2.1. Finding Optimal Filters
The auto-correlation function allows direct assessment of the diffusion constant
D. If only diffusion of a single species is considered, the amplitude at zero time of the autocorrelation function
G(0), allows one to determine the mean number of molecules
N in the detection volume,
Vdet, or the concentration,
c, if the parameters of detection volume are known [Eq.
(1)]:
Unfortunately, the situation becomes significantly more complicated if more than one molecular species are simultaneously present in solution. Considering a mixture of
n species, with corresponding brightnesses
, diffusion constants
D(i) and fractions
x(i),
i=1,…,
n, it is most convenient to define the autocorrelation function as
40 [Eq.
(2)]:
where
c(i) concentration,
brightness and
fraction of species
i, and
N corresponds now to the total number of molecules. The normalized diffusion term of species
i [i.e.
] is given by Equation
(3):
This model assumes a three-dimensional (3D) Gaussian-shaped volume element with spatial distribution of the detection probabilities
. The 1/
e2 radii in
x,
y and in
z directions are denoted by
ω0 and
z0, respectively. For one-photon excitation, the characteristic diffusion time
can be used to estimate the diffusion coefficient
by
.
A simple relation like Equation (
1) cannot be found for multiple species. Even if one knows the brightness
for each species, the diffusion coefficients
still have to be significantly distinct for successful extraction of
c(i) values, highlighting the need of a proper methodology that could differentiate species in a mixture. In ref.
22 it was proposed to use the differences in fluorescence lifetime to separate molecular species by time-correlated single-photon counting (TCSPC).
41, 42 For simplicity, we will describe how to generate optimal filters for separating two molecular species with mono color detection split for polarization. However, fFCS can be easily extended to account for spectral differences. In addition, fFCS can in principle be used to separate more than two species.
To fully use the anisotropy information a relationship between the total intensities in the parallel and perpendicular detection channels is kept. Therefore the methodology to find the proper filters by minimizing the relative errors proposed in ref.
22 needs to be optimized. Having this in mind, there are in principle three possible scenarios to generate filters: Scenario 1: single detector, single filter (
1d–sf). Scenario 2: two detectors, independent filters (
2d–inf). Scenario 3: two detectors, global filter (
2d–glf). We found that scenario 3 is most efficient in separating the species. It is described in more detail below. The theory behind scenario 1 was described in ref.
22, and in Section 4.1 we present the case for scenario 2.
In the case of MFD, where the situation is not as trivial as in the case of one-channel detection, the fluorescence signal from the mixture is divided into its parallel and perpendicular components and into two spectral ranges. Each registered photon emitted by
ith species is detected with probabilities
in either parallel or perpendicular detection channel, and certain relations between
and
exist due to specific anisotropy decays. If these relationships are disregarded (as done in scenarios 1 and 2), the anisotropy information is lost. To solve this problem, in scenario 3 we stack the TCSPC histograms of the perpendicular over the parallel detection channels and apply a global normalization to the stacked TCSPC histogram. As result, the corresponding probabilities of each
ith species are obtained as [Eq.
(4)]:
where the parallel channel probability is [Eq.
(5)]:
and similarly, the perpendicular channel probability is [Eq.
(6)]:
Thus,
and
represent the conditional probabilities to register a photon in the
jth bin of parallel or perpendicular TCSPC channels, provided that photon is emitted by
ith species. To differentiate from what is done in Section 4.1, here we define
as the stacked conditional probability of length 2
L. This is done to maintain the anisotropy information. To account for spectral differences (color detection) the histograms are stacked accordingly.
The total number of registered photons
is the sum of photon numbers emitted by all species in mixture solution [Eq.
(7)]:
where
is the photon counts of species
i. Instead of two measured decay histograms (
and
) of the two species in the mixture we consider one bimodal decay histogram
with total number of the stacked TCSPC channels 2
L and the conditional probability distribution
for the
ith species. This bimodal decay distribution can be defined as [Eq.
(8)]:
Using the global definition in Equation (
8), two bimodal filters
,
i=1, 2, will be generated with the property [Eq.
(10)]:
where the brackets denote averaging over long time of measurements. Filters
are obtained by minimizing relative errors
43 for parallel and perpendicular detection channels simultaneously [Eq.
(11)]:
The idea of one bimodal decay histogram
and two conditional probability distributions
, one for each species with total number of stacked TCSPC channels 2
L, makes it possible to calculate what we refer to as the species auto- (SACF) and cross-correlation (SCCF) function in a similar fashion to ref.
22. However, we need to transform our raw data streams into a modified format where the TCSPC channels of both detectors are stacked in a single array with length 2
L. In this way, the SCCF
between species
i and
m is [Eq.
(12)]:
where
is the signal in the
jth stacked TCSPC channel array of the total signal at measurement time
t and
is the signal in the stacked TCSPC channel
jth at measurement time
. The SACF is defined for the case of
i=m while SCCF corresponds to different species
i≠
m. In both cases, the orthonormality relationship [Eq.
(13)]:
is satisfied. Then, the species auto- and cross-correlation functions
averaged over an infinite number of measurements or over sufficiently long measurement time (
![[dbl greater-than sign]](/corehtml/pmc/pmcents/x226B.gif)
) are [Eq.
(14)]:
where
and
are pure fluorescence signals from the molecular species of
ith and
mth type. In a two-state system,
and
are equal. Thus, since the anisotropy differences are counted in addition to lifetimes, it becomes possible to highlight any dynamic process between two species in mixture solution if they differ in rotational correlation time or/and lifetime. In contrast to standard correlation curves the amplitude of dynamic term per molecule in species cross correlation function is equal to −1, like in an antibunching term that will be discussed in Section 2.2.1. One can understand the shape of SCCF as a probability distribution of interconversion between species: for very short
tc this probability is nearly equal to zero (SCCF starts at baseline level) and gets higher for larger
tc proportional to relaxation time of dynamics. Finally, everything is limited by diffusion time and probability to observe a transition is dropping to baseline level. If dynamics between species is missing then SCCF is showing no additional correlation amplitude above baseline.
In comparison to FCS, the species auto-correlation function (SACF) as defined in Equation (
12) with
i=
m assumes that other species are essentially non-fluorescent. In particular, it can be considered a background/scatter free autocorrelation function if a scatter filter was included. Then, the amplitude of the SACF is inversely related to the true number of molecules with the specific lifetime and polarization characteristics given by the filters. And in the same way as in standard FCS, at small lag times species are highly correlated and for longer correlation times everything is limited by diffusion showing a decrease of amplitude.
Let us consider an extreme case where two molecular species are simulated for typical conditions in single-molecule experiment: both species have the same fluorescence lifetime (
τG(1)=
τG(2)=4.0 ns), same brightness
Q(1)=
Q(2)=150 kHz, but differ only by rotational correlation times (
ρ(1)=0.1 ns,
ρ(2)=0.3 ns). Also, both species have the same diffusion time (
). The average number of simulated molecules in the observation volume was
(
and
). Details on the Brownian dynamics simulator can be found in Section 4.2. The chosen brightnesses are typical for FCS using confocal microscopy.
44–46 By applying additives like triplet or radical quenchers the signal can be further increased.
47This plausible experimental situation is simulated to emphasize that only differences in anisotropy can be used to separate species. The same simulation data set is used to determine the SACFs of the three different scenarios of filter generation. In all cases, exactly the same number of photons is correlated. The detection is always considered in two detection channels (parallel and perpendicular). shows the comparison of calculated SACF by Equation (
12) curves for the three difference scenarios. In all panels solid blue and red lines correspond to the SACF for species 1 and 2 respectively. On top the modeled correlation curves are shown as dashed lines.
Scenario 1: Single detector, single filter (
1d–sf). In this case, , there is no split by polarization, although polarized excitation was used. To mimic a single detector from the same simulation, we assumed that the total signal is the sum of the two simulated channels
. The same
decay pattern is used for both detection channels (parallel and perpendicular) and correspondingly the same filters are applied for parallel and perpendicular detection channels. The conditional probabilities
are obtained via Equations (
4–
6) using
and
. Filters are generated by simultaneously minimizing relative errors in both detection channels [Eq. (11)].
Scenario 2: Two detectors, independent filters (
2d–inf). shows the SACFs using independent filter generation for each detection channel and stacking them afterwards. The conditional probabilities
and
are obtained via Equation (
27).
and
filters are generated by simultaneously and independently minimizing relative errors in both detection channels [Eq. (29)].
Scenario 3: Two detectors, global filter (
2d–glf). When using global filter generation the conditional probabilities
and
are obtained via Equations (
4–
6).
and
filters are generated by simultaneously minimizing relative errors in both detection channels [Eq. (11)].
The case imitating one detection channel () shows poor statistics with some degree of separation between species because of polarized excitation. The separation gets better compared to the single detector imitating case for SACF from two detectors using independent filter generation (). Compared to , the contrast in is greater and a clear separation of species is possible, if the anisotropy is properly considered. This is only possible when polarized detection, splitting the fluorescence signal into parallel and perpendicular detection channels, is used like in single molecule MFD setups. In all following results scenario 3 of generating filters is used.
It is worth mentioning that in all cases, the background photons were considered as a third species. The normalized TCSPC histograms (
) for each species and their corresponding filters according to the three different scenarios are shown in . The use of the background filter is an additional standard methodology used for correctly recovering the amplitude of the correlation and was used in all simulated and experimental data analyses.
2.3. Experimental Results for Single-Molecule FRET Experiments
One exciting experimental application of fFCS in biophysics is the study of protein conformational dynamics. As an example, we have analyzed Syntaxin 1 (Sx); Sx forms part of the so-called SNARE (soluble NSF attachment receptor) proteins that regulate synaptic vesicle release.
54, 55 Sx is a transmembrane protein and the soluble domain consists of four long alpha helices that are known to undergo a conformational transition upon interacting with targets. One of the targets is SNAP25 located in presynaptic vesicles and it is known to open Syntaxin 1 allowing vesicle fusion and neurotransmitter release thereafter. On the other hand, Munc-18 is expected to act as a negative regulator of exocytosis.
55, 56 However, Munc-18 is also essential for exocytosis and maybe it also works as an activator instead of inhibitor.
57 Even when free in solution Sx had previously shown to be in equilibrium between two conformations,
34 in the following referred to as “open” and “closed”. Sx has two domains, the Habc domain consists of a three-helix bundle and a flexible H3 domain has a single long helix. H3 associates with the Habc domain in the closed and dissociates in the open conformation.
Herein, we used fFCS to extract the kinetic rates of this transition in a single-molecule FRET experiment combining lifetime and polarization-resolved information to maximize the contrast between conformations. The Sx double mutant (G105C/S225C) was labeled with donor dye Alexa488 and acceptor dye Alexa594 as described in Section 4.4.
34 A solution of double-labeled protein was diluted to ~10 p
m in PBS and placed on the MFD setup (Section 4.3). After careful calibration of the detection efficiencies of our experimental setup, we performed an analysis of the single-molecule bursts at first. The results are displayed in two-dimensional frequency histograms of the FRET indicator
FD/FA (ratio of donor fluorescence over acceptor fluorescence)
49 against fluorescence lifetime of the donor in the presence of acceptor,
τD(A). In , one can see a smeared population covering a broad range of fluorescence lifetimes. At least two FRET populations in addition to the third small
D-Only contribution at high
FD/
FA ratios with
τD(0) of 4 ns can be detected (). Such a presentation allows one to check for deviations from the static FRET line (orange line). The static FRET line in
FD/
FA versus
τD(A) plots is defined as [Eq.
(23)]:
where
ΦFD(0) and
ΦFA represent the quantum yields of the donor and acceptor fluorophores, respectively. The fluorescence lifetime of the donor without acceptor is
τD(0) and
τD(A) with an acceptor. As the populations are clearly off the static line, dynamics faster than the diffusion time is present.
58 Then, given the simplest case of a dynamic two-state system, a FRET line for this exchange (green dashed line) can be traced that follows the form [Eq.
(24)]:
where
τD(A), determined by the maximum likelihood estimator,
59 corresponds to the fluorescence-weighted average lifetime (
). For further details on the analysis and interpretation of MFD histograms, we refer the reader to Sisamakis et al.
49 The smeared population clearly follows the dynamic FRET line shown in green. The end points of the dynamic line correspond to
τD(A)(open)=3.6 ns and
τD(A)(closed)=0.8 ns, which were determined by sub-ensemble analysis of the dynamic FRET population. In combination with the average rotational correlation (
ρ=1.5 ns, also reported in ref.
34) we computed two fluorescence decays with Equation (
37) and the corresponding conditional probabilities (), which were used to generate the filters according to Equation (
11) using the global error minimization methodology described in Section 2.1. The SCCF computed by Equation (
12) is presented in . One can clearly see the enhanced negative correlation term as in the case of simulation data. When fitted with Equation (
19) one recovers a relaxation time
tR=0.6 ms which agrees nicely with values of
tR=0.8 ms
34 and
tR=0.6 ms
49 reported previously for another Sx double mutant (S91C/S225C), which also probes the exchange dynamics of the H3 domain with respect to the Habc domain.
In the previous section, we advised to use global target fit of the SACF and SCCF to stabilize the fit and reduce uncertainties. As indicated by the residuals of SACFs in , the global target fit of Sx revealed the need of an additional relaxation time found in all three correlation functions simultaneously with following relaxation times and formal amplitudes:
tR1=1.1 ms (84 %),
tR2=0.08 ms (16 %). The
tR=0.6 ms obtained with one free parameter recovers the average behavior, but only the global fit shows the need of the second faster relaxation time. The
tR2=0.08 ms term may be caused by a more complex kinetic scheme for the conformational transition. For example, Reiner et al.
60 have postulated that large-scale conformational motions require an intermediate structural unlocking step leading to a more flexible state which reacts to the final open state. Further measurements are needed to unambiguously solve this question. Finally, it is worth mentioning that the temporal resolution of fFCS is so powerful and provides such a good contrast that it is possible to distinguish additional kinetics not possible to observe otherwise, even with the 2D histograms from smFRET data.
If we assume an effective two state model for the open-closed transition, we can combine the results of the SACF and SCCF and extract the kinetic rates. Using the target fit Equation (
20) with the average relaxation time of 0.6 ms, the individual rate constants
k1,2=0.62 ms
−1 and
k2,1=0.94 ms
−1 and the corresponding equilibrium constant
Ko/c=1.5 (
N(open)=0.09,
N(closed)=0.06) are obtained.
To conclude, Sx was selected as an experimental realization where FRET adds the additional spectral and lifetime information to maximize contrast in fFCS. The combination of the SACFs and SCCF has shown to be a powerful tool to extract the kinetic rates and the equilibrium constant between conformers.
This article has been 
![[center dot]](/corehtml/pmc/pmcents/middot.gif)
[R] and rate constant of dissociation k2,1. The species fractions X in the dynamic equilibrium are defined by Equation 
![[congruent with]](/corehtml/pmc/pmcents/cong.gif)
7) and wrong NFCS=0.143. Moreover, regular FCS cannot detect the dynamics in this type of samples, and the decay is sufficiently described by a diffusion term (single gray bar at 1 ms in 
![[double less-than sign]](/corehtml/pmc/pmcents/x226A.gif)
τ, the two decays are nearly equal for any time t
![[perpendicular]](/corehtml/pmc/pmcents/x22A5.gif)
) polarizations relative to linearly polarized excitation laser beam respectively. To compute the normalized probabilities we follow ref. 
![[left angle bracket]](/corehtml/pmc/pmcents/lang.gif)
Δx2![[right angle bracket]](/corehtml/pmc/pmcents/rang.gif)
=
