Lecture 7 FCS, Autocorrelation, PCH, Cross-correlation

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1 Lecture 7 FCS, Autocorrelation, PCH, Cross-correlation Joachim Mueller Principles of Fluorescence Techniques Laboratory for Fluorescence Dynamics Figure and slide acknowledgements: Enrico Gratton Historic Experiment: 1 st Application of Correlation Spectroscopy (Svedberg & Inouye, 1911) Occupancy Fluctuation time Gold particles _ _ _ _ _ Svedberg and Inouye, Zeitschr. F. physik. Chemie 1911, 77:145 Collected data by counting (by visual inspection) the number of particles in the observation volume as a function of time using a ultra microscope Statistical analysis of raw data required Particle Correlation Graph of Raw Data: Particle Number time (s) 4 5 *Histogram of particle counts *Autocorrelation frequency experiment predicted Poisson number of particles Poisson statistics N =1.55 Autocorrelation not available in the original paper. It can be easily calculated today. 1 N = 1.56 G = ( ) 1

2 What we learn from the correlation function? Svedberg claimed: Gold colloids with radius R = 3 nm Experimental facts: Slit µm G(τ).6.4 characteristic diffusion time τ D 1.5s D Expected kt B µm = 7 6πηR s (Stokes-Einstein) D Exp µm 1 s R nm x Dτ D τ (sec) Conclusion: Bad sample preparation The ultramicroscope was invented in 193 (Siedentopf and Zsigmondy). They already concluded that scattering will not be suitable to observe single molecules, but fluorescence could. Fluorescence Correlation Spectroscopy (FCS) In FCS Fluctuations are in the Fluorescence Signal Diffusion Enzymatic Activity Phase Fluctuations Conformational Dynamics Rotational Motion Protein Folding Example of processes that could generate fluctuations Generating Fluctuations By Motion Sample What is Observed? 1. The Rate of Motion. The Concentration of Particles Observation Volume 3. Changes in the Particle Fluorescence while under Observation, for example conformational transitions Coverslip objective

3 Observation Volume in 1- & -Photon Excitation. 1-Photon: -Photon: Fluorescence excited everywhere along the beam Fluorescence only excited at focal volume 1-photon excitation beam Observation Volume: Defined by the confocal pinhole size, wavelength, magnification and numerical aperture of the objective -photon excitation beam Observation Volume: Defined by wavelength, and numerical aperture of the objective 1-photon pinhole needed to define a small volume -photon Brad Amos MRC, Cambridge, UK Approximately 1 µm 3 Two-Photon FCS Two-photon effect Fluorescence signal at detector F fluctuation time Example: Fluorescently labeled viral particles µm Intensity (cps) time (sec) 55 Time Histogram Data Treatment & Analysis Autocorrelation Intensity (cps) time (sec) Photon Counting Histogram (PCH) g(τ) τ (sec) data fit Autocorrelation Parameters: G() & k rate probability PCH Parameters: <N> & ε Counts per bin 3

4 Autocorrelation Function G(τ) = δf(t)δf(t + τ) F(t) Factors influencing the fluorescence signal: κq = quantum yield and detector sensitivity (how bright is our probe). This term could contain the fluctuation of the fluorescence intensity due to internal processes ( ) = κ r ( r ) ( r, ) F t Q d W C t Average fluorescence signal: F() t W(r) describes our observation volume C(r,t) is a function of the fluorophore concentration over time. This is the term that contains the physics of the diffusion processes Fluorescence fluctuation: δf( t) = F( t) F( t) F(t) in photon counts Calculating the Autocorrelation Function Fluorescence Fluctuation df() t = F() t F G(τ) Lag time τ (s) τ τ t + τ t + τ t time Average Fluorescence F δ Ft () δft ( + τ) G( τ ) = F The Autocorrelation Function t 3 t 4 t t 1 t 5 Detected Intensity (kcps) Time (s).4 G() 1/N As time (tau) approaches G(τ).3..1 Diffusion G(τ) = δf(t)δf(t + τ) F(t) Time(s) 4

5 The Effects of Particle Concentration on the Autocorrelation Curve.5.4 <N> =.3 G(t)..1 <N> = Time (s) Why Is G() Proportional to 1/Particle Number? A Poisson distribution describes the statistics of particle occupancy fluctuations. In a Poissonian system the variance is proportional to the average number of fluctuating species: Particle _ Number = Variance G(τ) δf ( t) δf ( t + τ ) G( τ ) = F ( t) F( t) G() = δ F( t) = ( F( t) F( t) ) F( t) Time(s) Variance G( ) = = N 1 N G(), Particle Brightness and Poisson Statistics Time Average =.75 Variance =.56 N Average.75 Variance = = Lets increase the particle brightness by 4x: Average = 1.1 Variance = 4.9 N.96 5

6 Effect of Shape on the (-Photon) Autocorrelation Functions: (for simple diffusion) For a 3-dimensional Gaussian excitation volume: 1 γ 8Dτ 8Dτ G( τ ) = N w z 1 Model of observation volume For a -dimensional Gaussian excitation volume: γ 8Dτ G( τ ) = 1+ N w 1 1-photon equation contains a 4, instead of 8 γ: shape factor (.354 for 3DG,.5 for DG) N: average number of particles inside volume D: Diffusion coefficient w o : radial beam waist of two-photon laser spot z o : axial beam waist of two-photon laser spot Additional Equations: 3D Gaussian Confocor analysis: G(τ) = N 1 + τ τ D 1 τ 1 + S τ D 1... where N is the average particle number, τ D is the diffusion time (related to D, τ D =w /8D, for two photon and τ D =w /4D for 1-photon excitation), and S is a shape parameter, equivalent to w/z in the previous equations. F( t+ τ ) F( t) Note: The offset of one is caused by a different definition of G(τ) : G( τ ) = F Triplet state term: T 1+ e 1 T τ τ ( T )..where T is the triplet state amplitude and τ T is the triplet lifetime. Orders of magnitude (for 1nM solution, small molecule, water) Diffusion Volume Device Size(µm) Molecules Time (s) milliliter cuvette 1 6x microliter plate well 1 6x1 9 1 nanoliter microfabrication 1 6x1 6 1 picoliter typical cell 1 6x femtoliter confocal volume 1 6x1 1-4 attoliter nanofabrication.1 6x

7 The Effects of Particle Size on the Autocorrelation Curve Diffusion Constants.5 3 um /s 9 um /s 71 um /s Stokes-Einstein Equation: D = and k T 6 π η r G(t) Fast Diffusion Slow Diffusion Time (s) MW Volume r 3 Monomer --> Dimer Only a change in D by a factor of 1/3, or 1.6 FCS inside living cells Two-Photon Spot Correlation Analysis D solution D nucleus = 3.3 inside nucleus Coverslip objective g(τ).4. in solution. 1E-5 1E-4 1E τ (sec) Measure the diffusion coefficient of Green Fluorescent Protein (GFP) in aqueous solution in inside the nucleus of a cell. 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 7

8 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( ). 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. Autocorrelation Adenylate Kinaseβ -EGFP Cytosol Plasma Membrane D 1 & 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. 8

9 Multiple Species Case 1: Species differ in their diffusion constant D Autocorrelation function can be used: G(τ) sample = M i=1 f i G() i 1 + 8Dτ w DG 1 (D-Gaussian Shape)! f i is the fractional fluorescence intensity of species i. G() sample = f i G() i G() sample is no longer γ/n! 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. Anti-Digoxin Antibody (IgG) Binding to Digoxin-Fluorescein Autocorrelation curves: triplet state Digoxin-Fl Digoxin-Fl IgG (99% bound) Digoxin-Fl IgG (5% Bound) 1 Binding titration from the autocorrelation analyses: F b = m S free + c K d + S free S. Tetin, K. Swift, &, E, Matayoshi, 3 Fraction Ligand Bound Kd=1 nm [Antibody] free (M) 9

10 Two Binding Site Model IgG + Ligand-Fl IgG Ligand-Fl + Ligand-Fl IgG Ligand-Fl % quenching Fraction Bound.6.4 K d IgG Ligand G() IgG Ligand 1. No quenching Binding sites Binding sites [Ligand]=1, G()=1/N, K d =1. Digoxin-FL Binding to IgG: G() Profile Y. Chen, Ph.D. Dissertation; Chen et. al., Biophys. J () 79: 174 Multiple Species Case : Species vary by a difference in brightness assuming that D1 D The quantity G() becomes the only parameter to distinguish species, but we know that: G() sample = f i G() i The autocorrelation function is not suitable for analysis of this kind of data without additional information. We need a different type of analysis 1

11 Photon Counting Histogram (PCH) Aim: To resolve species from differences in their molecular brightness Poisson Distribution for particle number: p(n ) = N N e N N! But distribution of photon counts is Non-Poissonian: p(k) = PCH(ε, N ) Single Species: p(k) is the probability of observing k photon counts Sources of Non-Poissonian Noise Detector Noise Diffusing Particles in an Inhomogeneous Excitation Beam* Particle Number Fluctuations* Multiple Species* Photon Counting Histogram (PCH) Aim: To resolve species from differences in their molecular brightness Molecular brightness ε : The average photon count rate of a single fluorophore PCH: probability distribution function p(k) where p(k) is the probability of observing k photon counts ε = 16cpsm N =.3 Single Species: p(k) = PCH(ε, N ) Note: PCH is Non-Poissonian! Sources of Non-Poissonian Noise Detector Noise Diffusing Particles in an Inhomogeneous Excitation Beam* Particle Number Fluctuations* Multiple Species* PCH Example: Differences in Brightness frequency (ε n =1.) (ε n =.) (ε n =3.7) Increasing Brightness Photon Counts 11

12 Single Species PCH: Concentration 5.5 nm Fluorescein Fit: ε = 16, cpsm N =.3 55 nm Fluorescein Fit: ε = 16, cpsm N = 33 As particle concentration increases the PCH approaches a Poisson distribution Photon Counting Histogram: Multiple Species Binary Mixture: p(k) = PCH(ε 1, N 1 ) PCH(ε, N ) Brightness Concentration Snapshots of the excitation volume Intensity Time Photon Counting Histogram: Multiple Species Sample : many but dim (3 nm fluorescein at ph 6.3) Sample 1: fewer but brighter fluors (1 nm Rhodamine) Sample 3: The mixture The occupancy fluctuations for each specie in the mixture becomes a convolution of the individual specie histograms. The resulting histogram is then broader than expected for a single species. p(k) = PCH(ε 1, N 1 ) PCH(ε, N ) 1

13 Resolve a protein mixture with a brightness ratio of two Alcohol dehydrogenase labeling experiments Singly labeled proteins Mixture of singly or doubly labeled proteins + residuals/σ log(pch) k residuals/σ log(pch) k Both species have same color fluorescence lifetime diffusion coefficient polarization c Sample A Sample B ε 1 kcpsm N 1 ε kcpsm N PCH in cells: Brightness of EGFP Molecular Brightness (cpsm) 3x1 4 x1 4 1x1 4 autofluorescence cytoplasm Excitation=895nm EGFP solution EGFP nucleus EGFP cytoplasm autofluorescence nucleus The molecular brightness of EGFP is a factor ten higher than that of the autofluorescence in HeLa cells Chen Y, Mueller JD, Ruan Q, Gratton E () Biophysical Journal, 8, 133. Brightness Encodes Stoichiometry Sample Fluorescence Brightness EGFP Protein F time ε N 1 ε F time N ε + F ε time N 1 N 13

14 Brightness and Stoichiometry Intensity (cps) x EGFP Brightness ε app (cpsm) 75 5 EGFP EGFP 5 EGFP Brightness 1 1 Concentration [nm] EGFP EGFP Brightness of EGFP is twice the brightness of EGFP Chen Y, Wei LN, Mueller JD, PNAS (3) 1, Distinguish Homo- and Hetero-interactions in living cells ECFP: EYFP: Apparent Brightness A B A B A B + A A λ γ = 95nm A B ε ε ε ε ε λ γ = 965nm A B ε ε ε ε single detection channel experiment distinguish between CFP and YFP by excitation (not by emission)! brightness of CFP and YFP is identical at 95nm (with the appropriate filters) you can choose conditions so that the brightness is not changed by FRET between CFP and YFP determine the expressed protein concentrations of each cell! PCH analysis of a heterodimer in living cells The nuclear receptors RAR and RXR form a tight heterodimer in vitro. We investigate their stoichiometry in the nucleus of COS cells. We expect: RAR RXR 3. λ γ = 95nm 3. λ γ = 965nm.5.5 ε app /ε monomer ε app /ε monomer RXR agonist.5 + RXR agonist total protein concentration [nm].5 - RXR agonist + RXR agonist RXRLBD-YFP [nm] Chen Y, Li-Na Wei, Mueller JD, Biophys. J., (5) 88,

15 Two Channel Detection: Cross-correlation Sample Excitation Volume 1. Isolate correlated signals.. Corrects for PMT noise Beam Splitter Detector 1 Detector Each detector observes the same particles Removal of Detector Noise by Cross-correlation Detector nm Fluorescein Detector Detector after-pulsing Cross-correlation Calculating the Cross-correlation Function Detector 1: F i time t τ t + τ G ( τ ) = ij df ( t) df ( t + τ ) i F ( t) F ( t) i j j Detector : F j time 15

16 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 ( τ ) 1 γ 8 D τ D τ 1 + = 1 1 N z 1 w 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 () or C() used for the cross-correlation. 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!! 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) 16

17 Two-color Cross-correlation Equations are similar to those for the cross correlation using a simple beam splitter: G ij (τ) = df (t) df (t + τ) i j F i (t) F j (t) Information Content Correlated signal from particles having both colors. Signal G 1 ( τ ) Autocorrelation from channel 1 on the green particles. G 1 ( τ ) Autocorrelation from channel on the red particles. G ( τ ) Uncorrelated Correlated Spectral Crosstalk G 1 (τ) A + B uncorrelated A + B correlated A + B uncorrelated, but crosstalk log τ 1 Uncorrelated+Crosstalk Ch. 8 6 Ch.1 %T Wavelength (nm) Two-Color FCS: Correct for Spectral Overlap Uncorrelated F ( f N + f N t) 8 Ch. Ch.1 = %T 1 4 F ( = f N + f N t) Wavelength (nm) For two uncorrelated species, the amplitude of the cross-correlation is proportional to: uncorrelated f11 f1 N1 + f1 f N G1 () f11 f1 N1 + ( f11 f + f1 f1) N1 N + f f N 1 G 1 correlated () G 1 uncorrelated () G 1 f ij : fractional intensity of species i in channel j A + B correlated A + B uncorrelated, but crosstalk log τ 17

18 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 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 A Monomer G 1 G G 1 B Dimer G 1 () G 1 () =. G(τ) 4 Minimum τ(s) G 1 () G 1 () =.71 6 Maximum 4 G(τ) G 1 G G τ(s) 18

19 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 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. G(τ) R1/R5 G 1 G G 1 Patel, R.C., et al., Ligand binding to somatostatin receptors induces receptorspecific oligomer formation in live cells. PNAS,. 99(5): p Molecular Dynamics What if the distance/orientation is not constant? (a) High FRET CFP YFP Fluorescence fluctuation can result from FRET or Quenching (b) -4 Ca Ca + calmodulin M13 FCS can determine the rate at which this occurs CFP Low FRET YFP This will yield hard to get information (in the µs to ms range) on the internal motion of biomolecules (c) CFP trypsin + YFP NO FRET Fluorescence Intensity (cps) trypsin-cleaved cameleon CFP cameleon Ca + -depleted cameleon Ca + -saturated YFP Wavelength (nm) 19

20 A) B) 16 1 G(τ) τ (s). A) Cameleon fusion protein consisting of ECFP, calmodulin, and EYFP. [Truong, 1 #193] Calmodulin undergoes a conformational change that allows the ECFP/EYFP FRET pair to get cl ose enough for efficient energy transfer. Fluctuations between the folded and unfolded states will yield a measurable kinetic component for the cross-correlation. B) Simulation of how such a fluctuation would show up in the autocorrelation and cross-correlation. Red dashed line indicates pure diffusion. In vitro Cameleon Data.1 Ca + Saturated.8.6 Donor Ch. Autocorrelation Acceptor Ch. Autocorrelation Cross-correlation G(τ).4.. Crystallization And Preliminary X-Ray Analysis Of Two New Crystal Forms Of Calmodulin, B.Rupp, D.Marshak and S.Parkin, Acta Crystallogr. D 5, 411 (1996) τ (s) Are the fast kinetics (~ µs) due to conformational changes or to fluorophore blinking? Dual-color PCH analysis (1) Fluorescence F(t) Time t δf A <F A > Fluorescence F(t) Time t Cross-Correlation Dual-Color PCH δf B <F B >

21 Signal A Signal B T sample Brightness in each channel: ε A, ε B Average number of molecules: N T sample Dual-color PCH analysis () Signal A T sample Signal B T sample Single Species: pk (, k ) = PCH( ε, ε, N) A B A B Brightness in each channel: ε A, ε B Average number of molecules: N Resolve Mixture of ECFP and EYFP in vitro fluctuations channels Dual-Color PCH species model χ = species model χ = ε A (cpsm) EYFP alone ECFP alone species 1 from fit species from fit ε B (cpsm) Chen Y, Tekmen M, Hillesheim L, Skinner J, Wu B, Mueller JD, Biophys. J. (5), ECFP & EYFP mixture resolved with single histogram. Note: Cross-correlation analysis cannot resolve a mixture of ECFP & EYFP with a single measurement! 1