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

1200020013241231021111311251110233133322111224221226122142345241141311423100100421123123201111000111-21100132000001001100010002322102110000201001-333122000231221024011102-12221123100011 033111021010010103011312121010121111211-1000322101230201212132111011002331 2242110001203010100221734410101002112211444421211440132123314313011222123310121111222412231113322132110000410432012120011322231200-25321203323311110 0210022013011321131200101314322112211223234422230321421532200202142123232043112312003314223452134110412322220221

The First FCS/PCH Measurement

0 5 10 15Time(s)

-0.2

0.0

0.2

0.4

0.6

0.8

G(t)

Poisson Distribution with ⟨N⟩ = 1.55

Svedberg and Inouye, Zeitschr f. Physik Chemie1911, 77:145-119

0.0001

0.001

0.01

0.1

1

0 2 4 6 8

Number of Particles

Freq

uenc

y

0

2

4

6

8

0 50 100 150 200 250

Time (s)

Par

ticle

s

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

0

0.2

0.4

0.6

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1

1.2

0 20 40 60 80 100Time

Phot

ons

0

0.2

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1

1.2

0 20 40 60 80 100Time

Am

plitu

de

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

0.00

0.05

0.10

0.15

0.20

-7 -6 -5 -4 -3 -2

Log [time (s)]

G(t

)

Other Analysis Approaches Include:

Histogram Analysis

Multidimensional Analysis

0

1

2

3

4

5

0 5 10 15 20Time (ms)

Sign

al

0

1

2

3

4

5

0 5 10 15 20Time (ms)

Sign

al

1

10

100

1000

10000

0 5 10 15 20Counts per bin

Num

ber o

f Occ

uran

ces

0.00.5

1.01.5

2.02.5

3.02015

105

00

1

2

3

4

Time (ns)Time (ms)

Cou

nts

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

0.00

1.00

2.00

3.00

4.00

5.00

-7 -6 -5 -4 -3 -2 -1

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

0.00

1.00

2.00

3.00

4.00

5.00

-7 -6 -5 -4 -3 -2 -1

Log [time (s)]

G(t)

-10

-5

0

5

-7 -6 -5 -4 -3 -2 -1

Log [Time (s)]

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

6

8

0

2

-6 -5 -4 -3 -2 -1

0

0.5

1.0

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

-0.5

0

0.5

1

1.5

2

2.5

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

0.1

1

10

100

1000

10000

1.E-07 1.E-06 1.E-05 1.E-04Time (s)

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