Figure 1.
Generalized scheme for analyzing time-binned smFRET data.
Bootstrapping can be used both in thermodynamic and kinetic analysis and is compatible with numerous data formats. Bold frames indicate functionalities available in BOBA FRET. a)As defined in the introduction, see also Gopich and Szabo [37]. b)Rayleigh criterion: two subpopulations are indistinguishable when their peak positions are separated by one standard deviation or less [2]. c)See [9]. d)See [10], [34], [35]. e)See [5], [8], [25]–[28], [32]. f)See [18], [19]. g)Multivariate tests (MANOVA) are conceivable to assess whether two or more outcome variables are significantly different at a time, for example the center and the width of a FRET distribution [66]. h)See [12]. i) j)See [34]. k)See [14], [64]. l)Typically used in fluorescence correlation spectroscopy (FCS) [78].
Figure 2.
Studying d3'EBS1*/IBS1* interaction by smFRET.
The d3'EBS1* hairpin is labeled with Cy3 and tethered to the surface of a quartz slide passivated with biotinylated BSA via a biotin-streptavidin linkage. Docking of a Cy5-IBS1* strand is characterized by the appearance of Cy5 fluorescence and a decrease in Cy3 emission due to FRET. Figure adapted from [70].
Figure 3.
Summary of the different analytical approaches performed in conjunction with bootstrapping to extract thermodynamic or kinetic parameters from time-binned smFRET data in this study.
The respective input and output variables are indicated as well. Please refer to the method section for a detailed mathematical description.
Figure 4.
Robustness of different approaches for thermodynamic analysis of smFRET data performed in conjunction with bootstrapping (method 1, 2 and 3, thermodynamics).
Simulated data for a two-state system with standard parameters as defined in the methods section. (A) Performance in response to trace length. As the number of data points increases from the left to the right, the mean docked fraction is estimated more precisely, while cross-sample variability decreases. (B–C) Performance in response to FRET spacing and SNR. A systematic downward bias is observed for threshold- and HMM-based approaches as the two FRET distributions show increasing overlap. Gaussian fitting performs well as long as the Rayleigh criterion is fulfilled (ΔFRET >0.144). (D) Performance in response to the ratio of rate constants. Threshold-based dwell time analysis easily breaks down, as noise in the docked state is mistaken for FRET transitions. At high kdocking/kundocking, Gaussian fitting and thresholding of FRET histograms underestimate the docked fraction because of slight overlap between the two FRET distributions. HMM yields the best results. (E) Performance in response to heterogeneously distributed SNR values. The results of the threshold-based analysis and Gaussian fitting are mostly unaffected by changes in the SNR distribution width, while HMM breaks down at σ(SNR) >2. All theoretical values were determined from the input parameters used of the simulations. Error bars (red and green swaths) were estimated by bootstrapping and cover 99.7% of the experimental variability (3σboba). Please refer to Figure S4 for representative simulated time traces and the text for further details.
Figure 5.
Robustness of thresholding and HMM approaches to analyze smFRET data performed in conjunction with bootstrapping (method 1, kinetics).
Simulated data for a two-state system as defined in the methods section. (A) Performance in response to trace length. Cross-sample variability decreases at long observation times, since the number of dwell times increases. (B) Performance in response to the ratio of rate constants. Two problems bias threshold- and HMM-based analysis: (i) false FRET transitions stemming from noise and (ii) irresolvable FRET transitions. (C–D) Performance in response to FRET spacing and SNR. A systematic downward bias is observed for threshold-based analysis as the two FRET distributions show increasing overlap. The result of the HMM-based analysis depends on the ratio of false and true dwell times. (E) Performance in response to heterogeneously distributed SNRs. The results of the threshold-based analysis and Gaussian fitting are mostly unaffected by changes in the SNR distribution width. All theoretical values were determined from the input parameters used of the simulations. Error bars (red and green swaths) correspond to the standard deviation estimated by bootstrapping (3σboba). Please refer to Figure S4 for representative simulated time traces and the text for further details.
Figure 6.
Representative time traces showing d3'EBS1*/IBS* interaction and thermodynamic analysis of FRET histograms (method 1).
(A) Fluorophore emission over time reveals abrupt anticorrelated changes in intensity (upper graphs). Calculation of FRET time traces reveals repetitive shuttling between a zero and a high FRET level (lower graphs). Based on the experimental design, these two states were assigned to the undocked and the docked state (Figure 1). The red lines correspond to the discretization by the Hidden Markov Model (vbFRET [25]). (B) FRET histograms built from the smFRET time traces shown in A. (C) Normalized cumulated FRET histograms built from individual time traces. The dashed green line depicts the threshold between the two FRET states used to determine the docked/undocked fractions and the normalized results are indicated. Solid green lines correspond to Gaussian approximation of the experimental data. The error (green swath) is the standard deviation associated with amplitude and width of the Gaussian fit functions as estimated by bootstrapping (3σboba).
Figure 7.
Statistical hypothesis testing using thermodynamic and kinetic smFRET data.
(A) Analysis of variance (ANOVA) of docked fractions determined by thresholding of normalized cumulated FRET histograms (Figure 6C) reveals that Ni2+ and Co2+ shift the conformational equilibrium significantly towards the docked state (*** P<0.001). The outcome of an ANOVA depends on the separation of the means (center values of the Gaussians) and how far the values are spread out (variance, σboba2, width of the Gaussians) and is given in form of a P-value, i.e. the probability that the null hypothesis is true (here: divalent metal ions do not significantly affect the equilibrium). The stronger the overlap between different groups, the greater the P-value. (B) Decay constants of the zero FRET state decrease in response to addition of Ni2+ or Co2+, leading to faster association (P<0.001). Data obtained by HMM analysis and single-exponential fitting (Figure 8). (C) Decay constants of the high FRET state significantly increase in the presence of Co2+ (P<0.001), which promotes stable association of d3'EBS1* and IBS1*. Data obtained by HMM analysis and stretched exponential fitting (Figure 8). Error bars correspond to the bootstrapped standard deviation (3σboba).
Table 1.
Thermodynamic analysis of the d3'EBS1*/IBS1* equilibrium by different methods.
Figure 8.
Kinetic analysis of smFRET data (method 1).
Docking is defined as the state transition from the undocked to the docked state, the undocking process is defined as the inverse reaction. (A) Transition density plots of HMM data show two clusters corresponding to the docking and the undocking reaction, respectively. According to the maximum evidence approach employed in vbFRET [25], a two-state system is therefore most likely to produce the experimental data, which is in agreement with the experimental design. Raw data were grouped via the weighted k-means clustering algorithm. Color code: occurrence in counts. (B–D) Dwell time histograms created from the normalized cumulative occurrence of dwell times in the docked and the undocked state as determined by HMM. The green lines correspond to a single-exponential fit to the experimental data, while the red lines represent a stretched exponential decay. Errors are indicated as a swath and correspond to 3σboba associated with the decay constants.
Table 2.
Kinetic analysis of d3'EBS1*/IBS1* association and dissociation using different methods to extract dwell times.