Single-Molecule Analysis beyond Dwell Times: Demonstration and Assessment in and out of Equilibrium

Abstract

We present a simple and robust technique for extracting kinetic rate models and thermodynamic quantities from single-molecule time traces. Single-molecule analysis of complex kinetic sequences (SMACKS) is a maximum-likelihood approach that resolves all statistically relevant rates and also their uncertainties. This is achieved by optimizing one global kinetic model based on the complete data set while allowing for experimental variations between individual trajectories. In contrast to dwell-time analysis, which is the current standard method, SMACKS includes every experimental data point, not only dwell times. As a result, it works as well for long trajectories as for an equivalent set of short ones. In addition, the previous systematic overestimation of fast over slow rates is solved. We demonstrate the power of SMACKS on the kinetics of the multidomain protein Hsp90 measured by single-molecule Förster resonance energy transfer. Experiments in and out of equilibrium are analyzed and compared to simulations, shedding new light on the role of Hsp90's ATPase function. SMACKS resolves accurate rate models even if states cause indistinguishable signals. Thereby, it pushes the boundaries of single-molecule kinetics beyond those of current methods.

Figure 1

Conformational dynamics of a DNA (left) and a protein system (right) measured by smFRET. (A) Schematic TIRF setup and close-up of a single Hsp90 molecule labeled with a FRET pair (donor dye, orange; acceptor dye, red) and immobilized within the evanescent field (see Materials and Methods). (B and E) FRET efficiency, E, trajectories (black) obtained from single-molecule fluorescence time traces (donor, green; acceptor, orange; arbitrary units). Fluorescence of the acceptor dye after direct laser excitation (gray) excludes photoblinking artifacts. The HMM-derived state sequence (Viterbi path) is displayed as overlays (low FRET, white; high FRET, gray). A zoom is included in (B). Emission PDFs in dimensions of donor and acceptor fluorescence are shown next to the corresponding time traces. The allowed “FRET line” (red) is determined for every molecule individually (cf. main text). (C and F) FRET histograms with Gaussian fits as indicated. (D and G) Cumulative dwell-time histograms with single-exponential fits (black; frequency weighted). Static traces (E, bottom) are described using the mean emission PDFs of the complete data set (cf. main text). Although the FRET histogram (F) shows two populations, the dwell-time distributions (G) clearly deviate from a single-exponential fit (total of 163 molecules). To see this figure in color, go online.

Figure 2

Accuracy of rates from dwell-time analysis compared to ensemble HMM. (A) Discrete state sequences were simulated for two-state models (left) with different rates, k01 and k10 (200 state sequences with 5 Hz sampling rate and 0.03 Hz bleach rate). The deviation of the determined k01 from the input k01, as obtained by dwell-time (DT) analysis and ensemble HMM, is shown as a function of both input rates (maximum relative deviation out of five simulations per data point). Here, only dwells with observed start and end points are considered by the dwell-time analysis. Including also dwells with one and zero observed limits (static traces) gives very similar results (see Fig. S3). Relative errors of the rates along the indicated lines are shown as a function of the mean number of dwell times per trace (right). Black lines serve as a guide to the eye. Clear systematic deviations occur already for the simplest model. The residual relative error of dwell-time analysis at growing dwell-time counts per trace vanishes only at higher sampling rates (see Fig. S4). These are accessible in the experiment only at the cost of not resolving the slower dwells due to photobleaching. (B) More complex models are more realistic for protein systems. The depicted four-state model (left) was used to generate synthetic data with equal FRET efficiencies for states 0/1 and 2/3, respectively. See Materials and Methods for details and Fig. S1 for example data. The size of the circles in the state models (A and B, left) is proportional to the state population. The arrow widths are proportional to the transition rates. Right: Comparison of the transition rates determined by dwell-time analysis and by SMACKS. Dwell-time analysis only provides half the transition rates, which are in addition far off the input values. To see this figure in color, go online.

Figure 3

SMACKS workflow. (A) The model parameters (start probability, π, transition matrix, A, and emission PDFs, B) and a set of donor (green) and acceptor (orange) fluorescence time traces constitute the basis of ensemble HMM (overlays represent the Viterbi path). To allow for intermolecule variations of the fluorescence signal, the individual emission PDFs are predetermined in a trace-by-trace HMM run and held fixed during optimization of the kinetic ensemble parameters. (B) Degenerate FRET states are included as multiples of experimentally discernible states (here, low FRET and high FRET). The BIC identifies the optimal model (here, four states). (C) The transition map (left) relates the mean FRET values before and after each transition found by the Viterbi algorithm (initial states 0, 1, 2, and 3 in red, green, blue, and orange, respectively). The most frequent transitions occur between the two least populated (short-lived) states. The transition histogram (middle) reveals the frequency of each transition in the data set. Excluding transitions that do not occur leads to a cyclic model (right). (D) The obtained model is critically evaluated in three ways. First, dwell-time distributions are reproduced by resimulating the model (left; experimental data, green; simulated data, gray). Second, random start parameters uncover potential local likelihood maxima, and random subsets reveal data-set heterogeneity (middle; subsets, green; complete set, black). Third, confidence intervals measure the precision of the obtained rates considering the finite data set, experimental noise, and data-set heterogeneity (right). A flow chart of SMACKS can be found in Fig. S5. To see this figure in color, go online.

Figure 4

Kinetic and thermodynamic results under varied nucleotide conditions. (A) Transition maps locate transitions in FRET space (initial states 0, 1, 2, and 3 in red, green, blue, and orange, respectively). FRET efficiencies scatter less in the presence of nucleotides. (B) Occurrence of transitions. (C) Derived kinetic state models: states 0/1 (2/3) show low (high) FRET efficiencies. Arrow widths represent the size of the transition rates. Circle sizes represent relative state populations. A four-link model is favored with ATP, whereas three links are sufficient for all other data sets. (Rates to remain in a given state are not depicted.) (D) The FRET histogram shows only small differences for apo (black and black-hatched region), ADP, or ATP data, whereas with AMP-PNP (purple) a shift to the closed conformation is observed. (E) A four-state model represents all four data sets best according to ΔBIC values; the color code is the same as in (F). Black lines are a guide to the eye. (F) Deduced rates and confidence intervals. (G) Qualitative cartoon of the 3D energy surface of Hsp90 in the presence of ATP. SMACKS reveals “hidden” states that are kinetically different but share the same FRET efficiency. A large energy barrier hinders transitions through the midpoint. (H) Quantitative 2D projection of the energy landscape shows the differences between the nucleotides; the color code is the same as in (F). Energy levels were calculated from transition rates, whereas well widths are arbitrary. A typical attempt frequency for proteins, 10⁸ Hz, was assumed. To see this figure in color, go online.

Figure 5

Quantifying energy coupling. (A and B) Two limit cases of systems driven by the hydrolysis of 1ATP. In (A), the external energy is absorbed between states 0 and 3. All remaining rates are set to 0.05 Hz. In (B), the external energy is introduced sequentially over four identical steps. The respective state models (top), energy scheme (center), and theoretical detection limit for free energies as a function of the forward rate (bottom) are shown. Simulated values (green) result from Eq. 6 applied to 200 discrete state sequences with 5 Hz sampling rate and 0.03 Hz bleach rate. They scatter about the expectation value of $\Delta G_{\text{obs}}$ (black line) calculated as explained in the Supporting Material. (C and D) State model (top), transition map (center), and transition histogram (bottom) for synthetic data (C), simulating the flow introduced by coupling to the hydrolysis of 1ATP = 30 kT, or for experimental data (D) of Hsp90+ATP stimulated by the cochaperone Aha1. To see this figure in color, go online.