Efficient use of single molecule time traces to resolve kinetic rates, models and uncertainties

Abstract

Single molecule time traces reveal the time evolution of unsynchronized kinetic systems. Especially single molecule Förster resonance energy transfer (smFRET) provides access to enzymatically important time scales, combined with molecular distance resolution and minimal interference with the sample. Yet the kinetic analysis of smFRET time traces is complicated by experimental shortcomings-such as photo-bleaching and noise. Here we recapitulate the fundamental limits of single molecule fluorescence that render the classic, dwell-time based kinetic analysis unsuitable. In contrast, our Single Molecule Analysis of Complex Kinetic Sequences (SMACKS) considers every data point and combines the information of many short traces in one global kinetic rate model. We demonstrate the potential of SMACKS by resolving the small kinetic effects caused by different ionic strengths in the chaperone protein Hsp90. These results show an unexpected interrelation between conformational dynamics and ATPase activity in Hsp90.

Fig. 1

(a) In smFRET experiments, there is an antagonistic relation between the signal-to-noise ratio (SNR), time resolution, and observation time. Increasing one of the three has a negative effect on the remaining two. (b) The empirical relation of observation time (limited by the time constant of photo-bleaching, τ_bl) and SNR derived from the experiment (see section titled Theory). A SNR of about 4 comes with τ_bl = 90 frames in typical alternating laser excitation (ALEX) experiments, yielding the empirical const = 1350. As indicated, a reduction of 25% in SNR may result in as much as 70% longer observation time.

Fig. 2

(a) Illustration of the TIRF experiment. An objective-type TIRF microscope was used to record the fluorescence of surface-immobilized protein molecules. FRET between two specifically attached dyes allows one to distinguish v-shaped, N-terminally open (left) from closed (right) conformations. (b) Experimental raw data recorded in real time under three different potassium chloride concentrations as indicated. Fluorescence intensities are color coded: donor (green), FRET sensitized acceptor (red), directly excited acceptor (gray), and FRET efficiency, E (black). Colored or white overlays indicate HMM-derived low-FRET or high-FRET states, respectively. [(c)–(e)] Cation dependence of Hsp90’s conformations for potassium chloride (c), including ADP (d), or sodium chloride (e) as specified.

Fig. 3

[(a) and (b)] The superior robustness of 2D HMM demonstrated by two example traces: FRET efficiency (FRET E, black), fluorescence intensity of the donor (green), acceptor (orange), Viterbi path (blue, right axis, state 0: low FRET, state 1: high FRET). (a) Blink events (highlighted in yellow) are misinterpreted by FRET efficiency based 1D HMM. (b) 1D HMM diverges under high noise conditions. Ergo, the Viterbi path is not defined. In contrast, 2D HMM still derives a suitable Viterbi path. (c) smFRET data as input for HMM: color code as in (a), plus directly excited acceptor (gray), and Viterbi path as gray and white overlays indicating high- and low-FRET states, respectively. (d) The FRET efficiency histogram of multiple traces provides only 1D information, although 2D information was originally recorded. (e) 2D histogram of donor fluorescence intensity vs. acceptor fluorescence intensity for every time point (black to light gray: minimal to maximal counts, white: no counts). The markers indicate the means of the low-FRET state (green) and the high-FRET state (orange) of 36 individual traces. Global Gaussians, as derived for the entire data set, are displayed as corresponding contours.

Fig. 4

(a) Semi-ensemble HMM optimizes a global kinetic model based on a complete data set (normally > 100 traces). While the kinetic parameters—start probabilities and transition matrix—are optimized globally, the predetermined, individual emission PDFs are held fixed. This allows further to identify states not only by a characteristic signal but also based on their kinetic behavior. For the example trace displayed, this results in a Viterbi path (overlays) with 4 kinetic states despite only 2 distinguishable FRET efficiencies. (b) Transition maps “before” and “after” optimization of the HMM. A fitting rate model generates well-defined clustering: the mean FRET efficiencies of the dwell preceding a transition (initial FRET E) are plotted against those of the following dwell (Final FRET E). The initial state is color-coded: state 0 (red), 1 (green), 2 (blue), 3 (pink), for further details see Ref. .

Fig. 5

(a) Three models that are mathematically equivalent as they all have 2 + 2 states and rank 1. (b) Determination of confidence bounds: every rate k is gradually moved away from its maximum likelihood estimator. The likelihood ratio between the old and new models is displayed as a function of the modified rate constant. The 95% confidence bounds are reached where the likelihood ratio crosses ${\chi }_{df=1}^2\left(\alpha =0.05\right)=\mathrm{3.84.}$

Fig. 6

Dissimilar cation dependence of Hsp90’s kinetics under varied potassium (a) or sodium concentrations (b), in linear (top) or logarithmic scale (bottom). The rates are labeled according to the state model in (c). State 0,1: low FRET; state 2,3: high FRET; circle sizes represent populations; arrow weights represent transition rate constants. (d) The average number of transitions observed per trace for each data set. [(a)–(d)] The number of molecules included in the dataset prev. 150/50/150/750 mM KCl is 154/107/102/129, respectively; and for 50/150/750 mM NaCl, it is 105/70/140, respectively. (e) Hsp90’s ATPase activity under varied cation conditions: KCl, left; NaCl, right. Measurements were performed with 3 different Hsp90 variants: wild-type (wt), 61C with C-terminal zipper (zip), and 385C with C-terminal zipper. Individual rates were normalized to the value obtained using 150 mM monovalent cation. For KCl, these values were 0.8/0.8/0.7 ATP/min/monomer; for NaCl: 1.2/0.7/0.4 ATP/min/monomer (in the above order). The lines connect the average measured rates.