 Letter
 Open Access
 Published:
Twophoton MINFLUX with doubled localization precision
eLight volumeÂ 2, ArticleÂ number:Â 5 (2022)
Abstract
Achieving localization with molecular precision has been of great interest for extending fluorescence microscopy to nanoscopy. MINFLUX pioneers this transition through point spread function (PSF) engineering, yet its performance is primarily limited by the signaltobackground ratio. Here we demonstrate theoretically that twophoton MINFLUX (2pMINFLUX) could double its localization precision through PSF engineering by nonlinear effect. CramÃ©rRao Bound (CRB) is studied as the maximum localization precision, and CRB of twophoton MINFLUX is halved compared to singlephoton MINFLUX (1pMINFLUX) in all three dimensions. Meanwhile, in order to achieve same localization precision with 1pMINFLUX, 2pMINFLUX requires only 1/4 of fluorescence photons. Exploiting simultaneous twophoton excitation of multiple fluorophore species, 2pMINFLUX may have the potential for registrationfree nanoscopy and multicolor tracking.
1 Introduction
Superresolution microscopy takes our vision from the conventional 200 nm diffraction limit down to 20 nm regime [1,2,3,4,5,6,7,8]. MINFLUX further extends the resolution to sub10 nm [9,10,11,12,13], through combining STEDlike coordinatetargeted excitation donut [1] and the coordinatestochastic singlemolecule localization microscopy (SMLM) [2, 3]. By engineering the point spread functions of the microscope, many techniques were introduced to break the precision confinement of localization [4,5,6, 14,15,16]. In MINFLUX [9,10,11,12,13], the excitation PSF is engineered to firstorder LaguerreGaussian beam. Using the intensity minimum of coordinatetargeted confocal excitation, MINFLUX reduces much of the required photons, as intensity minimum has much higher contrast of intensity than intensity maximum of Gaussian excitation, thus is less prone to Poissonian noise [9, 17]. Yet, the full potential of MINFLUX has not been reached regarding acquisition time and signaltobackground ratio. Multiphoton microscopy features with a nonlinear dependence of the incidence to the excited signal, hence bear potential to further increase the spatiotemporal resolution of MINFLUX. For example, in twophoton microscopy, the fluorescence intensity is proportional to the square of excitation intensity [18,19,20,21]. The contrast at intensity minima of MINFLUX donut could be enhanced by the square dependence. Intuitively, together with that twophoton excitation decrease the outoffocus fluorescence background, employing twophoton excitation to MINFLUX could significantly improve its localization precision. Here we give a theoretical framework of twophoton MINFLUX (2pMINFLUX). We show explicitly that the square of fluorescence intensity results in 2fold increase in maximum localization precision compared to 1pMINFLUX. This suggests that only 1/4 photons are needed for achieving the same localization precision. Taking advantage of spectral overlapping of absorption for different fluorophore [22,23,24], 2pMINFLUX may have the potential for registrationfree multicolor localizations through emission spectral separation, which paves new avenue for simultaneous single particle tracking of fluorophores of different spectra.
2 Results
2.1 Localization precision
For a quantitative illustration, suppose an ideal zerocenter donut excitation and a backgroundfree condition with only Poissonian noise. Representing localization estimations with confidence intervals (Fig.Â 1a), the width of confidence interval for twophoton fluorescence localization is 1/2 of width for singlephoton fluorescence, demonstrating improved localization precision.
To explicitly evaluate the improvement, CramÃ©rRao Bound (CRB) is calculated for maximum localization precision of 2pMINFLUX [9]. Modification is made on the Poissonian mean \(\lambda\) for twophoton fluorescence considering nonlinear effect:
where \(\vec {r}_{f}\) is the fluorophore position, \(f_{1}\) and \(f_{2}\) stand for, for simplicity, factors corresponding to absorption crosssection of fluorophore, quantum yield and collection efficiency of the system, and \(I_\mathrm{{1p}}\) and \(I_\mathrm{{2p}}\) are point spread functions (PSF) of donut excitations. All other parameters in the model are kept unchanged.
CRB is expressed in an intricate general form of \(\frac{\partial {\lambda }}{\partial x}\) and \(\frac{\partial {\lambda }}{\partial y}\). As we show in Code File 1 (Ref. [25]), for the typical fourpoint targeted coordinate pattern (TCP) [9,10,11,12,13], at the origin where localization precision is highest (i.e., minimum CRB), CRB can be expressed explicitly as [9]:
where L is the diameter of TCP circle, N is number of detected photons, fwhm is the full width at half maximum of the excitation PSF, and factor \(s=\sqrt{\left( \frac{1}{SBR}+1\right) \left( \frac{3}{4SBR}+1\right) }\), where SBR(L) is the (median) signaltobackground ratio (dependent on L) [9, 12]. Since \({L} \ll {fwhm_{1p}} \le {fwhm_{2p}}\), we can obtain precision increase slightly larger than twofold:
The reason for halving of CRB lies in that a factor of 2 appears when the square of \(I_{2p}\) is differentiated:
and that in CRB formula (see Eq.(S26) in [9]), the denominator has one more power of the above partial derivatives than the nominator, resulting in an additional factor of 2 in the denominator.
Maximum localization precision is compared for singlephoton and twophoton MINFLUX with respect to N (Fig.Â 1b). For L = 50 nm, SBR = 3, N = 100, and singlephoton and twophoton excitation wavelengths set to 647 nm and 800 nm, \(CRB_{2p}\) and \(CRB_{1p}\) are 1.15 nm and 2.31 nm respectively, with CRB enhancement ratio \(R_{CRB}\) = \({CRB_{1p}}\)/\({CRB_{2p}}\) = 2.01. In addition, 2pMINFLUX possesses same or slightly higher localization precision with only 1/4 photons compared to 1pMINFLUX (\(N_{2p}\) = 100 versus \(N_{1p}\) = 400, or \(N_{2p}\) = 400 versus \(N_{1p}\) = 1600).
Dependence on CRB of possible changes of parameters are considered (Fig.Â 2). Quite contrary to intuition, CRB decreases (i.e. localization precision increases) as excitation wavelength increases (Fig.Â 2a), which is different from traditional localization methods, where precision is proportional to excitation wavelength. CRB changes only slightly with different excitation wavelength, providing same SBR and L. We set singlephoton excitation wavelength as 647 nm since red/crimson dyes were most commonly used in previous works [9,10,11,12,13], and twophoton excitation wavelength as 800 nm corresponding to these dyes [23, 24]. Longer wavelength with less phototoxicity such as 1280 nm [26] may also be considered; note that with 1280 nm excitation, SBR may be compromised due to increased background from an enlarged PSF and decreased signal from extended intensity minima.
With \(L_0\) = 50 nm fixed, improved SBR results in increased localization precision (i.e., decreased CRB) (Fig.Â 2b). If SBR decreases to \(\le\) 1, then precision decreases sharply, due to the inverse proportion of SBR to CRB as in factor s. Thus circumstances with SBR \(\le\) 1 should be avoided as much as possible (SBR \(\le\) 1 is not a good parameter for any imaging technique).
We then consider the combined influence on CRB of SBR and L (Fig.Â 2c, d). SBR decreases as L decreases (Fig.Â 2d). Given a fixed SBR at \(L_0\) = 50 nm, there is a lower bound of CRB and a corresponding optimal L (denoted as \(L_{opt}\)) (Fig.Â 2c). We argue that given the limited SBR as MINFLUX uses donut minimum, decrease of L below 50 nm and to \(L_{opt}\) is not fruitful as it first seems. Note that SBR is not constant; median SBR is used to its distribution and ranges around 1.4 to 4.2 for biological samples at L = 50 nm [12]. For 1pMINFLUX, median SBR \(\approx\) 0.81 (Fig.Â 2d) at \(L_{opt}\), which suggests that SBR is \(\le\) 0.81 in 50 percent localizations and thus attainable CRB is impaired. For 2pMINFLUX, SBR decreases more rapidly with L compared to 1pMINFLUX (Fig.Â 2d), limiting further decrease of L (Fig.Â 2c). To make direct comparison, same L = 50 nm and median SBR = 3 are used for both 1pMINFLUX and 2pMINFLUX.
CRB across 2D xyplane for onephoton and twophoton MINFLUX is compared for L = 50 nm (Fig.Â 3). Of most interest is only CRB in center region of TCP circle instead of whole xyplane, since a previous round of iterative 1pMINFLUX with L = 100 nm already localizes the fluorophore with singledigit nanometer precision (e.g. 3.3 nm) [12]. A center region with radius of 3.3 nm has consistently \(R_{CRB}\) \(\ge\) 1.92. (Note that in iterative 2pMINFLUX, a secondlast round with L = 100 nm would likely obtain precision better than 3.3 nm as well, resulting in further improved \(R_{CRB}\).) CRB increases (precision decreases) with increasing r; 2pMINFLUX has faster increase of CRB than 1pMINFLUX (Additional file 1: Figure S1a, b), resulting in decrease of \(R_{CRB}\) with increasing r (Fig.Â 3c, d). This faster increase could be explained by the faster decrease of intensity of â€™signalâ€™ in 2pMINFLUX compared to 1pMINFLUX (Additional file 1: Supplementary note 1). At radius of 6.7 nm and 10.0 nm (representing 2\(\sigma\) and 3\(\sigma\)), average \(R_{CRB}\) are 1.67 and 1.31 respectively. In addition, 2pMINFLUX CRB (Fig.Â 3a, S1a) is much more anisotropic than 1pMINFLUX CRB (Fig.Â 3b, S1b); a trianglelike contour can be seen in CRB of 2pMINFLUX (Fig.Â 3a), but not in 1pMINFLUX CRB (Fig.Â 3b). This could also be explained qualitatively by plotting the different â€™signalsâ€™: in 2pMINFLUX, â€™signalsâ€™ decreases faster with r for angle 0 than angle 60 (Additional file 1: Supplementary note 1). Anisotropy also exists for CRB under L = 100 nm (Additional file 1: Supplementary note 2,Â Figure S2).
The enhancement of zlocalization precision is similar to xylocalization. 3Ddonut is modeled simply as a quadratic function [12]. The highest precision at the origin also increases by 2fold:
2.2 Localization reconstruction
Using maximum likelihood estimation (MLE), zaxis localization, as a 1D problem, can be solved analytically for 2pMINFLUX as well:
where \({n}_{0}\) and \({n}_{1}\) are number of photons detected with \(I\left( \frac{L}{2}\right)\) and \(I\left( \frac{L}{2}\right)\). Imaginary roots and a real root outside of \(\left( \frac{L}{2}, \frac{L}{2}\right)\) are neglected. In simulation, the above estimator agrees with ground truth, and improved localization precision can be seen (Fig.Â 4).
For localization estimation in xyplane, we investigated two methods: maximum likelihood estimation (MLE) and least mean square estimation (LMS), being unbiased and biased respectively. For LMS estimation, a same firstorder linearization is used [9]. The LMS estimator can be analytically solved:
where k is the number of exposures, and, \(\hat{{p}}_{i}= {n}_{i}/{N}\), where \({n}_{i}\) and \(\vec {{r}}_{i}\) are, respectively, number of collected photons and displacement of excitation beam for each exposure. Similar to the expression of CRB, denominator of LMS estimator is also multiplied by a factor of 2.
For MLE estimation, the loglikelihood function is maximized as classically done. The maximization is solved numerically. MLE estimation should be given a starting value for its convergence. In our simulation, LMS estimator served as this starting value thanks to its simple form. We simulated MINFLUX imaging in xyplane for L = 50 nm with numerically solved MLE estimation (Fig.Â 5). For photon number N = 250, 2pMINFLUX can already achieve 6 nm resolution, which is barely feasible in 1pMINFLUX (Fig.Â 5b, c). In addition, compared with 1pMINFLUX with N = 1000 (Fig.Â 5d), 2pMINFLUX with N = 250 achieved similar localization distributions, confirming the capability of 2pMINFLUX to reduce number of photons required.
We believe 2pMINFLUX would be capable of multicolor localizations (Fig.Â 5e, f). Because of the spectral overlapping of twophoton absorption peaks, it is possible to excite multiple fluorophores simultaneously. If emission spectra of the multiple fluorophores are not overlapped, then complete separation of different fluorescence can be achieved with simple dichromatic filters. Although it is not a must to use a single excitation for multicolor twophoton microscopy, a singlewavelength twophoton excitation is beneficial for multicolor MINFLUX, as it can be easily achieved with dichromatic beam splitters. In this configuration, multicolor 2pMINFLUX would be free of registration of different color channels, as they are excited with the same donut coordinates. Hence, this may enable simultaneous registrationfree multicolor MINFLUX tracking, which could be crucial for study of molecular interactions. Note that L could be adjusted dynamically, and that since CRB worsens with increased r (Fig.Â 3), simultaneous localization gains most benefit only when fluorophores are close enough to the coordinate origin of excitation pattern.
3 Discussion
In future experimental works, attention needs to be paid on choice of fluorophores, choice of wavelength of femtosecond laser, power of twophoton excitation, attainable fluorescence rate, possible faster bleaching, and attainable signaltobackground ratio.
3.1 Overview
Increase of excitation power is needed in MINFLUX to achieve enough fluorescence rate compared to confocal laser scanning microscopy (CLSM) [10]; as L decreases, excitation power should be increased as well [12]. Figure 6 compares the intensities of excitation PSFs and emission PSFs of 2pMINFLUX and 1pMINFLUX with L = 50 nm.
Assume under certain excitation power \(P_a\) and \(P_b\) for 1pdonut and 2pdonut respectively [with peak intensity of excitation PSF being \(I_a\) and \(I_b\) respectively (Fig.Â 6a)], the fluorescence intensities are the same for fluorophores located exactly at the PSF peaks (with normalization of 1) (Fig.Â 6b). Intensities at r = 25 nm are low for both 2pdonut and 1pdonut (0.0543 and 0.0832 respectively) (Fig.Â 6a). In order to maximize fluorescence signals at r = 25 nm, excitation intensity (and power) of the excitation donut should be increased. Increase of \(\approx\) 12.1fold and 18.4fold respectively for 1p/2pMINFLUX (Fig.Â 6c) would result in same fluorescence signals (with value of 1) at r = 25 nm for MINFLUX (Fig.Â 6d) compared with the original donut (Fig.Â 6a, b). For 2pMINFLUX, the fold of increase remains comparable with 1pMINFLUX; an additional 1.5fold only is sufficient.
Hypothetical emission donuts for 1p/2pMINFLUX could be derived as well (Fig.Â 6d). The intensity of peak for 1pMINFLUX emission donut is 12.1, whereas for 1pMINFLUX emission donut is \(18.4^2 = 338\) because of the square dependence. We argue that curves in Fig.Â 6(d) do not reflect reality because of fluorescence saturation (Additional file 1: Supplementary note 3) and fluorophores being in singlemolecule state (see subsection 4 â€™Backgroundâ€™ of Discussion).
3.2 Estimated fluorescence rate
The useful fluorescence rate (denoted as \(\eta _{MFX}\)) in 1pMINFLUX is \(\eta _{1pMFX} \le\) 50 kHz for the dyes AF647, CF680 and CF660C [12].
Before discussion on fluorescence rate and possible saturation in MINFLUX, basic characteristics of the relationship between fluorescence rate and excitation power for both onephoton and twophoton fluorescence should be explicated. The relationship is often shown with curves of fluorescence rate to excitation power. Each curve of fluorescence rate can be separated into two parts: a first part with strict linear (for onephoton) or quadratic (for twophoton) power dependence, and a second part that reaches plateau [27,28,29,30]. We use denotations \(\eta _{lin}\) (\(\eta _{1lin}\) or \(\eta _{2lin}\)) to describe the maximum fluorescence rate with linear (or quadratic, respectively) power dependence for singlephoton or twophoton fluorescence (i.e., at the transition point from linear to nonlinear (saturation) dependence); and \(\eta _{sat}\) (\(\eta _{1sat}\) or \(\eta _{2sat}\)) to describe the saturated fluorescence rate. A nonsaturated, strictly displacementdependent fluorescence should be ensured for either 1p/2pMINFLUX. Regions of interest (ROIs) should be quantified; for L = 50 nm (\(r_0\) = 25 nm), the radius r of the fluorophore can be described as \(r_0 \pm 3\sigma\), where \(\sigma\) is the CRB of a previous round of iteration with L = 100 nm. For a strictly linear/quadratic dependence within the ROIs, \(\eta _{r_0 \pm 3\sigma } \le \eta _{lin}\) should be satisfied (\(\eta _{r_0 \pm 3\sigma } = \eta _{lin}\) to maximize the fluorescence detection of MINFLUX). Thus MINFLUX fluorescence rate is estimated as
For 1pMINFLUX with \(\sigma _{1p}\) = 3.3 nm, \(\frac{I\left( r_0+k \sigma \right) }{I\left( r_0\right) }\) = 1.90; while for 2pMINFLUX with \(\sigma _{2p}\) = 1.6 nm (Additional file 1: Supplementary note 4), and \(\frac{I\left( r_0+k \sigma \right) }{I\left( r_0\right) }\) = 2.04.
Data of \(\eta _{2lin}\) for the red/crimson dyes were not found in literature. We assume that the ratios of \(\frac{\eta _{2lin}}{\eta _{1lin}}\) are the same for different fluorophores. For the dye TMR [27], \(\eta _{1lin}\) = 20 kHz, and \(\eta _{2lin}\) = 13 kHz. Thus the fluorescence rate for 2pMINFLUX could be estimated as \(\eta _{2pMFX} \approx \frac{\eta _{2lin}}{\eta _{1lin}} \frac{1.90}{2.04} \eta _{1pMFX}\) = 30 kHz. A smaller rate of 25 kHz is chosen, corresponding to collected photons N = 1000 in t = 40 ms exposure time. Indeed, the fluorescence rate would be smaller for twophoton fluorescence than singlephoton fluorescence. The realistic fluorescence rate achievable should be measured in experiment.
Although the outoffocus noise is almost absent for twophoton excitation, pulsed twophoton excitation may cause faster bleaching than continuouswave singlephoton excitation at the focal plane. This may limit the attainable number of repeats of localizations per molecule of 2pMINFLUX.
3.3 Estimated excitation power
The excitation power needed for 2pMINFLUX is calculated (Additional file 1: Supplementary Note 5). Equation (9) in ref. [20] is used for calculation. Twophoton crosssections of 100 GM is mainly considered; several red/crimson dyes (Atto647N, Silicon Rhodamine, STAR 635P, Atto594 and Atto590) all have > 100GM cross sections under 800 nm excitation [24]. For a fluorophore with 100 GM twophoton cross section, < 1 mW of confocal excitation power is enough for fluorescence rate to reach 25 kHz for a single molecule (Additional file 1: Figure S3a). With a 18.4fold decrease of intensities at r = 25 nm, excitation power of < 20 mW should be enough for 2pMINFLUX (Additional file 1: Figure S3b). According to the ref. [26], with a 775 nm laser of 82 MHz repetition rate, 60 mW and 52 mW excitation power at sample surface were used for ex vivo and in vivo deep brain twophoton microscopy respectively. Of course, because of tissue scattering, the focal excitation power (not estimated in the reference) is not as large as 52 mW, the reference still shows that 52 mW power of femtosecond laser is compatible with biological settings.
3.4 Background
We maintain that the background of 2pMINFLUX remain comparable to 1pMINFLUX. Indeed, increase of excitation intensity would result in increase of background of 1pMINFLUX compared to 1pconfocal, and also increase of background of 2pMINFLUX compared to 2pconfocal. We argue that background in 1p/2pMINFLUX consists of two kinds of background: (1) background originated from outer region of focal donut PSF, and (2) outoffocus background (originated from excitation outside focal PSF and arrived at detector due to scattering). Twophoton excitation would reduce the latter kind of background compared to 1pMINFLUX, because of the restriction of twophoton excitation in the focal spot. However, 2pMINFLUX may increase the first kind of background. Since the donut is larger for 2pMINFLUX, there is increased possibility of twophotoninduced luminescence of unstained features (excited with elevated excitation intensities although they have limited twophoton crosssections).
Importantly, both 1pMINFLUX and 2pMINFLUX are singlemolecule localization microscopy (SMLM) technique, not simple confocal laser scanning microscopy. There would not be excessive fluorescence background (no 12fold or 338fold for 1p/2pMINFLUX) originated from another molecule because (1) other molecules are in offstate; and (2) if an offstate molecule is turned on unwantedly, the molecule would be saturated because fluorescence saturation would occur at \(r >>\) 25 nm, and this particular data should be discarded. However, because of the larger radius compared to Gaussian PSFs in CLSM, 1pMINFLUX necessitates higher requirement of singlemolecule state; and 2pMINFLUX may necessitate even higher requirement than 1pMINFLUX because of its larger radius than 1pdonut as well.
4 Conclusion
In summary, MINFLUX can be enhanced with multiphoton excitation process, with 2fold increase of localization precision or 4fold decrease of required fluorescence photons compared to 1pMINFLUX. As different dyes can be excited simultaneously with twophoton excitation, 2pMINFLUX may have the potential for registrationfree multicolor localizations. This may be of use not only for nanometerprecision localization microscopy [11, 12], but also simultaneous tracking [10] of several fluorophores. This could be crucial for study of molecular interactions, for example, proteinprotein interactions, proteinnucleic acids interactions, or viruscell interactions.
Availability of data and materials
Data underlying the results presented in this paper are calculated or simulated with Code 1, Ref. [25].
Abbreviations
 CLSM:

Confocal laser scanning microscopy
 CRB:

CramÃ©rRao bound
 fwhm :

Full width at half maximum
 1pMINFLUX:

Singlephoton MINFLUX
 2pMINFLUX:

Twophoton MINFLUX
 MLE:

Maximumlikelihood estimation
 LMS:

Least mean square estimation
 NA:

Numerical aperture
 PSF:

Point spread function
 SBR:

Signaltobackground ratio
 SMLM:

Singlemolecule localization microscopy
 STED:

Stimulated emission depletion
 TCP:

Targeted coordinate pattern
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Funding
This work was supported by the Beijing Natural Science Foundation (JQ18019), the National Natural Science Foundation of China (62025501, 31971376, 61729501), the State Key Research Development Program of China (2017YFC0110202), and Shenzhen Science and Technology Program (KQTD20170810110913065).
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KZ and PX conceived the project. PX and DJ supervised the research. KZ, XX and WR performed the analysis and simulation, prepared the figures and wrote the manuscript. All authors participated in discussion and editing of the manuscript. All authors read and approved the final manuscript.
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Supplementary Information
Additional file 1: Figure S1
. CRB anisotropy for 2pMINFLUX. Figure S2. Comparison of CRB with L = 100 nm. Figure S3. Power dependencies of single molecule emission rates for EGFP (a) and TMR (b). Figure S4. Progression of spatially averaged CRB of iteration 2pMINFLUX and 1pMINFLUX in 2D xyplane. Figure S5. Excitation power needed for 2pMINFLUX. Table S1. Single molecule emission rates for TMR and EGFP for singlephoton and twophoton excitation of Figure S3. [2]. Table S2. Relationship of MINFLUX fluorescence rate Î·_{minflux} to Î·_{lin} and Î·_{sat}. Table S3. CRBs in each round of iteration MINFLUX in xyplane. Table S4. Parameters used in calculation of excitation power needed for 2pMINFLUX.
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Zhao, K., Xu, X., Ren, W. et al. Twophoton MINFLUX with doubled localization precision. eLight 2, 5 (2022). https://doi.org/10.1186/s4359302100011x
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DOI: https://doi.org/10.1186/s4359302100011x
Keywords
 Superresolution microscopy
 Singlemolecule localization
 MINFLUX
 Twophoton microscopy