Confocal-based fluorescence fluctuation spectroscopy with a SPAD array detector

The combination of confocal laser-scanning microscopy (CLSM) and fluorescence fluctuation spectroscopy (FFS) is a powerful tool in studying fast, sub-resolution biomolecular processes in living cells. A detector array can further enhance CLSM-based FFS techniques, as it allows the simultaneous acquisition of several samples–essentially images—of the CLSM detection volume. However, the detector arrays that have previously been proposed for this purpose require tedious data corrections and preclude the combination of FFS with single-photon techniques, such as fluorescence lifetime imaging. Here, we solve these limitations by integrating a novel single-photon-avalanche-diode (SPAD) array detector in a CLSM system. We validate this new implementation on a series of FFS analyses: spot-variation fluorescence correlation spectroscopy, pair-correlation function analysis, and image-derived mean squared displacement analysis. We predict that the unique combination of spatial and temporal information provided by our detector will make the proposed architecture the method of choice for CLSM-based FFS.


Supplementary Note 1: Theory
Let 0 and 1 be the 1/ 2 radii of two Gaussian detection volumes (often the detection volume is called point-spread-function, PSF, in the context of imaging, or collection-efficiency function, CEF, in the context of FCS) and 1 and 2 their respective 1/ 2 heights. Let the spatial shifts between the centers of the two focii be given by the vector ì . Assuming free diffusion with diffusion coefficient D and fluorophore concentration , the pair-correlation between the fluorescence signals 1 ( ) and 2 ( ) detected by the two detectors having the respective properties described above, can be written as .
Eq. 1 is the general correlation formula from which specific cases can be derived. For conventional FCS, or spot-variation FCS, only a single detector is assumed. Substituting ì = 0, 1 = 0 , and 1 = 0 , Eq. 1 simplifies to the conventional FCS formula: For fluorescence cross-correlation spectroscopy and two-focus FCS, Eq. 1 can be simplified assuming is the effective squared beam height, = 0 0 = 1 1 is the shape parameter of the PSFs (which are assumed to be equal), and is the distance between the centers of the two PSFs.
For the STICS approach with iMSD analysis, we assumed 0 to be equal for all pixels. The exponential factor in Eq. 3 can be rewritten as: Fitting the correlations with a 2D Gaussian function, = exp − 2 /2 2 , with 2 the variance results in the following relationship: The diffusion coefficient can thus be extracted from the slope of the 2 ( ) curve without needing information on the size of the PSF.

Supplementary Note 2: SPAD array detector properties
As SPAD technology is constantly improving, also the specifications of SPAD array detectors are continuously getting better. This section provides an overview of the key properties of the HVCMOS SPAD array detector used in this work. More details about the SPAD array detector used in this work can be found in 4 .

Physical dimensions
A sketch of the layout of the detector used in this work is shown in Figure S1. With a pixel size s , and pixel pitch p of 50 and 75 , respectively, the detector has a fill factor = 2 s / 2 p of about 44%. However, it is important to note that in this application the photons are not uniformly spread across the sensitive area of the detector, but instead follow a distribution that depends on the PSF of the system (Fig. 3(a)). In particular, photons are concentrated in the central area, thus the effective fill-factor, i.e., the number of photons effectively reaching the sensitive area of the detector, is higher than 44%. In future updates of this detector, the effective fill-factor will be further improved by introducing a set of micro-lenses in front of the SPAD array.

Timing properties
Each pixel of the array operates as an independent SPAD. Upon detection of a photon, a logical high voltage pulse is almost instantaneously generated in that pixel. The photon-time jitter of the pulse is ∼ 150 ps. The hold-off can be chosen by the operator and can range from 25 to 500 ns. During this hold-off time, no other photons can be detected by this pixel. Each pixel can be connected to a time-tagging platform for time-correlating single-photon counting experiments. is located near the detector edge (i.e., first row, second column) and therefore has only a minor influence on sum5x5 and no influence at all on sum3x3 and 12 . For this reason, we excluded the data collected by this pixel for the spot-variation measurements. It is important to note that for the CMOS SPAD the hot-pixel problem can be removed by a careful selection of the device.

Dark photon count rate
Alternatively, an active cooling system can be implemented, which, for silicon SPADs, guarantees approximately a decade in DCR decrease every 20 K of temperature reduction [41]. Figure S2 (b) shows the auto-correlation curves calculated for several pixels without illumination.

Afterpulsing
Pixels are numbered starting from 0 for the pixel in the upper left corner up to 24 for the pixel in the lower right corner. Except for pixel 1, all auto-correlation curves follow the same pattern, starting at a high value at short lag times and decreasing to 0 near 10 . This behavior is caused by detector afterpulsing, and, clearly, has to be taken into account in FCS measurements, as it will constitute a background in the auto-correlation curve.
There are various ways to correct for detector afterpulsing. The most straightforward method is cropping the FCS auto-correlation curve, thereby removing the spurious data points at short lag times. This strategy is justified in many biological applications, as the diffusion in biological samples is often relatively slow. However, when short lag time information is required, a different approach is needed. One method consists in characterizing the after-pulse component, e.g. by calculating the auto-correlation function (ACF) of the detector signal under constant white light illumination and fitting the data with a power law, as shown in Figure S2 (c). The same power law factor can then be added to the FCS fit model 5 . A second method is calculating FCS cross-correlations between the signal coming from different pixels. In this case, the afterpulsing effect can be completely filtered out, as the afterpulsing signals do not correlate, see Figure S2

Crosstalk
When a photon hits a pixel of the SPAD array detector and triggers an avalanche, the carriers flowing inside the diode may cause the emission of secondary photons 4 . These secondary photons can be detected by neighbouring pixels. This phenomenon is called optical crosstalk and degrades the signal-to-noise ratio (SNR). However, for the SPAD array detector used here, the crosstalk probability is less than 1.5% for first orthogonal neighbours and less than 0.2 % for first diagonal neighbours. In addition, the time difference between the detection of a genuine fluorescence photon and the detection of a crosstalk photon is less than 1 , i.e., much shorter than the 500 bin time of the FCS measurements. As a result, the cross-correlation curves of Figure S2

Supplementary Note 3: Validation of the setup
We validated our data acquisition and analysis platform by performing FCS on the Alexa 488 sample.
The SPAD array detector was replaced with a single-element SPAD (Micro Photon Devices, Bolzano, Italy), which was simultaneously connected to one of the 25 channels of our data acquisition platform and a Becker and Hickl time tagging card (SPC-830, Becker and Hickl GmbH, Berlin, Germany). In Figure S4    (shown in red). The 2 values for the normalized curves are 0.14 for the SPAD-array-detector and 0.087 for the single-element SPAD.

Supplementary Note 4: Lateral shifts in the field-of-view in pair-correlation analysis
In real space the distance between two neighbouring pixels is 75 . The magnification of the system is 500, so in the sample plane the pixel-pixel distance is 150 . Given that the overall detection volume (or system PSF) is the product of the excitation and the emission PSFs, and assuming a reassignment factor of 1/2 ref. 7 , the shifts in the field-of-view of two neighbouring pixels is 75 . ( 2 0 ) is a linear function. For free diffusion, the curve passes through the origin, and the diffusion coefficient can be derived from the slope. For anomalous diffusion, the intercept value (0) will be positive for diffusion in microdomains and negative for diffusion hindered by a meshwork 1 .
· The most commonly used fit model for G assumes a Gaussian PSF. This is a good approximation for small pinhole sizes. However, if the fieldof-view of the detector is more than 1 A.U., the PSF for pixels far away from the central pixel will significantly deviate from a Gaussian function 2 . Instead one can use a more complex numerical model, see e.g. 3 . Alternatively, the zoom of the optical system can be increased to reduce the field-of-view of the array detector to not more than 1 A.U. · The afterpulsing effect is still present when summing the signal from different pixels. This can be filtered out by cropping G, using lifetime information, or by calibrating the afterpulsing component.

Pair-correlation FCS / Twofocus FCS
The intensity trace from the central pixel is taken as a reference, and all 24 cross-correlations between the central pixel and the other pixels are calculated. All correlation curves that correspond to the same inter-pixel distance are averaged, resulting in 5 final cross-correlation curves. These can be fitted simultaneously, i.e., in a global fit, or separately, with the two-focus FCS model.
The diffusion coefficient can be extracted from the fits. The absence of afterpulsing in the cross-correlations makes this method suitable for measuring fast diffusion. The well-known distance between the foci yields a higher precision compared to conventional fluorescence crosscorrelation spectrosocpy.
· By averaging over all pixel-pairs with the same inter-pixel distance, any anisotropy in G remains undetected. E.g. the cross-correlations 12,11 and 12,13 are different when there is active transport in the horizontal direction. The loss of directionality hinders the study of anomalous diffusion. · The Gaussian model that is assumed is not valid for large inter-pixel distances, leading to poor fit results. · The information from only 24 crosscorrelations is used. This number can be doubled by calculating the cross-correlations in both directions, e.g. both 12,13 and 13,12 . However, this is still a small fraction of the total amount of information (25x25 = 625 correlations) contained in the data. spatio-temporal image correlation spectroscopy with mean squared displacement analysis Each 5x5 frame is treated like an image and the spatio-temporal correlations are calculated in similar way as in STICS: ( , , ) contains the correlation value averaged in space and time over all pairs ( ( , , ), ( − , − , − )). Note that ( , , ) is not necessarily equal to (− , − , ). The 9x9 pixel correlation images ( , ) are fitted with a 2D Gaussian function and the variance 2 ( ) and the peak location ( 0 ( ), 0 ( )) are analyzed.
Assuming equally sized Gaussian PSFs for all pixels, there is a linear relationship between 2 and : 2 = 2 + 2 0 /2. The diffusion coefficient can thus be derived from the slope of 2 without needing a-priori knowledge on the beam waist. In addition, if different values are found for 2 in the and direction, this indicates hindered Brownian motion (Gratton). The speed and direction of active transport (e.g. superimposed on Brownian motion) can be derived from the peak location. Knowing the geometry of the detector and the magnification of the system, this analysis method is completely calibration-free.
· All PSFs are approximated by a Gaussian function with the same width 0 . This approximation is less accurate for pixels near the edge of the detector, see Spot-variation FCS. This approximation is worse in STICS analysis than in pair-correlation analysis, since STICS also analyses pair-correlations between pixels located on opposite sides of the detector, thus with higher inter-pixel distances. · (0, 0, ) contains the afterpulsing component for small values. This effect can be filtered out by using a weighted fit with low (or zero) weights for (0, 0, < ), with the maximum lag time between a photon signal and the afterpulsing signal.