The Photon Counting Histogram: Statistical Analysis of Single Molecule Populations
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1 The Photon Counting Histogram: Statistical Analysis of Single Molecule Populations E. Gratton Laboratory for Fluorescence Dynamics University of California, Irvine
2 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
3 Fluorescence Trajectories counts Time sec Fluorescent Monomer: Intensity = 115, cps counts Time sec Aggregate: Intensity = 111, cps
4 Photon Count Histogram PCH 1.E+5 1.E+5 1.E+5 logoccurences 1.E+4 1.E+3 1.E+ 1.E+1 logoccurences 1.E+4 1.E+3 1.E+ 1.E+1 logoccurences 1.E+4 1.E+3 1.E+ 1.E+1 1.E counts 1.E counts 1.E counts Can we quantitate this? What contributes to the distribution of intensities?
5 Contribution from the detector noise Fixed Particle Noise Shot Noise Counts Time logoccurances counts Noise is Poisson Poi, exp!
6 The Point Spread Function PSF 3 exp, z z r z r I DG 4 exp 4, 4 4 z r z z r I GL 1 z R z z z R One Photon Confocal: Two Photon: Contribution from the profile of illumination
7 Single Particle PCH LogProbability Counts LogProbability Counts LogProbability Counts Have to sum up the poissonian distributions for all possible positions of the particle within the PSF p 1 1 V V Poi, PSF r dr
8 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!
9 Contribution from several particles of same brightness Combining Distributions Particle 1 Particle Together Time + = Time Time logoccurences + logoccurences = logoccurences counts counts counts
10 Combining Distributions Particle 1 Particle Together Time + = Time Time logoccurences logoccurences = logoccurences counts counts counts
11 Convolution Sum up all combinations of two probability distributions joint probability distribution Distributions particles must be independent 1 1 Log Probability Log Probability Intensity Intensity p 1 r r p 1 r p r
12 More Particles r r n n n r p r p p p p p p p p p p Contribution from particles of different brightness
13 How Many Particles Do We Have in the PSF? P n, N Poi n, N logoccurances particles in PSF Particle occupation fluctuates around average, N with a poissonian distribution Calculate poisson weighted average of n particle distributions PCH, N n p P n, N n
14 Multiple Species Species are independent so just convolute! 1 um Fluorescein 1 um R11 1 um Fl & 1uM R11 logoccurances logoccurances = logoccurances counts counts counts
15 Recap: Factors that contribute to the final broadening of the PCH initial distribution Fixed Particle Shot Noise Sum over PSF 1 ParticleConvolve 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
16 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.
17 Fitting PCH model PCH observed M M PCH observed 1 PCH observed d max M is number of observations d is number of fitting parameters Chen et al., Biophys. J., 1999, 77, 553.
18 Model Test 1.E+ 1.E+ 1.E-1 1.E-1 logoccurences 1.E- 1.E-3 1.E-4 logoccurences 1.E- 1.E-3 1.E-4 1.E-5 1.E-5 1.E E counts counts = 9,3 cpsm N = 1.8 = 91,33 cpsm N =.1
19 Hypothetical situation: Protein Interactions proteins are labeled with a fluorophore Proteins are soluble How do we assess interactions between these proteins?
20 Dimer has double the brightness + = monomer = x monomer All three species are present in equilibrium mixture Typical one photon monomer = 1, cpsm
21 Photon Count Histogram PCH E+5 1.E+5 1.E+5 logoccurances 1.E+4 1.E+3 1.E+ 1.E+1 logoccurances 1.E+4 1.E+3 1.E+ 1.E+1 logoccurances 1.E+4 1.E+3 1.E+ 1.E+1 1.E counts 1.E counts 1.E counts
22 Simulation Solution E+ 1.E+ 1.E-1 1.E-1 logoccurences 1.E- 1.E-3 1.E-4 1.E-5 logoccurences 1.E- 1.E-3 1.E-4 1.E-5 1.E E counts counts = 9, cpsm = 16, cpsm N = 1.3 N =.73
23 Global Fitting: Fit Data Sets Simultaneously E+ 1.E+ 1.E-1 1.E-1 logoccurences 1.E- 1.E-3 1.E-4 logoccurences 1.E- 1.E-3 1.E-4 1.E-5 1.E-5 1.E counts = 9, cpsm N = 1.3 Lin 1.E counts = 9, cpsm N 1 =.9 = 18,1 cpsm N =.5 or
24 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 / 3/ 3 z Two photon: w 4 V GL Definition is same as for FCS V PSF Can use FCS to determine w and maybe z w =.1 um, z = 1.1 um, V PSF =.91 um 3, C = 3 nm PSF r dr
25 How to Improve Accuracy Minimize sources of instrument noise PSF heterogeneity Shot noise Maximize particle burst amplitudes
26 Effect of Brightness Probability E-5 1E-6 1E-7 1E-8 PCH 1E-9 1E-1 Poisson 1E-11 1E Counts Probability PCH Poisson Counts = 1, cpsm = 1, cpsm
27 Saturation Effect Rhodamine 11 on the Zeiss Confocor 3 6 uw laser 1 uw laser 1.1 logoccurences weighted residuals counts Multi Species Fit logoccurences.1 weighted residuals counts Laser power is not an infinite source of brightness!
28 Concentration Effect Brightness increases by 1% Brightness increases by 1% Probability E-5 1E-6 1E-7 1E-8 PCH 1E-9 1E-1 Poisson 1E-11 1E N=1 N=1 Probability Counts Counts Poisson PCH Note: if N is too low, experiment becomes photon limited
29 Sampling Time Effect Gtau E-5 1.E-3 1.E-1 tau s Brightness cpsm E-5 1.E-3 1.E-1 Bin Size s Number E-5 1.E-3 1.E-1 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., 5, 89, 71.
30 PSF X,Y, and Z Dimensions Don t Matter V PSF =.8 fl 1.E+ 1.E-1 Z V PSF =.8 fl logoccurences 1.E- 1.E-3 1.E-4 1.E-5 1.E counts X
31 Functional Form DOES Matter 1.E+ 1.E-1 logoccurences 1.E- 1.E-3 1.E-4 1.E-5 1.E counts poisson 3DG GL
32 Functional Form Matters for PCH PSF z Profile 1.E+ 9.E-1 8.E-1 PCH 7.E-1 Intensity Intensity Log 6.E-1 5.E-1 4.E-1 3.E-1.E-1 1.E- 1.E E+ 1.E-1 1.E- 1.E-3 1.E-4 1.E-5 1.E-6 1.E-7 1.E-8 1.E-9 1.E-1 3D Gaussian GL Squared z um GL Squared 3D Gaussian 4 6 Probability Gauss- Lorentz Squared 3D Gaussian counts per time bin z um
33 Point Spread Function Effects p 1 1 V V Poi, PSF r dr This equation will wor for ANY PSF shape.
34 Alternative Methods Fluorescence Cumulant Analysis FCA Mueller Biophys. J. 4, 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 Kas et al. PNAS 1999, 96, Fits PSF in fourier transformed space Fits to non physical parameterized PSF
35 D PCH Red Channel Counts Green Channel Counts Red Channel Counts Green Channel Counts
36 Calculating the D PCH Function ;, / 1 /, ;,, N PCH N PCH A A A A A B A B A N x x N N x P 1,, the binomial distribution: We can find the D PCH function from the single channel PCH function! Chen et al., Biophys. J., 5, 88,
37 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
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