Improved resolution in fluorescence microscopy with the FRET pairs by time gating

Shangguo Hou; Jianfang Chen; Suhui Deng; Fei Wang; Qing Huang; Ya Cheng; Chunhai Fan

doi:10.1364/OE.23.013121

1. Introduction

The emerging of superresolution microscopy technique has offered a powerful tool for biologists to research the structure and dynamics of cells in the nanoscale [1]. Generally speaking, the superresolution techniques are realized either by temporally or spatially modulating the excitation or activation light [2]. Among these techniques, utilizing the spatially varied arrival times of the spontaneous emission, time-gated detection of the fluorescence signal has been explored to further decrease the emission spot [3–5]. Stimulated emission depletion (STED) microscopy with time gating is a paradigm of time-gating technique [3, 4]. In this method of gated-STED (g-STED), the lifetime of the spontaneous emission from the excited fluorophores varies spatially in the presence of the distribution of STED beam. The lifetime in the center of STED beam is much longer than that of periphery. Collecting photons mainly from the fluorophores in the doughnut center can be achieved by setting a time window which discards the early emitted photons at the periphery, and the contrast as well as the resolution can be further improved.

The fluorescence lifetime also changes during the occurrence of Förster resonance energy transfer (FRET) [6, 7]. FRET is a process in which radiationless transfer of energy occurs from an excited fluorophore (donor) to a chromophore (acceptor) in close proximity. Besides strongly depending on the spectral overlap between the donor emission and acceptor absorption spectra and the distance between the donor and the acceptor, the process of FRET also depends on the availability of the acceptors in the ground state [8]. If the acceptor is saturated when all acceptor molecules are in the excited state, FRET is largely blocked and the donor fluorescence will be enhanced. The lifetime will also approach the natural donor lifetime as well. Therefore, the optical response of the donor as well as the donor fluorescent lifetime is a nonlinear process. As many publications have reported, such FRET frustration could be induced through direct acceptor excitation [9, 10], excitation at high rates [11] or utilizing radical anion states of the acceptor [12]. Taking advantage of the nonlinear donor signal, the resolution enhancement in all three dimensions can be achieved [11–14]. The first realization of resolution improvement using such energy transfer probes experimentally was reported in 2010 [12]. A recent work in 2013 showed that frustrated FRET can improve the image contrast and the spatial resolution of the two-photon fluorescence microscopy [13]. Herein we present the spatial information is also encoded in the donor lifetime and it can be utilized to improve the resolution by extracting this temporal information.

2. Theory

2.1 Temporal dynamics in a saturated FRET microscopy

Donor or acceptor molecules can be driven into the excited state saturation by intense excitation in FRET blocking microscopy. When exposed to a Gaussian-like excitation with a high peak intensity, the acceptor molecules have a higher probability of being excited in the center than those at the periphery. The higher the excitation intensity is used, the more likely it becomes that central acceptor molecules can be excited directly into saturation and the donor will be de-quenched in this area. Therefore, the FRET incompetent fraction is the function of excitation rate, which increases monotonously with the intensity increases. When FRET saturation happens, the fluorescence lifetime of donor molecules displays a non-equilibrium distribution with a larger average donor lifetime in the center since the less effective FRET process occurs. Therefore, collecting the donor photons after a delay ensures the fluorescence light is recorded mainly from the center and enhances the contrast to attain a higher resolution.

To establish and test this concept, the probes of single acceptor-donor pairs are considered. Such FRET pairs can be easily constructed from dsDNA in experiments [12]. After the donor absorbs light, it relaxes from the excited state by emitting photons or transferring the energy to the acceptor. Then the acceptor is excited and decays to the ground state by spontaneous fluorescence. Since the FRET transition depends on the availability of the excited donors (D) and the acceptors in the ground state (a), the photophysics of the two coupled fluorophors should be modeled by a diagram between the four state pairs: Da, da, DA and dA [9]. The capital letters (D and A) denote the excited states of donor and accepter, whereas small letters (d and a) denote the respective ground states. We make several assumptions that the vibrational relaxation times are negligible compared to the fluorescence lifetime and the triplet states processes are also neglected since during these processes the temporal dynamics that we discussed here do not change.

\begin{array}{l} \frac{\partial N_{d a}}{\partial t} = K_{f}^{D} N_{D a} + K_{f}^{A} N_{d A} - K_{e x c} N_{d a} \\ \frac{\partial N_{D a}}{\partial t} = K_{e x c} N_{d a} + K_{f}^{A} N_{D A} - (K_{f}^{D} + K_{T}) N_{D a} \\ \frac{\partial N_{D A}}{\partial t} = K_{e x c} N_{d A} - (K_{f}^{A} + K_{f}^{D}) N_{D A} \\ \frac{\partial N_{d A}}{\partial t} = K_{f}^{D} N_{D A} + K_{T} N_{D a} - (K_{f}^{A} + K_{e x c}) N_{d A} \end{array}

Where

K_{f}^{D} = 1 / τ_{D}

and

K_{f}^{A} = 1 / τ_{A}

are the fluorescence rates of the donor and accepter, with

τ_{D}

and

τ_{A}

are the lifetimes of donor and accepter, respectively. K_T denotes the rate of energy transfer from donor to acceptor.

K_{e x c} = I_{e x c}^{D} σ_{D}

is the excitation rate of donor, where

I_{e x c}^{D}

is the excitation intensity and

σ_{D}

is the absorption cross section of donor.

In our scheme, a pulse excitation with a long duration compared with the lifetime of the fluorophores is considered for producing the FRET frustration. Thus, the system reaches to steady-state within the long pulse duration. Our analysis begins at $t = 0$ with the donor and the acceptor molecules are treated on their steady state after each pulse. Solving the four equations for the steady-state condition yielding the excited donor population at beginning:

\begin{array}{l} N_{D a 0} = \frac{1}{1 + \frac{K_{T}}{K_{f}^{A}} + \frac{K_{f}^{D} + K_{T}}{K_{e x c}} - \frac{K_{T}}{(K_{e x c} + K_{f}^{A} + K_{f}^{D})}} \\ N_{D A 0} = N_{D a 0} \frac{K_{T} K_{e x c}}{K_{f}^{A} (K_{e x c} + K_{f}^{A} + K_{f}^{D})} \\ N_{D 0} = N_{D a 0} + N_{D A 0} \end{array}

After each pulse, the population of excited donor fluorophores decays in time as

\begin{array}{l} \frac{\partial N_{D a} (r, t)}{\partial t} = - (K_{f}^{D} + K_{T}) N_{D a} (r, t) + K_{f}^{A} N_{D A} (r, t) \\ \frac{\partial N_{D A} (r, t)}{\partial t} = - (K_{f}^{D} + K_{f}^{A}) N_{D A} (r, t) \end{array}

As time progresses, the number of donor fluorescent photons arriving as a function of time can be expressed as

\begin{array}{l} \frac{\partial F_{D a} (r, t)}{\partial t} = K_{f}^{D} N_{D a} (r, t) \\ \frac{\partial F_{D A} (r, t)}{\partial t} = K_{f}^{D} N_{D A} (r, t) \end{array}

We calculate these equations via direct integration, yielding the total number of the donor fluorescent photons:

\begin{array}{l} F_{D} (r, t) = F_{D A} + F_{D a} \\ = N_{D 0} (r) [\frac{N_{D A 0} (r)}{N_{D 0} (r)} \frac{K_{f}^{D}}{K_{f}^{D} + K_{f}^{A}} (1 - \exp (- t (K_{f}^{D} + K_{f}^{A}))) \\ + \frac{K_{f}^{D}}{K_{f}^{D} + K_{T}} (1 - \frac{N_{D A 0} (r)}{N_{D 0} (r)} \frac{K_{T}}{K_{T} - K_{f}^{A}}) (1 - \exp (- t (K_{f}^{D} + K_{T}))) \\ + \frac{N_{D A 0} (r)}{N_{D 0} (r)} \frac{K_{f}^{A}}{K_{f}^{D} + K_{f}^{A}} \frac{K_{f}^{D}}{K_{T} - K_{f}^{A}} (1 - \exp (- t (K_{f}^{D} + K_{f}^{A})))] \end{array}

To explore the spatial information encoded in the photon arrival times of donor fluorescence, we investigate the temporal dependence of the donor fluorescent signal from the molecules located at different positions in the excitation spot. The positions are shown in Fig. 1(a) with different color circles. In the calculation, the lifetimes of individual donor and acceptor molecules are assumed to be 2ns and 4ns, respectively. A common FRET efficiency of 80% and an oil immersion objective with the numerical aperture of 1.4 are considered. Since the excitation power is responsible for the FRET blocking, we consider a specific example where the initial probability of the state N_DA0 equals to N_Da0 by adjusting the laser intensity $K_{e x c} = 0.44 K_{f}^{A}$ . As shown in Fig. 1(b), the photons from the molecules exposed to lower intensity arrive earlier on average than those from the central maximum of the excitation beam. After a time of about 1/10 of the donor fluorescent lifetime after the pulse, only 16% of the spontaneous emission from the center (r = 0, the red circle in Fig. 1(a)) has been emitted whereas 32% of the emission from the periphery (r = 180nm, the green circle in Fig. 1(a)) has been arrived. Thus, the time differences between the positions enable us to extract the central information with a time window.

Fig. 1 Temporal information in saturated FRET microscopy. (a) The normalized intensity distribution of excitation beam. (b) The decay rates of donor spontaneous emission from the donor molecules at the locations denoted by the circles in (a). The fluorescence is normalized by the total number of fluorescence $(F_{\inf} = F_{(t = \infty)})$ . The $x$ axis unit in (a) is the excitation wavelength $λ = 488 n m$ .

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2.2 Enhancement of the resolution by time gating

Now we will discuss the using of the temporal information to improve the spatial resolution in this section. As shown in Fig. 2(a), after setting a time window of t_g, only the late photons mainly emitted from the center of excitation spot can be used to form the image. Thus, deduced from Eq. (5), the donor fluorescence by time-gated detection is given as:

Fig. 2 Improving spatial resolution by time-gating. (a) Timing diagram for satFRET microscopy with time-gated detection. The spontaneous emission is collected during a time window (green) starting t_g after the completion of excitation pulse (blue) and lasting until the beginning of the next excitation pulse. Normalized transverse PSF (b-d) and the longitudinal PSF (e-g) for conventional microscopy (b, e), satFRET microscopy (c, f) and time-gated satFRET microscopy(d, g). The comparison between the PSF profiles along the X (h) and Z axial (i) for satFRET microscopy with (green line) and without (red line) time-gating. (j) The N-fold resolution improvement of time-gated satFRET microscopy compared to the conventional microscope as the function of t_g. The excitation wavelength $λ = 488 n m$ is used.

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\begin{array}{l} F (r) = F_{D} (r, \infty) - F_{D} (r, t_{g}) \\ = N_{D 0} (r) [\frac{N_{D A 0} (r)}{N_{D 0} (r)} \frac{K_{f}^{D}}{K_{f}^{D} + K_{f}^{A}} \exp (- t_{g} (K_{f}^{D} + K_{f}^{A})) \\ + \frac{K_{f}^{D}}{K_{f}^{D} + K_{T}} (1 - \frac{N_{D A 0} (r)}{N_{D 0} (r)} \frac{K_{T}}{K_{T} - K_{f}^{A}}) \exp (- t_{g} (K_{f}^{D} + K_{T})) \\ + \frac{N_{D A 0} (r)}{N_{D 0} (r)} \frac{K_{f}^{A}}{K_{f}^{D} + K_{f}^{A}} \frac{K_{f}^{D}}{K_{T} - K_{f}^{A}} \exp (- t_{g} (K_{f}^{D} + K_{f}^{A}))] \end{array}

Since the applying of time-gating reduces the image brightness, t_g = 0.5τ_D = 1ns is assumed in our calculation where the central signal drops to about 50% (Fig. 1(b)). Figure 2 indicates the spatial resolution is enhanced by time gating in all three dimensions. Compared with the conventional microscopy with probes of the single donor molecules (Fig. 2(b) and 2(e)), it is apparent that saturated FRET (satFRET) microscopy (Fig. 2(c) and 2(f)) increases the three-dimensional resolution by a factor of ~1.2 induced by the similar two-photon excitation of the coupled energy-transfer dyes [9]. Further, the time-gating (Fig. 2(d) and 2(g)) improves the resolution by ~1.2 times than that of the satFRET microscopy and by ~1.4 times than that of the conventional microscopy accompanied with the fluorescence weakening (Fig. 2(h) and 2(i)). To illustrate how the resolution is affected by the time-gating, Fig. 2(j) plots the improvements of the transverse resolution relative to the full width at half maximum (FWHM) of the point spread function (PSF) by the conventional microscopy at different time gates. With the increasing of t_g, the PSF decreases utile after a long time gate of t_g = 2τ_D, where nearly all the photons have been emitted (Fig. 1(b)).

2.3 Improvements of the resolution in STED microscopy with the FRET probes by time-gating

Without increasing the power of the depletion light, the resolution of the STED microscopy can be further increased by using the FRET pair as the probes [14]. When excited with a Gaussian-like focal spot, the acceptor molecules have a higher probability of being excited in the center than those at the periphery. The higher the excitation intensity is used, the more likely it becomes that central acceptor molecules can be excited directly into saturation and the donor in this area will be left to be fluorescent for the exited acceptors can't accept the energy. When a depletion beam with the doughnut-shape is added upon the acceptor molecules, the acceptor molecules in the outer region of the focus are forced to transit to the ground state by stimulated emission. This helps to eliminate FRET frustration at periphery and limits the FRET saturation in the center. Consequently, the fluorescent spot of donor is further decreased. In this method, combining the nonlinearity of the FRET pairs and the stimulated emission depletion, higher resolution than both FRET microscopy and STED microscopy is allowed [14]. Here, we expand our discussion by applying the time-gating to this FRET-assisted STED microscopy. We assume that the donor is excited by a pulse laser and the acceptor is depleted by stimulated emission with a continuous beam at the rate of $K_{s} = I_{s t e d}^{A} σ_{s t e d}^{A}$ , where $I_{s t e d}^{A}$ is the intensity of the depletion beam and $σ_{s t e d}^{A}$ is the stimulated emission cross sections. Similarly, after excitation, the donor population at beginning is given by

\begin{array}{l} {N'}_{D 0} (r) = N'_{D a 0} (r) + N'_{D A 0} (r) \\ = \frac{1}{1 + \frac{K_{T}}{K_{f}^{A} + K_{s}} + \frac{K_{f}^{D} + K_{T}}{K_{e x c}} - \frac{(K_{f}^{A} + K_{s})}{K_{e x c}} \frac{K_{T} K_{e x c}}{(K_{f}^{A} + K_{s}) (K_{e x c} + K_{f}^{A} + K_{s} + K_{f}^{D})}} [1 + \frac{K_{T} K_{e x c}}{(K_{f}^{A} + K_{s}) (K_{e x c} + K_{f}^{A} + K_{s} + K_{f}^{D})}] \end{array}

The number of donor fluorescent photons arriving as a function of time can be similarly derived, yielding

\begin{array}{l} F'_{D} (r, t) = N'_{D 0} (r) [\frac{N'_{D A 0} (r)}{N'_{D 0} (r)} \frac{K_{f}^{D}}{K_{f}^{D} + K_{f}^{A} + K_{s}} (1 - \exp (- t (K_{f}^{D} + K_{f}^{A} + K_{s}))) \\ + \frac{K_{f}^{D}}{K_{f}^{D} + K_{T}} (1 - \frac{N'_{D A 0} (r)}{N'_{D 0} (r)} \frac{K_{T}}{K_{T} - K_{f}^{A} - K_{s}}) (1 - \exp (- t (K_{f}^{D} + K_{T}))) \\ + \frac{N'_{D A 0} (r)}{N'_{D 0} (r)} \frac{K_{f}^{A} + K_{s}}{K_{f}^{D} + K_{f}^{A} + K_{s}} \frac{K_{f}^{D}}{K_{T} - K_{f}^{A} - K_{s}} (1 - \exp (- t (K_{f}^{D} + K_{f}^{A} + K_{s})))] \end{array}

The temporal dynamics is plotted in Fig. 3(b), which shows that the arrival time of fluorescent photons from the donor molecules at different positions in the excitation spot denoted by the circles with different colors in Fig. 3(a) can be distinguished. In this calculation, the excitation rate $K_{e x c} = 0.4 K_{f}^{A}$ and depletion beam power with a saturation factor of $ξ = I_{s t e d}^{A} / I_{s a t}^{A} = 20$ are assumed, in which $I_{s a t}^{A}$ is the saturation intensity for stimulated emission of the accepter. The data shows that photons from the molecules located at regions of lower excitation intensity arrive earlier on average than those from the center exposed to higher excitation intensity. After a time of about 1/10 of the donor fluorescent lifetime after the pulse, only 19% of the spontaneous emission from the center (r = 0, the black circle in Fig. 3(a)) has been emitted whereas 40% of the emission from the side (r = 50nm, the purple circle in Fig. 3(a)) has been arrived. Thus, the fluorescence signal of excited donor should be time-gated. Fluorescence can be collected after a duration after the excitation, as displayed in Fig. 4(a). Deduced from Eq. (8), the total collected fluorescence at position r with a gating-time t_g can be written as

Fig. 3 Temporal information in FRET assisted STED microscopy. The fluorescence as a function of time (b) emitted from the donor molecules at the locations denoted by the circles in (a). The fluorescence is normalized by the total number of fluorescence $(F_{\inf} = F_{(t = \infty)})$ . The $x$ axis unit in (a) is the excitation wavelength $λ = 488 n m$ .

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Fig. 4 Improving spatial resolution of FRET assisted STED microscopy with time-gating. (a) Timing diagram for time-gating. The normalized PSFs on the focal plane for FRET associated STED microscopy without (b) and with the time-gating (c). (d) The intensity profiles with different gating-time t_g (0, purple; 0.5τ, blue; τ, red; 2τ, black). (e) The N-fold improvement of resolution over that of a conventional microscope as a function of the strength of the STED beam with different gating-time (0, purple; 0.5τ, blue; τ, red; 2τ, black). (f) The N-fold improvement of resolution over that of a conventional microscope as a function of t_g with different STED beam power ( $ξ = 40$ , purple; $ξ = 20$ , blue; $ξ = 10$ , red). The excitation wavelength $λ = 488 n m$ is used.

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\begin{array}{l} F' (r) = F'_{D} (r, \infty) - F'_{D} (r, t_{g}) \\ = N'_{D 0} (r) [\frac{N'_{D A 0} (r)}{N'_{D 0} (r)} \frac{K_{f}^{D}}{K_{f}^{D} + K_{f}^{A} + K_{s}} \exp (- t_{g} (K_{f}^{D} + K_{f}^{A} + K_{s})) \\ + \frac{K_{f}^{D}}{K_{f}^{D} + K_{T}} (1 - \frac{N'_{D A 0} (r)}{N'_{D 0} (r)} \frac{K_{T}}{K_{T} - K_{f}^{A} - K_{s}}) \exp (- t_{g} (K_{f}^{D} + K_{T})) \\ + \frac{N'_{D A 0} (r)}{N'_{D 0} (r)} \frac{K_{f}^{A} + K_{s}}{K_{f}^{D} + K_{f}^{A} + K_{s}} \frac{K_{f}^{D}}{K_{T} - K_{f}^{A} - K_{s}} \exp (- t_{g} (K_{f}^{D} + K_{f}^{A} + K_{s}))] \end{array}

Both the PSFs with or without time-gating are calculated. Because the saturation effects are limited in the center by an additional depletion beam, the FWHM of 36nm is obtained with the method of FRET associated STED microscopy (Fig. 4(b)). Figure 4(c) displays that an apparent resolution enhancement is achieved by time-gating. The transverse resolution of 21nm is obtained and the resolution is increased by a factor of 1.71 at t_g = 0.5τ_D = 1ns when compared with FWHM in Fig. 4(b). It should be noted that this increase is at the cost of the decreasing of maximum amplitude by 60%, as shown by the blue curve in Fig. 4(d), which plots the intensity profiles of the PSF at different t_g. This figure reveals that with the increase of gating-time, the resolution can be continually enhanced while the signal intensity is continually decreased. Another important factor for the resolution is the power of depletion beam. Figure 4(e) and Fig. 4(f) plot the dependence of the resolution on the depletion power with different t_g and on the duration of t_g with different saturation factor $ξ$ . The data reveals that the resolution increases with increasing of the depletion power or the gating-time.

3. Conclusion

In summary, we have analyzed the temporal dynamics of the spontaneous emission of the donor molecule during the process of FRET and demonstrated that the distribution of the lifetime of the donor molecules varied with their positions within the excitation spot. By extracting the temporal information with a time-gate, the resolution enhancements in all three dimensions are achieved. Also, we show that the differences of the arrival times between different positions can be further enhanced in the case of adding a depletion beam upon the FRET probes. The improvement of the resolution is more obvious when the time-gating is applied in FRET assisted STED microscopy. The major benefit in this setup is the resolution can be significantly increased without the use of a higher power of depletion beam which reduces the risk of photobleaching.

However, in both cases, the applying of time-gating will result in a decrease of fluorescence signal to about 50%. If t_g is set longer, the signal to noise ratio (SNR) becomes poorer. Thus, the duration of the time gate should be optimized to balance the image brightness and the resolution in the practical experiments. This problem can be alleviated by offline analysis of time-correlated single-photon counting (TCSPC) recordings, allowing the experimentalist to collect all fluorescence and record arrival times using TCSPC hardware. One can also use the deconvolution method to further improve the image's SNR after time gating. And a longer acquisition time maybe helpful to compensate for the decrease in signal.

It is worth noting that such non-equilibrium distribution of the fluorescent lifetime of the donor molecules is a consequence of nonlinear response of the FRET pair to the excitation intensity. For higher energy transfer efficiency, the nonlinearity dependence increases and the distribution of fluorescence lifetime becomes sharper. The differences of the lifetime at different locations in the excitation spot increase, which means time-gating can be more adoptable. Therefore, for higher FRET efficiency, time-gated FRET microscopy allows a higher resolution.

Acknowledgments

This work is financially supported by the National Natural Science Foundation of China under Grand Nos 61378062 and 21227804. We would like to dedicate this paper to Professor Qing Huang, who unfortunately passed away just before the paper was submitted for publication. Qing Huang played an essential role in the research described here and he is greatly missed.

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Improved resolution in fluorescence microscopy with the FRET pairs by time gating

Abstract

1. Introduction

2. Theory

2.1 Temporal dynamics in a saturated FRET microscopy

2.2 Enhancement of the resolution by time gating

2.3 Improvements of the resolution in STED microscopy with the FRET probes by time-gating

3. Conclusion

Acknowledgments

References and links

Cited By

Figures (4)

Equations (9)

Optics Express