Fluorescence signal and resolution

2.2. Fluorescence signal and resolution

A STED resolution increase leads to a lower fluorescence signal because of higher photobleaching. To derive the link between STED resolution and the maximum fluorescence signal, the following assumptions apply.

The pixel size is much smaller compared to the size of the excitation and OFF-switching intensity distributions in the focus. In the middle area of the linear scan (steady-state conditions), each pixel experiences the very same total light dose that drives photobleaching (excitation and OFF-switching illumination). Before a spe-cific pixel is readout, half of the total bleaching dose has already been applied to the pixel during the imaging of earlier pixels. The same light dose is applied to the pixel after readout (Figure 2.1 (A)).

Assuming an exponential loss of the fluorescence level due to photobleaching

(Fig-Figure 2.1.: Bleaching of the fluorescence level in STED imaging. (A)At steady-state conditions, in a nanoscope with point-scanning illumination, half of the bleaching light dose is applied at a certain pixel before the readout is performed.

D_F is the dosage after the scan, andD_A= 0.5·D_F is the dosage a pixel experiences before and after readout. (B) The fluorescence level during the readout is FA and the level after the linear scan is completed is F_F. Both levels are defined by an exponential photobleaching. Figure reproduced from ref. [2].

ure 2.1 (B)), the remaining fluorescence levelsFAandFF are present during readout and after both illumination foci have passed, respectively. The fluorescence signal at a pixel is the product of the dwell time and the respective fluorescence levelF_Ain

the pixel, whereby F_A decreases non-linearly with the exponential photobleaching course. At a low bleaching light dose (e.g. short dwell times), the fluorescence signal increases linearly with the dwell time. However, also the bleaching light dose increases linearly with the dwell time, whereby the fluorescence level decreases due to photobleaching. At a certain (best) dwell time the fluorescence signal is at a maximum. A longer dwell time decreases the fluorescence level non-linearly which is not compensated by the longer linear integration of fluorescence photons, thus the fluorescence signal starts to fall.

To derive the best dwell time, the following considerations apply: a fluorophore undergoes a certain amount of excitation and de-excitation events (cycles) while it is scanned. During each cycle, a finite probability exists that the fluorophore will transitions into a dark-state without recovery (photobleaching). For simplicity, bleaching is primarily introduced linearely with the STED illumination, as shown for the red dye ATTO 647N (ATTO-TEC, Germany) [1]. The density function of the fluorophores present in the sample in respect to the photobleaching light dose is an exponential distribution for many fluorophores (continuous limit). The fluorescence signal Sf l that can be collected from a sample scales linearly with the start fluorescence level F₀, which is the initial number of fluorophores times the amount of fluorescence photons they emit. S_{f l} has a maximumS_{f l,max} at a certain dwell time. As mentioned before, this maximum is at the dwell time where a longer linear integration time cannot outperform the lower fluorescence level caused by the non-linear photobleaching which results from the longer illumination. To find the best dwell time (giving the maximum fluorescence signal), the number of photons leading to a photobleaching event ^PN_ph at a certain pixel before readout happens needs to be calculated:

XNph=timg·I·σ· 1 E_ph ·npx

2 , (2.1)

where timg is the dwell time, σ is a bleaching cross section per STED photon, E_ph is the STED photon energy, and n_px = A/a² is the number of pixels where photobleaching is driven. With a as the pixelsize, and A as the area of the focus where bleaching occurs.

The fluorescence signal is the product of the dwell time, a fluorescence photon-collecting efficiencyη, and the start fluorescence level which decreases exponentially with the photobleaching events:

S_{f l} =t_img·ηF₀·e^−C·PNph =t_img·ηF₀·e

−C·t_img·I

a² (2.2)

2.2. Fluorescence signal and resolution where the constant terms influencing the photobleaching have been combined into the constant factorC. Keeping the correct sampling,ais inverse proportional to the increase of resolution (between confocal and STED)ρ=d_conf/dx,y min=^p1 +I/Is, concrete a=d_conf/(ρ·ν). Whereby d_conf and d_{x,y min} are the confocal and STED resolution (defined as the FWHM), respectively. ν is an oversampling factor also known as Nyquist criterion. Using these expressions, the equation 2.2 can be written as:

S_{f l} =timg·ηF₀·e^{−C·timg·ρ2(ρ2−1)·ν2} (2.3) The constant terms again were moved into C for enhanced clarity. The maximum of equation 2.3 is at

timg= 1/(C·(ρ⁴−ρ²)·ν²). (2.4) This best dwell time differs for each dye and resolution (level of illumination), because C and ρ is changing. The exponent of the exponential function is mainly influenced by the ρ⁴ for high resolution enhancements. The maximum fluorescence signal S_{f l,max} which can be extracted is

Sf l,max= ηF₀

e·C·(ρ⁴−ρ²)·ν². (2.5) The equation 2.5 shows a 1/ρ⁴ dependency on the maximum fluorescence signal, since at higher resolutions more pixels have to be scanned with a higher STED illumination. Both, the STED intensity and pixel number, increase quadratically with the resolution enhancement. As a consequence, the signal drops rapidly when approaching very high resolutions. Please note that equation 2.5 does not take into account the bleaching by the excitation, that is why Sf l,max(ρ= 0) =∞.

In order to optimize the signal, ν should be kept as low as possible above the Nyquist criterion, since it has a quadratic dependence on the bleaching light dose.

In a typical scan (square pixels) the critical direction for the Nyquist frequency is the diagonal direction of the pattern. The diagonal extent of a pixel needs to be smaller than half the target resolution (FWHM) d_x,ymin/2. Assuming a STED resolution of 30 nm, the pixel size to fulfill the Nyquist criterion along the diagonal axis is a < d_{x,y min}/(2√

2)≈10.6 nm.

As can be seen in equation 2.5, a doubling of the resolution results in an approximately 16-times lower maximum signal. Keeping ρ and ν fixed, possibilities to increaseS_{f l,max}are: a higher detection efficiency (increasingη), a higher labeling

density and thus fluorescence level F₀, or a lowering of the constant parameters defining C.

One way to lower C is to reduce the bleaching cross section of the specific dye.

Another possibility is the promising approach to decrease the number of photons PNph leading to photobleaching events. Reducing ^PNph is the rationale of two different scan schemata presented in chapter 4 and 5. For instance, a photobleaching reduction of a factor κ will allow a higher S_{f l,max} by the same factor, if the dwell time is increased byκ. According to equation 2.5, the higher fluorescence signal can be converted into an approximately√⁴

κhigher resolution with the same fluorescence signal as before. As an example, a ten times higher maximum signal is convertible into a ∼45% higher resolution (i.e. 40 nm⇒22 nm), see figure 2.2.

Figure 2.2.: Resolution improvement by a 10-fold bleaching reduction.

Shown are plots of equation 2.5 for final resolutions below 50 nm with a different amount of photobleaching. The blue line represents the maximum signal for regular bleaching (e.g. C = 1), and is nominated to a resolution of 50 nm. The orange line shows the maximum signal with ten times lower bleaching (e.g. C= 1/10) compared to the blue line. The relative resolution improvement between both is ∼ 45% for any starting resolution.