and returns from ξ = ξmax to 1. The scanner repeats the same procedure
every Ty = 1/fz seconds. For each pixel, the intensity time traces Fξ(ti) at
time ti (i = 0,1,2,...,2n − 1) are used to calculate the temporal autocor-
relation curve (where n denotes the number of scans):()()GτδF tδFtτFtFtτ()=() +() +ξξξi ξiξξiξi ξwhere the angled brackets denote the average over a long time period
at each pixel position ξ. The fluorescence of fluctuation is
δFtξi()=(Ftξi)−〈〉Ftξi() and the lag-time is τξ.
The calculated autocorrelation function G(τ) was fitted similarly as
the conventional fixed point FCS with a 3D one-component model:
Gτ γ
NDτ
wDτ
w()=1+4
1+4(^2) xy z
−1
2
− 21
where γ is a geometric factor dependent on the shape of the focal vol-
ume (0.35 for 3D Gaussian), N is the number of molecules in the focal
volume, as calculated by NC= ww^2 xyz()π 2
3/ 2
(where C denotes the
average concentration of molecules in the PSF), D is a diffusion coef-
ficient in μm^2 s−1, wxy and wz are the radial and axial waist of the PSF. We
usually calculate the confocal volume as Vconf=()π 2 wwxy z
3/ (^22)
, enabling
us to interpret the factors wxy^2 wπz 3/^2 as effective volume Veff, where the
particles are actually detected for sources of fluorescence.
However, the original sFCS itself was based on the slow Galvo scan-
ner, which limited its applications to slowly diffusing molecules. We
therefore used a Galvo-resonant scanner for high-speed scanning.
Furthermore, slowly fluctuating noise was problematic during meas-
urement of living cells due to the movement of subcellular structures.
Such noise was statistically removed by the wavelet-based method
described in the following section.
Noise removal and data correction for FCS analysis
For correction of photobleaching effects, we averaged the intensity
time trace over all pixels Ft()iξ=(〈〉Ftiξ) and approximated it as an expo-
nential decay curve Ft()ii~(ft)=fe 0 ()−
ti
tb. When we assume that the
intensity Fξ(ti) obeys a Poisson distribution with mean μi = f(ti) and
variance σii^2 =(ft), we can correct the standard deviation of the intensity
and photobleach as Ftξc()iξ=(fF0)+(()tμi−)iσσ 0 /i, and we have the cor-
rected intensity Ftξc()i as in ref.^33 :
Ft
Ft
ftf
f
ft
()= f
()
()/(0)
+(0)1−
()
ξ (0)
c
i
ξi
i
i
For further correction, we applied wavelet-based smoothing^34 ,^35 to
subtract the inhomogeneous intensity fluctuation FtξW()i which is cal-
culated as follows. We applied a wavelet decomposition of the intensity
signal Ftξc()i which can be achieved by using a scale function φ and a
wavelet function ψ in multiresolution analysis. In this paper, we used
the Haar scaling and wavelet functions as
φt
t
()=
1,(0≤<1)
0,otherwise
ψt
t
()= t
1, (0≤<)
−1,(≤<1)
0, otherwise
1
2
1
2
If we represent the discrete intensity signal Ftξc()i={yy 01 ,,...,yn−1} with
the length n = 2 J (J > 0), the discrete wavelet transform gives the vector
of wavelet coefficients with the length n. The coarse approximation of
the signal is represented by a linear combination of the shifted scale
functions φj,k (t) = 2 j/2φ (2 jt – k), where j and k are the scale and location
of the scale function, respectively. The weights for this function are
scale coefficients cj,k. The residual details of the signal which are omit-
ted from the coarse approximation can be expressed by a linear com-
bination of the shifted wavelet functions ψj,k (t) = 2 j/2ψ (2 jt – k) with the
weights dj,k. If the coarse approximation level L < J is given, the signal
is represented by the wavelet coefficients with 2L scale and 2 j detail
coefficients, {ccLL,0,,,1...,cL,2L−1} and {ddjj,0,,,1...,dj,2j−1} (=jL,...,−J 1),
respectively.
In general, wavelet methods have been known to be advantageous
for statistical analyses including the removal of noise when the signal
is inhomogeneous in time. For example, when a series of inhomogene-
ous signal includes a long-term variation and noise, the variation can
be captured by large wavelet coefficients and the noise by small coef-
ficients. The successful removal of noise can be achieved by appropri-
ate threshold selection. However, in FCS measurements for living cells,
a long variation is usually caused by cell or organelle movement and
by noise arising from intensity fluctuation caused by molecular diffu-
sion within a confocal volume. Therefore, we use the term ‘fluctuation
signal’ for such ‘noise’. To derive the fluctuation signals of molecular
diffusion, we first applied discrete wavelet transform of the original
signal Ftξc()i={yy 01 ,,...,yn−1}, and removed noise (fluctuation signals)
by setting the detail coefficients dj,k to δSλj()djk, or δHλj()djk, , where δλSH()j
are the soft or hard thresholding functions, respectively,
δx
xλ xλ
xλ
xλ xλ
()=
−,(>)
0, (||≤ )
+,(<−)
λ
S
jj
j
jj
j
δx()= xx,(||>)λ
λ 0, otherwise
H j
j
Because the intensity signals obey Poisson distribution, the values of
λj are level-dependent thresholds for Poisson noise, and the translation-
invariant Poisson smoothing using Haar wavelets (TIPSH) algorithm
was used^35.
Second, we applied inverse wavelet transform of the wavelet coef-
ficients and to determine the estimated long-term variation
FtWξ()i={ yy 01 ,,...,y n−1}. The fluctuation signals are calculated by sub-
tracting this variation from the original inhomogeneous signals
((FtξC iξ)−FtW(i)). Finally, the absolute values of fluctuation signals can
be calculated by adding the time average of the variation FtξW()i, giving
FtξCW()iξ=(FtW iξ)+((FtCiξ)−FtW()i)
We have implemented the above method with the Python pro-
gramming language (Python Software Foundation, Python v.3.6)^36.
The autocorrelation function after noise removal was calculated by
a multiple-tau correlation algorithm^37 , followed by a nonlinear least-
squares fitting to the theoretical model. The radial and axial waist of
the PSF were determined by measuring fluorescent solutions and fixed
values on fitting.
Reconstitution and microscopy of protein-rich droplets
Liquid droplets of the Atg13–Atg17–Atg29–Atg31 complex and the Atg1
complex were formed by dilution of proteins from a stock solution into
buffer as follows; In Fig. 2b, liquid droplets of 3.3 μM Atg1 complex
(SNAP–Atg1(D211A), Atg13–SNAP, SNAP–Atg17–Atg29–Atg31) were
formed by dilution of proteins from a stock solution into buffer (final
concentrations: 50 mM HEPES, pH 7.0, 250 mM NaCl) with subsequent
incubation at 25 °C for the indicated times. In Fig. 2c, liquid droplets of