Introduction to FCS PCH analysis Cross-correlation

Transcription

1 Introduction to FCS PCH analysis Cross-correlation Enrico Gratton Laboratory for Fluorescence Dynamics University of California, Irvine

2 From cuvette to the microscope 1. Excitation & Emission Spectra Local environment polarity, fluorophore concentration. Anisotropy & Polarization Rotational diffusion 3. Quenching Solvent accessibility Character of the local environment 4. Fluorescence Lifetime Dynamic processes (nanosecond timescale) 5. Resonance Energy Transfer Probe-to-probe distance measurements 6. Fluorescence microscopy localization 7. Fluorescence Correlation Spectroscopy Translational & rotational diffusion Concentration Dynamics In the microscope, the spatial location matters: spatial correlations and distributions are a component of the experiment

3 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 on.

4 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

5 The fluctuation-dissipation principle 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. amplitude Equilibrium value amplitude Equilibrium value External perturbation Spontaneous fluctuation Synchronized time time Non-synchronized

6 First Application of Correlation Spectroscopy (Svedberg & Inouye, 1911) Occupancy Fluctuation Experimental data on colloidal gold particles: _ _ _ _ _ Collected data by counting (by visual inspection) the number of particles in the observation volume as a function of time using a ultra microscope

7 Particle Correlation Particle Number Frequency time (s) *Histogram of particle counts *Poisson behavior *Autocorrelation not available in the original paper. It can be easily calculated today. Number of Particles Comments to this paper conclude that scattering will not be suitable to observe single molecules, but fluorescence could

8 What can cause a fluctuation in the fluorescence signal??? 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 Each of the above processes has its own dynamics. FCS can recover that dynamics

9 Generating Fluctuations By Motion We need a small volume!! What is Observed? 1. The Rate of Motion. The Concentration of Particles Observation Volume Sample Space 3. Changes in the Particle Fluorescence while under Observation, for example conformational transitions

10 Data presentation and Analysis Counts (in khz) , ,000 Time (in us) The time series 600, Counts (in khz) ,365,000 1,370,000 1,375,000 Time (in us) 1,380,000 Detail of one time region Correlation plot (log averaged) Duration G(t) 3 PCH average 1,000, , ,000 1, E Tau (s) The autocorrelation function N and relaxation time of the fluctuation counts The histogram of the counts in a given time bin (PCH). N and brightness

11 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

12 The definition of the Autocorrelation Function F( t) = F( t) F( t) δ G(τ) = δf(t)δf(t + τ) F(t) Photon Counts time τ Average Fluorescence t t + τ

13 What determines the intensity of the fluorescence signal?? This is the fundamental equation in FCS F( t) = κq dr W ( r) C( r, t) 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 The value of F(t) depends on the profile of illumination!

14 What about the excitation (or observation) volume shape? F ( x, y, z) = I 0 I( z) e ( x + y w 0 ) I( z) I( z) Exp z = w0 z 1 = z 1+ ( w oz ) Gaussian z Lorentzian z More on the PSF in Jay s lecture For the -photon case, these expression must be squared

15 The Autocorrelation Function G(0) 1/N As time (tau) approaches 0 G(τ) Diffusion Time(s) 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

16 The Effects of Particle Concentration on the Autocorrelation Curve Observation volume <N> = 0.3 G(t) <N> = Time (s)

17 Why Is G(0) Proportional to 1/Particle Number? A Poisson distribution describes the statistics of particle occupancy fluctuations. For a Poisson distribution the variance is proportional to the average: < N >= Particle _ Number = Variance G(τ) δf ( t) δf ( t + τ ) G( τ ) = Definition F ( t) F( t) G(0) = δ F( t) = ( F( t) F( t) ) F( t) Time(s) Variance N G ( 0) = = N N = 1 N

18 G(0), Particle Brightness and Poisson Statistics Time Average = 0.75 Variance = 0.56 N Average Variance = = 0.96 Lets increase the particle brightness by 4x: Average = 1.1 Variance = 4.09 N 0.96

19 Effect of Shape on the (Two-Photon) Autocorrelation Functions: For a -dimensional Gaussian excitation volume: ) ( + = DG w D N G τ γ τ For a 3-dimensional Gaussian excitation volume: ) ( + + = DG DG z D w D N G τ τ γ τ -photon equation contains a 8, instead of ) ( + + = D D S N G τ τ τ τ γ τ 3D Gaussian time analysis: with τ D =w /4D and S=w/z

20 Blinking or other exponential processes: If the particle blinks during the times it goes through the illumination volume, an additional term appears in the fluctuation amplitude. How to account for this process?? Reasoning: let us assume that the particle is not moving and it is at the center of the PSF. The intensity will turn ON and OFF. The OFF time depends on the characteristic blinking time (triplet state lifetime). The ON time depends on the laser intensity. The larger the laser intensity, the lesser is the ON time. τ τ T T G( τ ) = (1 + e 1 T ) Triplet state term: T is the triplet state amplitude τ T is the triplet lifetime.

21 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. G Total ( τ ) = G ( τ ) G ( τ ) Blinking Diffusion G Binding ( τ ) = 1 + K f A f K B e λτ K = k f / k b is the equilibrium coefficient; λ = k f + k b is the apparent reaction rate coefficient; and fj is the fractional intensity contribution of species j

22 How different is G(binding) from G(diffusion)? Difference betw een ACF for diffusion and binding Data Diffusion Exponential G(t) e-5 1e-4 1e-3 1e- 1e-1 1e0 1e1 tau With good S/N it is possible to distinguish between the two processes. Most of the time diffusion and exponential processes are combined

23 Table of characteristic times for diffusion Orders of magnitude (for 1 µm solution, small molecule, water) Volume Device Size(µm) Molecules Time milliliter cuvette x microliter plate well x nanoliter microfabrication 100 6x picoliter typical cell 10 6x femtoliter confocal volume 1 6x attoliter nanofabrication 0.1 6x

24 The Effects of Particle Size on the Autocorrelation Curve Diffusion Constants 300 um /s 90 um /s 71 um /s Stokes-Einstein Equation: D = and k T 6 π η r G(t) Slow Diffusion Fast Diffusion Time (s) MW Volume 3 r Monomer --> Dimer Only a change in D by a factor of 1/3, or 1.6

25 Autocorrelation Adenylate Kinase -EGFP Chimeric Protein in HeLa Cells Examples of different Hela cells transfected with AK1-EGFP Fluorescence Intensity Examples of different Hela cells transfected with AK1β -EGFP Qiao Qiao Ruan, Y. Chen, M. Glaser & W. Mantulin Dept. Biochem & Dept Physics- LFD Univ Il, USA

26 G(τ) Autocorrelation of EGFP & Adenylate Kinase -EGFP EGFP-AK in the cytosol EGFP solution EGFP-AKβ in the cytosol EGFP cell Time (s) Normalized autocorrelation curve of EGFP in solution ( ), EGFP in the cell ( ), AK1-EGFP in the cell( ), AK1β-EGFP in the cytoplasm of the cell( ).

27 Autocorrelation of Adenylate Kinase EGFP on the Membrane Clearly more than one diffusion time A mixture of AK1b-EGFP in the cytoplasm and membrane of the cell.

28 Autocorrelation Adenylate Kinaseβ -EGFP Cytosol Plasma Membrane D 10 & D 13/ Diffusion constants (um /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.

29 Two Channel Detection: Cross-correlation Sample Excitation Volume 1. Increases signal to noise by isolating correlated signals.. Corrects for PMT noise Beam Splitter Detector 1 Detector Each detector observes the same particles

30 Removal of Detector Noise by Cross-correlation Detector nm Fluorescein Detector Detector after-pulsing Cross-correlation

31 Calculating the Cross-correlation Function Detector 1: F i time t τ t + t G ij ( τ ) = df ( t) df i F ( t) i j F ( t j + τ ) ( t) Detector : F j time

32 Cross-correlation calculations One uses the same fitting functions you would use for the standard autocorrelation curves. Thus, for a 3-dimensional Gaussian excitation volume one uses: G 1 ( τ ) γ + 4D τ 1 + 4D = N z 1 w τ 1 G 1 is commonly used to denote the cross-correlation and G 1 and G for the autocorrelation of the individual detectors. Sometimes you will see G x (0) or C(0) used for the crosscorrelation.

33 Two-Color Cross-correlation The cross-correlation ONLY if particles are observed in both channels Sample Red filter Green filter Each detector observes particles with a particular color The cross-correlation signal: Only the green-red molecules are observed!!

34 Two-color Cross-correlation Equations are similar to those for the cross correlation using a simple beam splitter: G ij (τ) = df i (t) df j (t + τ) F i (t) F j (t) Information Content Correlated signal from particles having both colors. Autocorrelation from channel 1 on the green particles. Signal G 1 ( τ ) G 1 ( τ ) Autocorrelation from channel on the red particles. G ( τ )

35 Experimental Concerns: Excitation Focusing & Emission Collection We assume exact match of the observation volumes in our calculations which is difficult to obtain experimentally. Excitation side: (1) Laser alignment () Chromatic aberration (3) Spherical aberration Emission side: (1) Chromatic aberrations () Spherical aberrations (3) Improper alignment of detectors or pinhole (cropping of the beam and focal point position)

36 Two-Color Fluctuation Correlation Spectroscopy Uncorrelated G ij < Fi ( t ) F j ( t + τ ) > ( τ ) = 1 < F ( t ) >< F ( t) > i j 100 Correlated F ( = f N + f N t) Ch. 1 1 %T F ( = f N + f N 1 t) Ch Wavelength (nm) Interconverting G 1 For two uncorrelated species, the amplitude of the cross-correlation is proportional to: (0) f f 1 N 1 f + ( f 11 f f + N f 1 + f f 1 ) f N 1 N N + f 1 f N

37 Applications: Cross-correlation Ramesh C Patel, Ujendra Kumar, Don C Lamb, John S Eid, Magalie Rocheville, Michael Grant, Aruna Rani, Theodore L Hazlett, Shutish C Patel, Enrico Gratton, and Yogesh C Patel. Ligand binding to somatostatin receptors induces receptor-specific oligomer formation in live cells. Proc Natl Acad Sci USA. 00; 99(5): PMCID: PMC151

38 Does SSTR1 exist as a monomer after ligand binding while SSTR5 exists as a dimer/oligomer? Collaboration with Ramesh Patel* and Ujendra Kumar* *Fraser Laboratories, Departments of Medicine, Pharmacology, and Therapeutics and Neurology and Neurosurgery, McGill University, and Royal Victoria Hospital, Montreal, QC, Canada H3A 1A1; Department of Chemistry and Physics, Clarkson University, Potsdam, NY Fluorescein Isothiocyanate (FITC) Somatostatin Texas Red (TR) Somatostatin Cell Membrane R1 R1 R5 R5 Three Different CHO-K1 cell lines: wt R1, HA-R5, and wt R1/HA-R5 Hypothesis: R1- monomer ; R5 - dimer/oligomer; R1R5 dimer/oligomer

39 SSTR1 CHO-K1 cells with SST-fitc + SST-tr Green Ch. Red Ch. Very little labeled SST inside cell nucleus Non-homogeneous distribution of SST Impossible to distinguish co-localization from molecular interaction

40 A Monomer G 1 G G 1 G 1 (0) G 1 (0) = 0. G(τ) Minimum 4 0 B Dimer τ(s) 8 G 1 (0) G 1 (0) = 0.71 Maximum G(τ) 6 4 G 1 G G τ(s)

41 Experimentally derived auto- and cross-correlation curves from live R1 and R5/R1 expressing CHO-K1 cells using dual-color two-photon FCS. G(τ) R1 G 1 G G 1 G(τ) R1/R5 G 1 G G τ(s) τ(s) The R5/R1 expressing cells have a greater cross-correlation relative to the simulated boundaries than the R1 expressing cells, indicating a higher level of dimer/oligomer formation. Patel, R.C., et al., Ligand binding to somatostatin receptors induces receptorspecific oligomer formation in live cells. PNAS, (5): p

42 Discussion 1.The PSF: how much it affects our estimation of the processes?.models for diffusion, anomalous? 3. Binding? 4.FRET (dynamic FRET)? 5. Bleaching? 6. and many more questions

43

44 Box size=6.4 µm Diffusion coefficient D=3 µm /s Periodic boundary conditions τ D =w /8D=.6 ms 100 red and 100 blue particles in the box. The detector is sensitive only to the blue particles. The particles perform a random motion in 3D. At random times after excitation, the blue particle (in the singlet state) can convert into the red particle (in the triplet state). After about 10-5 s, the triplet state decays and the particle returns to be blue (singlet state). The particle is only detected when inside the illumination volume (in pink). The intensity is properly weighted according to a 3-D Gaussian intensity model

45 Correlation function for pure diffusion Correlation function for diffusion and excited-state reaction (triplet state) Panel 1: 100 particles in a box of approximately 6.4 µm side and a PSF of 0.5 µm waist and 1.5 µm axial waist. Panel : 00 particles in a box. All particles undergo an excited state reaction with a decay rate of 10-5 s. The system is at equilibrium with half the particles in the triplet excited state. What is the apparent G(0) in panel? Why are the two correlation functions different?

46 Photon counting histogram for the sample with 100 particles in a box (panel 1) and with 00 particles (panel ) undergoing an excited state reaction at a rate of 10-5 s. The system is at equilibrium and half of the particles are in the triplet excited state. Why are the two histograms identical (within noise)?

47

48 Multiple Species Case 1: Species vary by a difference in diffusion constant, D. Autocorrelation function can be used: G(τ) sample = M i=1 f i G(0) i 1 + 8Dτ w DG 1 (D-Gaussian Shape)! G(0) sample = f i G(0) i G(0) sample is no longer g/n!

49 Antibody - Hapten Interactions Binding site Binding site carb Mouse IgG: The two heavy chains are shown in yellow and light blue. The two light chains are shown in green and dark blue..j.harris, S.B.Larson, K.W.Hasel, A.McPherson, "Refined structure of an intact IgGa monoclonal antibody", Biochemistry 36: 1581, (1997). Digoxin: a cardiac glycoside used to treat congestive heart failure. Digoxin competes with potassium for a binding site on an enzyme, referred to as potassium-atpase. Digoxin inhibits the Na-K ATPase pump in the myocardial cell membrane.

50 Anti-Digoxin Antibody (IgG) Binding to Digoxin-Fluorescein triplet state Autocorrelation curves: Digoxin-Fl IgG (99% bound) Digoxin-Fl IgG (50% Bound) Digoxin-Fl 10 Binding titration from the autocorrelation analyses: F b = m S free K d + S free + c S. Tetin, K. Swift, &, E, Matayoshi, 003 Fraction Ligand Bound K d =1 nm [Antibody] free (M)

51 Two Binding Site Model IgG + Ligand-Fl IgG Ligand-Fl+ Ligand-Fl IgG Ligand-Fl % quenching Fraction Bound K d IgG Ligand G(0) IgG Ligand 1.00 No quenching Binding sites Binding sites [Ligand]=1, G(0)=1/N, K d =1.0

52 Digoxin-FL Binding to IgG: G(0) Profile Y. Chen, Ph.D. Dissertation; Chen et. al., Biophys. J (000) 79: 1074

53 Case : Species vary by a difference in brightness assuming that D1 D The quantity G(0) becomes the only parameter to distinguish species, but we know that: G(0) sample = f i G(0) i The autocorrelation function is not suitable for analysis of this kind of data without additional information. We need a different type of analysis

54 The Photon Counting Histogram: Statistical Analysis of Single Molecule Populations

55 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)

56 Fluorescence Trajectories k (counts) Time (sec) Fluorescent Monomer: Intensity = 115,000 cps k (counts) Time (sec) Aggregate: Intensity = 111,000 cps

57 Photon Count Histogram (PCH) 1.E+05 1.E+05 1.E+05 log(occurences) 1.E+04 1.E+03 1.E+0 1.E+01 log(occurences) 1.E+04 1.E+03 1.E+0 1.E+01 log(occurences) 1.E+04 1.E+03 1.E+0 1.E+01 1.E k (counts) 1.E k (counts) 1.E k (counts) Can we quantitate this? What contributes to the distribution of intensities?

58 Contribution from the detector noise Fixed Particle Noise (Shot Noise) Counts Time log(occurances) k (counts) Noise is Poisson Poi( k, k ) = exp( k ) k k! k

59 The Point Spread Function (PSF) = exp ), ( z z r z r I DG ω = ) ( 4 exp ) ( 4 ), ( z r z z r I GL ω ω π ω + = 0 1 ) ( z R z z ω ω λ = πω 0 z R One Photon Confocal: Two Photon: Contribution from the profile of illumination

60 Single Particle PCH Log(Probability) Counts Log(Probability) Counts Log(Probability) Counts Have to sum up the poissonian distributions for all possible positions of the particle within the PSF p (1) ( k) = 1 V 0 V 0 Poi r r ( k, PSF r ) ε ( ) dr

61 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!

62 Contribution from several particles of same brightne Combining Distributions Particle 1 Particle Together k k k Time 0 + = Time 0 Time log(occurences) + log(occurences) = log(occurences) k (counts) k (counts) k (counts)

63 Combining Distributions Particle 1 Particle Together k k k Time 0 + = Time 0 Time log(occurences) log(occurences) = log(occurences) k (counts) k (counts) k (counts)

64 Convolution Sum up all combinations of two probability distributions (joint probability distribution) Distributions (particles) must be independent Log (Probability) Log (Probability) Intensity Intensity p (1+ ) ( k) = r = k r= 0 p (1) ( k r) p () ( r)

65 More Particles = = = = k r r n n n r p r k p k p k p k p 0 1) ( (1) 1) ( (1) ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( (1) (1) () k p k p k p = ) ( ) ( ) ( () (1) (3) k p k p k p = Contribution from particles of different brightn

66 How Many Particles Do We Have in the PSF? P ( n, N) = Poi( n, N ) log(occurances) particles in PSF Particle occupation fluctuates around average, N with a poissonian distribution Calculate poisson weighted average of n particle distributions PCH ( k, N ) = p ( n) ( k) P( n, N ) n

67 Multiple Species Species are independent so just convolute! 1 um Fluorescein 1 um R110 1 um Fl & 1uM R110 log(occurances) log(occurances) = log(occurances) k (counts) k (counts) k (counts)

68 Recap: Factors that contribute to the final broadening of the PCH initial distribution Fixed Particle Shot Noise Sum over PSF 1 Particle Convolve PCH w/ Self Particle PCH Conv. with 3 Particle 1 particle PCH PCH Average weighted by number probability Species 1 PCH Species PCH convolution total broadening Final PCH

69 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.

70 Fitting PCH model ( k) PCH observed ( k) M k M PCH observed ( k) (1 PCH observed ( k)) χ = k d max M is number of observations d is number of fitting parameters Chen et al., Biophys. J., 1999, 77, 553.

71 Model Test 1.E+00 1.E+00 1.E-01 1.E-01 log(occurences) 1.E-0 1.E-03 1.E-04 log(occurences) 1.E-0 1.E-03 1.E-04 1.E-05 1.E-05 1.E E k (counts) k (counts) ε = 9,030 cpsm N = 1.8 ε = 91,330 cpsm N = 0.1

72 Hypothetical situation: Protein Interactions proteins are labeled with a fluorophore Proteins are soluble How do we assess interactions between these proteins?

73 Dimer has double the brightness + ε = ε monomer ε = x ε monomer All three species are present in equilibrium mixture Typical one photon ε monomer = 10,000 cpsm

74 Photon Count Histogram (PCH) E+05 1.E+05 1.E+05 log(occurances) 1.E+04 1.E+03 1.E+0 1.E+01 log(occurances) 1.E+04 1.E+03 1.E+0 1.E+01 log(occurances) 1.E+04 1.E+03 1.E+0 1.E+01 1.E k (counts) 1.E k (counts) 1.E k (counts)

75 Simulation Solution 1.E+00 1.E E-01 1.E-01 log(occurences) 1.E-0 1.E-03 1.E-04 1.E-05 log(occurences) 1.E-0 1.E-03 1.E-04 1.E-05 1.E E k (counts) k (counts) ε = 9,000 cpsm ε = 16,000 cpsm N = 1.3 N = 0.73

76 Global Fitting: Fit Data Sets Simultaneously 1.E+00 1.E E-01 1.E-01 log(occurences) 1.E-0 1.E-03 1.E-04 log(occurences) 1.E-0 1.E-03 1.E-04 1.E-05 1.E-05 1.E k (counts) ε = 9,000 cpsm N = 1.3 Link 1.E k (counts) ε 1 = 9,000 cpsm N 1 = 0.9 ε = 18,100 cpsm N = 0.50 or

77 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: V DG = w ( π / ) 3/ 3 0 z0 Two photon: πw 4 0 V GL = λ Definition is same as for FCS V PSF Can use FCS to determine w 0 (and maybe z 0 ) w 0 = 0.1 um, z 0 = 1.1 um, V PSF = um 3, C = 3 nm r r = PSF( ) dr

78 How to Improve Accuracy Minimize sources of instrument noise PSF heterogeneity Shot noise Maximize particle burst amplitudes

79 Effect of Brightness Probability E-05 1E-06 1E-07 1E-08 PCH 1E-09 1E-10 Poisson 1E-11 1E Counts Probability PCH Poisson Counts ε = 10,000 cpsm ε = 100,000 cpsm

80 Saturation Effect Rhodamine 110 on the Zeiss Confocor 3 60 uw laser 10 uw laser log(occurences) weighted residuals k (counts) Multi- Species Fit log(occurences) weighted residuals k (counts) Laser power is not an infinite source of brightness!

81 Concentration Effect Brightness increases by 100% Brightness increases by 10% Probability E-05 1E-06 1E-07 1E-08 PCH 1E-09 1E-10 Poisson 1E-11 1E N=1 N=10 Probability Counts Counts Poisson PCH Note: if N is too low, experiment becomes photon limited

82 Sampling Time Effect G(tau) E-05 1.E-03 1.E-01 tau (s) Brightness (cpsm) E-05 1.E-03 1.E-01 Bin Size (s) Number E-05 1.E-03 1.E-01 Bin Size (s) Again, shorter sampling leads to photon limited acquisition In general sample as long as possible without diffusion averaging Wu and Mueller, Biophys. J., 005, 89, 71.

83 PSF X,Y, and Z Dimensions Don t Matter V PSF = 0.08 fl 1.E+00 1.E-01 Z V PSF = 0.08 fl log(occurences) 1.E-0 1.E-03 1.E-04 1.E-05 1.E k (counts) X

84 Functional Form DOES Matter 1.E+00 1.E-01 log(occurences) 1.E-0 1.E-03 1.E-04 1.E-05 1.E poisson k (counts) 3DG GL

85 Functional Form Matters for PCH PSF z-profile 1.E+00 9.E-01 8.E-01 PCH 7.E-01 Intensity Intensity (Log) 6.E-01 5.E-01 4.E-01 3.E-01.E-01 1.E-01 0.E E+00 1.E-01 1.E-0 1.E-03 1.E-04 1.E-05 1.E-06 1.E-07 1.E-08 1.E-09 1.E-10 GL Squared 3D Gaussian z (um) GL Squared 3D Gaussian Probability Gauss- Lorentz Squared 3D Gaussian k (counts per time bin) z (um)

86 Point Spread Function Effects p (1) ( k) = 1 V 0 V 0 ( r ) Poi k, ε PSF( r ) dr r This equation will work for ANY PSF shape.

87 Alternative Methods Fluorescence Cumulant Analysis (FCA) Mueller Biophys. J. 004, 86, Similar to method of moments Any distribution can be described by a sum of moments Simple algebraic formulas for cumulants Fluorescence Intensity Distribution Analysis (FIDA) Kask et al. PNAS 1999, 96, Fits PSF in fourier transformed space Fits to non-physical parameterized PSF

88 D PCH Red Channel Counts Green Channel Counts Red Channel Counts Green Channel Counts

89 Calculating the D PCH Function ) ;, ( ) / (1 ) / ( ), ;,, ( k N PCH k k k k N PCH A A k k A k A A B A B A ε ε ε ε ε ε ε = k N k x x k N N k x P = ) (1 ),, ( the binomial distribution: We can find the D PCH function from the single channel PCH function! Chen et al., Biophys. J., 005, 88,

90 Summary The photon count histogram can be modeled by integration of component noise sources Heterogeneous samples can be resolved through global analysis Accuracy is related to magnitude of particle fluctuations relative to instrument fluctuations