Fluorescence intensity multiple distributions analysis: concurrent determination of diffusion times and molecular brightness
A method for distinguishing samples having flourescent particles includes (a) monitoring intensity fluctuations of fluorescence emitted by the particles in at least one measurement volume by detecting sequences of photon counts by at least one photon detector, (b) determining, from the sequences of photon counts, intermediate statistical data involving at least two probability functions of the number of photon counts detected in different sets of counting time intervals, and (c) determining from the intermediate statistical data a distribution of particles as a function of at least two arguments, wherein one argument is a specific brightness of the particles, or a measure thereof, and another argument is a diffusion coefficient of the particles, or a measure thereof.
1. A method for characterizing samples having fluorescent particles, comprising the steps of:
a) monitoring intensity fluctuations of fluorescence emitted by the particles in at least one measurement volume by detecting sequences of photon counts by at least one photon detector,
b) determining from the sequences of photon counts intermediate statistical data comprising at least two probability functions, {circumflex over (P)} 1 (n 1 ), {circumflex over (P)} 2 (n 2 ), . . . , of the number of photon counts, n 1 , n 2 , . . . , detected in different sets of counting time intervals,
c) determining from said intermediate statistical data a distribution of particles as a function of at least two arguments, wherein one argument is a specific brightness of the particles, or a measure thereof, and another argument is a diffusion coefficient of the particles, or a measure thereof,
wherein said distribution function of particles is determined by fitting the experimentally determined probability functions {circumflex over (P)} 1 (n 1 ), {circumflex over (P)} 2 (n 2 ), . . . by corresponding theoretical probability functions P 1 (n 1 ), P 2 (n 2 ), . . . and
wherein said theoretical probability distributions P 1 (n 1 ), P 2 (n 2 ), . . . are calculated as functions of apparent concentrations and apparent brightness which depend on the widths of the counting time intervals in the different sets.
2. The method according to claim 1 wherein, in calculations of the theoretical distributions P 1 (n 1 ), P 2 (n 2 ), . . . , an optical spatial brightness function B(r) is accounted for by a separately determined relationship between brightness B and volume elements dV.
3. The method according to claim 2 , wherein the relationship between the spatial brightness B and volume elements dV is expressed through a variable x=ln(B 0 /B) by a relationship
ⅆ
V
ⅆ
x
=
A
0
(
1
+
a
1
x
+
a
2
x
2
)
x
a
3
,
where B 0 is maximum brightness and A 0 , a 1 , a 2 and a 3 are empirical parameters of the spatial brightness function.
4. The method according to claim 1 wherein each set of counting time intervals consists of intervals of equal width while different probability functions {circumflex over (P)} T 1 (n 1 ), {circumflex over (P)} T 2 (n 2 ), . . . correspond to counting time intervals of different widths T 1 , T 2 , . . . .
5. The method according to claim 4 , wherein the apparent concentration is calculated as
c
app
(
T
)
=
c
app
(
0
)
Γ
(
T
)
,
the apparent brightness is calculated as
q app ( T )= q app (0)Γ( T ),
and Γ(T) is calculated as
Γ
(
T
)
=
1
c
app
(
0
)
q
app
2
(
0
)
T
2
∫
0
T
ⅆ
t
1
∫
0
T
ⅆ
t
2
G
(
t
2
-
t
1
)
,
where G(t) denotes autocorrelation function of fluorescence intensity and T denotes the width of the counting time interval.
6. The method according to claim 1 wherein the counting time intervals in each set are consecutive in time.
7. The method according to claim 1 wherein, counting time intervals in each set overlap.
8. The method according to claim 1 wherein said intermediate statistical data are processed applying inverse transformation with regularization and/or constraints.
9. The method according to claim 1 wherein the theoretical distributions P 1 (n 1 ), P 2 (n 2 ), . . . are calculated through their generating functions
G
p
(
n
)
(
ξ
_
)
=
∑
n
ξ
_
n
P
(
n
)
.
10. The method according to claim 9 , wherein the generating function is calculated using the expression G(ξ)=exp[∫dqc(q)∫d 3 r(e (ξ−1)qTB(r) −1)]where c(q) is the apparent density of particles with specific brightness q, T is the length of the counting time interval, and B(r) is the spatial brightness profile as a function of coordinates.
11. The method according to claim 9 , wherein said generating functions are calculated using the formula
G
P
(
n
)
(
ξ
)
=
exp
[
∑
i
c
i
∫
(
ⅇ
(
ξ
-
1
)
q
i
B
(
r
)
T
-
1
)
ⅆ
V
]
in which c is an apparent concentration and q is an apparent brightness which both depend on the width of the counting time interval T.
12. The method according to claim 1 wherein said distribution function of particles is determined by fitting the experimentally determined probability functions {circumflex over (P)} 1 (n 1 ), {circumflex over (P)} 2 (n 2 ), . . . by corresponding theoretical probability functions P 1 (n 1 ), P 2 (n 2 ), . . . .
13. The method according to claim 12 wherein the theoretical probability functions P 1 (n 1 ), P 2 (n 2 ), . . . are calculated through their generating functions
G
P
(
n
)
(
ξ
)
=
∑
n
ξ
n
P
(
n
)
.
14. The method according to claim 1 wherein concentrations of particles are selected to be approximately one or less molecules per measurement volume.
15. The method according to claim 1 wherein said at least one photon detector is either an avalanche photodiode or a photomultiplier.
16. The method according to claim 1 wherein said at least one photon detectors are at least two photon detectors which are used for monitoring fluorescence of different wavelengths or polarization.
17. The method according to claim 1 wherein said fluorescent particles are characterized by applying an homogeneous fluorescence assay.
18. The method according to claim 1 , wherein the intensity fluctuation of fluorescence emitted by the particles in said at least one measurement volume are monitored by detecting said sequences of photon counts using said at least one photon detector, said at least one photon detector being part of a confocal apparatus.
19. The method according to claim 18 wherein said confocal apparatus comprises:
a) a radiation source ( 12 ) for providing excitation radiation ( 14 ),
b) an objective ( 22 ) for focusing the excitation radiation ( 14 ) into a measurement volume ( 26 ) of said at least one measurement volume,
c) a detector ( 42 ) of said at least one photon detector for detecting emission radiation ( 30 ) that stems from the measurement volume ( 26 ), and
d) an opaque means ( 44 ) positioned in the pathway ( 32 ) of the emission radiation ( 30 ) or excitation radiation ( 14 ) for erasing the central part of the emission radiation ( 30 ) or excitation radiation ( 14 ).