Correction of the afterpulsing effect in fluorescence correlation spectroscopy using time symmetry analysis

Kunihiko Ishii; Tahei Tahara

doi:10.1364/OE.23.032387

1. Introduction

Fluorescence correlation spectroscopy (FCS) is widely used in biophysical or physicochemical studies for investigating molecular diffusion processes, photophysical dynamics such as triplet state formation, and spontaneous dynamics of complex molecules [1]. In particular, the ability of FCS to detect conformational fluctuations with a sub-microsecond time resolution [2] makes it a unique tool in biophysical studies of proteins and nucleic acids.

A well-known problem of FCS in the short time region is the afterpulsing effect of photon detectors [3]. The afterpulsing effect is an intrinsic property of photon-counting devices such as Geiger-mode avalanche photodiodes (APD), which are commonly employed in FCS: Upon a photon detection event, APD generates a fake signal with a nano- to microsecond delay time after the true signal with a certain probability (~1%). Although its effect in an ensemble experiment is usually negligible, the afterpulsing effect is a serious problem in FCS because of its temporally correlated nature. The characteristic correlation signal of afterpulsing appears in the nano- to microsecond time region. Therefore, it interferes with the true correlation signals that represent fast conformational dynamics as well as the triplet state formation.

The simplest way to eliminate the afterpulsing effect is to subtract the afterpulse correlation that is separately estimated [3,4]. However, this method relies on the reproducibility of the afterpulsing effect, which is actually known to be sensitive to the temperature, supply voltage, and count rates [4,5], and calls for careful measurement of the reference correlation data. Instead, two independent detectors are usually employed to eliminate the afterpulsing effect [6]. In this case, one obtains correlation curves that are free from the afterpulsing effect by splitting the photon signal into two, detecting them with separate detectors, and calculating the cross-correlations between them. Although this method is robust and reliable, one loses information brought about by photon pairs that are detected by the same detector (50% of all photon pairs), and the optical setup requires an additional detector. These drawbacks hamper performing advanced measurements such as multi-focus [7,8] or multi-parameter [9] FCS in the nano- to microsecond time region.

A sophisticated approach for removing the afterpulsing effect with a single detector was proposed by Enderlein and Gregor [10]. It utilizes fluorescence lifetime information measured with the time-correlated single photon counting (TCSPC). TCSPC-based FCS is an extension of the ordinary FCS and now finding wide application because one can obtain and utilize the information of fluorescence lifetime [11]. The method of Enderlein and Gregor is based on the fact that the delay time of an afterpulsing signal is usually much longer than the fluorescence lifetime of ordinary fluorophores, so that the afterpulse contribution can be regarded as a constant in the TCSPC histogram on the time scale of typical fluorescence lifetime (~ns). They showed that the afterpulsing effect can be eliminated under this condition using an analysis called fluorescence lifetime correlation spectroscopy (FLCS) [12–15]. It was reported that this method works for a pure sample [10] or a mixture sample with known fluorescence lifetimes [16]. However, it has also been pointed out that this method does not work well for the case of multiple species mixture [9].

In this work, we introduce a new analysis method for solving the afterpulsing problem in FCS by using TCSPC and time symmetry analysis. This method takes advantage of the time symmetry of the dynamics of an equilibrated system. We first describe the basic idea of this analysis by using two-dimensional fluorescence lifetime correlation maps [17–20], which clearly illustrate the temporally asymmetric nature of the afterpulsing effect. Then, we apply this analysis to the correlation data of fluorescent dye solutions and demonstrate that the afterpulsing effect is completely eliminated. We show that the present method works correctly even in the case that the previous FLCS-based method [10] gives a distorted correlation, and provide a clear explanation for this difference. We also show that the time symmetry analysis can be performed in combination with FLCS, so that one can utilize the existing software environment of FLCS [21,22] for applying the time symmetry analysis.

2. Method

2.1 Time symmetry analysis

In an FCS experiment using TCSPC, we measure the arrival time of a photon p with two different time scales [23]. One is macrotime (T_p), which is the absolute arrival time measured from the start of the experiment, and the other is microtime (t_p), which is the relative delay time measured from the nearest excitation pulse [Fig. 1(a)]. The FCS decay curve, G(∆T), is calculated by using the macrotime data of all detected photons. In this calculation, the number of photon pairs detected with a temporal separation of ∆T is counted as

\begin{matrix} G (Δ T) \equiv \frac{〈 I (T) I (T + Δ T) 〉}{{〈 I (T) 〉}^{2}} \\ = \frac{\sum_{p, q} {\begin{matrix} 1 & Δ T - Δ Δ T / 2 < T_{q} - T_{p} < Δ T + Δ Δ T / 2 \\ 0 & otherwise \end{matrix}}{N^{2} T_{\max}^{- 2} \cdot (T_{\max} - Δ T) \cdot Δ Δ T} . \end{matrix}

Here, I(T) is the fluorescence intensity at T, N is the number of detected photons, ∆∆T is an arbitrary window size, and T_max is the total measurement time. p and q run over all photons. In this work, ∆T and ∆∆T are chosen to fill the gap between consecutive data points in the correlation curves, i.e., ∆T_k + ∆∆T_k /2 = ∆T_k₊₁−∆∆T_k₊₁ /2, where k represents the k-th data point.

Fig. 1 Schematic illustration of the principle of the time symmetry analysis. (a) A timing chart of photon signals obtained with a TCSPC-FCS setup. T: macrotime, t: microtime. (b,c) 2D emission-delay correlation maps of the true signal (b) and the afterpulsing effect (c). 1D decay curves on top and left are made by integrating the 2D maps along the vertical and horizontal axes, respectively. The asymmetry of the 2D maps is evaluated as the difference of the integrated 1D decay curves (d).

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When afterpulses exist, the FCS curve will be affected. Here, we represent the probability of afterpulsing as p(∆T) as a function of the delay time ∆T from a true pulse. Typically, p(∆T) is a decaying function in nano- to a few microsecond time region, and it vanishes after ~10 µs. The contribution of the afterpulsing effect in the correlation function is represented as

G^{†} (Δ T) = \frac{p (Δ T)}{〈 I (T) 〉} .

A notable feature of p(∆T) is that it is zero for ∆T < 0, i.e., an afterpulse comes only after the true pulse. In the following time symmetry analysis, we utilize this feature by referring to the microtime information in the correlation calculation.

Suppose that our TCSPC module has n microtime channels. The fluorescence intensity in the i-th TCSPC channel (the corresponding microtime is t⁽ⁱ⁾) at macrotime T is represented as I(T; t⁽ⁱ⁾). Temporal correlation of I(T; t⁽ⁱ⁾) can be thoroughly characterized by making two-dimensional emission-delay correlation maps [17–20]. The (unnormalized) two-dimensional emission-delay correlation map for a delay time ∆T is defined as

\begin{matrix} M (Δ T; t^{(i)}, t^{(j)}) = 〈 I (T; t^{(i)}) I (T + Δ T; t^{(j)}) 〉 \\ = \frac{1}{(T_{\max} - Δ T) \cdot Δ Δ T} \sum_{p, q} {\begin{matrix} δ_{t_{p} t^{(i)}} δ_{t_{q} t^{(j)}} & Δ T - Δ Δ T / 2 < T_{q} - T_{p} < Δ T + Δ Δ T / 2 \\ 0 & otherwise \end{matrix} . \end{matrix}

In the time symmetry analysis, we consider the asymmetry of this 2D map through a function defined as follows:

C (Δ T; t^{(i)}) \equiv C^{″} (Δ T; t^{(i)}) - C^{'} (Δ T; t^{(i)})

C^{'} (Δ T; t^{(i)}) \equiv \sum_{j = 1}^{n} M (Δ T; t^{(i)}, t^{(j)})

C^{″} (Δ T; t^{(i)}) \equiv \sum_{j = 1}^{n} M (Δ T; t^{(j)}, t^{(i)}) .

M(∆T; t⁽ⁱ⁾, t⁽^j⁾) consists of the true correlation signal, M⁰(∆T; t⁽ⁱ⁾, t⁽^j⁾), and the afterpulsing effect, M^†(∆T; t⁽ⁱ⁾, t⁽^j⁾):

M (Δ T; t^{(i)}, t^{(j)}) = M^{0} (Δ T; t^{(i)}, t^{(j)}) + M^{†} (Δ T; t^{(i)}, t^{(j)}) .

In an equilibrated system, because the detailed balance condition is satisfied, the correlation map of the true signal, M⁰(∆T; t⁽ⁱ⁾, t⁽^j⁾), is symmetric [Fig. 1(b)], i.e.,

\begin{matrix} C^{0} (Δ T; t^{(i)}) \equiv \sum_{j = 1}^{n} M^{0} (Δ T; t^{(j)}, t^{(i)}) - \sum_{j = 1}^{n} M^{0} (Δ T; t^{(i)}, t^{(j)}) \\ = 0. \end{matrix}

On the other hand, the correlation map corresponding to the afterpulsing effect, M^†(∆T; t⁽ⁱ⁾, t⁽^j⁾), is not symmetric in general [Fig. 1(c)]. This is because afterpulsing occurs only after the true signal. Actually, the shape of M^†(∆T; t⁽ⁱ⁾, t⁽^j⁾) can be obtained from the following consideration: By its definition, the first photon plotted in M^†(∆T; t⁽ⁱ⁾, t⁽^j⁾) is the photon that generates an afterpulse, whereas the second “photon” in M^†(∆T; t⁽ⁱ⁾, t⁽^j⁾) is the false signal due to the afterpulse. Therefore, t-distribution of the first photon is equal to the ensemble-averaged decay curve,

\bar{I} (t^{(i)}) \equiv 〈 I (T; t^{(i)}) 〉

, because afterpulsing occurs after any initial photons with equal probability. On the other hand, the second false photons due to afterpulsing distribute along the t-axis uniformly [10], because p(∆T) is regarded as a constant on the time scale of t, provided that ∆T is much longer than the fluorescence lifetime. Then, the probability for the second false photon (due to afterpulsing) to be detected at the i-th channel is

p (Δ T; t^{(i)}) = n^{- 1} p (Δ T)

, where n is the number of the channels. Taken together,

\begin{matrix} M^{†} (Δ T; t^{(i)}, t^{(j)}) = 〈 I (T; t^{(i)}) 〉 \cdot p (Δ T; t^{(j)}) \\ = \bar{I} (t^{(i)}) \cdot n^{- 1} p (Δ T) \end{matrix}

and

\begin{matrix} {C^{'}}^{†} (Δ T; t^{(i)}) \equiv \sum_{j = 1}^{n} M^{†} (Δ T; t^{(i)}, t^{(j)}) \\ = p (Δ T) \bar{I} (t^{(i)}), \end{matrix}

\begin{matrix} {C^{″}}^{†} (Δ T; t^{(i)}) \equiv \sum_{j = 1}^{n} M^{†} (Δ T; t^{(j)}, t^{(i)}) \\ = p (Δ T) n^{- 1} \bar{I}, \end{matrix}

where

\bar{I} = 〈 I (T) 〉

is the average fluorescence intensity. From Eqs. (4-11), the asymmetry of the observed two-dimensional emission-delay correlation map is derived as

\begin{matrix} C (Δ T; t^{(i)}) = {C^{″}}^{†} (Δ T; t^{(i)}) - {C^{'}}^{†} (Δ T; t^{(i)}) \\ = p (Δ T) {n^{- 1} \bar{I} - \bar{I} (t^{(i)})} \end{matrix}

[Fig. 1(d)]. C(∆T; t⁽ⁱ⁾) is evaluated from the experimental data using Eqs. (3-6), and

\bar{I}

and

\bar{I} (t^{(i)})

are easily obtained from the ensemble decay curve. Therefore, we can obtain the afterpulsing probability p(∆T) using Eq. (12). Then, we can evaluate the contribution from the afterpulsing effect, G^†(∆T) in Eq. (2), and hence can eliminate it from the correlation curve obtained by the measurement.

The above analysis is valid for all the system in equilibrium, for which Eq. (8) is held. For judging the effect of non-equilibrium dynamics, one can obtain C(∆T; t⁽ⁱ⁾) at a large delay time (e.g., ∆T = 100 µs), where the afterpulsing effect is negligible. If non-equilibrium dynamics take place, C(∆T; t⁽ⁱ⁾) would deviate from the flat distribution that is expected for the true correlation of an equilibrium system [Fig. 1(d)].

2.2 Experimental

To confirm the feasibility of the above-described method, we analyzed the photon data of dye solutions taken with a TCSPC-FCS setup [19]. We briefly describe the experimental condition in the following.

The output of an optical parametric oscillator (Coherent Mira-OPO with intracavity frequency doubling, 532 nm, 76 MHz), which was pumped by a Ti: sapphire oscillator laser (Coherent Mira 900-F), was used as the excitation light source. The excitation pulse was focused onto the sample solution through a glass coverslip and an oil-immersion objective lens (Nikon S Fluor 100 × H, N.A. = 1.3). The fluorescence signal was collected with the backscattering geometry and was split into two by a non-polarizing beamsplitter after passing through a confocal pinhole and a bandpass filter (Chroma Technology D585/40m), and then detected by two APDs (id Quantique id100-20-ULN). We used the output of one of these APDs for applying the time symmetry analysis. For comparison, we also obtained the cross-correlations between the two APDs. The arrival time of each photon (macrotime and microtime) was recorded by a TCSPC module (Becker & Hickl SPC-140) in the time-tagging mode [24]. Data acquisition, calculation of the correlations, and the time symmetry analysis were carried out by using Igor Pro (Wavemetrics) and its external module written in C.

Dye solutions of tetramethylrhodamine-5-maleimide (TMR, Invitrogen) and mixture of TMR and Cy3 monofunctional NHS-ester (GE healthcare) were prepared with milli-Q water. The concentration of the solutions was approximately 5 nM. Nonionic surfactant (0.01% Triton X-100) was added to prevent adsorption of the dye molecules on the glass surface. Under the measurement condition employed, no appreciable photobleaching was observed.

3. Results and discussion

3.1 Time symmetry analysis

Figure 2(a) shows $Δ \bar{I} (t^{(i)}) \equiv n^{- 1} \bar{I} - \bar{I} (t^{(i)})$ [see Eq. (12)] obtained from the photon data of a pure TMR solution. The asymmetry functions, C(∆T; t⁽ⁱ⁾), were calculated using Eq. (4) at various ∆T as shown in Fig. 2(b). The shape of C(∆T; t⁽ⁱ⁾) matches well with $Δ \bar{I} (t^{(i)})$ at all ∆T, which indicates that Eq. (9) obtained in the previous section is appropriate. Spiky dispersive features at around the time origin are due to timing instability of APD. The temporal response of APD shows a shift depending on the time interval from the preceding photon [25]. Such timing shift leads to a dispersive feature in C(∆T; t⁽ⁱ⁾), because the time origins of C″(∆T; t⁽ⁱ⁾) and C′(∆T; t⁽ⁱ⁾) become different. The intensity of the dispersive feature changes with ∆T, reflecting the dependence of the timing shift on the time interval from the preceding photon. For simplicity, we neglect this feature in the following analysis. However, if necessary, it can be corrected by using the photon interval analysis that was recently developed by us [25,26].

Fig. 2 Decay functions obtained from the pure TMR solution. (a) $Δ \bar{I} (t^{(i)}) \equiv n^{- 1} \bar{I} - \bar{I} (t^{(i)})$ calculated from ensemble decay data. (b) The asymmetry functions C(∆T; t) evaluated at selected ∆T values. (c) Afterpulsing probability function p(∆T) determined by Eq. (14). The vertical axis is scaled as afterpulsing probability per unit time (52.7 ns).

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The amplitude of C(∆T; t⁽ⁱ⁾) decays with ∆T, reflecting the time-dependent behavior of the afterpulsing probability, p(∆T). From this data, p(∆T) is derived using Eq. (12) as follows,

\begin{matrix} p (Δ T) = C (Δ T; t^{(i)}) / {n^{- 1} \bar{I} - \bar{I} (t^{(i)})} \\ \equiv C (Δ T; t^{(i)}) / Δ \bar{I} (t^{(i)}) . \end{matrix}

This relation holds for all i. Therefore, to obtain the optimum value of p(∆T) which equally satisfy Eq. (13) for all i, we adopt the least square solution:

p (Δ T) = \frac{\sum_{i = 1}^{n} C (Δ T; t^{(i)}) Δ \bar{I} (t^{(i)}) / σ^{2} (t^{(i)})}{\sum_{i = 1}^{n} {Δ \bar{I} (t^{(i)})}^{2} / σ^{2} (t^{(i)})} .

σ(t⁽ⁱ⁾) in this equation is the standard error of C(∆T; t⁽ⁱ⁾). The obtained p(∆T) is shown in Fig. 2(c). We use a simple form of σ(t⁽ⁱ⁾) in this work, i.e., the square root of the ensemble decay

\bar{I} (t^{(i)})

. Alternatively, when a higher accuracy is needed, another form of σ(t⁽ⁱ⁾) may be used in order to approximate the afterpulsing effect on C(∆T; t⁽ⁱ⁾) more precisely (see Appendix).

Since p(∆T) was obtained from the experimental data, we can subtract the afterpulsing effect using the following formula that is obtained from Eq. (2),

G^{0} (Δ T) \equiv G (Δ T) - p (Δ T) {\bar{I}}^{- 1} .

Here, G(∆T) and G⁰(∆T) are the autocorrelation curves before and after the afterpulse correction, respectively. Figure 3 shows the two autocorrelation curves of the TMR solution. It is clearly seen that the strong decaying feature in the submicrosecond region due to the afterpulsing effect is completely eliminated after the afterpulse correction. Note that the sudden drop of the correlation amplitude at ~100 ns reflects the dead time of the TCSPC system. In Fig. 3, the correlation curve obtained by the cross-correlation method using two detectors is also shown. In order to accurately compare the autocorrelation with cross-correlation, we employed the two-detector setup in our measurements and obtained the cross-correlation between two detectors, whereas we used only one of these detectors to calculate the autocorrelation. The cross-correlation curve perfectly matches the autocorrelation curve (G⁰(∆T)) obtained after the afterpulse correction, except for the initial time region (~100 ns) that is affected by the TCSPC dead time. Hence, it is unambiguously demonstrated that the time symmetry analysis works properly as expected.

Fig. 3 Correlation curves of the TMR solution. Blue solid line: raw autocorrelation without afterpulse correction, red solid line: autocorrelation after afterpulse correction using the time symmetry analysis, black dashed line: cross correlation between two detectors, green dashed line: autocorrelation after afterpulse correction using FLCS.

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We also applied the time symmetry analysis to a mixture solution of TMR and Cy3, whose fluorescence lifetimes are 2.4 ns and 0.18 ns, respectively [19]. As shown in Fig. 4, the obtained correlation after the afterpulse correction completely matches the cross-correlation curve, except for the initial time region (~100 ns), in the same way as the correlation of the pure solution. This indicates that the time symmetry analysis is also applicable to inhomogeneous samples that contain multiple species with different fluorescence lifetimes.

Fig. 4 Correlation curves of the mixture solution of Cy3 and TMR. Blue solid line: raw autocorrelation without afterpulse correction, red solid line: autocorrelation after afterpulse correction using the time symmetry analysis, black dashed line: cross correlation between two detectors, green dashed line: autocorrelation after afterpulse correction using FLCS. Inset: differences between the raw autocorrelation and the autocorrelations after afterpulse correction using the time symmetry analysis (red) or FLCS (green).

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3.2 Comparison with FLCS-based analysis

Next, we examine the previous method based on FLCS [10] for comparison with the present method. First, we briefly summarize the procedure of FLCS. In FLCS, one defines “filter functions,” with which one can extract the contribution of a certain species from the photon data of a mixture of multiple species, referring to the fluorescence decay curve of each species. The filter functions are calculated as follows. First, one assumes the decay curves of the individual species in the sample (d_α(t⁽ⁱ⁾); α is the species index). The fluorescence decay curve of the mixture sample is represented as a sum of contributions from m independent species:

I (t^{(i)}) = \sum_{α = 1}^{m} I_{α} d_{α} (t^{(i)}) .

Here, d_α(t⁽ⁱ⁾) are normalized to be

\sum_{i = 1}^{n} d_{α} (t^{(i)}) = 1

and I_α are the integrated fluorescence intensity of species α. The contribution of species α to the observed decay curve, I(t⁽ⁱ⁾), is obtained by solving Eq. (16) as

I_{α} = \sum_{i = 1}^{n} f_{α} (t^{(i)}) I (t^{(i)}) .

f_α(t⁽ⁱ⁾) is the filter function for the species α. The filter functions for all species are derived all together using a matrix notation [10]:

F = {D^{T} {\bar{I}}^{- 1} D}^{- 1} D^{T} {\bar{I}}^{- 1} .

Here, F_ij = f_i(t⁽^j⁾), D_ij = d_j(t⁽ⁱ⁾), and

\bar{I}

is an n × n diagonal matrix with

{\bar{I}}_{i j} = \bar{I} (t^{(i)}) δ_{i j}

. A filter function can selectively extract a single species because

\sum_{i = 1}^{n} f_{α} (t^{(i)}) d_{β} (t^{(i)}) = δ_{α β}

. The species-specific correlation function between species A and B is derived by using the corresponding filter functions as

\begin{matrix} g_{AB} (Δ T) \equiv 〈 I_{A} (T) I_{B} (T + Δ T) 〉 \\ = \sum_{i, j = 1}^{n} 〈 I_{A} (T; t^{(i)}) I_{B} (T + Δ T; t^{(j)}) 〉 \\ = \sum_{i, j = 1}^{n} f_{A} (t^{(i)}) 〈 I (T; t^{(i)}) I (T + Δ T; t^{(j)}) 〉 f_{B} (t^{(j)}) \\ = \sum_{i, j = 1}^{n} f_{A} (t^{(i)}) M (Δ T; t^{(i)}, t^{(j)}) f_{B} (t^{(j)}) . \end{matrix}

In the last line of this equation, we show the correspondence of FLCS [10,12–16] and 2D emission-delay correlation map that we introduced with 2D FLCS [17–20]. This correlation function is evaluated from the photon data as

\begin{array}{l} g_{AB} (Δ T) \\ = \frac{1}{(T_{\max} - Δ T) \cdot Δ Δ T} \sum_{p, q} {\begin{matrix} f_{A} (t_{p}) f_{B} (t_{q}) & Δ T - Δ Δ T / 2 < T_{q} - T_{p} < Δ T + Δ Δ T / 2 \\ 0 & otherwise \end{matrix} . \end{array}

Equation (19) or (20) represents how FLCS calculates species-specific correlation functions.

In the previous work of Enderlein and Gregor [10], they used FLCS to evaluate the afterpulsing effect. As afterpulses are expected to show a flat TCSPC histogram, a flat distribution is chosen for the afterpulse contribution:

d_{ap} (t^{(i)}) = n^{- 1} .

For representing the temporally decaying real fluorescence signal, the averaged decay curve is used as d_α(t⁽ⁱ⁾) after subtracting a constant background corresponding to the afterpulses, which is assumed to be the same as the minimum

\bar{I} (t^{(i)})

value,

{\bar{I}}_{\min}

:

d_{fl} (t^{(i)}) = {\bar{I} (t^{(i)}) - {\bar{I}}_{\min}} / {\bar{I} - n {\bar{I}}_{\min}} .

Here, note that d_ap(t⁽ⁱ⁾) and d_fl(t⁽ⁱ⁾) are normalized to be

\sum_{i = 1}^{n} d_{ap} (t^{(i)}) = \sum_{i = 1}^{n} d_{fl} (t^{(i)}) = 1

. If the system is homogeneous and the decay curve consists of only single species, one can safely apply the filter function calculated from Eq. (21) and (22) using Eq. (18) to discriminate afterpulses. We tested this analysis using the photon data of the pure TMR solution. As a result, the proper correlation curve can be retrieved as shown in Fig. 3. Note that this FLCS-based method also discriminates the effect of constant background signals, including afterpulsing, on the overall correlation amplitude. However, the error of the correlation amplitude due to afterpulsing is not so large (e.g., less than 1% for the data shown in this paper), and the relevant correction can be made also based on the present time symmetry analysis (see Appendix).

On the other hand, if the system consists of multiple species having different fluorescence lifetimes (inhomogeneous case), the situation is different. In the case of the mixture solution of TMR and Cy3, FLCS gives a corrected autocorrelation curve that is systematically biased toward the lower side and significantly deviates from the cross-correlation (Fig. 4). It deviates from the raw autocorrelation even at ∆T > ~20 µs, where afterpulsing effect does not exist (see Fig. 2). This is because the action of the filter function to each species is not uniform. The contribution of a longer lifetime species is more strongly suppressed compared to a shorter lifetime species, because of resemblance of its decay curve to the flat afterpulse distribution. The deviation would make it difficult to precisely evaluate the autocorrelation amplitude. Therefore, this method is not directly applicable to an inhomogeneous sample consisting of multiple species with different fluorescence lifetimes. This problem of the FLCS-based method has been also pointed out by another group [9].

To solve this problem, the fluorescence decay curves of all the species should be obtained in advance to determine the unbiased filter function. This is not always possible when one studies complex systems. In contrast, the time symmetry analysis is robust against inhomogeneity and allows us to correct the afterpulsing effect more efficiently, in particular for investigating equilibrium reactions and spontaneous molecular fluctuations taking place in the sample. This is because it only uses the symmetry of the correlation signal [Eq. (4)], so that it is not affected by the number of independent species in the sample.

3.3 Implementation of the time symmetry analysis to FLCS

Although the original FLCS-based method does not work well in the inhomogeneous case as shown above, it can be improved by combining the time symmetry analysis through simple modification. First, one puts the ensemble-averaged decay curve and a flat distribution, which represents the afterpulse contribution, into D:

d_{en} (t^{(i)}) = \bar{I} (t^{(i)}) / \bar{I},

d_{ap} (t^{(i)}) = n^{- 1} .

Then, the filter functions for these decay components are determined using Eq. (18). The obtained filter functions are applied to the photon data as Eq. (19) to evaluate the cross-correlations g_en,ap(∆T) and g_ap,en(∆T):

\begin{matrix} g_{en, ap} (Δ T) = \sum_{i, j = 1}^{n} f_{en} (t^{(i)}) M (Δ T; t^{(i)}, t^{(j)}) f_{ap} (t^{(j)}) \\ = \sum_{i, j = 1}^{n} f_{en} (t^{(i)}) M^{0} (Δ T; t^{(i)}, t^{(j)}) f_{ap} (t^{(j)}) + \sum_{i, j = 1}^{n} f_{en} (t^{(i)}) M^{†} (Δ T; t^{(i)}, t^{(j)}) f_{ap} (t^{(j)}), \end{matrix}

\begin{matrix} g_{ap, en} (Δ T) = \sum_{i, j = 1}^{n} f_{ap} (t^{(j)}) M (Δ T; t^{(j)}, t^{(i)}) f_{en} (t^{(i)}) \\ = \sum_{i, j = 1}^{n} f_{ap} (t^{(j)}) M^{0} (Δ T; t^{(j)}, t^{(i)}) f_{en} (t^{(i)}) + \sum_{i, j = 1}^{n} f_{ap} (t^{(j)}) M^{†} (Δ T; t^{(j)}, t^{(i)}) f_{en} (t^{(i)}) . \end{matrix}

By taking the difference of each side of these equations and using

M^{0} (Δ T; t^{(i)}, t^{(j)}) = M^{0} (Δ T; t^{(j)}, t^{(i)})

, one obtains

\begin{matrix} g_{en, ap} (Δ T) - g_{ap, en} (Δ T) = \sum_{i, j = 1}^{n} f_{en} (t^{(i)}) {M^{†} (Δ T; t^{(i)}, t^{(j)}) - M^{†} (Δ T; t^{(j)}, t^{(i)})} f_{ap} (t^{(j)}) \\ = \sum_{i, j = 1}^{n} f_{en} (t^{(i)}) {\bar{I} (t^{(i)}) \cdot n^{- 1} p (Δ T) - \bar{I} (t^{(j)}) \cdot n^{- 1} p (Δ T)} f_{ap} (t^{(j)}) \\ = p (Δ T) \bar{I} . \end{matrix}

Here, we used

\sum_{i = 1}^{n} f_{α} (t^{(i)}) d_{β} (t^{(i)}) = δ_{α β}

and Eqs. (23), (24). Thus, it is found that the afterpulsing probability p(∆T) is obtainable using the symmetry of the cross-correlations, g_en,ap(∆T) and g_ap,en(∆T). Then, the afterpulsing effect can be subtracted from the autocorrelation curve using Eq. (15). We have confirmed that this method gives the exactly same result as the time symmetry analysis based on Eq. (14) (data not shown).

In the case of a multi-component system, one can also determine the afterpulse-free species-specific correlation functions, g⁰_AB(∆T), by using the obtained p(∆T) as follows:

\begin{matrix} g_{AB}^{0} (Δ T) = \sum_{i, j = 1}^{n} f_{A} (t^{(i)}) M^{0} (Δ T; t^{(i)}, t^{(j)}) f_{B} (t^{(j)}) \\ = \sum_{i, j = 1}^{n} f_{A} (t^{(i)}) {M (Δ T; t^{(i)}, t^{(j)}) - M^{†} (Δ T; t^{(i)}, t^{(j)})} f_{B} (t^{(j)}) \\ = g_{AB} (Δ T) - p (Δ T) {\sum_{i = 1}^{n} f_{A} (t^{(i)}) \bar{I} (t^{(i)})} {\sum_{j = 1}^{n} n^{- 1} f_{B} (t^{(j)})} . \end{matrix}

Here, f_A(t⁽ⁱ⁾) and f_B(t⁽ⁱ⁾) are the ordinary filter functions for species A and B computed from a set of the reference decay curves as Eq. (18). Note that d_en(t⁽ⁱ⁾) [Eq. (23)] and d_ap(t⁽ⁱ⁾) [Eq. (24)] are not used in the calculation of these filter functions. The first term in the right-hand side of Eq. (28) is the correlation calculated using FLCS without considering the afterpulsing effect. Quantities in brackets in the second term can be readily evaluated using the corresponding filter functions.

4. Conclusion

In this work, we have shown that the time symmetry analysis of TCSPC-FCS data is a reliable method to correct the afterpulsing effect of APDs. It can be used not only for a pure sample but also for an inhomogeneous mixture sample for which the previous method does not work well. The only assumption we put in the time symmetry analysis is the equilibrium of the system, which is valid in most FCS experiments. The numerical procedure is straightforward and does not rely on arbitrary parameters or reference data. Therefore, it is easily implemented in an automated FCS measurement system.

More notably, the time symmetry analysis is fully compatible with a recently developed new method, 2D FLCS [18–20], which is an ideal tool for investigating microsecond biomolecular dynamics. Actually, the time symmetry analysis is an extension of 2D FLCS. In the original 2D FLCS, one assumes the symmetry of 2D maps and applies further analysis, because in the case of an equilibrated system, the true correlation signal should have time-reversal symmetry. In the present work, we utilize this property of 2D maps for evaluating the afterpulsing effect in FCS experiments using a single detector. We believe that the time symmetry analysis will extend the potential of 2D FLCS in study of microsecond dynamics by facilitating an advanced measurement scheme with a relatively simple optical setup.

5 Appendix

5.1 Precise evaluation of σ(t⁽ⁱ⁾)

In Eq. (14), the standard error σ(t⁽ⁱ⁾) of C(∆T; t⁽ⁱ⁾) is required to calculate p(∆T). In the main text, we use a simple form, i.e., $σ (t^{(i)}) = {\bar{I}}^{1 / 2} (t^{(i)})$ . Alternatively, one can evaluate σ(t⁽ⁱ⁾) more precisely as follows. The variance of C(∆T; t⁽ⁱ⁾) is separated into two parts:

σ^{2} [C (Δ T; t^{(i)})] (t^{(i)}; Δ T) = σ^{2} [C^{″} (Δ T; t^{(i)})] (t^{(i)}; Δ T) + σ^{2} [C^{'} (Δ T; t^{(i)})] (t^{(i)}; Δ T) .

From Eqs. (3-6) and (14),

\begin{matrix} C^{″} (Δ T; t^{(i)}) = \sum_{j = 1}^{n} M^{0} (Δ T; t^{(j)}, t^{(i)}) + \sum_{j = 1}^{n} M^{†} (Δ T; t^{(j)}, t^{(i)}) \\ \approx \sum_{j = 1}^{n} G^{0} (Δ T) \bar{I} (t^{(j)}) \bar{I} (t^{(i)}) + \sum_{j = 1}^{n} p (Δ T) n^{- 1} \bar{I} (t^{(j)}) \\ = \sum_{j = 1}^{n} {G (Δ T) - p (Δ T) I^{- 1}} \bar{I} (t^{(j)}) \bar{I} (t^{(i)}) + \sum_{j = 1}^{n} p (Δ T) n^{- 1} \bar{I} (t^{(j)}) \\ = \sum_{k, l = 1}^{n} M (Δ T; t^{(k)}, t^{(l)}) \bar{I} (t^{(i)}) {\bar{I}}^{- 1} - p (Δ T) \bar{I} (t^{(i)}) + p (Δ T) n^{- 1} \bar{I} \end{matrix}

and

\begin{matrix} C^{'} (Δ T; t^{(i)}) = \sum_{j = 1}^{n} M^{0} (Δ T; t^{(i)}, t^{(j)}) + \sum_{j = 1}^{n} M^{†} (Δ T; t^{(i)}, t^{(j)}) \\ \approx \sum_{j = 1}^{n} G^{0} (Δ T) \bar{I} (t^{(i)}) \bar{I} (t^{(j)}) + \sum_{j = 1}^{n} p (Δ T) n^{- 1} \bar{I} (t^{(i)}) \\ = \sum_{j = 1}^{n} {G (Δ T) - p (Δ T) {\bar{I}}^{- 1}} \bar{I} (t^{(i)}) \bar{I} (t^{(j)}) + \sum_{j = 1}^{n} p (Δ T) n^{- 1} \bar{I} (t^{(i)}) \\ = \sum_{k, l = 1}^{n} M (Δ T; t^{(k)}, t^{(l)}) \bar{I} (t^{(i)}) {\bar{I}}^{- 1} . \end{matrix}

Here, we approximated the true correlation signal M⁰(∆T; t⁽ⁱ⁾, t⁽^j⁾) using the ensemble-averaged decay curve,

\bar{I} (t^{(i)})

. Note that

G (Δ T) = \frac{〈 I (T) I (T + Δ T) 〉}{{〈 I (T) 〉}^{2}} = \sum_{k, l = 1}^{n} M (Δ T; t^{(k)}, t^{(l)}) {\bar{I}}^{- 2} .

The variance of correlation signal is considered to be equal to the number of observed photon pairs (Poisson distribution). Therefore, combining Eqs. (29-31), one obtains

\begin{matrix} σ^{2} [C (Δ T; t^{(i)})] (t^{(i)}; Δ T) = C^{″} (Δ T; t^{(i)}) + C^{'} (Δ T; t^{(i)}) \\ = 2 \sum_{k, l = 1}^{n} M (Δ T; t^{(k)}, t^{(l)}) \bar{I} (t^{(i)}) {\bar{I}}^{- 1} - p (Δ T) \bar{I} (t^{(i)}) + p (Δ T) n^{- 1} \bar{I} . \end{matrix}

In this formula, p(∆T) is not known in advance. One can simply replace it with a crude estimate of p(∆T) obtained from Eq. (14) by setting σ(t⁽ⁱ⁾) = 1.

5.2 ∆T-independent afterpulsing effect

In this work, we have focused on the afterpulsing effect that appears as a ∆T-dependent artificial correlation at short delay time (<~20 µs). However, it is known that afterpulsing also affects FCS data as a ∆T-independent small decrease of the correlation amplitude [16]. This effect is explained as follows. The correlation function [Eq. (1)] is rewritten by using the fluctuation of the fluorescence intensity around its mean value, δI(T), as

G (Δ T) = 1 + \frac{〈 δ I (T) δ I (T + Δ T) 〉}{{〈 I (T) 〉}^{2}} .

The denominator of the second term (

{〈 I (T) 〉}^{2}

) becomes larger when afterpulsing occurs. This causes the reduction of the second term in a ∆T-independent manner. In the cross-correlation method using two-detectors, this effect persists and reduces the correlation value. An advantage of the previous FLCS-based method is that it can remove this effect by applying the filter function to the denominator [10]. Although we have not considered this effect in the present time symmetry analysis, if necessary, one can introduce a correction factor to take this effect into account. After evaluating p(∆T) values over the time range of afterpulsing (<~20 µs), one obtains the total afterpulsing probability,

\bar{p} = \int_{0}^{\infty} p (Δ T) d Δ T

. Then, the corrected denominator of Eq. (34) is obtained as

{〈 I (T) 〉}^{2} / {(1 + \bar{p})}^{2}

. For the detector used in this work,

\bar{p}

was calculated from the data shown in Fig. 2(c) and was 4.7 × 10⁻³. Therefore, the ∆T-independent afterpulsing effect reduced the correlation amplitude by 0.94%, which is negligible in the present case.

Acknowledgments

This work was supported in part by Grant-in-Aid for Scientific Research on Innovative Areas (No. 25104005) from The Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan.

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Correction of the afterpulsing effect in fluorescence correlation spectroscopy using time symmetry analysis

Abstract

1. Introduction