The SPLIT method

Modulating the sample illumination and measuring the temporal dynamics of the fluorescence can alleviate the stringent condition of silencing all the fluorophores located on a specific portion of the DL-PSF. The N photons observed at each pixel can still originate from fluorophores located at any position within the DL-PSF but they could be emitted with different temporal dynamics according to the position of the generating fluorophore in the DL-PSF. The maximum achievable spatial resolution is ultimately determined by the ability to distinguish between different temporal dynamics. The key point here is that the issue of resolving spatial features is translated into the spectroscopy problem of resolving temporal dynamics components. The scheme of the method that we call SPLIT (Separation of Photons by LIfetime Tuning) is depicted in Fig. 1a. Suppose that within the DL-PSF of the microscope, we can distinguish two spatial components 1 and 2 characterized by different temporal dynamics. We make use of the phasor analysis of lifetime data31,32,33 to represent the two different temporal dynamics as two vectors in the phasor plot. The total number of photons detected at one pixel is the sum of the photons originating in the two spatial components plus the uncorrelated background (BKGD) N=N 1 +N 2 +N BKGD , where only N 1 represents the ‘wanted’ part of all the photons. Following the rules of phasors, the vector P=(g,s) associated with the intensity decay at one pixel can be expressed as the linear combination of the vectors P 1 =(g 1 ,s 1 ) and P 2 =(g 2 ,s 2 ) associated with the two components, with weights f 1 and f 2 given by the corresponding fractions of detected photons: P=(N 1 P 1 +N 2 P 2 )/N=f 1 P 1 +f 2 P 2 . This is a linear system of equations in the unknowns f 1 and f 2 . We can write this system in the form P=M f, where f=(f 1 ,f 2 ) is the vector of the fractional components and M ij is the matrix

Figure 1: Schematic principle of the SPLIT method. (a) It is assumed that the photons are emitted within the DL-PSF with a different dynamics (1 or 2) according to the emitter position. The goal is to separate the photons emitted from 1, those emitted from 2 and those with no temporal dynamics (uncorrelated background, BKGD). This is obtained in the phasor plot expressing the experimental phasor P as a linear combination of the phasors P 1 and P 2 plus the phasor of the background (P BKGD ). (b) Schematic of the image formation process in SPLIT. The SPLIT method uses the temporal information of the signal at each pixel to generate a set of g and s images. These images are then processed to obtain the final SPLIT image. Full size image

which describes the two different temporal dynamics in the phasor domain. For a given n × n matrix M, the solution of this system is given by f=M−1P. Once we find f the images N i (x,y) (i=1,…,n) of the photons emitted in each of the n subdiffraction volumes and the image N BKGD (x,y) of the background can be calculated as N i (x,y)=f i (x,y)N(x,y) and . As a result the original image N(x,y) has been split into n+1 images based on the assumption that we can observe and distinguish, within our observation volume, n different dynamics, associated with n linearly independent vectors in the n-dimensional phasor space. These dynamics, using the RESOLFT concept6, are seen as generalized reversible states of an ensemble of molecules, as they do not correspond necessarily to a specific state of the molecule but rather to a temporal fingerprint. The overall dynamics observed in the DL-PSF is described here using the linear combination properties of phasors: if we assume that there are only two components, the phasor will fall on the line connecting the phasors from pure components (P 1 and P 2 ). If we take into account the uncorrelated background as a third component, the phasor P will fall in a triangle where the vertices are the phasors P 1 and P 2 and the phasor P BKGD of the uncorrelated background (Fig. 1a). To separate more than two dynamics components (n>2) and the uncorrelated background, we extend the analysis to phasors obtained at multiple harmonic frequencies34,35. The image formation process in SPLIT is depicted schematically in Fig. 1b. Shortly, the temporal information of the signal at each pixel is used to generate the g and s images. These images are then processed to obtain the final SPLIT images.

The SPLIT method in time-resolved CW-STED

We focus now on the specific case of SE-induced lifetime variations and on the CW-STED microscopy architecture, that is, a Gaussian excitation beam and a doughnut-shaped STED beam (Fig. 2a). However, the proposed approaches can be easily adapted to other configurations and other state transitions. The first ingredient is a model to describe the n dynamics components into which to split the measured intensity pixel-by-pixel, namely, we need the matrix M. For simplicity, we assume (i) a Gaussian profile of the conventional DL-PSF h(x′,y′,z′)=exp(−2r2/w2)exp(−2z′2/w z 2), with w and w z being the beam waists along the radial and axial direction, respectively; and r2=x′2+y′2 the radial distance from the focal point (x=0, y=0) (ii) a parabolic approximation for the doughnut-shaped STED beam I STED (r)=I STED (w)r2/w2, with I STED (w) the STED beam intensity at position r=w; (iii) a single exponential decay rate for the unperturbed fluorophores γ 0 =1/τ 0 , where τ 0 is the unperturbed excited-state lifetime. Under these assumptions the spatial distribution of the decay rate is approximated by a parabolic function γ(r2)=γ 0 +γ 0 k S r2/w2, where k S =I STED (w)/I SAT is the ratio between I STED (w) and the saturation value I SAT for which the probability of decay due to SE and spontaneous emission are equal (see Supplementary Note 1). Importantly, the value of k S determines the relative variation of decay rate values within the E-PSF of the CW-STED microscope (Fig. 2a).

Figure 2: The SPLIT method in time-resolved CW-STED. (a) A doughnut-shaped STED beam overlapped with a confocal spot generates a continuous distribution of dynamics within the DL-PSF. The STED beam intensity determines the relative variation of decay rate γ/γ 0 (solid green (k S =1) and red (k S =10) line) within a Gaussian DL-PSF (solid black line) or the corresponding E-PSF (dashed green (k S =1) and red (k S =10) line). (b,c) Simulated average time-resolved confocal and STED images of two point-like particles plus a uniform level of uncorrelated background (confocal FWHM=200 nm, particles distance=104 nm, k S =10, τ 0 =2.5 ns, S=105, B=104) and horizontal profile. In the time-gated STED image (T g =τ 0 ) the signal becomes very low compared with the background level. In the SPLIT series, the photons of the super-resolved component 1 are efficiently separated from component 2 and from the background. The colourmap represents the simulated intensity normalized to the maximum value of the confocal image. Scale bar, 100 nm. (d) Resolution and noise propagation in the SPLIT method versus the number of components. Resolution and noise are quantified, respectively, as the FWHM of the SPLIT E-PSF and the condition number k cond obtained for k S =10 (FWHM of the STED E-PSF is shown for comparison as the first point). Full size image

The time-dependent fluorescence signal F(x,y,t) at each pixel can be obtained by integrating the contribution of all the fluorophores located in the E-PSF centred in the pixel position (x,y) (see Supplementary Note 1)

where C(r2) describes the concentration of the fluorophores in a concentric region of radius r around the pixel position and K is a constant that depends on the quantum yield of the fluorophore, the maximum of the excitation intensity and the detection efficiency. In contrast to deconvolution methods, the proposed method does not try to reassign photons to the original position, thereby the position at which the fluorophores are located within each concentric cylinder is not important. In other words, this approach does not need prior knowledge of the E-PSF of the CW-STED system, which makes this approach suitable also for non-expert users. The temporal dynamics of F(x,y,t) encodes nanoscale spatial information in the distribution of exponential decay components. We split the integral and calculate n dynamics components defined uniquely by the parameters γ 0 and k S (see Methods), from which the decoding matrix M is derived.

We tested the proposed method on synthetic time-resolved CW-STED images obtained with known γ 0 and k S . Figure 2b shows the ability of the SPLIT method in separating the photons coming from the inner subdiffraction volume from those of the periphery and the uncorrelated background, whereas time gating is affected by an increasing fraction of background (Fig. 2b,c). Notably, the spatial features appearing on the background image are due to the approximation of the continuous distribution of dynamics to only two components (see Supplementary Fig. 1). The spatial resolution of the SPLIT image can be further increased using a higher number n of components (Fig. 2d and Supplementary Fig. 2a). In gated CW-STED microscopy this is done by increasing the time-delay T g (Supplementary Fig. 2b). For instance, using n=4 it is possible to get the same spatial resolution of CW-STED but at a STED beam intensity, which is 1 order of magnitude lower (Supplementary Fig. 2c). Note that the separation of dynamics obtained in the SPLIT method is conceptually different from time gating. The separation in SPLIT is based on the analysis of variations of the signal over the whole time range (Fig. 1b). For this reason, a SPLIT image could be obtained at increasing values of n even when T g is limited by the period T (the reciprocal of the repetition rate, typically in the order of 107 Hz) (Supplementary Fig. 2d).

The SPLIT image exploits the additional spatial information potentially encoded in the g(x,y) and s(x,y) images (see Supplementary Note 2). This additional amount of spatial information is available on a STED image but not on a confocal image (Supplementary Fig. 3). The improvement in spatial resolution in a SPLIT image at increasing values of n comes from the analysis of increasingly higher temporal frequencies in the signal (see Supplementary Note 2). Thus, it comes from a better sorting of photons as a function of dynamics/locations. However, when noise is taken into account, the larger the number n of components the higher will be the noise propagated to the final images, quantified as the condition number k cond of the matrix M to invert (Fig. 2d and Supplementary Fig. 4; see Supplementary Note 3). For a given level of depletion and for a given level of noise, there is a finite number of values of n for which the noise in the final image is below a desired threshold.

Experimental determination of unknown decoding parameters

To decode the spatial information hidden in the gradients of dynamics induced by the STED beam, we need to know, according to our model, only the two parameters γ 0 and k S. The parameter τ 0 =1/γ 0 is usually known for a specific fluorophore or can be easily measured from the sample with the very same instrumentation by setting the STED beam power to zero. The parameter k S =I STED (w)/I SAT is proportional to the STED beam power but its precise value depends on the optical configuration and on the properties of the sample. It is interesting that, using our analytical model of the SE-induced lifetime variations, we are able to estimate the value of k S from the same image F(x,y,t) by considering the average time-resolved decay of all the pixels of an image (see Supplementary Note 1),

where B denotes the uncorrelated background. To validate the model, we imaged 40 nm fluorescent beads at several STED beam powers (Fig. 3a). The two-dimensional (2D) histogram of the values g(x,y) and s(x,y) associated with each pixel is represented in the phasor plot (Fig. 3b). The phasor of the confocal image (zero STED power) is centred to the position corresponding to a single exponential decay with τ 0 =4.5 ns. The same value τ 0 =1/γ 0 is found by fitting the average photon-arrival time histogram to equation (3) with k S =0 (Fig. 3c). With the increasing of the STED beam power the phasor becomes elongated as different dynamics are sampled in the image. The precise value of k S at each STED power can be determined by fitting the average photon-arrival time histogram to equation (3) with τ 0 fixed (τ 0 =4.5 ns). The good agreement with the model is confirmed by the linearity between k S and the STED beam power. The phasor associated with the theoretical decay expressed by equation (3) for τ 0 =4.5 ns and increasing the value of k S describes the expected trajectory of the average phasor of the image as a function of the STED power. To assess the validity of the method for the imaging of non-point-like structures, we also performed simulations using more convoluted structures similar to those found in cytoskeletal networks (see Supplementary Fig. 5). Also in this case, by using the values of k S obtained by fitting the average time-resolved STED decay of the image, we were able to separate the images of the super-resolved components and the background.

Figure 3: STED phasors and average dynamics at different STED powers. (a) Time-resolved STED images of 40 nm yellow–green fluorescent beads at several STED beam powers. Numbers indicate STED beam power in mW (measured at the back aperture of the objective lens). The colourmap in a represents the time-integrated intensity detected at one pixel normalized to the maximum value of each image (threshold set to 20% of the maximum value). (b,c) Phasor plots (b) and average time-resolved decays (c) associated to the images in a. Increasing STED powers induce an increasing spread of the phasor and an increasing stretching of the average decay from an exponential (k S =0, τ 0 =4.5 ns) into the trend described by equation (3) with k S >0. This equation describes a trajectory in the phasor plot for increasing values of k S (solid line), which overlaps with experimental phasor. The values of k S obtained from the fit of the average decay scale linearly with the STED beam power. Scale bar, 1 μm. Full size image

SPLIT imaging of subcellular structures

We finally applied the SPLIT method to the imaging of biological structures, namely microtubules on fixed HeLa cells, as reported in Fig. 4. We compare results obtained by labelling tubulin with two different dyes, Alexa Fluor 488 (Fig. 4a) and Oregon Green 488 (Fig. 4b). The parameters γ 0 and k S are found from the same experimental time-resolved data sets by fitting the average decay of all the pixels (Supplementary Fig. 6). According to the model and the simulations, the expected full-width at half-maximum (FWHM) of the SPLIT image (n=2) is of the order of 100 nm, as confirmed by the experimental intensity profiles (Fig. 4 and Supplementary Fig. 7). The SPLIT image is compared with the confocal and the gated CW-STED image (gating time is set to T g =1 ns), showing the improvement in spatial resolution with simultaneous efficient removal of background. Under the same experimental conditions (λ STED =560 nm) the Oregon Green 488 fluorophores exhibit more uncorrelated background due to STED beam-induced excitation, thereby in the relative gated STED image, the improvement in resolution is totally masked by the strong background induced by direct STED beam excitation. Even though specific methods for background subtraction27,28,29 have been developed recently, it is remarkable how the uncorrelated background photons are automatically separated in the calculation of the SPLIT image (Fig. 4c). This is possible because we are operating a separation of the signal in the frequency domain (where the uncorrelated background is well-separated from the other components) rather than in the time domain (where the uncorrelated photons are evenly distributed). Here in addition to the different temporal dynamics of the excitation and STED beams, sufficient to remove the uncorrelated background, we are also decoding the spatial frequencies hidden in the gradient of dynamics induced by the STED beam.