Introduction to FCS · 2018. 4. 12. · Introduction to Fluorescence Correlation Spectroscopy (FCS) Fluctuation analysis goes back to the beginning of the century •Brownian motion

Introduction to FCS

What is diffusion? Diffusion of molecules Diffusion of light Generalization of the concept of diffusion Translational diffusion Rotational diffusion Some general rules about diffusion Diffusion in restricted spaces Diffusion in presence of flow Anomalous diffusion How to measure diffusion? Macroscopic methods Molecular level methods Fluctuation based methods Single particle tracking methods

Original observation by Brown: the Brownian motion (From A Fowler lectures) In 1827 the English botanist Robert Brown noticed that pollen grains suspended in water jiggled about under the lens of the microscope, following a zigzag path. Even more remarkable was the fact that pollen grains that had been stored for a century moved in the same way. In 1889 G.L. Gouy found that the "Brownian" movement was more rapid for smaller particles (we do not notice Brownian movement of cars, bricks, or people). In 1900 F.M. Exner undertook the first quantitative studies, measuring how the motion depended on temperature and particle size. The first good explanation of Brownian movement was advanced by Desaulx in 1877: "In my way of thinking the phenomenon is a result of thermal molecular motion in the liquid environment (of the particles)." This is indeed the case. A suspended particle is constantly and randomly bombarded from all sides by molecules of the liquid. If the particle is very small, the hits it takes from one side will be stronger than the bumps from other side, causing it to jump. These small random jumps are what make up Brownian motion.

In 1905 A. Einstein explained Brownian motion using energy equipartition: the kinetic theory of gases developed by Boltzmann and Gibbs could explain the randomness of the motion of large particles without contradicting the Second Principle of Thermodynamics. This was the first “convincing” proof of the particle nature of matter as declared by the adversaries of atomism. The role of fluctuations Explanation in terms of fluctuations

Why measure diffusion with optical methods? Diffusion in microscopic spaces Diffusion in biological materials

Difference between diffusion, “hopping” and binding

Introduction to Fluorescence Correlation Spectroscopy (FCS) Fluctuation analysis goes back to the beginning of the century

•Brownian motion (1827). Einstein explanation of Brownian motion (1905) Noise in resistors (Johnson, Nyquist), white noise {Johnson, J.B., Phys. Rev. 32 pp. 97 (1928), Nyquist, H., Phys. Rev. 32 pp. 110 (1928)}

•Noise in telephone-telegraph lines. Mathematics of the noise. Spectral analysis

•I/f noise

•Noise in lasers (1960)

•Dynamic light scattering

•Noise in chemical reactions (Eigen)

•FCS (fluorescence correlation spectroscopy, 1972) Elson, Magde, Webb

•FCS in cells, 2-photon FCS (Berland et al, 1994)

•First commercial instrument (Zeiss 1998)

Manfred Eigen

Elliot Elson

Keith Berland

Why we need FCS to measure the internal dynamics in cell??

Methods based on perturbation Typically FRAP (fluorescence recovery after photobleaching) Methods based on fluctuations Typically FCS and dynamic ICS methods

There is a fundamental difference between the two approaches, although they are related as to the physical phenomena they report.

Dan Axelrod

Introduction to “number” fluctuations

In any open volume, the number of molecules or particles fluctuate according to a Poisson statistics (if the particles are not-interacting) The average number depends on the concentration of the particles and the size of the volume The variance is equal to the number of particles in the volume This principle does not tell us anything about the time of the fluctuations

The fluctuation-dissipation principle (systems at equilibrium)

If we perturb a system from equilibrium, it returns to the average value with a characteristic time that depends on the process responsible for returning the system to equilibrium Spontaneous energy fluctuations in a part of the system, can cause the system to locally go out of equilibrium. These spontaneous fluctuations dissipate with the same time constant as if we had externally perturbed the equilibrium of the system.

External perturbation

time

ampl

itude

Equilibrium value

Spontaneous fluctuation

time

ampl

itude

Equilibrium value

Synchronized Non-synchronized

First Application of Correlation Spectroscopy (Svedberg & Inouye, 1911) Occupancy Fluctuation

Collected data by counting (by visual inspection) the number of particles in the observation volume as a function of time using a “ultra microscope”

120002001324123102111131125111023313332211122422122612214 2345241141311423100100421123123201111000111_2110013200000 10011000100023221002110000201001_333122000231221024011102_ 1222112231000110331110210110010103011312121010121111211_10 003221012302012121321110110023312242110001203010100221734 410101002112211444421211440132123314313011222123310121111 222412231113322132110000410432012120011322231200_253212033 233111100210022013011321113120010131432211221122323442230 321421532200202142123232043112312003314223452134110412322 220221

Experimental data on colloidal gold particles:

1

2

3

456

10

2

3

456

100

Fre

qu

en

cy

86420

Number of Particles

7

6

5

4

3

2

1

0

Parti

cle

Num

ber

5004003002001000

time (s)

*Histogram of particle counts *Poisson behavior *Autocorrelation not

available in the original paper. It can be easily calculated today.

Particle Correlation and spectrum of the fluctuations

Comments to this paper conclude that scattering will not be suitable to observe single molecules, but fluorescence could

• Number of fluorescent molecules in the volume of observation, diffusion or binding

• Conformational Dynamics • Rotational Motion if polarizers are used either in emission

or excitation • Protein Folding • Blinking • And many more

Example of processes that could generate fluctuations

What can cause a fluctuation in the fluorescence signal???

Each of the above processes has its own dynamics. FCS can recover that dynamics

Generating Fluctuations By Motion

Sample Space

Observation Volume

1. The rate of motion 2. The concentration of particles 3. Changes in the particle fluorescence while under observation, for example conformational transitions

What is Observed? We need a small volume!!

Note that the spectrum of the fluctuations responsible for the motion is “flat” but the spectrum of the fluorescence fluctuations is not

Defining Our Observation Volume under the microscope: One- & Two-Photon Excitation.

1 - Photon 2 - Photon

Approximately 1 um3

Defined by the wavelength and numerical aperture of the

objective

Defined by the pinhole size, wavelength, magnification and

numerical aperture of the objective

Why confocal detection? Molecules are small, why to observe a large volume? • Enhance signal to background ratio • Define a well-defined and reproducible volume Methods to produce a confocal or small volume (limited by the wavelength of light to about 0.1 fL) • Confocal pinhole • Multiphoton effects 2-photon excitation (TPE) Second-harmonic generation (SGH) Stimulated emission Four-way mixing (CARS) (not limited by light, not applicable to cells) • Nanofabrication • Local field enhancement • Near-field effects

Brad Amos MRC, Cambridge, UK

1-photon

2-photon Need a pinhole to define a small volume

22

)(λ

πτ fhc

Apdna ≈

na Photon pairs absorbed per laser pulse p Average power τ pulse duration f laser repetition frequency A Numerical aperture λ Laser wavelength d cross-section

From Webb, Denk and Strickler, 1990

Laser technology needed for two-photon excitation Ti:Sapphire lasers have pulse duration of about 100 fs Average power is about 1 W at 80 MHz repetition rate About 12.5 nJ per pulse (about 125 kW peak-power) Two-photon cross sections are typically about

δ=10-50 cm4 sec photon-1 molecule-1 Enough power to saturate absorption in a diffraction limited spot

Orders of magnitude (for 1 μM solution, small molecule, water) Volume Device Size(μm) Molecules Time milliliter cuvette 10000 6x1014 104 microliter plate well 1000 6x1011 102 nanoliter microfabrication 100 6x108 1 picoliter typical cell 10 6x105 10-2 femtoliter confocal volume 1 6x102 10-4 attoliter nanofabrication 0.1 6x10-1 10-6

Determination of diffusion by fluctuation spectroscopy

Time (in us)600,000400,000200,0000

Cou

nts

(in k

Hz)

550500450400350300250200150100

500

The time series

Time (in us)1,380,0001,375,0001,370,0001,365,000

Cou

nts

(in k

Hz)

80

70

60

50

40

30

20

10

0

Detail of one time region

Correlation plot (log averaged)

Tau (s)1E-5 0.0001 0.001 0.01 0.1 1 10

G(t)

6

5

4

3

2

1

0

The autocorrelation function N and relaxation time of the fluctuation

PCH average

counts1614121086420

0.1

1

10

100

1,000

10,000

100,000

1,000,000

The histogram of the counts in a given time bin (PCH). N and brightness

Data presentation and Analysis

Duration

How to extract the information about the fluctuations and their characteristic time?

Distribution of the amplitude of the fluctuations Distribution of the duration of the fluctuations

To extract the distribution of the duration of the fluctuations we use a math based on calculation of the correlation function To extract the distribution of the amplitude of the fluctuations, we use a math based on the PCH distribution

The definition of the Autocorrelation Function

)()()( tFtFtF −=δ G(τ) =δF(t)δF(t + τ)

F(t) 2

Phot

on C

ount

s

t

τ

t + τ

time

Average Fluorescence

),()()( tCWdQtF rrr∫= κ

kQ = quantum yield and detector sensitivity (how bright is our

probe). This term could contain the fluctuation of the

fluorescence intensity due to internal processes

W(r) describes the profile of illumination

C(r,t) is a function of the fluorophore concentration over time. This is the term that contains the “physics” of the diffusion processes

What determines the intensity of the fluorescence signal??

The value of F(t) depends on the profile of illumination!

This is the fundamental equation in FCS

What about the excitation (or observation) volume shape?

( ) 2022 )(2

0 )(,,w

yx

ezIIzyxF+

−

=

−= 2

0

22)(zw

zExpzI

2)(1

1)(

ozwzzI

+=

Gaussian z

Lorentzian z

More on the PSF in Jay’s lecture

For the 2-photon case, these expression must be squared

The Autocorrelation Function

10-9 10-7 10-5 10-3 10-1

0.0

0.1

0.2

0.3

0.4

Time(s)

G(τ)

G(0) ∝ 1/N As time (tau) approaches 0

Diffusion

In the simplest case, two parameters define the autocorrelation function: the amplitude of the fluctuation (G(0)) and the characteristic relaxation time of the fluctuation

The Effects of Particle Concentration on the Autocorrelation Curve

= 4

= 2

0.5

0.4

0.3

0.2

0.1

0.0

G(t)

10 -7 10 -6 10 -5 10 -4 10 -3

Time (s)

Observation volume

Why Is G(0) Proportional to 1/Particle Number?

NNN

NVarianceG 1)0( 22 ===10-9 10-7 10-5 10-3 10-1

0.0

0.1

0.2

0.3

0.4

Time(s)

G(τ

)

A Poisson distribution describes the statistics of particle occupancy fluctuations. For a Poisson distribution the variance is proportional to the average:

VarianceNumberParticleN =>=< _

( )2

2

2

2

)(

)()(

)(

)()0(

tF

tFtF

tF

tFG

−==

δ

2)()()(

)(tF

tFtFG

τδδτ

+= Definition

G(0), Particle Brightness and Poisson Statistics

1 0 0 0 0 0 0 0 0 2 0 1 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 1 0 0

Average = 0.275 Variance = 0.256

Variance = 4.09

4 0 0 0 0 0 0 0 0 8 0 4 4 4 0 0 0 0 0 0 4 0 0 0 0 0 0 0 4 0 4 0 0 0 4 0 0 4 0 0

Average = 1.1 0.296

296.0256.0275.0 22 ==∝ VarianceAverageN

Lets increase the particle brightness by 4x:

Time

∝N

Effect of Shape on the (Two-Photon) Autocorrelation Functions:

For a 2-dimensional Gaussian excitation volume: 1

22

41)(−

+=

DGwD

NG τγτ

For a 3-dimensional Gaussian excitation volume:

21

23

1

23

4141)(−−

+

+=

DGDG zD

wD

NG ττγτ

2-photon equation contains a 8, instead of 4

21

21

11)(−−

⋅+⋅

+=

DD

SN

Gττ

ττγτ

3D Gaussian “time” analysis: with τD=w2/4D and S=w/z

Blinking and binding processes

Until now, we assumed that the particle is not moving. If we assume that the blinking of the particle is independent on its movement, we can use a general principle that states that the correlation function splits in the product of the two independent processes.

)()()( τττ DiffusionBlinkingTotal GGG ⋅=

−+= −λττ e

KffKG BABinding

2

1)(

K = kf / kb is the equilibrium coefficient; λ = kf + kb is the apparent reaction rate coefficient; and fj is the fractional intensity contribution of species j

How different is G(binding) from G(diffusion)?

With good S/N it is possible to distinguish between the two processes. Most of the time diffusion and exponential processes are combined

DataDiffusionExponential

Difference betw een ACF for diffusion and binding

tau1e-5 1e-4 1e-3 1e-2 1e-1 1e0 1e1

G(t)

5.5

5

4.5

4

3.5

3

2.5

2

1.5

1

0.5

0

The Effects of Particle Size on the Autocorrelation Curve

300 um2/s 90 um2/s 71 um2/s

Diffusion Constants

Fast Diffusion

Slow Diffusion

0.25

0.20

0.15

0.10

0.05

0.00

G(t)

10 -7 10 -6 10 -5 10 -4 10 -3

Time (s)

D = k ⋅T6 ⋅π ⋅ η ⋅ r

Stokes-Einstein Equation:

3rVolumeMW ∝∝

and

Monomer --> Dimer Only a change in D by a factor of 21/3, or 1.26

Autocorrelation Adenylate Kinase -EGFP Chimeric Protein in HeLa Cells

Qiao Qiao Ruan, Y. Chen, M. Glaser & W. Mantulin Dept. Biochem & Dept Physics- LFD Univ Il, USA

Examples of different Hela cells transfected with AK1-EGFP

Examples of different Hela cells transfected with AK1β -EGFP

Fluorescence Intensity

Time (s)

EGFPsolution

EGFPcell

EGFP-AKβ in the cytosol

EGFP-AK in the cytosol

Normalized autocorrelation curve of EGFP in solution (•), EGFP in the cell (• ), AK1-EGFP in the cell(•), AK1β-EGFP in the cytoplasm of the cell(•).

Autocorrelation of EGFP & Adenylate Kinase -EGFP

Autocorrelation of Adenylate Kinase –EGFP on the Membrane

A mixture of AK1b-EGFP in the cytoplasm and membrane of the cell.

Clearly more than one diffusion time

Diffusion constants (um2/s) of AK EGFP-AKβ in the cytosol -EGFP in the cell (HeLa). At the membrane, a dual diffusion rate is calculated from FCS data. Away from the plasma membrane, single diffusion constants are found.

10 & 0.1816.69.619.68

10.137.1

11.589.549.12

Plasma Membrane Cytosol

Autocorrelation Adenylate Kinaseβ -EGFP

13/0.127.97.98.88.2

11.414.4

1212.311.2

D D

egfp

countswG(0)NBright

akb1093egfp1000600000.360.0055.790.21

akb1097egfp1001160000.270.0122.20.16

akb1098egfp1002180000.330.0132.260.16

akb1099egfp1003220000.330.0142.40.19

akb1100egfp1004140000.390.0111.850.16

akb1101egfp1005220000.330.0152.410.18

average is 0,33

Sheet2

w=0.33

countsDG(0)Nbrightness

akb1000b100016000section4.280.19

akb1006b100620000head10/0.233.780.23akb1on the border 50mV

akb1007b100716000tail10/0.174.50.19akb1on the border 50mV

akb1026b102614000section10/0.264.260.17akb3 on the border, 50mV

akb1027b102711000perfect10/0.133.480.16akb3 on the border, 50mV

akb1028b102810000head10/0.293.160.15akb3 on the border, 50mV

akb1029b102912000tail10/0.163.830.16akb3 on the border, 50mV



akb1032b103210000perfect remove a little head10/0.133.430.14akb3 on the border, 50mV



akb1035b103510000good remove little10/0.23.880.14akb3 on the border, 50mV


akb1037b10378000good14.90.0162.680.13akb3, near the border, 300mV

akb1038b10387000head19.50.0192.820.13akb3, near the border, 300mV

akb1039b10397000head110.0182.740.12akb3, near the border, 300mV

akb1040b10405600good remove a peak6.610.0162.380.11akb3, very close to the border, 300mV

akb1041b10415600good5.170.172.170.12akb3, very close to the border, 300mV

akb1042b10425600good remove head10/0.362.290.12akb3, very close to the border, 300mV

akb1043b10435600good10/0.092.380.12akb3, very close to the border, 300mV

akb1044b10445000good remove a little5.140.172.160.12akb3, very close to the border, 300mV

akb1046b10465600cut head10/0.122.260.12akb3, 10mW from the border to the center , see image 300mV

akb1047b10475000ok3.120.0192.030.11akb3, 10mW from the border to the center , see image 300mV


akb1049b10494000good4.780.0261.460.14akb3, 10mW from the border to the center , see image 300mV







akb1057b105714000section10 & 0.182.980.23akb3, 10mW from the border to the center , see image 300mV another set 188, (77--245)

akb1058b10589000head16.60.0192.320.18akb3, 10mW from the border to the center , see image 300mV another set 188, (77--245)

akb1059b10598000section9.610.022.030.19akb3, 10mW from the border to the center , see image 300mV another set 188, (77--245)

akb1061b10616400good9.680.0242.080.16akb3, 10mW from the border to the center , see image 300mV another set 188, (77--245)




akb1065b10656400good9.540.01920.16akb3, 10mW from the border to the center , see image 300mV another set 188, (77--245)

akb1066b10666400ok9.120.172.340.14akb3, 10mW from the border to the center , see image 300mV another set 188, (77--245)

akb1067b106770002.540.14akb3, 10mW from the border to the center , see image 300mV another set 188, (77--245)

akb1068b10687000ok2.560.14akb3, 10mW from the border to the center , see image 300mV another set 188, (77--245)

akb1083b10838000good10/0.211.980.2akb3 @1000mV, the computer is rebooted.vertical (156-78),(30-195), 10 points

akb1084b10843200ok150.0291.260.13akb3 @1000mV, the computer is rebooted.vertical (156-78),(30-195), 10 points

akb1085b10853200ok12.90.0341.240.13akb3 @1000mV, the computer is rebooted.vertical (156-78),(30-195), 10 points



Sheet3

egfp

countswG(0)NBright

akb1093egfp1000600000.360.0055.790.21

akb1097egfp1001160000.270.0122.20.16

akb1098egfp1002180000.330.0132.260.16

akb1099egfp1003220000.330.0142.40.19

akb1100egfp1004140000.390.0111.850.16

akb1101egfp1005220000.330.0152.410.18

average is 0,33

Sheet2

w=0.33

countsDG(0)Nbrightness

akb1000b100016000section4.280.19

akb1006b100620000head10/0.233.780.23akb1on the border 50mV

akb1007b100716000tail10/0.174.50.19akb1on the border 50mV




akb1029b102912000tail10/0.163.830.16akb3 on the border, 50mV



akb1032b103210000perfect remove a little head10/0.133.430.14akb3 on the border, 50mV



akb1035b103510000good remove little10/0.23.880.14akb3 on the border, 50mV


akb1037b10378000good14.90.0162.680.13akb3, near the border, 300mV

akb1038b10387000head19.50.0192.820.13akb3, near the border, 300mV

akb1039b10397000head110.0182.740.12akb3, near the border, 300mV

akb1040b10405600good remove a peak6.610.0162.380.11akb3, very close to the border, 300mV

akb1041b10415600good5.170.172.170.12akb3, very close to the border, 300mV

akb1042b10425600good remove head10/0.362.290.12akb3, very close to the border, 300mV

akb1043b10435600good10/0.092.380.12akb3, very close to the border, 300mV

akb1044b10445000good remove a little5.140.172.160.12akb3, very close to the border, 300mV

akb1046b10465600cut head13/0.122.260.12akb3, 10mW from the border to the center , see image 300mV







akb1053b10534000good120.0251.60.12akb3, 10mW from the border to the center , see image 300mV



akb1057b105714000section10/0.182.980.23akb3, 10mW from the border to the center , see image 300mV another set 188, (77--245)

akb1058b10589000head16.60.0192.320.18akb3, 10mW from the border to the center , see image 300mV another set 188, (77--245)

akb1059b10598000section9.610.022.030.19akb3, 10mW from the border to the center , see image 300mV another set 188, (77--245)





akb1065b10656400good9.540.01920.16akb3, 10mW from the border to the center , see image 300mV another set 188, (77--245)

akb1066b10666400ok9.120.172.340.14akb3, 10mW from the border to the center , see image 300mV another set 188, (77--245)

akb1067b106770002.540.14akb3, 10mW from the border to the center , see image 300mV another set 188, (77--245)

akb1068b10687000ok2.560.14akb3, 10mW from the border to the center , see image 300mV another set 188, (77--245)

akb1083b10838000good10/0.211.980.2akb3 @1000mV, the computer is rebooted.vertical (156-78),(30-195), 10 points





Sheet3

Two Channel Detection: Cross-correlation

Each detector observes the same particles

Sample Excitation Volume

Detector 1 Detector 2

Beam Splitter 1. Increases signal to noise by isolating correlated signals.

2. Corrects for PMT noise

lfd

Removal of Detector Noise by Cross-correlation

11.5 nM Fluorescein

Detector 1

Detector 2

Cross-correlation

Detector after-pulsing

lfd

)()(

)()()(

tFtF

tdFtdFG

ji

jiij ⋅

+⋅=

ττ

Detector 1: Fi

Detector 2: Fj

t

τ

Calculating the Cross-correlation Function

t + t

time

time

lfd

Thus, for a 3-dimensional Gaussian excitation volume one uses:

21

212

1

212

1212

4141)(−−

+

+=

zD

wD

NG ττγτ

Cross-correlation calculations

One uses the same fitting functions you would use for the standard autocorrelation curves.

G12 is commonly used to denote the cross-correlation and G1 and G2 for the autocorrelation of the individual detectors. Sometimes you will see Gx(0) or C(0) used for the cross-correlation.

lfd

Two-Channel Cross-correlation

Each detector observes particles with a particular color

The cross-correlation ONLY if particles are observed in both channels

The cross-correlation signal:

Sample

Red filter Green filter

Only the green-red molecules are observed!!

lfd

Two-color Cross-correlation

Gij(τ) =dFi (t) ⋅dFj (t + τ)

Fi(t) ⋅ Fj (t)

)(12 τG

Equations are similar to those for the cross correlation using a simple beam splitter:

Information Content Signal

Correlated signal from particles having both colors.

Autocorrelation from channel 1 on the green particles.

Autocorrelation from channel 2 on the red particles.

)(1 τG

)(2 τG

lfd

Experimental Concerns: Excitation Focusing & Emission Collection

Excitation side: (1) Laser alignment (2) Chromatic aberration (3) Spherical aberration Emission side: (1) Chromatic aberrations (2) Spherical aberrations (3) Improper alignment of detectors or pinhole (cropping of the beam and focal point position)

We assume exact match of the observation volumes in our calculations which is difficult to obtain experimentally.

lfd

Uncorrelated

Correlated

Two-Color FCS

1)()()()(

)( −>>+<=

tFtFtFtF

Gji

jiij

ττ

Interconverting

2211111 )( NfNftF +=2221122 )( NfNftF +=

Ch.2 Ch.1

450 500 550 600 650 7000

20

40

60

80

100

Wavelength (nm)

%T

For two uncorrelated species, the amplitude of the cross-correlation is proportional to:

+++

+∝ 2

2222121122122112

11211

222211121112

)()0(

NffNNffffNffNffNff

G

lfd

Simulation of cross-correlation Same diffusion =10 µm2/s

Channel 1 Channel 2

Molecules 100

Molecules 50

Molecules 20 20

1/70

1/120

Correlation plot (log averaged)

Tau (s)1E-5 0.0001 0.001 0.01 0.1 1

G(t)

7

6

5

4

3

2

1

0

fraction

The fraction cannot be larger than the smallest of the G(0)’s

Discussion

1. The PSF: how much it affects our estimation of the processes?

2. Models for diffusion, anomalous? 3. Binding? 4. FRET (dynamic FRET)? 5. Bleaching?

6. ……and many more questions

The Photon Counting Histogram: Statistical Analysis of Single Molecule

Populations

Laboratory for Fluorescence Dynamics University of California, Irvine

Transition from FCS

• The Autocorrelation function only depends on fluctuation duration and fluctuation density (independent of excitation power)

• PCH: distribution of intensities (independent of time)

Fluorescence Trajectories

Intensity = 115,000 cps

Intensity = 111,000 cps

Fluorescent Monomer:

Aggregate:

0

5

10

15

20

0 0.1 0.2 0.3 0.4 0.5Time (sec)

k (c

ount

s)

0

5

10

15

20

0 0.1 0.2 0.3 0.4 0.5Time (sec)

k (c

ount

s)

Photon Count Histogram (PCH)

Can we quantitate this?

What contributes to the distribution of intensities?

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

0 3 6 9 12 15 18 21 24k (counts)

log(

occu

renc

es)

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

0 3 6 9 12 15 18 21 24k (counts)

log(

occu

renc

es)

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

0 3 6 9 12 15 18 21 24k (counts)

log(

occu

renc

es)

Fixed Particle Noise (Shot Noise)

0

10

20

30

40

50

60

70

0 2000 4000 6000 8000 10000

Time

Cou

nts

Noise is Poisson ( )kkk

kkPoik

−= exp!

),(

1

10

100

1000

0 20 40 60k (counts)

log(

occu

ranc

es)

Contribution from the detector noise

The Point Spread Function (PSF)

−−= 2

0

2

20

2

322exp),(zzrzrI DG ω

−=

)(4exp

)(4),( 2

2

42

40

2 zr

zzrIGL ωωπ

ω

+=

220

2 1)(Rzzz ωω

λπω20=Rz

One Photon Confocal:

Two Photon:

Contribution from the profile of illumination

Single Particle PCH

+ +

Have to sum up the poissonian distributions for all possible positions of the particle within the PSF

Counts

Log(

Prob

abili

ty)

Counts

Log(

Prob

abili

ty)

Counts

Log(

Prob

abili

ty)

( ) rdrPSFkPoiV

kpV

∫=

0

)(,1)(0

)1( ε

+…

+ +… +

• What if I have two particles in the PSF? • Have to calculate every possible position of

the second particle for each possible position of the first!

0

10

20

30

Time

k8

0

10

20

30

Time

k

8

Combining Distributions

+ =

Particle 1 Particle 2 Together

0

10

20

30

Time

k

8

0 10 20 30 40k (counts)

log(

occu

renc

es)

0 10 20 30 40k (counts)

log(

occu

renc

es)

0 10 20 30 40k (counts)

log(

occu

renc

es)

Contribution from several particles of same brightness

+ =

0 10 20 30 40k (counts)

log(

occu

renc

es)

0

10

20

30

Time

k8

0

10

20

30

Time

k

8

Combining Distributions

+ =

Particle 1 Particle 2 Together

0

10

20

30

Time

k

8

= 0 10 20 30 40

k (counts)

log(

occu

renc

es)

0 10 20 30 40k (counts)

log(

occu

renc

es)

⊗

Convolution • Sum up all combinations of two probability distributions

(joint probability distribution) • Distributions (particles) must be independent

1

10

100

1000

10000

0 20 40 60Intensity

Log

(Pro

babi

lity)

1

10

100

1000

0 50 100Intensity

Log

(Pro

babi

lity)

∑=

=

+ ⋅−=kr

rrprkpkp

0

)2()1()21( )()()(

More Particles

∑=

=

−− ⋅−=⊗=kr

r

nnn rprkpkpkpkp0

)1()1()1()1()( )()()()()(

)()()( )1()1()2( kpkpkp ⊗=+ +…

+ +… )()()( )2()1()3( kpkpkp ⊗=

Contribution from particles of different brightness

How Many Particles Do We Have in the PSF?

∑ ⋅=n

n NnPkpNkPCH ),()(),( )(

),(),( NnPoiNnP =Particle occupation fluctuates around average, N with a poissonian distribution

Calculate poisson weighted average of n particle distributions

0 10 20 30 40particles in PSF

log(

occu

ranc

es)

Multiple Species

• Species are independent so just convolute!

0 10 20 30 40k (counts)

log(

occu

ranc

es)

= 0 10 20 30 40

k (counts)

log(

occu

ranc

es)

0 10 20 30 40k (counts)

log(

occu

ranc

es)

⊗

1 uM Fluorescein 1 uM R110 1 uM Fl & 1uM R110

Chart1

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

k (counts)

log(occurances)

0

0

4

3

22

47

58

100

123

142

129

96

83

64

62

35

28

12

9

3

2

1

1

hist1

BinFrequency

0240

4247

894

1276

1647

2039

2429

2831

3227

3620

4021

4417

4817

5215

5610

6010

6411

688

7210

765

808

845

882

924

962

1000

1044

1081

1120

1160

1201

More0

hist1

Intensity

Log (Probability)

hist2

BinFrequency

013253

4183430

8241335

12260336

16157204

2084123

244372

281546

32431

36121

40021

44017

48017

52015

56010

60010

64011

6808

72010

7605

8008

8405

8802

9204

9602

10000

10404

10801

11200

11600

12001

More0

hist2

Intensity

Log (Probability)

hist3

Intensity

Log (Probability)

convolute

BinFrequency

09

481

8117

12134

16134

2099

2479

2850

3240

3633

4032

4424

4825

5217

5616

6013

6414

6810

7210

7615

807

8411

885

924

966

1005

1042

1082

1123

1161

1201

More1

convolute

Intensity

Log (Probability)

traj

BinFrequencydivider

0240155.84415584420000000006500.02

42472193.8061938062160.3896103896000000009150

8942889.11088911092257.792207792261.038961039000000012050

12763116.88311688312973.3766233766859.240759240849.350649350600000013000

16471882.11788211793207.79220779221131.5684315684694.705294705330.5194805195000007850

20391006.9930069931937.0129870131220.7792207792914.8851148851429.620379620425.324675324700004200

2429515.48451548451036.3636363636737.1628371628987.012987013565.7842157842356.493506493518.83116883120002150

2831179.8201798202530.5194805195394.4055944056596.003996004610.3896103896469.4805194805265.084915084920.129870129900750

322747.952047952185.0649350649201.8981018981318.8811188811368.5814185814506.4935064935349.1008991009283.366633366617.53246753250200

362011.98801198849.350649350670.4295704296163.2367632368197.2027972028305.8441558442376.6233766234373.1768231768246.803196803212.98701298750

4021012.337662337718.781218781256.9430569431100.9490509491163.6363636364227.4225774226402.5974025974325.024975025182.81718281720

4417004.695304695315.184815184835.214785214883.7662337662121.6783216783243.1068931069350.6493506494240.75924075920

48170003.79620379629.390609390629.220779220862.2877122877130.0699300699211.7382617383259.74025974030

521500002.34765234777.792207792221.728271728366.5834165834113.2867132867156.84315684320

5610000001.94805194815.794205794223.226773226857.99200799283.91608391610

60100000001.44855144866.193806193820.229770229842.9570429570

641100000001.54845154855.394605394614.9850149850

688000000001.34865134873.9960039960

72100000000000.9990009990

76500000000000

80800000000000

84500000000000

88200000000000

92400000000000

96200000000000

100000000000000

104400000000000

108100000000000

112000000000000

116000000000000

120100000000000

12400000000000

12800000000000

13200000000000

13600000000000

14000000000000

14400000000000

14800000000000

15200000000000

15600000000000

traj

Intensity

Log (Probability)

Sheet9

00018324180

10001419614400.135335283200

20001832418810.27067056650.27067056650.2706705665

302415225171220.27067056651.08268226590.5413411329

400013169131630.18044704431.62402339880.5413411329

500074972040.09022352221.44357635450.3608940886

600020400202450.03608940890.90223522160.1804470443

700021441212860.0120298030.43307290640.0721788177

800021441213270.00343708660.16841724140.0240596059

900014196143680.00085927160.05499338490.0068741731

1001112144134090.00019094930.01546688950.0017185433

11011121441344100.00003818990.00381898510.0003818985

120416141961848110.00000694360.00084017670.0000763797

13000981952120.00000115730.00016664660.0000138872

14011111211256130.0000001780.0000300890.0000023145

150001936119605.99999412681.9999995853

16000152251564

170002248422681.9999957856

18000214412172

19024172891976

20011121441380

21000141961484

220636214412788

230981266763592

24012144204003296

2508641419622100

260101002248432104

270224841214434108

280204001112131112

290121441522527116

300111522516120

310247499

320001419614

330242248424

340391728920

350001214412

36041686412

370391316916

380111316914

390115256

40000242

410005255

420008648

430245257

4401316998122

450246368

4604164168

4702486410

480394167

49074941611

50098152514

51086452513

52074941611

530244166

540101001112121

5501214486420

560141961214426

570172891419631

580224841419636

590152251625631

600141961419628

610172891522532

6208642772935

630141962144135

640162562248438

650287841728945

660193611419633

67034115686442

6803713691010047

6903814441214450

7003210241522547

7103310892144154

7203411562667660

7305252144126

740101001316923

750152251316928

760183241112129

7707491112118

78063686414

790256251126

800309001010040

8102878474935

82038144498147

83042176498151

8403814441112149

8503310891134

860266762428

8703090063636

8803210243935

89038144498147

90035122598144

9105025001214462

9203310891419647

930277291419641

9401419686422

95052598114

9604161112115

970247499

980117498

990116367

100024113

1010005255

1020008648

1030001522515

1040001625616

1050005255

1060002352923

1070001214412

1080001316913

1090001832418

1100112040021

1110001214412

1120001832418

1130111214413

1140001214412

1150001112111

1160001625616

1170009819

1180006366

1190007497

1200008648

1210007497

1220004164

1230117498

1240005255

125000000

126000242

1270006366

128024395

129011001

130000393

1310114165

1320114165

1330006366

1340395258

13504161112115

1360416246

1370111112112

1380001316913

1390241214414

1400391625619

1410391522518

1420241010012

1430117498

1440009819

1450241522517

1460241419616

1470001010010

1480116367

149000393

1500005255

1510008648

1520007497

1530009819

1540008648

1550009819

1560001625616

1570008648

1580001010010

1590008648

1600009819

1610001214412

1620006366

1630001214412

1640002144121

1650001522515

1660002248422

1670001522515

1680001936119

1690001419614

1700001832418

1710006366

1720009819

1730001010010

1740009819

1750004164

1760001010010

1770001010010

1780008648

1790001112111

1800001419614

1810001728917

1820001522515

1830002352923

1840001728917

1850004164

1860007497

187000242

1880007497

1890001112111

1900005255

1910001010010

1920001010010

1930001112111

1940001522515

1950006366

1960002144121

1970111832419

19804161214416

1990112562526

20005251832423

20102486410

2020001419614

2030006366

20409811112120

2050525116

206063674913

2070395258

208041663610

2090193613922

21001112163617

21101728963623

212074998116

2130101001010020

21401832498127

21501316952518

21601522563621

21701112186419

2180101003913

2190416397

220098141613

2210101001111

2220749249

22301112163617

224052563611

22501214441616

22601214463618

2270131693916

22809811110

2290636117

2300162561117

2310141963917

2320235293926

23305934812461

23407860842480

23509081003993

23605126011152

23707657762478

238081656141685

239094883612144106

240065422598174

24102984141633

242042176452547

2430287843931

24401010041614

24507493910

246063641610

24701214441616

248098152514

24902878474935

25002457641628

25102878441632

25202984198138

253045202574952

25404217641419656

25503196198140

256035122598144

25703612961522551

25804116811419655

25904621161625662

260044193686452

26105631361112167

262010511025749112

26305631361214468

2640204001010030

2650193611214431

26602772974934

26702984152534

2680204001419634

26901728986425

2700235291214435

2710111211010021

27202040098129

27301522586423

27401316998122

2750111211010021

27601214441616

27708641214420

2780214411625637

2790121441522527

28006361112117

2810131691112124

282086486416

283086441612

284011001

28509812411

2860416397

2870749249

288024113

289024113

29005254169

29104164168

29201112152516

29309813912

29401214463618

295039003

296011243

297011112

298011394

2990006366

3000004164

3010118649

3020009819

3030009819

3040008648

3050118649

3060009819

3070004164

3080244166

3090391112114

3100391214415

3110394167

3120395258

3130117498

3140001112111

3150111010011

3160241112113

3170001522515

3180241214414

3190007497

3200008648

3210115256

322039396

3230416246

324011112

325039245

326039114

327000111

328000242

329011243

330000111

331000393

332000111

333000242

334000111

3350004164

3360394167

337011394

338011112

339000393

3400114165

341000393

342011394

343011243

3440007497

3450395258

346011112

3470244166

34804164168

349052563611

3500006366

35102486410

3520391010013

35304161010014

3540111010011

35504165259

35604161010014

35704161010014

35808641010018

3590121441522527

3600101002878438

36107491625623

36207491728924

36304161316917

36404161936123

3650111112112

3660242040022

3670001214412

3680111214413

3690002667626

3700112144122

37104161936123

3720001625616

3730002040020

3740002040020

3750241112113

3760001728917

3770001316913

37801198110

37904161214416

38001198110

38102498111

3820111010011

38303998112

38404161214416

38505251522520

3860391522518

387041698113

3880241214414

3890111211316924

3900111211316924

3910131691010023

3920121441625628

39309811522524

39405251112116

39504161522519

3960395258

39701010074917

398063698115

39901010074917

4000141962040034

40104161112115

402063663612

40305251316918

40404161728921

40503986411

406074998116

4070246368

4080008648

409041674911

41004164168

411074952512

4120395258

413074941611

414039396

415052563611

4160193611112130

417033108941637

418032102463638

41902248474929

42001316986421

421074941611

42205251010015

4230246368

42403986411

4250114165

426052552510

4270101000010

428086498117

429098152514

43003974910

43106361010016

432041663610

43301522541619

43408642410

43508642410

4360141962416

43702667686434

43801625663622

43901832452523

4400256251214437

44102248474929

44201214486420

4430245762426

4440277291112138

44502878452533

446035122598144

4470319612433

44804419363947

44905227042454

45004217643945

45103814441139

452047220952552

45304520253948

454033108941637

455045202541649

456051260198160

457045202552550

458035122563641

45904419361316957

4600266763929

4610266761112137

46205530252144176

46307251841419686

46406846241316981

46506643561522581

46605934811522574

46705126011936170

46805227041522567

46905732491419671

47004520252144166

47105732491419671

47206137211936180

47304621161728963

47404924012040069

47506238441214474

476042176474949

47702667686434

47801832474925

4790121441214424

48001625652521

48102562541629

48202562586433

48302352998132

484039152174946

485049240141653

48604722091728964

48703713691625653

48804419362144165

4890204001728937

4900287841832446

4910298411010039

49201936174926

4930204001112131

4940277292457651

4950266761728943

4960319611625647

497052270498161

49804116811419655

49904924011010059

500038144463644

501055302574962

50207759291214489

50307251841112183

50403713691214449

50504016001316953

506045202598154

50703310891112144

50804924012040069

50904823041112159

51004520251832463

51103411561214446

512034115698143

5130309001522545

5140298411112140

51502040052525

51602144186429

5170214411010031

5180141961936133

5190131691112124

52003974910

5210111211214423

522074963613

5230111211832429

52405251522520

52501522586423

5260183241010028

5270131691316926

52803713691112148

529032102463638

53001832498127

53105251316918

5320131691625629

5330266761522541

53405934812040079

53504217641522557

536061372186469

53707454761625690

538042176486450

53906036001010070

540092846410100102

541055302586463

542061372174968

543074547674981

54407251841112183

545053280941657

546037136986445

5470319611214443

54801316998122

54901112186419

5500111211316924

5510141961010024

5520111211419625

55301832498127

5540193611832437

5550131691316926

556052598114

55702498111

55805251214417

55907491112118

560074932102439

5610242667628

56204162457628

56308641936127

5640112144122

5650241214414

56605251728922

56707491419621

56807492040027

5690162561936135

5700309002248452

57104621163090076

572039152133108972

5730298411625645

5740214411728938

57507491936126

5760101001625626

57706361112117

57804161522519

57905251316918

5800241316915

58105252144126

58204161936123

583098186417

58404161728921

5850241832420

5860001625616

5870241625618

5880001728917

5890111625617

59001198110

5910111010011

5920002144121

5930391214415

59404161316917

5950246368

59602486410

5970241010012

5980116367

59904161010014

600041686412

60101010063616

602074986415

603086498117

604098174916

605086463614

606074941611

60702498111

608074963613

6090193613922

6100204002422

6110162563919

6120193612421

61301832452523

6140277293930

61501936141623

61603210241112143

6170214411010031

61803210241112143

61901010074917

62001214498121

62101936152524

6220287841214440

62301214474919

6240162561625632

62502562574932

62609811832427

62704162144125

62809811419623

6290121441936131

6300131692667639

63108641625624

6320111522516

63304161112115

6340241419616

6350111419615

6360111010011

6370007497

6380392040023

6390242352925

6400001625616

6410241522517

6420001832418

6430001832418

6440002144121

6450391214415

64605251112116

6470392352926

6480391214415

6490241010012

6500111214413

6510111419615

65201198110

6530001214412

6540009819

6550111010011

6560001010010

6570112562526

6580001936119

6590001832418

6600003196131

6610002144121

6620002040020

6630002667626

6640001936119

6650002457624

6660001522515

6670001214412

6680005255

6690005255

6700008648

6710007497

6720117498

6730006366

6740005255

6750008648

6760116367

6770007497

6780006366

6790009819

680000393

6810118649

6820009819

6830001010010

6840111010011

6850111522516

6860007497

6870001010010

6880006366

6890001419614

6900005255

69101198110

6920008648

6930004164

6940116367

6950247499

69607491316920

697086474915

69807491316920

69906361316919

70006361316919

70103986411

7020112144122

70305251522520

70404161936123

70504162040024

7060391316916

7070391214415

7080131691625629

7090121441316925

7100152251112126

7110131691316926

712074963613

7130319611625647

7140214411112132

71501832486426

716086498117

7170001010010

7180115256

7190246368

72001198110

7210005255

7220111214413

7230001010010

72401198110

7250008648

7260008648

7270009819

72802486410

7290117498

7300116367

7310391419617

73202498111

7330115256

7340241214414

7350242248424

7360111010011

7370001316913

7380241522517

7390241214414

7400111936120

74104161214416

74208641010018

74308641112119

7440162561010026

74504419361112155

74605025001316963

74708470561316997

74806440961316977

74908064001522595

750084705686492

751066435698175

75204924011214461

75308165611214493

754069476186477

75506036001316973

756048230498157

757046211698155

75807049001316983

75904318491936162

76004016001832458

7610235291832441

7620224841316935

7630224842040042

7640214412562546

7650224841625638

7660193611010029

7670183242248440

76803512252352958

7690162562248438

77001625686424

7710172891728934

77203310892248455

7730214412144142

7740235293090053

77504161728921

77605252248427

7770121441832430

77807492667633

77906362562531

7800002562525

7810241832420

7820111419615

78304161832422

7840241316915

7850111728918

7860111316914

7870111936120

78802486410

789041674911

7900183241625634

7910277291010037

7920235292248445

7930235292040043

79404823041419662

79506947611728986

796092846419361111

79706643561522581

798041168163647

79903814441010048

800061372186469

801069476163675

802078608441682

803062384452567

804050250063656

805049240186457

806053280998162

80706137211214473

808075562598184

80906036001112171

81007150411010081

81107251841112183

812065422586473

81305631361112167

814053280941657

81506440963967

816034115698143

8170319611010041

81802457674931

8190131691010023

82001214452517

8210241112113

822098198118

8230394167

824086474915

825035122563641

8260266761010036

8270287841010038

82802144186429

8290319611316944

83005934811936178

831058336486466

832039152152544

83302248474929

83406339691214475

835067448998176

83601011020124103

837085722598194

83808267241112193

83908979211010099

840088774474995

841080640020400100

84201041081615225119

84307454761316987

844038144486446

8450309001010040

84606339691316976

847070490063676

8480948836636100

84901011020111121112

85001181392439121

85101011020115225116

852045202586453

85303612961625652

8540277292040047

85503210241112143

8560287841214440

8570245761316937

8580193611316932

8590183241522533

8600101001522525

8610183241010028

862063686414

863063674913

86401419698123

86509811936128

8660111211010021

8670101001214422

868086474915

8690636399

87003974910

87104161112115

872041663610

87301010063616

874052598114

87507491214419

87601112186419

8770241316915

87809813912

879098198118

8800111112112

88108641214420

88204161316917

88305251112116

8840111419615

88504161522519

88605251112116

8870241112113

8880111112112

8890111112112

8900391112114

8910111728918

8920391522518

8930241112113

8940001316913

8950111214413

8960009819

897000393

8980115256

8990001010010

9000001832418

9010117498

9020117498

9030001010010

9040001625616

905000393

9060001112111

9070006366

908011243

9090006366

9100115256

9110004164

912000000

913000393

914000242

915000242

916000000

917000242

918000393

919000000

920000242

921000111

922000242

923000393

924000111

925000393

9260004164

9270007497

9280006366

9290004164

930000242

931000393

932000393

933000111

934000242

935000111

936000000

937000242

938000242

939000242

940000000

941000000

942000242

943000242

944000111

945011112

946000000

947000393

9480416115

9490416397

950000242

951000242

952000111

953000111

954011243

9550009819

9560004164

957000000

958000393

9590114165

960000111

9610007497

962000393

9630001728917

9640118649

9650009819

9660116367

967000393

9680118649

969011394

970098174916

9710121441214424

972086474915

973041674911

9740141961625630

9750172891316930

97603196186439

97703814441316951

97805429162040074

979080640026676106

980077592923529100

98107860841419692

98206542252144186

98304016001625656

98404318491522558

98503090063636

9860245761214436

98703713692439

98802772974934

9890152253918

990063674913

99101522552520

9920121441214424

9930396369

994098141613

995052552510

996063641610

99701112141615

998098141613

999098152514

15.186691.97210.085137.56725.271

461.35740435.8597750

5.00E-012.8362482754

1.75E-019.93E-01

Sheet9

13

183

241

260

157

84

43

15

4

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Log(Probability)

Counts

0

4

8

12

16

20

24

28

32

36

40

44

48

52

56

60

64

68

72

76

80

84

88

92

96

100

104

108

112

116

120

poisson

Time

Counts

Sheet10

BinFrequency

00

40

80

120

161

202

243

2827

3279

36185

40266

44196

48145

5264

5628

604

641

680

More0

0340

10494

20478

304612

404116

503720

604024

704028

803732

903236

1004840

1104044

1204748

1304452

1404156

1504660

1604764

1703868

18045

19051

20040

21033

22043

23035

24041

25045

26043

27039

28040

29023

30039

31030

32039

33038

34039

35039

36045

37042

38038

39045

40044

41049

42034

43030

44032

45051

46049

47046

48030

49048

50041

51036

52050

53045

54041

55036

56050

57034

58035

59040

60040

61038

62028

63043

64037

65043

66026

67046

68037

69031

70041

71048

72038

73039

74029

75032

76036

77047

78040

79043

80047

81036

82035

83042

84047

85042

86046

87033

88035

89037

90044

91048

92046

93041

94037

95044

96035

97044

98045

99034

100040

101049

102037

103027

104044

105046

106036

107045

108043

109034

110047

111034

112046

113036

114046

115032

116027

117051

118045

119039

120037

121048

122039

123042

124033

125046

126042

127037

128041

129031

130041

131034

132038

133035

134039

135043

136037

137035

138042

139031

140043

141049

142037

143045

144060

145036

146034

147043

148034

149035

150045

151038

152043

153042

154033

155054

156039

157041

158031

159043

160041

161038

162032

163040

164037

165046

166029

167039

168039

169033

170040

171038

172046

173034

174056

175033

176047

177032

178040

179043

180039

181036

182041

183045

184020

185044

186046

187029

188041

189044

190038

191045

192040

193047

194054

195039

196044

197037

198038

199039

200046

201037

202037

203042

204037

205047

206030

207032

208040

209050

210045

211042

212046

213044

214045

215033

216036

217039

218046

219050

220037

221041

222048

223033

224031

225034

226046

227041

228035

229053

230040

231046

232044

233039

234038

235035

236037

237046

238053

239053

240040

241039

242034

243035

244037

245041

246041

247043

248036

249049

250036

251038

252036

253035

254039

255036

256034

257036

258043

259040

260041

261040

262036

263034

264036

265052

266042

267037

268056

269052

270037

271044

272047

273035

274034

275045

276039

277047

278035

279045

280031

281030

282040

283034

284037

285038

286038

287030

288039

289040

290041

291048

292030

293034

294037

295030

296037

297037

298042

299048

300052

301038

302018

303035

304032

305046

306033

307036

308033

309041

310041

311031

312038

313034

314049

315051

316052

317047

318032

319044

320030

321039

322052

323038

324043

325037

326032

327039

328038

329052

330055

331027

332036

333038

334045

335044

336045

337048

338042

339037

340039

341056

342040

343030

344038

345041

346041

347049

348036

349038

350028

351038

352035

353044

354037

355040

356039

357033

358034

359046

360035

361034

362055

363037

364038

365047

366032

367046

368029

369039

370043

371043

372042

373036

374039

375036

376050

377040

378046

379047

380043

381038

382056

383035

384047

385038

386040

387051

388041

389039

390030

391039

392039

393034

394037

395039

396038

397048

398032

399040

400036

401040

402036

403040

404043

405036

406042

407041

408034

409051

410053

411038

412037

413039

414036

415039

416034

417039

418037

419041

420051

421046

422041

423043

424042

425047

426052

427052

428047

429034

430034

431039

432053

433049

434051

435041

436035

437047

438038

439037

440036

441032

442041

443044

444044

445033

446030

447040

448036

449046

450041

451047

452044

453047

454038

455055

456044

457050

458043

459044

460026

461042

462041

463030

464039

465031

466033

467036

468037

469055

470039

471034

472038

473036

474052

475041

476030

477047

478046

479040

480035

481038

482034

483048

484042

485041

486034

487051

488040

489033

490046

491034

492038

493039

494041

495044

496036

497031

498028

499032

500044

501046

502033

503034

504055

505047

506032

507039

508039

509036

510032

511037

512038

513036

514040

515030

516041

517041

518038

519044

520034

521027

522024

523045

524038

525035

526039

527056

528030

529039

530042

531043

532035

533046

534039

535044

536041

537043

538041

539043

540047

541047

542054

543043

544044

545036

546044

547038

548057

549036

550047

551038

552041

553033

554032

555043

556042

557036

558031

559043

560043

561036

562033

563037

564037

565032

566049

567050

568040

569040

570038

571040

572034

573045

574037

575050

576030

577039

578035

579036

580036

581048

582025

583053

584045

585045

586049

587036

588047

589044

590033

591041

592050

593037

594030

595037

596049

597026

598036

599046

600040

601043

602045

603041

604039

605037

606042

607037

608040

609036

610040

611040

612036

613031

614035

615040

616051

617034

618032

619028

620038

621054

622049

623042

624038

625035

626038

627050

628030

629047

630037

631046

632053

633051

634045

635045

636039

637040

638043

639027

640049

641036

642038

643039

644041

645034

646057

647039

648035

649045

650038

651046

652033

653033

654040

655042

656044

657042

658042

659048

660044

661041

662044

663037

664040

665048

666040

667031

668028

669036

670041

671045

672024

673051

674044

675046

676042

677044

678033

679036

680044

681044

682033

683035

684034

685040

686026

687036

688039

689042

690038

691033

692037

693038

694053

695045

696048

697046

698031

699040

700041

701043

702038

703041

704040

705036

706043

707042

708039

709047

710045

711029

712048

713038

714047

715036

716047

717038

718034

719055

720030

721035

722037

723039

724044

725031

726045

727037

728035

729044

730036

731032

732043

733045

734038

735039

736042

737043

738043

739029

740042

741045

742038

743036

744058

745025

746037

747034

748050

749039

750027

751051

752035

753040

754052

755033

756049

757032

758046

759037

760026

761046

762042

763045

764035

765061

766040

767037

768033

769042

770045

771037

772039

773038

774036

775050

776037

777037

778040

779029

780041

781044

782038

783035

784032

785054

786037

787050

788040

789032

790038

791044

792045

793043

794033

795035

796035

797041

798043

799045

800034

801036

802043

803044

804043

805035

806051

807041

808036

809038

810037

811051

812039

813035

814033

815051

816044

817046

818029

819038

820042

821038

822046

823036

824042

825045

826044

827035

828039

829042

830048

831040

832039

833035

834052

835047

836045

837037

838044

839053

840048

841043

842034

843042

844045

845037

846033

847048

848030

849036

850045

851025

852033

853027

854031

855039

856040

857036

858041

859043

860042

861038

862036

863042

864043

865033

866043

867042

868033

869036

870037

871037

872033

873040

874034

875048

876041

877042

878028

879039

880041

881048

882046

883036

884040

885029

886049

887039

888037

889054

890027

891038

892039

893035

894041

895051

896031

897039

898052

899031

900046

901045

902042

903046

904052

905054

906048

907041

908039

909036

910051

911025

912049

913040

914039

915046

916038

917047

918043

919035

920035

921052

922046

923036

924025

925043

926029

927045

928038

929038

930047

931034

932038

933039

934040

935040

936037

937032

938045

939043

940038

941038

942039

943053

944043

945043

946038

947044

948041

949042

950036

951026

952043

953046

954049

955039

956032

957040

958040

959038

960035

961032

962040

963029

964027

965037

966038

967038

968035

969035

970043

971031

972038

973041

974037

975041

976028

977043

978032

979035

980037

981046

982033

983041

984034

985043

986037

987047

988034

989043

990032

991035

992051

993039

994039

995031

996042

997033

998037

999039

39.98

Time

Counts

0

4

8

12

16

20

24

28

32

36

40

44

48

52

56

60

64

68

k (counts)

log(occurances)

0

0

0

0

1

2

3

27

79

185

266

196

145

64

28

4

1

0

00.9048374180.36787944120.0497870684

10.09048374180.36787944120.1493612051

20.00452418710.18393972060.2240418077

30.00015080620.06131324020.2240418077

40.00000377020.015328310.1680313557

50.00000007540.0030656620.1008188134

60.00000000130.00051094370.0504094067

700.0000729920.0216040315

800.0000091240.0081015118

Counts

Log(Probability)

Counts

Log(Probability)

Counts

Log(Probability)

Conv. with

1 particle PCH … 3 Particle PCH

Species 1 PCH

Average weighted by number probability

Species 2 PCH …

Final PCH

convolution

total broadening

initial distribution

Recap: Factors that contribute to the final broadening of the PCH

Convolve

w/ Self

1 Particle PCH

2 Particle PCH

Sum over PSF

Fixed Particle Shot Noise

Method

• Sum up Poisson distributions from all possible arrangements and number of fluorophores in excitation volume (PSF) – Intensity weighted sum of all possible single

particle histograms (Poisson functions) – Convolution to get multiple particle histograms – Number probability weighted sum of multiple

particle histograms – Convolution to get multi-species histograms

Chen et al., Biophys. J., 1999, 77, 553.

Fitting

dkkPCHkPCHM

kPCHkPCHMk observedobserved

observedmodel

−

−⋅⋅−

=∑

max

2

2 ))(1()()()(

χ

M is number of observations

d is number of fitting parameters

Chen et al., Biophys. J., 1999, 77, 553.

Model Test

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

0 10 20 30k (counts)

log(

occu

renc

es)

ε = 91,330 cpsm

N = 0.12

ε = 9,030 cpsm

N = 1.28

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

0 10 20 30k (counts)

log(

occu

renc

es)

Hypothetical situation: Protein Interactions

• 2 proteins are labeled with a fluorophore • Proteins are soluble • How do we assess interactions between these

proteins?

Dimer has double the brightness

+ ε = εmonomer ε = 2 x εmonomer

All three species are present in equilibrium mixture

Typical one photon εmonomer = 10,000 cpsm

Photon Count Histogram (PCH)

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

0 2 4 6 8 10 12 14k (counts)

log(

occu

ranc

es)

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

0 2 4 6 8 10 12 14k (counts)

log(

occu

ranc

es)

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

1.E+05

0 2 4 6 8 10 12 14k (counts)

log(

occu

ranc

es)

+ +

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

0 5 10 15k (counts)

log(

occu

renc

es)

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

0 5 10 15k (counts)

log(

occu

renc

es)

Simulation Solution

ε = 9,000 cpsm

N = 1.3

ε = 16,000 cpsm

N = 0.73

+ +

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

0 5 10 15k (counts)

log(

occu

renc

es)

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

0 5 10 15k (counts)

log(

occu

renc

es)

Global Fitting: Fit Data Sets Simultaneously

ε = 9,000 cpsm N = 1.3

ε1 = 9,000 cpsm N1 = 0.29 ε2 = 18,100 cpsm N2 = 0.50

Link

+ +

or

What we measure is the number of particles in the PSF. How Do We Get Concentrations?

• N is defined relative to PSF volume • One photon:

• Two photon:

• Definition is same as for FCS • Can use FCS to determine w0 (and maybe z0)

rdrPSFVPSF

∫= )(( ) 2/30203 2/πzwV DG =

λπ 40

2wVGL =

w0 = 0.21 um, z0 = 1.1 um, VPSF = 0.091 um3, C = 23 nM

How to Improve Accuracy

• Minimize sources of instrument noise – PSF heterogeneity – Shot noise

• Maximize particle burst amplitudes

1E-121E-11

1E-101E-091E-08

1E-071E-061E-05

0.00010.0010.01

0.11

0 2 4 6 8 10

Counts

Prob

abili

ty

0.0000001

0.000001

0.00001

0.0001

0.001

0.01

0.1

1

0 2 4 6 8 10

Counts

Prob

abili

ty

Effect of Brightness

ε = 10,000 cpsm ε = 100,000 cpsm

Poisson PCH

PCH

Poisson

-200

0

200w

eigh

ted

resi

dual

s-200

0

200

wei

ghte

d re

sidu

als

Saturation Effect Rhodamine 110 on the Zeiss Confocor 3

10 uW laser

Laser power is not an infinite source of brightness!

0.00000010.000001

0.000010.0001

0.0010.01

0.11

0 2 4 6k (counts)

log(

occu

renc

es)

60 uW laser

0.000001

0.00001

0.0001

0.001

0.01

0.1

1

0 2 4 6 8 10k (counts)

log(

occu

renc

es)

Multi-Species Fit

0.0000001

0.000001

0.00001

0.0001

0.001

0.01

0.1

1

0 2 4 6 8 10

Counts

Prob

abili

ty

1E-121E-11

1E-101E-091E-08

1E-071E-061E-05

0.00010.0010.01

0.11

0 2 4 6 8 10

Counts

Prob

abili

ty

Concentration Effect

N=1 N=10

Poisson PCH PCH Poisson

Brightness increases by 100%

Brightness increases by 10%

Note: if N is too low, experiment becomes photon limited

Sampling Time Effect

Wu and Mueller, Biophys. J., 2005, 89, 2721.

00.05

0.10.15

0.20.25

0.30.35

1.E-05 1.E-03 1.E-01tau (s)

G(ta

u)

02000400060008000

1000012000

1.E-05 1.E-03 1.E-01Bin Size (s)

Brig

htne

ss (c

psm

)

0

510

15

2025

30

1.E-05 1.E-03 1.E-01Bin Size (s)

Num

ber

Again, shorter sampling leads to photon limited acquisition

In general sample as long as possible without diffusion averaging

PSF X,Y, and Z Dimensions Don’t Matter Z

X

VPSF = 0.08 fL

VPSF = 0.08 fL 1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

0 5 10 15k (counts)

log(

occu

renc

es)

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

0 5 10 15k (counts)

log(

occu

renc

es)

Functional Form DOES Matter

3DG

GL2 poisson

Functional Form Matters for PCH

PSF z-Profile

1.E-101.E-091.E-081.E-071.E-061.E-051.E-041.E-031.E-021.E-011.E+00

0 2 4 6

z (um)

Inte

nsity

(Log

)

3D Gaussian

GL Squared

0.E+00

1.E-01

2.E-01

3.E-01

4.E-01

5.E-01

6.E-01

7.E-01

8.E-01

9.E-01

1.E+00

0 2 4 6

z (um)

Inte

nsity

3D GaussianGL Squared

PCH

0.000001

0.00001

0.0001

0.001

0.01

0.1

1

0 2 4 6 8

k (counts per time bin)

Prob

abili

ty 3D Gaussian

Gauss-Lorentz Squared

Chart1

00

0.050.05

0.10.1

0.150.15

0.20.2

0.250.25

0.30.3

0.350.35

0.40.4

0.450.45

0.50.5

0.550.55

0.60.6

0.650.65

0.70.7

0.750.75

0.80.8

0.850.85

0.90.9

0.950.95

11

1.051.05

1.11.1

1.151.15

1.21.2

1.251.25

1.31.3

1.351.35

1.41.4

1.451.45

1.51.5

1.551.55

1.61.6

1.651.65

1.71.7

1.751.75

1.81.8

1.851.85

1.91.9

1.951.95

22

2.052.05

2.12.1

2.152.15

2.22.2

2.252.25

2.32.3

2.352.35

2.42.4

2.452.45

2.52.5

2.552.55

2.62.6

2.652.65

2.72.7

2.752.75

2.82.8

2.852.85

2.92.9

2.952.95

33

3.053.05

3.13.1

3.153.15

3.23.2

3.253.25

3.33.3

3.353.35

3.43.4

3.453.45

3.53.5

3.553.55

3.63.6

3.653.65

3.73.7

3.753.75

3.83.8

3.853.85

3.93.9

3.953.95

44

4.054.05

4.14.1

4.154.15

4.24.2

4.254.25

4.34.3

4.354.35

4.44.4

4.454.45

4.54.5

4.554.55

4.64.6

4.654.65

4.74.7

4.754.75

4.84.8

4.854.85

4.94.9

4.954.95

55

5.055.05

5.15.1

5.155.15

5.25.2

5.255.25

5.35.3

5.355.35

5.45.4

5.455.45

5.55.5

5.555.55

5.65.6

5.655.65

5.75.7

5.755.75

5.85.8

5.855.85

5.95.9

5.955.95

66

6.056.05

6.16.1

6.156.15

6.26.2

6.256.25

6.36.3

6.356.35

6.46.4

6.456.45

6.56.5

6.556.55

6.66.6

6.656.65

6.76.7

6.756.75

6.86.8

6.856.85

6.96.9

6.956.95

77

7.057.05

7.17.1

7.157.15

7.27.2

7.257.25

7.37.3

7.357.35

7.47.4

7.457.45

7.57.5

7.557.55

7.67.6

7.657.65

7.77.7

7.757.75

7.87.8

7.857.85

7.97.9

7.957.95

88

8.058.05

8.18.1

8.158.15

8.28.2

8.258.25

8.38.3

8.358.35

8.48.4

8.458.45

8.58.5

8.558.55

8.68.6

8.658.65

8.78.7

8.758.75

8.88.8

8.858.85

8.98.9

8.958.95

99

9.059.05

9.19.1

9.159.15

9.29.2

9.259.25

9.39.3

9.359.35

9.49.4

9.459.45

9.59.5

9.559.55

9.69.6

9.659.65

9.79.7

9.759.75

9.89.8

9.859.85

9.99.9

9.959.95

1010

3D Gaussian

GL Squared

3DG

GL^2

z (um)

Intensity

1

1

0.9992003199

0.997423129

0.9968051145

0.9897519974

0.9928258579

0.9771620197

0.9872815716

0.9599357243

0.9801986733

0.9384489084

0.9716107672

0.9131529325

0.9615583782

0.8845545671

0.9500886338

0.8531948884

0.9372548956

0.8196286389

0.9231163464

0.7844052603

0.9077375355

0.748052508

0.8911878885

0.7110632203

0.873541186

0.6738854883

0.8548750167

0.636916185

0.8352702114

0.6004975874

0.8148102622

0.5649166752

0.793580734

0.5304066038

0.7716686739

0.49714983

0.7491620228

0.4652823867

0.7261490371

0.4348988611

0.7027177229

0.4060576977

0.6789552903

0.3787865283

0.6549476305

0.353087306

0.6307788205

0.3289410871

0.6065306597

0.3063123657

0.5822822402

0.285152911

0.5581095554

0.2654050949

0.5340851478

0.2470047214

0.5102777978

0.2298833902

0.486752256

0.2139704338

0.4635690175

0.1991944766

0.4407841408

0.1854846634

0.4184491089

0.1727716045

0.3966107333

0.1609880829

0.3753110989

0.1500695624

0.3545875486

0.1399545335

0.3344727057

0.1305847284

0.3149945317

0.1219052322

0.2961764167

0.1138645116

0.2780373005

0.1064143829

0.260591821

0.0995099316

0.2438504869

0.0931094

0.2278198714

0.0871740494

0.2125028243

0.0816680087

0.1978986991

0.0765581127

0.184003592

0.0718137378

0.1708105895

0.0674066365

0.1583100226

0.0633107759

0.1464897235

0.0595021803

0.1353352832

0.0559587801

0.124830308

0.052660269

0.1149566705

0.0495879676

0.1056947559

0.0467246969

0.0970237

0.0440546584

0.0889216175

0.0415633237

0.0813658197

0.0392373313

0.0743330209

0.0370643912

0.0677995311

0.0350331971

0.0617414361

0.0331333447

0.0561347628

0.0313552578

0.0509556313

0.0296901187

0.0461803914

0.0281298059

0.0417857461

0.0266668359

0.0377488601

0.0252943106

0.0340474547

0.0240058682

0.0306598898

0.0227956393

0.0275652323

0.0216582061

0.0247433132

0.0205885652

0.0221747727

0.0195820935

0.0198410947

0.0186345173

0.0177246314

0.0177418837

0.0158086187

0.0169005346

0.0140771833

0.0161070831

0.0125153423

0.0153583913

0.0111089965

0.0146515509

0.009844917

0.0

Documents

Introduction to FCS · 2018. 4. 12. · Introduction to Fluorescence Correlation Spectroscopy (FCS) Fluctuation analysis goes back to the beginning of the century •Brownian motion