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Tracking-FCS: Fluorescence correlation spectroscopy of individual particles

Andrew J. Berglund and Hideo Mabuchi

Open Access Open Access

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

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Figures (4)
Fig. 1.
Fig. 1. (Left) Schematic diagram of the optics and electronics for tracking-FCS. Key: TIA = time-interval analyzer, APD = avalanche photodiode, MC = microcontroller, PZT = piezoelectric translation stage, D550 = dichroic filter with 550 nm cutoff. (Right) Detail of the sample volume. 60-nm diameter fluorescent beads suspended in water diffuse freely in the xy plane, but are confined by glass coverslips in the z direction. The coverslips are mounted on a piezoelectric sample stage, so that the entire bulk fluid volume can be translated. A tracked particle diffuses freely (in the axially confined geometry) in the reference frame of the bulk fluid while the feedback control translates the entire sample volume in order to hold the particle on the laser axis defined by x = y = 0. Because the sense of the laser rotation, and therefore the sign of the feedback controller, reverses upon crossing the focal plane of the microscope optics, the sample mount is adjusted in the z direction so that the focal plane lies just outside of the sample volume.

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Fig. 2.
Fig. 2. Fluorescence data and motion of the sample stage during tracking of a 60 nm microsphere in water. The upper plot shows fluorescence data, and the lower plot shows the x (solid) and y (dotted) positions of the sample stage during the fluorescence trace. Just before 7.5 s, a particle diffused into the capture region and the controller correspondingly responded by moving the sample stage to track this particle. The irregular motion of the sample stage at 7.6 s resulted from an (expected) arithmetic overflow in the microcontroller. The residual fluorescence fluctuations during tracking arise from the competition between diffusion and feedback control and also from the uncontrolled motion of the particle in the z direction. The fluctuations are recorded in Fig. 4 and a detailed theory is given in the next section.

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Fig. 3.
Fig. 3. Time-converging estimate of the diffusion coefficient D for the microsphere tracking data in Fig. 2 with bin time Δt = 20 ms. The dotted lines are error estimates calculated for the estimator D̂, assuming underlying Brownian statistics. Inset: Final estimate of D as a function of bin time Δt. For bin times larger than ~ 10 ms, the estimates are roughly constant with mean value D = 6.2 μm2/s. See the text for an explanation of the estimator convergence with Δt.

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Fig. 4.
Fig. 4. Fluorescence correlation functions recorded during the tracking period in Fig. 2, normalized to the mean fluorescence. The noisy curves were measured from the tracking data in Fig. 2, averaged over coarse-grained time bins of 100 (dotted) and 200 (solid) μs. At higher time resolution, the oscillations due to the deterministic laser rotation make it difficult to resolve the overall shapes of the autocorrelation curve. The smooth solid curve is a fit to Eq. (18). The fit parameters are γxy = 134 Hz, D = (6.2 s-1)wxy2, ρ 0 = 1.4wxy , γz = 11.3 Hz, wz = 4.5μm, and z 0 = 2.8wxy . γxy is the tracking controller bandwidth. All fit parameters are scaled by the true beam waist wxy , which is approximately 1μm. For this value, the diffusion coefficient D determined by the statistical estimate from Fig. 3 and the value from a fit to Eq. 18 are identical.

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Equations (25)

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Φ t ( ρ , θ ) = exp ⁡ { − 2 w xy 2 [ ( ρ cos ⁡ θ − r 0 cos ⁡ ω 0 t ) 2 + ( ρ sin ⁡ θ − r 0 sin ⁡ ω 0 t ) 2 ] } .
f c ( ρ , θ ) = h ( ρ ) cos ⁡ θ , fs ( ρ , θ ) = h ( ρ ) sin ⁡ θ
h ( ρ ) = 2 π Γ 0 ω 0 exp ⁡ [ − 2 w xy 2 ( ρ 2 + r 0 2 ) ] I 1 [ 4 r 0 ρ w xy 2 ] .
h ( ρ ) ≈ 4 π r 0 ρ w xy 2 exp ⁡ [ − 2 r 0 2 w xy 2 ] .
D ̂ = [ 〈 ( Δ x ) 2 〉 + 〈 ( Δ y ) 2 〉 ] ( 4 Δ t )
d X t = − γ x X t dt + 2 D d W t
P 0 ( x ) = 1 2 π x ¯ 2 exp ⁡ [ − x 2 2 x ¯ 2 ]
P τ ( x 2 ∣ x 1 ) = 1 2 π b τ 2 exp ⁡ [ − ( x 2 − a τ x 1 ) 2 2 b τ 2 ]
G ( τ ) = 〈 ∫∫ d x 1 d x 2 P τ ( x 2 ∣ x 1 ) P 0 ( x 1 ) Φ x ( x 2 − w x χ t + τ ) Φ x ( x 1 − w x χ t ) 〉 t
ζ x = γ x τ x 1 + γ x τ x = w x 2 w x 2 + 4 x ¯ 2
G x ( τ ; χ t ) = ζ x 1 − λ τ , x 2 exp ⁡ [ − 2 ζ x ( χ t + τ 2 − 2 λ τ , x χ t χ t + τ + χ t 2 1 − λ τ , x 2 ) ]
λ τ , x = ( 1 − ζ x ) e − γ x τ .
G ( τ ) = 〈 G x ( τ ; x ̂ · r t w x ) G y ( τ ; y ̂ · r t w y ) G z ( τ ; z ̂ · r t w z ) 〉 t
g ( τ ) = ( ζ x y 2 1 − λ τ , xy 2 ) ( ζ z 1 − λ τ , z 2 ) exp ⁡ [ − 4 ρ 0 2 ( ζ xy 1 − λ τ , xy 2 ) − 4 z 0 2 ( ζ z 1 + λ τ , z ) ] ,
G ( τ ) = g ( τ ) exp ⁡ [ 4 ρ 0 2 ( ζ xy λ τ , xy 1 − λ τ , xy 2 ) cos ⁡ ω 0 τ ]
G ˜ ( τ ) = 1 T ˜ ∫ τ − T ˜ 2 τ + T ˜ 2 dτ ′ G ( τ ′ ) dτ ′ ≈ g ( τ ) T ˜ ∫ τ − T ˜ 2 τ + T ˜ 2 dτ ′ exp ⁡ [ 4 ρ 0 2 ( ζ xy λ τ , xy 1 − λ τ , xy 2 ) cos ⁡ ω 0 τ ′ ] dτ ′
1 Nπ ∫ 0 Nπ exp ⁡ [ z cos ⁡ θ ] d θ = I 0 ( z )
G ˜ ( τ ) ≈ g ( τ ) I 0 [ 4 ρ 0 2 ( ζ xy λ τ , xy 1 − λ τ , xy 2 ) ]
P 0 ( x ) = lim t → ∞ P ( X t = x )
P τ ( x 2 ∣ x 1 ) = lim t → ∞ P ( X t + τ = x 2 ∣ X t = x 1 )
G ( τ ) = 〈 σ t σ t + τ 〉 t = 〈 Φ t ( X t ) Φ t + τ ( X t + τ ) 〉 t .
G ( τ ) = 〈 ∫∫ d q 1 2 π d q 2 2 π e − i q 1 X t − i q 2 X t + τ Φ ˜ t ( q 1 ) Φ ˜ t + τ ( q 2 ) 〉 t
= ∫∫ d q 1 2 π d q 2 2 π 〈 e − i q 1 X t − i q 2 X t + τ 〉 t 〈 Φ ˜ t ( q 1 ) Φ ˜ t + τ ( q 2 ) 〉 t
〈 e − i q 1 X t − i q 2 X t + τ 〉 t = ∫∫ d x 1 d x 2 e − i q 1 x 1 e − i q 2 x 2 P 0 ( x 1 ) P τ ( x 2 ∣ x 1 ) .
G ( τ ) = 〈 ∫∫ d x 1 d x 2 P 0 ( x 1 ) P τ ( x 2 ∣ x 1 ) Φ t ( x 1 ) Φ t + τ ( x 2 ) 〉 t .
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