Fundamentals of Fluctuation
Spectroscopy IV:
Photon Counting Histogram
Analysis
Don C. Lamb
LMU
Department of Physical
Chemistry
Amplitude Fluctuations
A photon counting histogram analysis investigates
the amplitude of the fluctuations
Number of Particles
8
6
4
2
0
0
100
200
Time (s)
Svedberg and Inouye, Zeitschr f. Physik Chemie 1911, 77:145-119
The measured probability function for
detecting N photons in a time bin is a
renormalization of the histogram of the
photon counting data.
For a Poisson Distribution:
Pexp ( N ) =
P( N ) =
N
N
e
freq( N )
NTotal photons
− N
∆N 2 = N
N!
Poissonian Statistics
∆N 2 < N
super-Poissonian Statistics
∆N 2 > N
sub-Poissonian Statistics
Photon Counting Statistics
The number of detected photons from a constant intensity
light source is governed by Poisson statistics
p(k , k
(
η E E )k e −η E
)=
E
k!
≡ Poi(k , k
)
where: k is the number of detected photons
⟨k⟩ = ηE E is the average number of detected photons
ηE is the detection efficiency
E is the energy impinging on the detector
e.g. A non-diffusing 500 nm fluorescent bead in the
excitation volume:
From: Chen et al. 1999
Biophys J 77:553
Mandel’s Formula
For a fluctuating intensity source, the photon counting
distribution is given by Mandel’s formula:
Mandel, Proc. Phys. Soc. (1958) 72:1037-1048
(
η E E (t , T ))k e −η E
P(E (t , T ))dE
p(k , t , T ) = ∫
0
k!
∞
E
where: P(E(t,T)) is the energy probability distribution
T is the integration time of the measurement
E (t , T ) =
t +T
∫I
D
(t )dt
t
where: ID is the intensity reaching the detector
Effect of binning:
For T Æ 0: power fluctuations tract intensity fluctuations
For T Æ ∞: intensity fluctuations average out,
p(E) Æ δ(E – ⟨E⟩)
Choose T small enough to tract intensity fluctuations:
E(t) = ID(t)T
p (k , t , T ) = ∫
∞
0
where: ηI = ηET
(η I I D (t ) )k e −η I
I D (t )
k!
P (I D (t ) ) dI D
Diffusing Particle in a Confocal Volume
ID depends upon the position of the particle
The PSF gives the measured fluorescence intensity of a point
particle at the position r within the probe volume
The intensity at the detector from a fluorophore at
position r is given by:
I exn β PSF (r )
I D (r ) =
n
where n = number of absorbed photons per excitation
β includes corresponding scale factors between
excitation and detection intensity
n
We define the Molecular Brightness to be the measured
intensity of a molecule at the center of the PSF:
I 0n βη I kQW n (0)
ε=
=
n
n
η I I D (r ) = ε PSF (r )
n
(k ; ε ) = ∫ [ε PSF (r )] e
P (r ) dr
k!
(k ; ε ) = ∫ Poi (k , ε PSF (r ) )P(r ) dr
n
p
(1)
p (1)
k
−ε PSF
n
n
(r )
PCH for Particles in a Box
The probability of detecting k photons from a single molecule
in a box of volume V0 is given by:
(
)
p (1) (k ;V0 , ε ) = ∫ Poi k , ε PSF (r ) P (r ) dr
(
n
)
n
1
=
Poi k , ε PSF (r ) dr
V0 V∫0
The average count rate is:
n
1
k =
ε PSF (r ) dr
V0 V∫0
=
ε VPSF
V0
For multiple particles in a box:
(
)
p ( 2 ) (k ;V0 , ε ) = ∫∫ Poi k , ε PSF (r1 ) + ε PSF (r2 ) P (r1 )P (r2 )dr1dr2
p
(N )
n
n
N
n
⎛
⎞ N
(k ;V0 , ε ) = ∫1...∫N Poi⎜ k , ε ∑ PSF (ri ) ⎟∏ P(ri )dri
i
⎝
⎠ i
The expression can also be written as a convolution:
p
p
( 2)
(k ;V0 , ε ) = ( p
(N )
(1)
⊗p
)(k ,V , ε ) = ∑ p
∞
(1)
0
r =0
⎛ (1) N times (1) ⎞
(k ;V0 , ε ) = ⎜ p ⊗ L ⊗ p ⎟(k ,V0 , ε )
⎝
⎠
(1)
( r ;V0 , ε ) p (1) ( k − r ;V0 , ε )
PCH in an Open Volume
Particles can enter and leave the subvolume V0
The probability of having N particles in the subvolume V0 is
given by:
p (1) ( N ) = Poi (N , N
)
The probability of observing k photons is given by the
product of the probability of observing k photons
with N particles in the volume multiplied by the
probability of having N particles in the volume:
∞
∏(k ; N PSF , ε ) = ∑ p ( N ) (k ;V0 , ε )Poi (N , N PSF
)
N =0
where
p ( 0 ) (k ;V0 , ε ) = δ (k )
The average count rate is given by:
k = ε N PSF
Information available from analysis: ε , N PSF
Two key assumptions for PCH:
!
1) The molecule does not move significantly during
a time bin
2) The molecular brightness is constant in time and
follows the spatial profile of the excitation
volume (no reactions, photophysics, etc . . .)
PCH versus Concentration
Fits to Poisson distribution
From: Chen et al. 1999
Biophys J 77:553
At high concentration, Poission statistics dominate
The super-Poisson nature of the distribution is seen in the tail
of lower concentration measurements
PCH versus Concentration
Determination of ⟨NPSF⟩ and ε by fitting to the probability
function
∏ (k ; N PSF , ε ) =
∞
∑ p (k ;V , ε )Poi(N ,
(N )
0
N =0
N PSF
From: Chen et al. 1999
Biophys J 77:553
c (nM)
c (nM)/
5.5 nM
⟨k⟩
⟨k⟩/
0.28
ε
⟨N⟩
⟨N⟩/
0.347
χ2
550
100
26.25
93.8
0.807
32.53
93.7
1.14
55
10
2.71
9.7
0.807
3.36
9.7
0.98
5.5
1
0.28
1
0.807
0.347
1
0.84
)
PCH versus ε
From: Chen et al. 1999
Biophys J 77:553
The brighter the molecule, the more clearly the nonPoissonian statistics are observable
Bright, Slow Molecules
PCH vs Concentration
100
Fraction
10
c
5*c
25*c
-1
10-2
10-3
10-4
10-5
0
200
400
Photon Counts/bin
PCH with and without many dim molecules
Fraction
100
10-1
10-2
10-3
10-4 Few Bright
Many Dim
10-5
Mixture
10-6
0.1
1.0
10.0
100.0
Photon Counts/bin
1000.0
Fluorescent Intensity Distribution Analysis
The differences between PCH and FIDA are:
1) Treatment of the excitation volume
2) Mathematical approach
The number of photons for a single species in a small volume
element is given by:
(
∞
pdVi (k ) = ∑ Poi(m, N )Poi k , mε PSF (ri )
m =0
)
N = cdVi
where:
∞
pdVi (k ) = ∑
m =0
(cdVi )m e cdV
i
m!
(mε PSF (r)) e
k
− mε PSF ( r )
k!
The total probability function is given by the convolution of
all of the small volume elements
∏ (k ; N , ε ) = ∑ pdVi (k − m; N , ε ) pdV j≠i (m; N , ε )
∞
m =0
∞
pdV j≠i (m; N , ε ) = ∑ pdV j≠i (m − n; N , ε ) pdVk ≠i , j (n; N , ε )
n =0
This leads to a large number of convolutions that is
numerically ‘clumsy and slow’ to calculate
FIDA
The generating function is defined as:
∞
G (v ) = ∑ p(n)v n
n =0
The generating function has the following properties:
1) Under certain conditions, the generating function
completely determines the distribution
2) The generating function of the sum of independent
variables is the product of the generating functions
(or sum of the logarithm of the generating functions)
3) Moments can be determined from the derivates of the
generating function
For v = e 2πiξ , G (ξ ) and p(n ) are Fourier transform pairs
The generating function for a volume element dVi is given by:
∞
G (ξ ; dVi ) = ∑ pdVi (n ) e 2πiξ n
n =0
∞
∞
G (ξ ; dVi ) = ∑ ∑
n =0 m =0
(cdVi )m e −cdV
i
(mε PSF (r)) e
n
m!
G (ξ ; dVi ) = exp[cdV (e (e
− mε PSF ( r )
n!
2 πiξ
)
−1 ε PSF ( r )
)]
−1
e 2πiξ n
Treatment of Volume in FIDA
The generating function is given by integrating over the dV:
[
)]
(
2 πiξ
G (ξ i ) = exp ∫ cdV e (e −1)ε PSF ( r ) − 1
V
FIDA reduces the 3D integral over volume to a 1D integral.
For Example: 3D Gaussian.
Each concentric surface has the same brightness on the
detector. Perform a transformation:
) (
(
)
2 x2 + y 2 2z 2
u ≡ − ln PSF (r ) =
+ 2
2
wr
wz
u
dV
u
= π wr2 wz
du
2
The 3D volume integral becomes a 1D integral
(
)
⎡ π wr2 wz
⎤
(
e 2 πiξ −1)εe −u )
G (ξ i ) = exp ⎢ ∫ c
u du e
−1 ⎥
2
⎣u
⎦
FIDA defines the PSF empirically using:
(
)
dV = a1u + a2u 2 + a3u 3 du
where the coefficients a1, a2 and a3 are determined from the
PCH/FIDA analysis of a known fluorescent standard
The photon counting distribution is determined from the discrete
inverse Fourier transform of the generating function
∞
p(n ) = ∑ G (ξ )e − 2πiξ n
ξ =0
Distributions of Molecular Brightnesses
So far we have assumed each species has a well defined
molecular brightness, εi
A distribution of molecular brightnesses can be fit to the
PCH.
Adapted from:
Kask, et al., 1999
PNAS 96:13756.
Warning! There are more parameters than data
points. Criteria other than minimum χ2
are needed to fit to the data.
Multiple Species
PCH distinguishes between different species via the
molecular brightness, independent of the diffusion
time.
p
( N1 , N 2 )
N1 + N 2
(k ;V0 , ε1 , ε 2 ) = ∫1...∫N + N ∏ P(ri )dri
1
2
i
N1
N2
⎛
⎞
Poi⎜⎜ k , ε1 ∑ PSF (ri ) + ε 2 ∑ PSF (r j ) ⎟⎟
i
j
⎝
⎠
∏ (k ; N1 , ε1 , N 2 , ε 2 ) = ∏ (k ; N1 , ε1 ) ⊗ ∏ (k ; N 2 , ε 2 )
From: Chen et al. 1999
Biophys J 77:553
Background, Dark counts, scattered laser light, etc. can be
treated as an additional species
Multiple Species
Mixture of 20 % rhodamine and 80 % coumarin
Molecular brightness versus dilution
From: Müller, Chen,
Gratton 2000 Biophys J
78:474
Measuring Labeling Efficiencies
PCH of alcohol dehydrogenase for (a) singly labeled protein
and (b) a mixture of singly labeled and doubly labeled protein
From: Müller, Chen,
Gratton 2000 Biophys J
78:474
PCH can also be used to investigate the amount of
aggregation, formation of dimers, trimers, . . .
Resolvability
χ2 misfit contour map for 1.6 × 107 photons with εA = 1.5 and
εB = 6.0 (solid lines) and εA = 0.25 and εB = 1.0
scaled by 61.6 (dashed lines)
From: Müller, Chen,
Gratton 2000 Biophys J
78:474
!
The small number of data points in the fit
limitations the number of parameters
one can reliably fit to.
2D PCH
With two channel detection, 2D PCH can be analyzed
Green Species Only
Red Species Only
25
Photons, Red Channel
Photons, Red Channel
25
0
0
0
25
0
25
Photons, Green Channel
Photons, Green Channel
2 non-interacting species
Doubled Labeled Species
25
Photons, Red Channel
Photons, Red Channel
25
0
0
25
Photons, Green Channel
0
0
25
Photons, Green Channel
2D PCH analysis provides additional data points for
parameter determination with multiple species