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Data Analysis Issues in Time-Resolved Fluorescence Anisotropy Bill Laws Department of Chemistry University of Montana Support from NSF EPSCoR, NIH, and University of Montana Nick Vyleta, Abe Coley, Dawn Paul, Sandy Ross

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The Issues The Project Large number of iterated parameters Differentiate numerous kinetic models that are very similar Apply techniques to macromolecular systems More complex systems even more parameters and kinetic models Requires a very fast measuring technique such as time-resolved fluorescence anisotropy Expand/develop methods in both data collection and data analysis Enhance understanding of macromolecular structure and function Emphasis on rapid dynamics of small regions that is, the kinetics of local motions Correlations between parameters i e -t/ Multiple sets of acceptable values - - which is correct? j e -t/ j

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Fluorescence Ground state: all electrons of molecule in their lowest energy orbitals Absorption: an electron acquires energy (E 1 ) from a photon, moving it to a higher orbital Excited state: the molecule with an electron in a higher orbital energy lost from molecular reorganization (IC) photochemistry may occur Emission: molecule returns to ground state as electron loses energy by emitting a photon A molecule has electronic, vibrational, rotational, and translational energies At room temperature, only rotational and vibrational higher energy states will be populated S0S0 S1S1 S2S2 S3S3 SiSi Absorption E = hv Internal Conversion Time Scales Abs.: ~ 10 –15 s IC.: 10 –12 s Fluor: ~ 10 –9 s Frank-Condon state Fluorescence Jablonski Diagram = h/

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Wavelength (nm) Relative Intensity or Absorbance Absorbance Fluorescence Representative Spectra

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Steady-State Fluorescence Time per collection of a data point is long ( s or greater) Provides value proportional to the number of photons emitted in that time period No kinetic information about the mechanism of fluorescence Fluorescence spectrum obtained by steady-state fluorescence Time-Resolved Fluorescence Photon counting Phase and modulation Kinetics on the nanosecond time scale Therefore need picosecond resolution for data

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Time-Resolved Fluorescence A A* hv k nr kfkf hv Typical rates k nr : 10 7 – 10 10 s -1 k f : 10 7 – 10 9 s -1 Typical lifetime of 1 to 100 ns

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Representation of the Collection of a Decay Curve ( = 5 ns)

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‘Time broadening’ due to response time of the instrumentation Instrument Response Function (IRF) Counts Time (ns)

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Counts Time (ns) ‘Time broadening’ due to response time of the instrumentation F(t) = ∫ I RF(t - s) I (s) ds 0 t Instrument Response Function (IRF) Counts Time (ns)

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22 Data Fitting F(t) = ∫ IRF(t - s) I(s) ds 0 t Collect both F(t) and IRF(t), and kinetic model yields decay function, I(t) With ‘guestimates’, generate an initial fit by convolving I(t) with IRF(t) parameter 22 Also use weighted residuals, autocorrelation of residuals, and parameter confidence limits to evaluate fits Iterate parameters via non-linear least squares, minimizing reduced 2 2 = [F c (k) – F(k)] 2 k = [F(k)] 1/2 1k2 1k2 1 n-p k=1 n

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3 exp 2 exp 1 exp AC(j) = w i 1/2 i (w i+j ) 1/2 +j / w i i 2 1m1m 1n1n m n i=1 Resid(t) = [F c (t) - F(t)] / [F(t)] 1/2 j from 1 to (n-m) 1n1n i = F c (i) - F(i)w i = [1/F(t)] / 1/F(t)] i=1 n

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Fluorescence Anisotropy Photon absorption requires its E vector to align (cos 2 ) with absorption dipole moment Pulse of polarized light (vertical V) excites a defined population If molecules ‘frozen’, only detect emitted photons with V polarization If free to rotate, depolarization occurs and detect both V and H intensities V and H intensities dependent on rates of depolarizing motions Steady-state anisotropyTime-resolved anisotropy = I V – I H I V + 2 I H r(t) = I V (t) – I H (t) I V (t) + 2 I H (t) provides information on size, shape, and stability r(t) provides the same information as well as mechanism(s) of depolarization by kinetics laser sample polarizer detector rotate

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IM(t)IM(t) IV(t)IV(t) IH(t)IH(t) = = 5 ns Our Anisotropy Method Collect I M (t), I V (t), and I H (t) decays (along with each IRF) These three decays constitute a dataset I V (t) = e {1 + 2 r(t)} -t/ I H (t) = e {1 - r(t)} -t/ Note: exponential times exponential One lifetime and one rotational correlation time Iterating 12 parameters: , , , , 2 scalars, and 3 x 2 instrumental factors Simultaneous (global) analysis of the I M, I V, and I H dataset Over determination of common parameters , , , and I M (t) = e 1313 -t/ r(t) = e -t/ = I V (t) – I H (t) I V (t) + 2 I H (t)

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Real World Complexities Typical to require more than one lifetime (n > 1) Likely to have more than one rotational correlation time (m > 1) I M (t) = i e i = 1 n -t/i-t/i Sums of exponentials for intensity decay Lifetimes Correlation Times Ground-state heterogeneity from multiple fluorophores and/or environments i terms a function of several factors, including concentrations and spectral overlap Resolution capabilities require 2 ~ 2 1 ; with a global analysis need ~20% difference Although fits decay curve, correct kinetic model might not yield sums of exponentials r(t) = ij e j = 1 m -t/j-t/j Sums of exponentials for anisotropy decay Different j values for macromolecular rotations, segmental motions, and local motions i term is limiting anisotropy for fluorophore i at t = 0 - - must be between -0.2 and 0.4 Resolution capabilities about a factor of 2; global analysis resolves ~40% difference

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Sums of exponentials times sums of exponentials Large number of correlated parameters Yields multiple sets of acceptable values Multiple kinetic models possible ij terms represent the association of a fluorophore (i) with a depolarizing motion (j) If a ij term equals zero, fluorophore i not depolarized by motion j For our systems, all fluorophores depolarized by macromolecular rotations Difference in kinetic models will be from segmental and local motions Multiple models with different sets of parameters can fit the data n m 1 2 3 4 1 1 3 7 15 2 1 9 49 225 3 1 27 343 3375 4 1 81 2401 50625 Number of Kinetic Models (2 n – 1) m I V (t) = i e {1 + 2 r(t)} i =1 n -t/i-t/i I M (t) = i e i =1 n -t/i-t/i 1313 I H (t) = i e {1 - r(t)} i =1 n -t/i-t/i r i (t) = ij e -t/ j m j =1 r(t) = i =1 n -t/i-t/i { i e ij e } j =1 m -t/ j i =1 n -t/i-t/i i e

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Kinetic Models for a 2 x 2 Profile 1111 1111 1111 1111 1111 1111 1111 1111 2222 2222 2222 2222 2222 2222 2222 2222 f1m1f2m2f1m1f2m2 f1m1f2m2f1m1f2m2 f1m1f2m2f1m1f2m2 f1m1f2m2f1m1f2m2 f1m1f2m2f1m1f2m2 f1m1f2m2f1m1f2m2 f1m1f2m2f1m1f2m2 f1m1f2m2f1m1f2m2 f1m1f2m2f1m1f2m2 1111 2222 11 0 0 22 0 12 21 0 11 12 0 22 11 12 21 0 11 12 0 0 21 22 11 0 21 22 0 12 21 22 11 12 21 22 Model 5 4 0 1 8 3 2 6 7 Physical Associations Parameter Associations m r i (t) = ij e j = 1 -t/j-t/j j i 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 Model Variables

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Able to establish experimental and analysis methods to differentiate models and recover parameters Experimental verification of kinetic approach and techniques for models 0, 1, 2, and 7 First, used simulated data Iterating a minimum of 58 parameters with a maximum of 8 common parameters Needed multiple (6) datasets collected as a function of an independent variable Best to vary i, with i, ij, and j common Analysis of all datasets simultaneously (globally) Kinetic Models for a 2 x 2 Profile j Modeli 1 2 51 11 12 2 21 0 41 0 0 2 21 22 01 11 12 2 21 22 11 11 0 2 0 22 81 11 0 2 21 22 31 11 12 2 0 0 21 0 12 2 21 0 61 11 12 2 0 22 71 0 12 2 21 22

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Kinetic Models for a 2 x 3 Profile j M i 1 2 3 j M i 1 2 3 j M i 1 2 3 0 1 11 12 13 2 21 22 23 1 1 11 0 0 2 0 22 23 2 1 11 12 0 2 0 0 23 3 1 11 0 13 2 0 22 0 4 1 11 12 13 2 0 0 0 5 1 0 12 13 2 21 0 0 6 1 0 0 13 2 21 22 0 7 1 0 12 0 2 21 0 23 8 1 0 0 0 2 21 22 23 9 1 11 0 13 2 0 22 23 10 1 11 12 0 2 0 22 23 11 1 11 12 13 2 0 0 23 12 1 11 12 13 2 0 22 0 13 1 11 0 0 2 21 22 23 14 1 11 0 13 2 21 22 0 15 1 11 12 0 2 21 0 23 16 1 11 12 13 2 21 0 0 17 1 0 0 13 2 21 22 23 18 1 0 12 0 2 21 22 23 19 1 0 12 13 2 21 0 23 20 1 0 12 13 2 21 22 0 21 1 11 12 13 2 21 22 0 22 1 11 12 13 2 21 0 23 23 1 11 12 13 2 0 22 23 24 1 11 12 0 2 21 22 23 25 1 11 0 13 2 21 22 23 26 1 0 12 13 2 21 22 23

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Minimum of 84 iterated parameters with a maximum of 12 common (assumes 6 datasets) NSF proposal submitted to explore experimental approaches Current project needs at least a 2 x 3 profile Cannot differentiate the 9 remaining models and non-physical parameters recovered Reduce number of iterated parameters Fix values if known from another experiment Scale parameters wrt to an iterated one based on known ratio or model prediction Use to restrict ij and j values Multiple datasets as a function of an independent variable But some experimental systems do not have an independent variable Use of more than one independent variable Data Analysis Issues Large number of iterated parameters Correlations between parameters Resolution of similar kinetic models HELP??? Use = 0 to eliminate anisotropy of one fluorophore

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