LMU Fundamentals of Fluctuation Spectroscopy I: The basics of
Transcript of LMU Fundamentals of Fluctuation Spectroscopy I: The basics of
Don C. Lamb
LMU
Department of Physical Chemistry
Fundamentals of Fluctuation Spectroscopy I:
The basics of Fluorescence Correlation Spectroscopy
A Poisson Process
Experiments that result in counting the number of events in a given time or in a given object can be described by a Poisson process provided: a) Number changes on nonoverlapping intervals are independent.
b) The probability of exactly one change occurring in a sufficiently short interval of length h is approximately λh
c) The probability of two or more changes in a sufficiently short interval is essentially zero.
e.g. Fluctuations in molecular number in a small volume.
n = 6 4 7 5
!)(
NeN
NPNN −
=Poisson Distribution
Poissonian StatisticsNN =∆ 2
The First FCS Measurement
Observation of gold colloids using an ultra-microscope (Svedberg and Inouye, Zeitschr f. Physik Chemie 1911,77:145-119)
Measurement of the Equilibrium Thermodynamic Fluctuations in Molecular Number
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The First FCS/PCH Measurement
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Poisson Distribution with ⟨N⟩ = 1.55
Svedberg and Inouye, Zeitschr f. Physik Chemie1911, 77:145-119
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ticle
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Experimental Setup
Objective
Sample
Dichroic Mirror
Ar-Kr Ion Laser
ConfocalPinhole
DPA
PiezoStage
Probe Volume
Typical FCS setups use either a confocal geometry or
two-photon excitation
Data Acquisition
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plitu
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Counter
Photon Counts
Clock
APD
Hardware Correlator
Hard Disk Software Correlator
Time Mode Photon Mode
APDCounter
Signal
SyncAPD
Counter
Signal
Sync
Advantageous for average count rates higher than 1 count per bin
Advantageous for average count rates lower than 1 count per bin
Autocorrelation AnalysisFreely diffusing, non-interacting particles in an open volume.
Photon are not detected stochastically, but in bursts when a molecule transverses the probe volume.
We can determine:Excited state lifetimeRotational Diffusion ConstantTriplet-State LifetimeTriplet-State AmplitudeTranslational Diffusion ConstantConcentrationMolecular Brightness
Schematic Autotcorrelation Function:
Delay Time (ms)
Aut
ocor
rela
tion
Am
plitu
de From: Schwille, Haustein, Fluorescence Correlation Spectroscopy, In: Single Molecule Techniques, Biophysics Textbook Online
Analysis Techniques
The autocorrelation function is one of many techniques to analyze the data
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G(t
)
Other Analysis Approaches Include:
Histogram Analysis
Multidimensional Analysis
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Autocorrelation Analysis
2
2
)(
)()()()(
tA
tAtAtAG
−+=
ττ
The normalized autocorrelation function (ACF) is given by:
2)(
)()(
tA
tAtA τδδ +=
22
)()0(
)(
)()(
A
AA
tA
tAtA τδδτδδ=
+
For processes that are:
Stationary: i.e. the average parameters do not change with time
the ACF is independent of the absolute time
Ergodic: i.e. every sizeable sampling of the process is representative of the whole
the time average is equal to the ensemble average
)()()( tAtAtA −=δwhere
Occasionally in the literature, the ACF is defined as:
)()()( ττ += tAtAg
A(t)
Time t
τ
Properties of the Autocorrelation Function
G(τ) has maximum at G(0)
2
1
12
)()0()(
⎟⎠
⎞⎜⎝
⎛==
∑
∑
=
+=
l
l
l
l
ii
ii
i
A
AA
A
AAG
τδδτδδτ
( ) 02 ≥iAδ
can be < 0τδδ +ii AA
The autocorrelation function (ACF) measures the self similarity of the observable A as a function of τ
Properties of the Autocorrelation Function
For non-conserved, non-periodic signals
G(τ) → 0 as τ → ∞
The amplitude is proportional to the size of the fluctuations
( )2
1
1
2
2
)0()0()0(
⎟⎠
⎞⎜⎝
⎛
−==
∑
∑
=
=
l
l
l
l
ii
ii
A
AA
A
AAG
δδ
2
2
)0(µσ
=G
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Log [time (s)]
G(t)
NG γ=)0(s125
4
2
µ
τ
=
=D
wrD
230 pM Rhodamine 6G in buffer
G(τ) is the probability distribution of detecting a photon at delay τ when a photon was detected at τ = 0
Autocorrelation Function, Single Species
),()()( tCWdQtF rrr∫= κ
Where κ is the detection efficiency
Q = σφ ; Effective Quantum Yield
W(r) = In(r)S (r)Χ (r) ; Probe Volume
In(r) = laser intensity profilefor n-photon excitation
S(r) = Sample extent
Χ(r) = Detection efficiency
C(r,t) = Number Density
The fluorescence intensity is given by:
The ACF is given by:
2
)()0()(
F
FFG
τδδτ =
2)(
)0,(),()()()(
⎥⎦⎤
⎢⎣⎡ ∫
= ∫∫rr
r'rr'rr'r
WdC
CCWWddG
δτδτ
Point-Spread Functions (PSF)
1-photon excitation, confocal detection:
Approximate the PSF by a 3 dimensional Gaussian
2-photon excitation:
Approximate the PSF by the expression for the Gaussian Beam Waist of the laser:
( ) ( )⎥⎦
⎤⎢⎣
⎡−
+−= 2
2
2
22
022exp0,0,0),,(
zr wz
wyxIzyxW
where wr and wz the radial and axial distance from the center to where the intensity has decayed by (1/e)2
respectively
( ) ( )⎥⎦
⎤⎢⎣
⎡ +−= 2
22
2
20
0 )(2exp
)(0,0,0),,(
zwyx
zwwIzyxI
( ) ⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+=
2
20
20
2 1)(λπ w
zwzw
w0 is the beam waist and λ is the excitation laser wavelength
where
( ) ( )⎥⎦
⎤⎢⎣
⎡ +−= 2
22
4
402
0 )(4exp
)(0,0,0),,(
zwyx
zwwIzyxW
The PSF is the measured fluorescence intensity of a point particle at the position r within the excitation volume
Autocorrelation Function, Single Species( ) ( )tCDt
tC ,, 2 rr δδ∇=
∂∂
( ) ( )( ) ( ){ }
( ) ( ){ }( ){ }
( )
∫
∫ ∫
∫∫
∫
∫
−−⋅
−⋅−⋅
−⋅
−⋅
⋅
⋅
=
⎭⎬⎫
⎩⎨⎧=
=
=
−=∂
∂
=
tDi
tDii
tDf
i
tDf
i
fi
fi
edC
eCCeded
eCCedCtC
eCedtC
tCDedt
tC
tCedtC
2
2
2
2
'
''
2
2
)0,(0,21
)0,(0,)0,(),(
0,,
,,
,,
ρ
ρ
ρ
ρ
π
δδπ
δδδδ
δδ
δρδ
δδ
rrρ
rρrρ
rρ
rρ
rρ
rρ
ρ
r''r''r'ρ
r'ρρr'r
ρρr
ρρr
ρρr
For a freely diffusing species:
( )2
1
2 /11
/11),,( ⎟⎟
⎠
⎞⎜⎜⎝
⎛
+⎟⎟⎠
⎞⎜⎜⎝
⎛+
=DzrD
DD wwNNG
ττττγττ
[ ]2)(
)0,(),()()()(
∫= ∫∫
rr
r'rr'rr'r
WdC
CCWWddG
δτδτ
Dwr
D 4
2
=τwhere
For a 3-Dimensional Gaussian Probe Volume:
( )⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ −−
=Dt
eDtC
CtC n
4' 2
24)0,(),(
rr
r'rπ
δδ
)0,(),( r'r CC δτδ
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G(t)
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Res
idua
lsAutocorrelation Function, Single Species
( )2
1
2 /11
/11),,( ⎟⎟
⎠
⎞⎜⎜⎝
⎛
+⎟⎟⎠
⎞⎜⎜⎝
⎛+
=DzrD
DD wwNNG
ττττγττ
NG γ=)0(s125
4
2
µ
τ
=
=D
wrD
230 pM Rhodamine 6G in buffer
⟨N⟩ = 0.085 molecules
D = 280 µm2/s (wr = 374 nm)
Freely Diffusing Particles with Flow
( ) ( ) ( )y
tCvtCDt
tC∂
∂−∇=
∂∂ ,,, 2 rrr δδδ
For flow (or scanning) in the y direction in a 3-Dimensional Gaussian Probe Volume :
( )( )
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛
+⎟⎟⎠
⎞⎜⎜⎝
⎛+
= +−
Drwv
DzrDDD e
wwNNG ττ
τ
ττττγττ 1
21
2
2
22
/11
/11),,(
( )⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ +−+−−
=Dt
tvyy
eDtC
CtC4
'2' 2222
23
4)0,(),(
rr
r'rπ
δδ
G(τ
)
2
0
2
4
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8
0
2
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Log [Time (s)]
v = 0 µm/s
v = 420 µm/s
v = 700 µm/s
v = 1,350 µm/s
TMR Labeled Endothelin
Gamma Factor
)0(
)0()(
),()()(
QW
WdQ
tCWdQtF
κ
δκ
κ
=
=
=
∫∫
rr
rrr
For a single particle fixed at the center of the PSF:
The fluorescent intensity is given by:
For molecules freely diffusing in solution:
)0(),( δ=tC r
We define the molecular brightness as ε = κQW(0)
∫
∫
∫
==
=
=
=
=
)0()(;
)0()(
),()0()(
)(
WWdVVCN
VCWWdC
tCWWd
NtF
effeff
eff
rr
rr
rrr
ε
ε
ε
ε
Veff is the volume of uniform illumination at maximum intensity that yields the same overall detection rate
Hence the molecular brightness can be determined from the ACF and the average intensity
NF=ε
Gamma Factor
2)(
)0,()0,()()()0(
⎥⎦⎤
⎢⎣⎡ ∫
= ∫∫rr
r'rr'rr'r
WdC
CCWWddG
δδ
( )r'rr'r −= δδδ CCC )0,()0,(
( )
[ ]( )
( )[ ] NWWdC
WWd
WdC
WdC
WdC
CWWddG
γ
δ
==
=
∫
−=
∫∫
∫∫
∫∫⎥⎦⎤
⎢⎣⎡
2
2
22
2
)0()(
)0()(
)(
)(
2)(
)()()0(
rr
rr
rr
rr
rr
r'rr'rr'r
( )( )[ ]
( )( )[ ]
( )( )[ ]∫
∫∫∫
∫∫
=
=
=
)0()(
)0()(
)0()(
)0()(
)0()(
)0()(
2
2
2
2
2
WWd
WWd
WWd
WWd
NVN
WWd
WWd
CN
eff
rr
rr
rr
rr
rr
rrγ
Afterpulsing
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3
1.E-07 1.E-06 1.E-05 1.E-04Time (s)
G(t)
I = 43940 cpsI = 2773 cpsI = 406 cps
The amplitude of the Autocorrelation Function is inversely proportional to the intensity
A Power Law is used to empirically fit the afterpulsing
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G(t)
*I
ValueStretched ExpPower Law
Autocorrelation from Reflected Laser Light
Signal-to-Noise Considerations
High Intensity Limit:
Uncertainty dominated by number of fluctuations:
Low Intensity Limit:
Uncertainty dominated by number of photons:
21
exp⎟⎟⎠
⎞⎜⎜⎝
⎛≈
C
tNS
τ
( )N
ItNS
Tγ
21
exp≈
NI T ε=
( ) εγ21
exptNS
≈
where texp is the measurement time of the experiment and τC is the correlation time of the fluctuations
Only possibilities to improve the S/N ratio are: extend the measurement timeincrease the counts per molecule secondchange the geometry
S/N is independent of sample concentration!!!
Limitations
Time Scale: ns/µs → ms/s/hrs
Early time limit:Detector afterpulsing (100 ns - 5 µs)Detector deadtime: (2 ns - 30 ns)Numbers of available photons: (10 ns - 100 ns)
Long time limit:Time molecule remains in the excitation volume
(Typically ~ 1 ms)
Increase the long time limit by:Increasing the excitation volume: (10 ms)Place sample in viscous solvents or gels: (s)
Slow reactions can be measured by changes in the autocorrelation function with time. (hrs)
Concentration Limits: ~ 1 µM → 10 pM
Maximum Concentration: (1 µM)The change in signal from thermodynamic
fluctuations becomes comparable to other sourcesof noise in the system
Minimum Concentration: (10 pM)Limit statisticsImpurities