Frustrated FRET for high-contrast high-resolution two-photon imaging

Two-photon fluorescence microscopy has become increasingly popular in biomedical research as it allows high-resolution imaging of thick biological specimen with superior contrast and penetration than confocal microscopy. However, two-photon microscopy still faces two fundamental limitations: 1) image-contrast deterioration with imaging depth due to outof-focus background and 2) diffraction-limited spatial resolution. Herein we propose to create and detect high-order (more than quadratic) nonlinear signals by harnessing the frustrated fluorescence resonance energy transfer (FRET) effect within a specially designed donor-acceptor probe pair. Two distinct techniques are described. In the first method, donor fluorescence generated by a two-photon laser at the focus is preferentially switched on and off by a modulated and focused one-photon laser beam that is able to block FRET via direct acceptor excitation. The resulting image, constructed from the enhanced donor fluorescence signal, turns out to be an overall three-photon process. In the second method, a two-photon laser at a proper wavelength is capable of simultaneously exciting both the donor and the acceptor. By sinusoidally modulating the two-photon excitation laser at a fundamental frequency ω, an overall four-photon signal can be isolated by demodulating the donor fluorescence at the third harmonic frequency 3ω. We show that both the image contrast and the spatial resolution of the standard two-photon fluorescence microscopy can be substantially improved by virtue of the high-order nonlinearity. 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Introduction
Modern optical microscopy has greatly opened up exciting opportunities for imaging biological samples. For high-resolution (sub-cellular) imaging in live tissues and organisms, two-photon (2P) fluorescence microscopy is the most appealing tool [1][2][3]. It uses localized nonlinear excitation to generate fluorescence signal within the focus volume where the probability of absorbing two simultaneous incident photons in a single event is the highest [4]. In comparison to one-photon confocal microscopy, such spatially confined excitation (i.e., optical sectioning) of 2P microscopy enables higher image contrast and hence deeper penetration for scattering samples.
However, 2P microscopy will finally lose its image contrast when the optical sectioning ability eventually breaks down when focusing deep into scattering samples. Due to the scattering loss, the laser power is attenuated exponentially along the light path. Increasing the total excitation laser power with imaging depth helps to maintain the same excitation intensity at the focal plane, but inevitably also raises the probability of exciting the out-of-focus fluorophores (especially those located near the sample surface). As a result, the out-of-focus background will grow and eventually overwhelm the in-focal signal when approaching the fundamental imaging-depth limit [5][6][7][8]. Obviously, such loss of image contrast cannot be overcome with a higher excitation laser power which will enhance the signal and background equally. A number of strategies have been devised to address this challenge, including using longer excitation wavelength [9], adaptive optics [10,11], differential aberration [12], structured illumination [13], optical phase conjugation [14], spatial and temporal focusing [15] and focal modulation [16].
In addition to the loss of image contrast, 2P microscopy also suffers from relatively coarse diffraction-limited spatial resolution [17,18]. Due to the long excitation wavelength and moderate numerical aperture of the objective, the lateral resolution of 2P microscopy is only about 400 nm which obscures many interesting features such as dendritic spines. Although several super-resolution techniques (PALM [19], STORM [20], SIM [21], SOFI [22] and STED [23]) have been developed to break the diffraction limit, only STED [17,23,24] is compatible with 2P imaging because STED is equipped with single-element detectors rather than CCD cameras. Unfortunately, it is technically challenging for two-photon STED to spatially shape a doughnut beam with clean intensity null deep inside highly scattering samples. Herein we propose a new concept of harnessing higher-order (more than quadratic) nonlinear fluorescence to improve both the spatial resolution and image contrast of 2P microscopy. Instead of involving more virtual states, the super nonlinearity is created by frustrated FRET, i.e., blocking energy transfer from the donor to the acceptor in a FRET pair. In brief, if the donor alone is excited, FRET will occur between the excited donor and the ground state acceptor, which quenches the donor fluorescence. In contrast, if the donor and the acceptor are both excited, energy transfer will be largely blocked (due to the mismatch of spectral overlap) and thus the donor fluorescence will be dequenched and enhanced [ Fig. 1] [25]. Therefore, frustrated FRET is inherently a nonlinear process requiring excitation of both donor and acceptor. The concept of frustrated FRET was originally proposed to enhance the resolution of confocal microscopy [26,27], and was recently applied in selective synchronously amplified fluorescence image recovery from a high background [28]. Here we a b exploit it for enhancing image contrast and the spatial resolution of 2P microscopy in scattering samples.
In this report, we propose two distinct but related experimental schemes to generate and detect such super-nonlinearity associated with frustrated FRET. The first technique utilizes a modulated one-photon (1P) laser beam to directly excite the acceptor, preferentially switching on/off the FRET process. The enhanced donor fluorescence is proved to be an overall threephoton process. The second technique employs only one 2P laser that could excite both the donor and acceptor. The 2P laser is modulated at a fundamental frequency ω and an overall four-photon donor fluorescence signal is deciphered at the third harmonic frequency 3ω through demodulation [29]. We present the expected performance of both new 2P imaging techniques with analytical theory and numerical simulations. Synchronized donor/acceptor excitation pulse trains. The 1P laser pulses are temporally followed by the 2P laser pulses with a time gap shorter than 1ns, which avoids the potential stimulated emission of the donor but still blocks FRET efficiently. The final image is reconstructed from the enhanced fluorescence signal which is demodulated from the lock-in amplifier.

Frustrated FRET implemented with two excitation beams
In the first approach, as shown in Fig. 2(a), a two-photon (2P) laser for donor excitation is collinearly combined with a one-photon (1P) laser beam for direct acceptor excitation. Both 2P and 1P excitation beams are pulsed lasers (synchronized at the same frequency ~80MHz) with pulse widths of ~100fs and ~1ns, respectively. As the fluorescence lifetime (~3ns) is much shorter than the pulse spacing (~13ns), the donor and the acceptor are treated on their ground states right before each pulses. Note that to avoid the potential donor stimulated emission caused by the 1P excitation beam, the 1P pulse train is temporally ahead of the 2P train for each pulse by sub-nanoseconds [ Fig. 2(c)]. The 1P laser is intensity modulated at a high frequency (~5MHz). When the 1P laser is blocked, an efficient FRET occurs from the 2P-excited donor to the ground state acceptor, resulting in a low donor fluorescence signal. In contrast, when 1P laser is unblocked, the acceptor will be brought to the excited state right before the action of 2P pulses, which inhibits FRET process and enhances the donor fluorescence. The final image is thus reconstructed from the enhanced donor fluorescence signal detected by a lock-in amplifier at the modulation frequency. In principle, all regular FRET pairs can work for this technique. In practice, we prefer donors to have high 2P absorption cross-section and high quantum yield. Besides a fluorophore, the acceptor could also be a non-fluorescent quencher. Fluorophores or quenchers with red-shifted excitation spectra are preferred as the FRET acceptors, in which case the wavelength of the 1P acceptor excitation laser (λ 1P ) will be close to that of the 2P donor excitation laser (λ 2P ), ensuring similar attenuation effects.
Quantitatively, under illumination by a train of 2P laser pulses, the excitation rate constant of the donor within the short laser pulses is related to 2P I through Similarly, the excitation rate constant of the acceptor within the 1P laser pulse is related to is the two-photon absorption cross section (GM) of the donor at wavelength 2P λ , ,1 a P σ is the one-photon absorption cross section (cm 2 ) of the acceptor at δ is 2P pulse width (~100 fs) and 1P δ is 1P pulse width (~1 ns).
The probability of the donor or the acceptor being excited to the excited state after each laser pulse is determined by the first-order kinetics 1 exp( ) Because the excitation of the donor and the acceptor are independent processes, after each pulse, the probabilities of the donor and the acceptor being excited to the excited states are When the 1P acceptor excitation laser is blocked (off) during the modulation procedure, energy will be transferred between the excited donor and ground state acceptor, quenching the donor fluorescence. The donor fluorescent emission rate in the absence of 1P excitation beam is where N is the number of fluorophores, ε is the collection efficiency, .
fl d k is the fluorescence emission rate of the donor, τ is the intrinsic donor fluorescence lifetime, and FRET k is the energy transfer rate. We have assumed the original donor fluorescence quantum yield as unity. We then consider the case when the 1P acceptor excitation beam is on. As schematized in Fig. 3, the donor fluorescence quantum yield is determined with two possible scenarios after each pulse: the acceptor remains in the ground state, which has a probability of 1 A P − , and the acceptor is also excited to the excited state, which has a probability of A P . For the first scenario, the donor fluorescence quantum yield for such a D-a pair is the same as that in Eq.
For the second scenario of the excited D-A pair, if assuming comparable fluorescence lifetimes for the donor and the acceptor, there exist two subscenarios with certain chances (we assume the chance to be 50:50). If the acceptor is relaxed to the ground state earlier than the donor, the donor fluorescence quantum yield is given by Eq. (5) as well. On the other hand, if the acceptor is relaxed to the ground state later than the donor, the donor fluorescence quantum yield is then 1. Therefore, we can calculate the weighted donor fluorescence quantum yield for D P population as: [ ] where ( )  is the energy transfer efficiency E t . The detected donor fluorescence emission rate ' fl S in the presence of 1P beam thus becomes: Subtracting Eq. (5) from Eq. (7) which is equivalent to the electronic demodulation of the lock-in amplifier at the modulation frequency [ Fig. 2(c) original two-photon process has now been turned into an overall three-photon effect after harnessing the frustrated FRET effect. It is worth noting that the FRET efficiency (E t ) does not affect the super-nonlinearity generated by frustrated FRET. On the other hand, the detected fluorescence signal depends on E t , thus the FRET pairs with high energy transfer efficiencies are desired to achieve high photon flux.

Frustrated FRET implemented with one excitation beam
The modulated probability of the donor or the acceptor being excited to the excited state after each laser pulse is . If a moderate excitation probability (β = 0.1) is assumed, the 3ω harmonic fluorescent signal is about 9.4% of the regular 2P signal. Therefore, a sizable four-photon fluorescent signal can be detected at the 3ω demodulated harmonic frequency.

Enhancement of signal-to-background contrast deep inside scattering samples
Now we numerically estimate how the above two new frustrated FRET techniques could enhance the signal-to-background contrast at fundamental imaging-depth limit of the regular two-photon microscopy, which is defined as [5][6][7][8] where V in is the focal volume, V out is the total volume of the sample along the beam path except V in , τ is the pixel dwell time during the imaging, C is the local fluorophore concentration, I is the 2P laser intensity, r is the distance from the optical axis, and z is the axial distance from the sample surface. We assume that the signal and the background share the same fluorescence collection efficiency at the large-area non-descanned detector and the fluorophores are uniformly stained ( S ) throughout the volume. The wavelengths are set as 1000 nm for the two-photon laser and 700 nm for the one-photon laser, both of which lie within the transparent optical window (650 ~1300 nm) of biological tissues. To simplify the calculation, we only consider the ballistic photons for samples whose anisotropy factors are low or moderate, and set the mean free path length as 200 μm, the value for brain tissues of the near IR region [3]. The intensity is shown as where l s is the mean free path (200 μm), ω 0 is the beam waist at the focus (0.35µm), z R is the Rayleigh range (0.5µm), and z focal is the depth of the focal plane. Numerical criteria of S/B = 1 determines the 2P imaging-depth limit to be z focal = 1023μm (as shown in Fig. 6), which is very close to the experimental result of ~1 mm on mouse brain tissues [6]. We can further calculate the fluorescence photon flux (in the unit of number of photons per microsecond) within the focal volume (V in ) and the background volume (V out ) respectively. The fluorescence photon flux of the conventional 2P microscopy is given by According to Eqs. (8) and (20), the fluorescence photon fluxes of the newly proposed higherorder nonlinear techniques are derived respectively as: Signal-to-background ratio 1 46.2 294 *The signal and background fluorescence photon fluxes were calculated with a fluorophore concentration of 1.66 × 10 −5 M, a collection efficiency of 0.5, and a FRET efficiency of 80%. The powers were set as 200mW for the 2P laser and 20mW for the 1P laser. Both the two lasers had a repetition rate of 80MHz and the pulse width were 100fs and 1ns of the 2P and 1P lasers respectively. The two photon cross section of both the donor and acceptor fluorophores were assumed as 100GM and the one photon cross section of the acceptor were assumed as 2 × 10 −16 cm 2 .
Along with the substantially improved signal-to-background ratio, the super nonlinearity created by frustrated FRET is inevitably associated with a lower fluorescence photon signal than the standard 2P microscopy (Table 1). Hence, the issue of shot noise becomes relevant here. In order for the new techniques to deliver equal signal-to-noise levels as that of the traditional 2P microscopy, prolonged pixel dwell times are needed. For instance, if a 512 pixel × 512 pixel image is taken at 200 Hz with a standard 2P microscopy, which means the pixel dwell time is around 10 µs, 2.06 × 10 4 photons in total will be collected from a single pixel within the focal volume. To collect a similar number of signal photons, the two-laser technique requires around 85 µs pixel dwell time and the one-laser technique requires around 1250 µs pixel dwell time, which correspond to about 20 Hz and 2 Hz respectively.

Improvement of the spatial resolution in 3D
In addition to enhancing the signal-to-background contrast deep inside the scattering samples, the super-nonlinearity created by frustrated FRET could also contribute to the enhancement of the spatial resolution. For regular 2P microscopy, the point-spread-function (PSF 2P ) is the square of the illumination PSF of the 2P excitation laser: 2 Figure 7 indicates the apparent resolution enhancements by both the frustrated FRET techniques. Compared with the regular 2P microscopy, it is apparent that the one-laser technique (four-photon effect) increases the spatial resolution by a factor of sqrt (2) in all three dimensions. Meanwhile, the two-laser technique (three-photon effect) also improves the resolution by ~1.4 times owing to the relatively shorter wavelength of the 1P laser than that of the 2P laser. Fig. 7. Both the two-laser and one-laser frustrated FRET techniques can improve the spatial resolution of the regular 2P microscopy in all three dimensions. We assume that the images are taken with an air-objective of N.A. = 0.7, refraction index of 1, and the 2P and 1P laser wavelengths to be 1000 nm and 700 nm, respectively. The color bar linearly depends on the fluorescence intensity: red represents the highest intensity and back represents the lowest intensity.

Conclusion
In summary, a new concept of super-nonlinear fluorescence microscopy based on the frustrated FRET effect is proposed to significantly improve image contrast, as well as to enhance the diffraction-limited spatial resolution, of two-photon microscopy deep inside scattering samples. In this article, we described two techniques to generate and detect the higher-order nonlinear signals. In the first approach, we excite the donor and acceptor respectively with two collinearly combined laser beams, and detect the resulting enhanced fluorescence signal. In the second approach, we exploit a single 2P laser beam to excite both donor and acceptor and separate the four-photon signal through harmonic demodulation. Each of these two techniques has its own advantages. The two-laser method allows more efficient and flexible excitation arrangement and hence has a wider range of applicable FRET probes; while the one-laser method requires a relatively simple experimental apparatus. Moreover, by prolonging the collection dwell time accordingly, a signal-to-noise level similar to that of conventional 2P microscopy can be achieved.
Finally, the current approach can be compared with the recently developed deep-imaging methods using photo-activatable fluorophores [32-34]. Both frustrated FRET techniques and mutiphoton activation and imaging of photo-activatable fluorophores represent moleculebased (rather than wave-based) strategies of harnessing special imaging probes with inherent nonlinear response. Owing to the faster temporal response of energy transfer process over chemical bond breaking required in photo-activatable fluorophores, the frustrated FRET techniques should be advantageous in terms of imaging speed especially in in vivo applications.