocal three dimensional tracking of a single nanoparticle with concurrent spectroscopic readout,” Appl. Phys. Lett. **88**, 223901 (2006). [CrossRef]

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

08. A. E. Cohen, “Control of Nanoparticles with arbitrary two-dimensional force fields,” Phys. Rev. Lett. **94**, 118102 (2005). [CrossRef] [PubMed]

09. A. E. Cohen and W. E. Moerner, “Suppressing Brownian motion of individual biomolecules in solution,” Proc. Natl. Acad. Sci. USA **103**, 4362–4365 (2006). [CrossRef] [PubMed]

10. S. Chaudhary and B. Shapiro, “Arbitrary steering of multiple particles independently in an electro-osmotically driven microfluidic system,” IEEE Trans. Contr. Syst. Technol. **14**, 669–680 (2006). [CrossRef]

11. M. D. Armani, S. V. Chaudhary, R. Probst, and B. Shapiro, “Using feedback control of microflows to independently steer multiple particles,” IEEE J. Microelectromech. Syst. **15**, 945–956 (2006). [CrossRef]

03. A. J. Berglund and H. Mabuchi, “Tracking-FCS: Fluorescence Correlation Spectroscopy of individual particles,” Opt. Express **13**, 8069–8082 (2005). [CrossRef] [PubMed]

04. A. J. Berglund, K. McHale, and H. Mabuchi, “Feedback localization of freely diffusing fluorescent particles near the optical shot-noise limit,” Opt. Lett. **32**, 145–147 (2007). [CrossRef]

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

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

15. S. B. Andersson, “Tracking a single fluorescent molecule in a confocal microscope,” Appl. Phys. B **80**, 809–816 (2005). [CrossRef]

16. A. J. Berglund and H. Mabuchi, “Performance bounds on single-particle tracking by fluorescence modulation,” Appl. Phys. B **83**, 127–133 (2006). [CrossRef]

10. S. Chaudhary and B. Shapiro, “Arbitrary steering of multiple particles independently in an electro-osmotically driven microfluidic system,” IEEE Trans. Contr. Syst. Technol. **14**, 669–680 (2006). [CrossRef]

11. M. D. Armani, S. V. Chaudhary, R. Probst, and B. Shapiro, “Using feedback control of microflows to independently steer multiple particles,” IEEE J. Microelectromech. Syst. **15**, 945–956 (2006). [CrossRef]

17. D. Montiel, H. Cang, and H. Yang, “Quantitative characterization of changes in dynamical behavior for single-particle tracking studies,” J. Phys. Chem. B **110**, 19763–19770 (2006). [CrossRef] [PubMed]

*i.e*., in the time domain). A detailed model is presented for the general case of a Brownian particle tracked by an arbitrary linear control law and subject to Gaussian position-sensor noise. We give a prescription for calculating statistical quantities such as the tracking error and various correlation functions and give explicit results for first-and second-order systems. For a fluorescent particle tracked (or trapped) in the focus of a spatially-modulated Gaussian excitation laser, fluorescence fluctuations depend on the competition between diffusion and control through the statistics of the tracking error. In Sect. 3, we exploit this fact to calculate generalized equations of closed-loop fluorescence correlation spectroscopy (FCS), which reduce to familiar results in the appropriate limits. Our model completely characterizes any linear fluorescent-particle tracking control system that uses a Gaussian excitation laser. In Sect. 4, we give an extensive comparison between the theory presented here and our own experimental results. In particular, we use the theoretical model to infer tracking control parameters. We then give two applications of such an analysis. First, we consider the problem of determining a particle’s localization,

*i.e*. the standard deviation in the tracking error, and show how this parameter can be determined from typical tracking data in at least three complementary ways. Second, we use the statistics of the tracking control signal to develop a particle classification procedure for distinguishing between species in a binary mixture based on very little information collected over small spatial and temporal scales. The results presented in this paper are applicable not only to our own two-dimensional tracking results, but to all optical, linear-feedback particle tracking and trapping control systems, which differ (theoretically) only in the choice of reference frame.

## 2. Linear control system model

*C*(

*s*) and the control actuator (for example, the piezoelectric stage in our experiment) is represented by

*P*(

*s*). We assume that the control system does not couple the different Cartesian coordinates of a particle’s diffusion, so we may consider each coordinate separately. The tracked particle moves by Brownian motion with “velocity”

*U*(

*t*):

*D*is the diffusion coefficient and d

*W*(

_{p}*t*) is a stochastic Wiener increment. (As discussed in most textbook treatments of stochastic processes [20], the Wiener process

*W*(

_{p}*t*) is continuous but nowhere differentiable, and strictly speaking, the “velocity”

*U*(

*t*) is undefined. In our analysis, however, this term will always drive a

*linear*differential equation, where it serves as convenient shorthand notation for a more carefully written, and well-defined, stochastic differential equation.) The time-integral of

*U*(

*t*) is the position of the particle,

*X*(

_{p}*t*), at time

*t*:

*X*(

*t*), and the error signal

*X*(

*t*) and the particle’s position

*X*(

_{p}*t*). The control objective is to lock this error signal to zero,

*i.e*., to achieve perfect tracking. In the experimental scenarios considered here, the tracked particle’s position

*X*(

_{p}*t*) is sensed by optical methods. In order to capture the noisy nature of such a sensor, Gaussian white noise

*n*(with dimensions of, for example, nm/

*W*(

_{n}*t*) is another Wiener process statistically independent of

*W*(

_{p}*t*). For optical detection methods, the noise density

*n*arises from photon counting statistics and no amount of technical sophistication or signal processing can suppress it. For otherwise optimal tracking, this sensor noise places a fundamental limit on the performance of the tracking controller [4

04. A. J. Berglund, K. McHale, and H. Mabuchi, “Feedback localization of freely diffusing fluorescent particles near the optical shot-noise limit,” Opt. Lett. **32**, 145–147 (2007). [CrossRef]

16. A. J. Berglund and H. Mabuchi, “Performance bounds on single-particle tracking by fluorescence modulation,” Appl. Phys. B **83**, 127–133 (2006). [CrossRef]

*n*and need not be considered in detail until we return to a discussion of fluorescence fluctuations.

### 2.1. Specification of transfer functions

*U*(

*t*) and

*N*(

*t*), drive the outputs,

*X*(

*t*) and

*E*(

*t*). The two output functions

*X*(

*t*), the sample stage position, and

*E*(

*t*), the deviation of the particle from the laser centroid, play central roles in analyzing a tracking experiment. In particular, during tracking, we cannot access the particle’s position

*X*(

_{p}*t*) directly; rather, we can only measure the sample stage position

*X*(

*t*), which tracks

*X*(

_{p}*t*) but is not identically equal to it. Similarly, we measure the particle’s

*fluorescence*, which is a function of

*E*(

*t*), the deviation of the particle’s position from the excitation laser centroid. Therefore, we are particularly interested in the statistics of these two signals. In this section, we will calculate the joint, two-time probability distributions of the processes

*X*(

*t*) and

*E*(

*t*) for a generic stable control system of arbitrary order, and we will explicitly record the results for particular parameterizations of first-and second-order systems.

^{~}represent the Laplace transform of a time-domain function, so that, for example,

*X̃*(

*s*) is the Laplace transform of

*X*(

*t*). In the Laplace domain, we now find a linear algebraic system

*L*(

*s*) =

*C*(

*s*)

*P*(

*s*) denote the loop transfer function, we find

*T*(

*s*) is

*strictly proper*if it is the ratio of two polynomials in

*s*with the order of the denominator greater than the order of the numerator. It is

*stable*if all of its poles,

*i.e*., the zeros of its denominator polynomial, lie in the left half of the complex s-plane. Note that

*T*(

_{EU}*s*) and

*T*(

_{XU}*s*) as defined in Eq. (5) have poles at the origin

*s*= 0, and it is at least possible (if this pole is not canceled by a zero in the numerator) that these are unstable. However, a pole at the origin represents a particularly innocuous form of instability,

*marginal*stability. Because the transfer function representing the time derivative of such a process is stable, we will be able to handle these marginally stable cases with little difficulty.

*X̃*(

_{U}*s*) =

*T*(

_{XU}*s*)

*Ũ*(

*s*) and similarly for

*X̃*(

_{N}*s*),

*X̃*(

_{U}*s*), and

*Ẽ*(

_{N}*s*), we have

*X̃*(

_{U}*s*),

*X̃*(

_{N}*s*),

*Ẽ*(

_{U}*s*), and

*Ẽ*(

_{N}*s*) separately and sum them to find the desired output statistics.

### 2.2. State-space realizations and the Fokker-Planck equation

*T*(

*s*) driven by an input

*U*(

*t*) ↔

*Ũ*(

*s*) with output

*X*(

*t*) ↔

*X̃*(

*s*). It is a straightforward task to find a

*state-space realization*of the system [18] consisting of three matrices

**A**,

**B**,

**C**, satisfying

*X*(

*t*) defined by Eq. (11) of Sect. 2.2, for τ ≥ 0. Δ

*X*

_{Δt}(

*t*) =

*X*(

*t*+ Δ

*t*) -

*X*(

*t*) is the discrete time-derivative of

*X*(

*t*) when data is sampled at time intervals Δ

*t*and E[∙] denotes an expectation value.

**D**representing the direct feedthrough of input signal to output signal. We assume that the system under consideration is strictly proper, so that we may take

**D**= 0.) If

*T*(

*s*) is of order

*m*, then

**A**,

**B**, and

**C**have sizes

*m*×

*m*,

*m*×1 and 1×

*m*respectively. The specification of these matrices is not unique, but as long as they satisfy Eq. (10), they represent a valid realization. Letting

**q**(

*t*) be an

*m*-component internal state vector, the system’s dynamics can now be written in the form

*U*(

*t*)

*dt*= √α

*dW*(

*t*). The equation of motion for

**q**(

*t*) then becomes the multivariate Ornstein-Uhlenbeck process

**q**(

*t*) is a vector-valued random process, whose statistics are given by the time-dependent probability distribution

*p*(

**q**,

*t*) satisfying a linear Fokker-Planck equation:

*j*and

*k*represent the components of their corresponding vectors or matrices. When the system begins in state

**q**

_{0}at time

*t*= 0, the full time-dependent solution to time

*t*= 0, the full time-dependent solution to Eq. (14) is given by the conditional probability distribution

*p*(

_{t}**q**∣

**q**

_{0}):

*m*-dimensional multivariate Gaussian distribution with mean vector

**m**and (symmetric) covariance matrix ∑. The covariance matrix ∑

_{t}in Eq. (15) satisfies

**A**all have negative real part [corresponding to left-half-plane poles of

*T*(

*s*)], Eq. (17) admits a finite stationary solution ∑

_{∞}defined algebraically by the Lyapunov equation

**A**even when ∑

_{t}becomes unbounded; for example, in the simple uncontrolled Brownian motion case where

**A**= 0 we find ∑

_{t}= 2

*Dt*.

**q**(

*t*) is easily found with Eqs. (15) and (20) and a little manipulation. Denoting the joint probability that

**q**(

*t*+

*t*) =

**q**

_{2}and

**q**(

*t*) =

**q**

_{1}by

*p*

_{τ}(

**q**

_{2},

**q**

_{1}), we find for

*τ*> 0,

*X*(

*t*) can be read off of the mean and covariance of Eq. (23). It will be useful to record the statistics of a few functions of

*X*(

*t*) as well. The results for various first- and second-order moments are recorded in Table 1.

*T*(

*s*) driven by a stochastic input signal of the form

*dU*(

*t*) = √

*αdW*(

*t*), the results summarized in Table 1 give the statistics of the resulting output signal in terms of the state-space realization of

*T*(

*s*) and the solution of the Lyapunov equation, Eq. (18). For low-order systems, we may calculate these quantities explicitly (see the first- and second-order examples below). However, much of the analysis here is included in standard numerical analysis software. With these tools, it is a straightforward task to investigate quite complicated systems, including (for example) multiple resonances and time delays using polynomial (Padé) approximations.

### 2.3. Marginally stable systems

*marginally stable*because they have a pole at the origin

*s*= 0. In this section, we will modify the analysis of section 2.2 to account for this slight technical complication. To begin, consider the system

*X̃*(

*s*) =

*T*(

*s*)

*Ũ*(

*s*). If

*T*(

*s*) is marginally stable, then we can rewrite this system as

*T̄*(

*s*) =

*sT*(

*s*) is the closed-loop transfer function of a

*stable system*. For this case, we can simply consider the time derivative

*X*(

*t*) defined by the state-space realization

**A̅**,

**B̅**, and

**C̅**as in Sect. 2.3.

*X˙*(

*t*) as before.

*X˙*(

*t*) in terms of a state-space realization

**A̅**,

**B̅**,

**C̅**and

_{∞}of

*T̅*(

*s*), with

*U*(

*t*)

*dt*= √α

*dW*(

*t*) as in Sect. 2.2. In this case, the statistics of

*X*(

*t*) can be found by integration:

*X*(0). We quickly see that the mean is given by

### 2.4. First- and second-order systems

*X*(

*t*) and tracking error

*E*(

*t*) driven by the particle’s motion

*U*(

*t*) and measurement noise

*N*(

*t*) as described in Sect. 2. We consider two systems, specified by the transfer functions

*C*(

*s*) and

*P*(

*s*):

*C*(

*s*) with unity-gain frequency

*γ*/2

_{c}*π*. However, in the first case given by Eq. (30), the plant has a flat transfer function that can be driven arbitrarily hard with no amplitude or phase rolloff. This first-order system corresponds to the ideal tracking case, in which the bandwidth is set by the level of aggression of the control law, via

*γ*. The second case given by Eq. (31) has the same control law, but the plant transfer function is now a low-pass filter, exhibiting both amplitude and phase rolloff at frequencies above

_{c}*γ*/2

_{p}*π*. We will use the second-order model to analyze our own experimental results in later sections.

*C*(

*s*) and

*P*(

*s*) and can be checked for consistency with Eq. (10). Using Tables 1 and 2 and the realizations given in the Appendix, we can explicitly calculate expectation values.

*E*(

*t*), in the first-order model. For

*T*(

_{EU}*s*), we find

**A**= -

*γ*,

_{c}**C**= 1, and ∑

_{∞}=

*D*/

*γ*so that, for example,

_{c}*optimal*and Eq. (33) represents the upper bound on tracking performance derived in Ref. [16

16. A. J. Berglund and H. Mabuchi, “Performance bounds on single-particle tracking by fluorescence modulation,” Appl. Phys. B **83**, 127–133 (2006). [CrossRef]

*E*(

*t*) for the second-order system. Following the same procedure as for the first-order case, we find

*γ*>

_{c}*γ*/4, where

_{p}*ν*becomes imaginary, but they remain stable for all

*γ*,

_{p}*γ*> 0. The first-order tracking results are reproduced in the limit

_{c}*γ*→ ∞, that is, in the limit that the plant rolloff becomes much larger than the controller bandwidth.

_{p}## 3. Closed-loop fluorescence correlation spectroscopy

*X*(

*t*) and tracking error

*E*(

*t*). The former signal can be measured directly during a tracking experiment. The latter cannot be measured directly, but it can be sensed through the statistics of the fluorescence photon count rate. To see this, consider a one-dimensional system with a particle at position

*X*(

_{p}*t*) when the sample stage is at position

*X*(

*t*). For a fluorescence detectivity profile Φ(

*x*) centered at the sample stage position, the rate of photon arrivals from this particle is given by

22. 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]

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

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

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

*closed-loop*particle tracking probes the particle’s motion through the tracking error

*E*(

*t*) where the fluctuations arise from competition between free diffusion and feedback-assisted damping. In this section, we will calculate these fluorescence autocorrelation functions for Gaussian Φ(

*x*), including the contribution from a deterministic spatial modulation pattern of the laser. These closed-loop calculations are a generalization of the open-loop FCS case, and we will show that they reduce to those results in the appropriate (weak feedback) limit. We will only calculate the expectation values of the fluorescence correlation functions; the variances could in principle be calculated using the same methods, because we have already derived the full Gaussian distribution of the relevant statistical quantities. As an example, see Ref. [26

26. S. Saffarian and E. L. Elson, “Statistical Analysis of Fluorescence Correlation Spectroscopy: The Standard Deviation and Bias,” Biophys. J. **84**, 2030–2042 (2003). [CrossRef] [PubMed]

### 3.1. Calculation of the fluorescence autocorrelation function

*x*-direction only. In many tracking scenarios, the laser moves in a deterministic (often circular) modulation pattern around a centroid position

*X*(

*t*). Denote the time-dependent offset of the laser from the beam centroid by

*x*(

_{L}*t*), and let the detectivity function be Gaussian with beam waist

*w*,

03. A. J. Berglund and H. Mabuchi, “Tracking-FCS: Fluorescence Correlation Spectroscopy of individual particles,” Opt. Express **13**, 8069–8082 (2005). [CrossRef] [PubMed]

**x**

_{L}is deterministic, and

**E**is stochastic with Gaussian probability distribution

*p*(

**E**) characterized by its mean and covariance

*σ*

_{0}

^{2}and

*σ*

_{τ}

^{2}were calculated in Sect. 2:

σ ¯

_{0}

^{2}=

*σ*

_{0}

^{2}+

*w*

^{2}/4, we find for the normalized fluorescence correlation function

*σ*

_{0}

^{2}and

*σ*

_{τ}

^{2}of Eq. (42).

*x*

_{L}as long as the lock point of the tracking control moves with it. For tracked diffusion in higher dimensions, even with asymmetric tracking and beam profiles, the full fluorescence autocorrelation function is just a product of terms of the form of Eq. (43), calculated along each Cartesian coordinate,

### 3.2. Recovery of open-loop results in the weak-tracking limit

*G*(

*t*;

*τ*) when the tracking is very weak. Because the system damping is represented by the matrix

**A**, the weak-tracking limit can be found by letting this matrix approach 0. This limit is a bit tricky, however, because we must simultaneously let the standard deviation in the particle’s position

*σ*

_{0}

^{2}→ ∞. That is, the damping becomes infinitesimally small while the particle’s confinement becomes correspondingly large. If we just blindly try to take the limit

**A**→ 0, we can easily find nonsensical results in which the unnormalized correlation function goes to 0 and the normalized version diverges, as the mean and variance of the fluorescence signal both go to zero at different rates. Furthermore, the closed-loop case includes only a single tracked particle, whereas the open-loop case is formulated for an ensemble of particles characterized by their concentration.

**A**tend to zero and

*σ*

_{0}

^{2}approach infinity. To accomplish this, note that for very small

**A**, we have

_{∞}is symmetric, and in the last line we took

**CB**=

**A**= 0 for the system in Eq. (11).

*G*(

*t*;

*τ*) at

*τ*= 0. Using Eq. (47) and taking the large

*σ*

_{0}

^{2}limit, we find:

### 3.3. Two-dimensional tracking in a rotating laser

*x*and

*y*directions, with the excitation laser rotating at angular frequency

*ω*

_{0}and radius

*r*, so that

*t*for this case:

*ω*

_{0}

*τ*may be distracting in the measured value of

*g*(

*τ*), but we can suppress it by averaging

*g*(

*τ*) over the rotation period. Denoting this averaged correlation function by

*g̅*(

*τ*), we have

*I*

_{0}is the zeroth-order modified Bessel function, and the approximation holds when

*ω*

_{0}is much larger than the feedback tracking bandwidth (the largest eigenvalue of

**A**).

### 3.4. Behavior of g(τ)for τ ≈ 0

*g*(

*τ*) at

*τ*= 0 is a measure of the fraction of the beam which is filled with particles, or equivalently, it is a measure of the overlap between the beam profile and the distribution of particles in the sample. That is, it is a measure of the sample concentration. In that situation, a lower concentration leads to greater fluctuations, relative to the mean intensity. The one-dimensional fluorescence correlation function takes the form [25

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

*N̅*is the average number of particles in the laser focus and the sample concentration can be determined from the relation

*g*(0) = 1/

*N̅*.

*g*(

*τ*) near

*τ*= 0 is also a measure of the overlap of the trapped particle’s position distribution with the beam profile. However, in closed-loop tracking, there is only one particle in the laser focus at any time, and the concentration becomes difficult to define. Furthermore, as the tracking becomes better, the fluctuations decrease and

*g*(

*τ*) tends to 0. However, for the two-dimensional rotating laser case in our experiment, the value of the correlation function near

*τ*= 0 still gives a measure of the particle’s confinement,

*i.e.*, the steady-state variance of the tracking error,

*σ*

_{0}

^{2}.

*r*= 0. From Eq. (51), we may define

*g*

_{0}=

*g*(

*τ*= 0) to find

*g*(0), the variance of the fluorescence fluctuations.

*g*(

*τ*) takes the form of Eq. 45 as discussed in Sect. 3.3. We suppose that the laser rotation frequency

*ω*

_{0}is much larger than any diffusion or control timescale (

*i.e*.

*ω*

_{0}/2

*π*is much larger than the largest eigenvalue of

**A**). We may then assume that at

*τ*=

*π*/

*ω*

_{0}(

*i.e*., at the first minimum of cos

*ω*

_{0}

*τ*) we have

*σ*

_{τ=π/ω0}

^{2}≈

*σ*

_{0}

^{2}. Now define two quantities

*g*

_{0}

^{±}by

*σ*

_{τ=π/ω0}

^{2}we find

*r*/

*w*, we can determine a tracked particle’s confinement from the fluorescence correlation function through the value of

*g*(

*τ*) near

*τ*= 0.

### 3.5. Relation to other literature results

**x**

_{L}constant and

**A**→ 0, Eq. (46) reproduces the standard open-loop FCS result as shown above. For

**A**→ 0 and

**x**

_{L}describing a two-dimensional circular orbit with the radius of rotation

*r*much larger than the beam waist

*w*, Eq. (46) reproduces the “fluorescence particle counting” results of Ref. [27

27. 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]

**x**

_{L}and

**A**, but with an arbitrary radius of rotation, we find the recent results of Ref. [28

28. 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. **90**, 1317–1327 (2005). [CrossRef]

**A**= 0) models.

## 4. Experimental results

29. A. J. Berglund, “Feedback Control of Brownian Motion for Single-Particle Fluorescence Spectroscopy,” Ph.D. thesis, California Institute of Technology (2006), http://etd.caltech.edu/etd/available/etd-10092006-165831/.

*μ*m) liquid layer between two microscope coverslips. When a fluorescent nanoparticle diffuses away from the laser focus, its deviation from the laser centroid is detected, filtered by analog controller circuits and used to drive a piezoelectric stage in order to translate the sample and bring the particle back to the laser centroid. The 532 nm excitation laser beam is deflected in a circular pattern at angular frequency

*ω*

_{0}= 2

*π*×8 kHz, and a fluorescent particle’s

*x*and

*y*positions are detected in real time by phase-sensitive demodulation of the fluorescence signal at the rotation frequency. A CCD camera detects elastically scattered excitation light; the diffraction patterns caused by interference between the tightly focused incident and reflected beams [30

30. L. Novotny, R. D. Grover, and K. Karrai, “Reflected image of a strongly focused spot,” Opt. Lett. **26**, 789–791 (2001). [CrossRef]

^{-1}and we typically lock to total photon count rates between 5 and 15 kHz (fluorescent count rates between 4 and 14 kHz). Because of this servo, nanoparticles with different fluorescence characteristics exhibit

*no difference in brightness*during a tracking trajectory.

### 4.1. Tracking error and fluorescence fluctuations

*x*position of the sample stage during a tracking trajectory by

*X*(

*t*), then the mean-square deviation calculated over time intervals Δ

*t*provides an estimate of the diffusion coefficient:

*D*, we expect

*D̂*(Δ

*t*) =

*D*, independent of Δ

*t*; however, for a realistic tracking control system,

*D̂*(Δ

*t*) will not be equal to

*D*for time intervals Δ

*t*that are small compared to the tracking bandwidth.

*D̂*(Δ

*t*) for each of the three trajectories displayed in Fig. 3. These plots are shown in Fig. 4, together with least-squares fits to the second-order model of Sect. 2.4, including a noise density

*n*. We also calculated the fluorescence autocorrelation function

*g*(

*τ*) for each trajectory [31

31. T. A. Laurence, S. Fore, and T. Huser, “Fast, flexible algorithm for calculating photon correlations,” Opt. Lett. **31**, 829–831 (2006). [CrossRef] [PubMed]

*t*= 10 ms in

*D̂*(Δ

*t*) and as a revival near

*τ*= 10 ms in

*g*(

*τ*). We find outstanding agreement between the theory and experiment, an indication that our model successfully accounts for fluctuations in both the tracking error and fluorescence count rate. The fit parameters for each set of curves are shown in Table 3, where we find good agreement between parameters determined through these two separate data channels.

### 4.2. Particle localization

*D̂*(Δ

*t*) and

*g*(

*τ*) to infer the standard deviation in the tracking error, that is, the particle’s

*localization*due to feedback control. Recall that the tracking error

*E*(

*t*) is the difference between the particle’s position and the laser centroid position. We define the

*localization L*(along one axis) to be

*D̂*(Δ

*t*) in Ref. [4

**32**, 145–147 (2007). [CrossRef]

*L*from both spatial information, through

*D̂*(Δ

*t*), and fluorescence fluctuations, through

*g*(

*τ*). For our investigation of the tracking localization

*L*, we return to the same data set presented in Ref. [4

**32**, 145–147 (2007). [CrossRef]

*D̂*(Δ

*t*) and

*g*(

*τ*), fit these curves to the second-order-plus-noise model of Sects. 2-3 then calculate the localization using the theoretical expression of Sects. 2-3 then calculate the localization using the theoretical expression

*τ*= 0. We can determine the parameters

*D*,

*γ*,

_{c}*γ*, and

_{p}*n*in two independent ways, using fits to either

*D̂*(Δ

*t*) or

*g*(

*τ*). However, the value of

*D*can be found much more reliably from the spatial information in

*D̂*(Δ

*t*), so we constrain the value of

*D*in

*g*(

*τ*) according to the value found from

*D̂*(Δ

*t*). Finally, we can also determine

*L*from the value of

*g*(

*t*) near

*t*= 0 using Eq. (57) together with our calibrated values of the beam waist

*w*= 1.0

*μ*m and rotation radius

*r*= 0.6

*μ*m. In Fig. 6, we compare the value of

*L*determined from these three methods, with the results summarized in Table 4. Note the unacceptably large spread in the measurement noise

*n*determined from

*D̂*(Δ

*t*) for the smaller particles. We find the most reliable localization values to be those given by fitting

*g*(

*t*) to find

*γ*,

_{c}*γ*, and

_{p}*n*with the diffusion coefficient

*D*constrained by the asymptotic value of

*D̂*(Δ

*t*) at large Δ

*t*; this method combines the sensitivity of

*D̂*(Δ

*t*) at long times with the high resolution of

*g*(

*τ*) at short times.

### 4.3. Fast classification through hypothesis testing

*ex post facto*reconstruction of individual particle trajectories via off-line analysis of a sequence of images captured with a fluorescence microscope [32

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

33. S. Bonneau, M. Dahan, and L. D. Cohen, “Single quantum dot tracking based on perceptual grouping using minimal paths in a spatiotemporal volume,” IEEE Trans. Image Process. **14**, 1384–1395 (2005). [CrossRef] [PubMed]

34. E. Meijering, I. Smal, and G. Danuser, “Tracking in Molecular Bioimaging,” IEEE Signal Processing Mag. **23**, 46–53 (2006). [CrossRef]

*real-time*information about a particle’s diffusional motion through the feedback tracking signal. We may therefore seek to classify particles based on the tracking feedback signal rapidly and with high fidelity. We may form an estimate of the diffusion coefficient along each direction by calculating the variance in the trajectory step size over

*N*time intervals of length Δ

*t*, where the choice of

*N*and Δ

*t*will determine the statistical accuracy of the estimate as well as the total estimation time

*T*=

*N*Δ

*t*. Over each time interval

*T*, we may form an unbiased estimator of the diffusion coefficient

*D*from

*D̂*(Δ

*t*) as defined in Eq. (58). Other estimators of

*D*may be defined, emphasizing Bayesian analysis [35

35. K. McHale, A. J. Berglund, and H. Mabuchi, “Bayesian estimation for species identification in Single-Molecule Fluorescence Microscopy,” Biophys. J. **86**, 3409–3422 (2004). [CrossRef] [PubMed]

*D*[17

17. D. Montiel, H. Cang, and H. Yang, “Quantitative characterization of changes in dynamical behavior for single-particle tracking studies,” J. Phys. Chem. B **110**, 19763–19770 (2006). [CrossRef] [PubMed]

*D̂*(Δ

*t*) specifically for its ease of implementation and potential for real-time applications.

*D*Δ

*t*, while the estimate

*D̂*(Δ

*t*) obeys

*χ*

^{2}statistics with mean value

*D*[36]. Because our feedback control system is

*linear*,

*D̂*(Δ

*t*) still obeys χ

^{2}statistics even for small Δ

*t*, albeit with a mean value that deviates from the asymptotic (underlying) value of the particle diffusion coefficient

*D*. This fact is confirmed in Fig. 7 for each type of particle (60 and 210 nm), where we show the distribution of

*D̂*(Δ

*t*) for Δ

*t*= 10 ms together with the expected

*χ*

^{2}distributions for varying sample numbers

*N*.

*λ*

_{1}of particles of type 1 (diffusion coefficient

*D*

_{1}) and a fraction

*λ*

_{2}= 1 -

*λ*

_{1}of particles of type 2 (diffusion coefficient

*D*

_{2}≥

*D*

_{1}). We wish to find a threshold value

*D*such that we may assign a particle to class 1 if

_{th}*D̂*(Δ

*t*)

*D*and class 2 for

_{th}*D̂*(Δ

*t*) ≥

*D*. Let

_{th}*P*denote the probability that a classification is correct under this thresholding algorithm. A straightforward calculation shows that, for the expected

_{corr}*χ*

^{2}statistics of

*D̂*(Δ

*t*), the value of

*P*is maximized by choosing

_{corr}*D*

^{*}

_{th}is a weak function of

*λ*

_{1}and

*N*for even moderately large

*N*, and if

*λ*

_{1}=

*λ*

_{2},

*i.e*. if the particles occur with equal likelihood (or we have no prior knowledge of their distribution), then

*D*

^{*}

_{th}does not depend on

*N*at all.

*D*

^{*}

_{th}given by Eq. (61) may become negative or unbounded, but these limits simply indicate regimes in which a measurement is too noisy to warrant any correction to the prior distribution.

*hypothesis testing*[36], which we briefly review. In the binary form of this problem, an

*m*-component measurement is made with result

*θ*, which may represent a single measurement or a sequence of measurements. The experimenter knows that this measurement result was drawn from one of two distributions,

*p*

_{1}(

*θ*) or

*p*

_{2}(

*θ*), with corresponding probabilities

*λ*

_{1}and

*λ*

_{2}= 1 -

*λ*

_{1}, and wishes to decide which of these distributions was most likely to have produced the observed value. Let

*H*

_{1}represent the hypothesis that the underlying was

*p*

_{1}(

*λ*) and similarly for

*H*

_{2}. Then we define a test procedure by the following threshold criterion: we accept hypothesis

*H*

_{1}if

*λ*

_{1}

*p*

_{1}(

*θ*)

*p*

_{1}(

*θ*) >

*λ*

_{2}

*p*

_{2}(

*θ*), and reject it otherwise. A standard theorem states that this test procedure minimizes the probability of an incorrect classification; furthermore, the Neyman-Pearson Lemma states that any other test procedure that

*decreases*the probability of incorrectly accepting

*H*

_{1}necessarily

*increases*the probability of incorrectly accepting

*H*

_{2}[36].

*D̂*(Δ

*t*), the test described above divides the positive real line into regions corresponding to particles of type 1 (

*D̂*(Δ

*t*) <

*D*

^{*}

_{th}) and type 2 (

*D̂*(Δ

*t*) ≥

*D*

^{*}

_{th}) where

*D*

^{*}

_{th}is simply the point where

*D*and

_{x}*D*along two Cartesian directions, even for the case that these are not identically distributed,

_{y}*i.e*., the diffusion is not isotropic. In that case, the measurement vector

*θ*= [

*D̂*(Δ

_{x}*t*),

*D̂*(Δ

_{y}*t*)] lies in a

*D*-

_{x}*D*plane, and the threshold criterion is a

_{y}*line*dividing the plane into regions corresponding to each type of particle. For higher-dimensional measurement vectors, the hypothesis testing criterion is a surface partitioning the measurement space into regions corresponding to

*H*

_{1}and

*H*

_{2}.

*T*, Δ

*t*and

*N*. Because we form our estimate using short segments of very long trajectories, we may confirm whether a particular sample correctly identified a particle by comparing it to a high-fidelity identification based on the entire trajectory. In this way, we retain the ability to calculate the success probability,

*P*. These results are shown in Fig. 8. At fixed estimation time

_{corr}*T*,

*P*strictly increases with decreasing Δ

_{corr}*t*down to about 10 ms. Beyond this point, the nanoparticles used here move (on average) only 250–500 nm and we are no longer able to make fast determinations of position with the accuracy required to form a good estimate

*D̂*(Δ

*t*).

*T*= 60 ms we collect only (on average) 275 fluorescence photons but still identify particles with > 75% success; over this interval the larger (smaller) particles move on average only 600 (1150) nm. When the observation time is doubled to

*T*= 120 ms, the success rate reaches 90%. At

*T*= 1 s, the success rate is > 99% and even the faster particles move less than 5

*μ*m. This method exhibits impressive fidelity based on information from very few photons collected over small spatial volumes and should be applicable to experiments involving single quantum dots or fluorescent biomolecules. Furthermore, these success rates are limited by the tracking error and feedback bandwidth, which are in turn nearly limited by photon counting shot-noise in our experiment [4

**32**, 145–147 (2007). [CrossRef]

*changes*in the diffusive behavior of an individual nanoparticle caused,

*e.g.*, by binding events or conformational switching (see also Ref. [17

17. D. Montiel, H. Cang, and H. Yang, “Quantitative characterization of changes in dynamical behavior for single-particle tracking studies,” J. Phys. Chem. B **110**, 19763–19770 (2006). [CrossRef] [PubMed]

## 5. Conclusion

## A. State-space realizations for the first- and second-order models

**A**,

**B**, and

**C**, (or

**A̅**,

**B̅**,

**C̅**for the marginally stable

*T*(

_{XU}*s*) case), together with Tables 1 and 2, give a prescription for calculating the statistics in both models. In each case, the transfer input-output transfer function is given by

*T*(

*s*)=

**C**(

*s*

**I**-

**A**)

^{-1}

**B**.

## Acknowledgments

## References and links

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

02. | V. Levi, Q. Ruan, M. Plutz, A. S. Belmont, and E. Gratton, “Chromatin dynamics in interphase cells revealed by tracking in a two-photon excitation microscope,” Biophys. J. |

03. | A. J. Berglund and H. Mabuchi, “Tracking-FCS: Fluorescence Correlation Spectroscopy of individual particles,” Opt. Express |

04. | A. J. Berglund, K. McHale, and H. Mabuchi, “Feedback localization of freely diffusing fluorescent particles near the optical shot-noise limit,” Opt. Lett. |

05. | 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. |

06. | H. Cang, C. M. Wong, C. S. Xu, A. H. Rizvi, and H. Yang, “Confocal three dimensional tracking of a single nanoparticle with concurrent spectroscopic readout,” Appl. Phys. Lett. |

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

08. | A. E. Cohen, “Control of Nanoparticles with arbitrary two-dimensional force fields,” Phys. Rev. Lett. |

09. | A. E. Cohen and W. E. Moerner, “Suppressing Brownian motion of individual biomolecules in solution,” Proc. Natl. Acad. Sci. USA |

10. | S. Chaudhary and B. Shapiro, “Arbitrary steering of multiple particles independently in an electro-osmotically driven microfluidic system,” IEEE Trans. Contr. Syst. Technol. |

11. | M. D. Armani, S. V. Chaudhary, R. Probst, and B. Shapiro, “Using feedback control of microflows to independently steer multiple particles,” IEEE J. Microelectromech. Syst. |

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

13. | J. Enderlein, “Positional and temporal accuracy of single molecule tracking,” Sing. Mol. 1 , |

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

15. | S. B. Andersson, “Tracking a single fluorescent molecule in a confocal microscope,” Appl. Phys. B |

16. | A. J. Berglund and H. Mabuchi, “Performance bounds on single-particle tracking by fluorescence modulation,” Appl. Phys. B |

17. | D. Montiel, H. Cang, and H. Yang, “Quantitative characterization of changes in dynamical behavior for single-particle tracking studies,” J. Phys. Chem. B |

18. | O. L. R. Jacobs, |

19. | N. G. Van Kampen, |

20. | C. W. Gardiner, |

21. | H. Risken, |

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

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

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

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

26. | S. Saffarian and E. L. Elson, “Statistical Analysis of Fluorescence Correlation Spectroscopy: The Standard Deviation and Bias,” Biophys. J. |

27. | T. Meyer and H. Schindler, “Simultaneous measurement of aggregation and diffusion of molecules in solutions and in membranes,” Biophys. J. |

28. | 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. |

29. | A. J. Berglund, “Feedback Control of Brownian Motion for Single-Particle Fluorescence Spectroscopy,” Ph.D. thesis, California Institute of Technology (2006), http://etd.caltech.edu/etd/available/etd-10092006-165831/. |

30. | L. Novotny, R. D. Grover, and K. Karrai, “Reflected image of a strongly focused spot,” Opt. Lett. |

31. | T. A. Laurence, S. Fore, and T. Huser, “Fast, flexible algorithm for calculating photon correlations,” Opt. Lett. |

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

33. | S. Bonneau, M. Dahan, and L. D. Cohen, “Single quantum dot tracking based on perceptual grouping using minimal paths in a spatiotemporal volume,” IEEE Trans. Image Process. |

34. | E. Meijering, I. Smal, and G. Danuser, “Tracking in Molecular Bioimaging,” IEEE Signal Processing Mag. |

35. | K. McHale, A. J. Berglund, and H. Mabuchi, “Bayesian estimation for species identification in Single-Molecule Fluorescence Microscopy,” Biophys. J. |

36. | M. H. DeGroot, |

**OCIS Codes**

(180.2520) Microscopy : Fluorescence microscopy

(180.5810) Microscopy : Scanning microscopy

(300.6280) Spectroscopy : Spectroscopy, fluorescence and luminescence

**ToC Category:**

Microscopy

**History**

Original Manuscript: May 4, 2007

Revised Manuscript: June 6, 2007

Manuscript Accepted: June 6, 2007

Published: June 7, 2007

**Virtual Issues**

Vol. 2, Iss. 7 *Virtual Journal for Biomedical Optics*

**Citation**

Andrew J. Berglund, Kevin McHale, and Hideo Mabuchi, "Fluctuations in closed-loop fluorescent particle tracking," Opt. Express **15**, 7752-7773 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-12-7752

Sort: Year | Journal | Reset

### References

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- V. Levi, Q. Ruan, M. Plutz, A. S. Belmont, and E. Gratton, "Chromatin dynamics in interphase cells revealed by tracking in a two-photon excitation microscope," Biophys. J. 89, 4275-4285 (2005). [CrossRef] [PubMed]
- A. J. Berglund and H. Mabuchi, "Tracking-FCS: Fluorescence Correlation Spectroscopy of individual particles," Opt. Express 13, 8069-8082 (2005). [CrossRef] [PubMed]
- A. J. Berglund, K. McHale, and H. Mabuchi, "Feedback localization of freely diffusing fluorescent particles near the optical shot-noise limit," Opt. Lett. 32, 145-147 (2007). [CrossRef]
- 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]
- H. Cang, C. M. Wong, C. S. Xu, A. H. Rizvi, and H. Yang, "Confocal three dimensional tracking of a single nanoparticle with concurrent spectroscopic readout," Appl. Phys. Lett. 88, 223901 (2006). [CrossRef]
- A. E. Cohen and W. E. Moerner, "Method for trapping and manipulating nanoscale objects in solution," Appl. Phys. Lett. 86, 093109 (2005). [CrossRef]
- A. E. Cohen, "Control of Nanoparticles with arbitrary two-dimensional force fields," Phys. Rev. Lett. 94, 118102 (2005). [CrossRef] [PubMed]
- A. E. Cohen and W. E. Moerner, "Suppressing Brownian motion of individual biomolecules in solution," Proc. Natl. Acad. Sci. USA 103, 4362-4365 (2006). [CrossRef] [PubMed]
- S. Chaudhary and B. Shapiro, "Arbitrary steering of multiple particles independently in an electro-osmotically driven microfluidic system," IEEE Trans. Contr. Syst. Technol. 14, 669-680 (2006). [CrossRef]
- M. D. Armani, S. V. Chaudhary, R. Probst, and B. Shapiro, "Using feedback control of microflows to independently steer multiple particles," IEEE J. Microelectromech. Syst. 15, 945-956 (2006). [CrossRef]
- J. Enderlein, "Tracking of fluorescent molecules diffusing within membranes," Appl. Phys. B 71, 773-777 (2000). [CrossRef]
- J. Enderlein, "Positional and temporal accuracy of single molecule tracking," Sing. Mol. 1, 225-230 (2000).
- A. J. Berglund and H. Mabuchi, "Feedback Controller design for tracking a single fluorescent molecule," Appl. Phys. B 78, 653-659 (2004). [CrossRef]
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- A. J. Berglund and H. Mabuchi, "Performance bounds on single-particle tracking by fluorescence modulation," Appl. Phys. B 83, 127-133 (2006). [CrossRef]
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- A. J. Berglund, "Feedback Control of Brownian Motion for Single-Particle Fluorescence Spectroscopy," Ph.D. thesis, California Institute of Technology (2006), http://etd.caltech.edu/etd/available/etd-10092006-165831/.
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- T. A. Laurence, S. Fore, and T. Huser, "Fast, flexible algorithm for calculating photon correlations," Opt. Lett. 31, 829-831 (2006). [CrossRef] [PubMed]
- M. J. Saxton and K. Jacobson, "Single-particle tracking: applications to membrane dynamics," Annu. Rev. Biophys. Biomolec. Struct. 26, 373-399 (1997). [CrossRef]
- S. Bonneau, M. Dahan, and L. D. Cohen, "Single quantum dot tracking based on perceptual grouping using minimal paths in a spatiotemporal volume," IEEE Trans. Image Process. 14, 1384-1395 (2005). [CrossRef] [PubMed]
- E. Meijering, I. Smal, and G. Danuser, "Tracking in Molecular Bioimaging," IEEE Signal Processing Mag. 23, 46-53 (2006). [CrossRef]
- K. McHale, A. J. Berglund, and H. Mabuchi, "Bayesian estimation for species identification in Single-Molecule Fluorescence Microscopy," Biophys. J. 86, 3409-3422 (2004). [CrossRef] [PubMed]
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