The threshold value, τ_1−α, indicating statistically significant differences between two sections of intensity ratio points with a total length L at various levels of confidence 1−α, is shown in Figure 2. The threshold value τ_1−α does depend very weakly on both the average intensity ratio, 〈ρ〉, between the two detection channels, and the average number of photons per bin, 〈N_photon〉, and is shown here only for equal intensity in the two channels, 〈ρ〉= 1, and 〈N_photon〉= 25 photons per bin. We fit the slow increase of the threshold value τ_1−α as a function of the total number of sample points, L, with the empirical fit function:

where A represents an amplitude, s a scaling factor, and λ a power law exponent. Table 1 lists the resulting best-fit parameters for the various levels of confidence, 1−α, for the fits displayed in Figure 2.

If the analysis is performed with the additional safeguard that N_min = 10 consecutive points have to surpass the threshold value, τ_1−α, the effective probability, α_eff, of false positives is further suppressed, for example to α_eff = 0.05% in the case of α = 1%, as listed in Table 1. This safeguard might seem overly cautious, but in our particular application we are interested in an accurate identification of large intensity ratio changes, ρ_i/ρ_i+1, and the waiting times, t_w, in between. Break points missed because of this additional safeguard are either characterized by small intensity ratio changes, ρ_i/ρ_i+1, or a short waiting time, t_w, leading up to the break point (see Section 2.3), neither one of which constitutes a shortcoming in the analysis of our single molecule experiments [38].

Trial break points, k′, near the beginning (k′ → 0) or the end (k′ → L) of the investigated sequence of the trajectory generate one section with a very small sample size. The resulting large fluctuations in the mean and standard deviation of this small section lead to an increased probability of false positive identification, as shown in Figure 3 for N = 100,000 simulations for sequences of length L = 100. The increased probability of false positives near one of the limits of the sequence does not depend on the length of the remainder of the sequence, L − k′. As this increased likelihood for the identification of false break points significantly exceeds the ideal, targeted, probability of α, as indicated by the dashed line in Figure 3, we exclude N_excl = 10 points at the beginning and the end of the investigated sequence of intensity ratio points from the analysis.

To illustrate the strength of the analysis method, Figure 4a displays an example with L = 1000 simulated intensity ratio points with 〈N_photon〉 = 25 photons per bin. The sequence features a break point in the center, k = L/2, with a change in the intensity ratio of 30%. The corresponding likelihood measure, ℒ(k′), calculated at each trial break point, k′, for this example is displayed in panel (b) of Figure 4. The maximum likelihood value, ℒ_max, significantly exceeds the threshold value τ_1−α (added as a dashed line for the 1 − α = 99% level of confidence) for this sample length, leading to a clear identification of the simulated break point. We use the location of the maximum likelihood value, ℒ_max = L(k̂), as the best estimate, k̂, for the location of the actual break point, k.

We determine the sensitivity of the analysis, that is the probability of false negatives, or break points missed, by testing the method described above on two types of sequences of simulated intensity ratio points. The first set of tests is performed on sequences that include a single break point at location k, the second set of tests uses simple model sequences with multiple break points (N_jumps = 200) of identical spacing and constant changes in the intensity ratio (square waves). The probability of false negatives in sequences with a single break point depends strongly on the change in intensity ratio, ρ_i/ρ_i+1, the average number of photons per bin, 〈N_photon〉, and the length of the investigated sequence, L. Figure 7 shows these dependences for a bin size of 〈N_photon〉 = 25 photons per bin both as a contour plot and as cuts for a variety of intensity ratio changes. As expected, closely spaced break points with small changes in the intensity ratio are likely missed, but both closely spaced large jumps as well as well-separated small jumps can be detected reliably.

The algorithm employed to search for multiple break points in photon arrival trajectories (Section 4.4) was separately tested on simulated model trajectories with multiple break points. The resulting probability for false negatives is very similar to the diagram depicted in Figure 7, except for a reduced sensitivity for break points in very short sequences (L < 40 points) caused by the additional safeguards added to protect against false positive identifications (Section 4.1).

For a given sequence of photons, smaller bin sizes, 〈N_photon〉, lead to higher photon counting noise per bin for the intensity ratio, which, despite producing more bins to analyze for the sequence, reduces the sensitivity significantly (Figure 8). This observation is in contrast to results for statistical single-channel photon trajectory analysis, where the largest information content is revealed using a photon-by-photon approach [10]. On the other hand, large bin sizes can reduce the number of bins per break point below the additional safeguards introduced to guard against false positives (Section 4.1), such that actual break points could be rejected.

Returning to the challenge of analyzing single molecule photon arrival trajectories containing frequent blinking events (Figure 1) we now compare the results of the method described in this paper to those from the separate identification of intensity break points in each channel in combination with a subsequent coincidence analysis. A maximum likelihood analysis [29] of the photon arrival times in a single detection channel can identify many of the “dark” of “bright” periods, but due to their high frequency and very short duration nevertheless misses a significant fraction randomly in one of the two channels, as illustrated in Figure 1. A subsequent test for coincidences of these breaks points in both channels [47] to distinguish changes in the intensity ratio from changes in the total intensity therefore yields false-positive change points for the ratiometric measure whenever a break point is missed in one of the two channels.

Figure 9 illustrates the result of this coincidence analysis for the photon arrival trajectory shown in Figure 1. As the trajectories were collected for a single molecule in a rigid polymer matrix, no molecular reorientations and therefore no changes in the fluorescence intensity ratio occur in the experiment. As shown, the single-channel analysis leads to a dramatic overestimation of the single molecule orientational dynamics. In comparison, the statistical analysis of the photon frequency, as suggested in this paper with a very short bin width on the order of the average blinking duration, yields the expected result of a constant ratio without any break points.

As an example of the application of the proposed method to a single molecule trajectory with reorientations, Figure 10 illustrates the strength of the method by comparing an experimental trajectory with the corresponding reconstructed single molecule orientational dynamics. The reconstructed trajectory follows the experimental intensity ratio very well, capturing all significant reorientations of the probe molecule. Even more importantly, dynamical quantities such as the correlation function for the orientational motion of the probe molecule are represented equally well with the reconstructed trajectory [38]. In analogy to the discussion in the proceeding sections, we tested the accuracy and sensitivity of the analysis method applied to simulated single molecule trajectories where jump times and amplitudes were known. The simulated trajectories very closely resemble those recorded in single molecule experiments in a glass matrix at the glass transition temperature [38]. The dynamics in a glass is characterized by a very broad distribution of waiting times, spanning several orders of magnitude. In addition, the assumed exponential distribution of jump sizes generates a large number of break points with small changes of the intensity ratio. Even though both of these factors present a challenge to the analysis routine, the overall performance is very satisfactory, as basically all simulated break points with a change in the intensity ratio of ρ_i/ρ_i+1 > 1.25 are identified with 63% of the extracted jump times exhibiting an error in the location of less than 2 bins. This error in the location is well within the intrinsic timing error expected for single molecule experiments for a given recorded photon rate and can be controlled by choosing the excitation intensity appropriately.

For normally distributed populations the student t-test is traditionally used to find statistically significant differences between the means of two samples [53]. We determine the actual threshold values indicating significant differences between the means of two samples (that is, between two consecutive sections of binned intensity ratio values) through random number simulations. To this end we simulate N = 100,000 trials of photon arrival time sequences of varying lengths, L = 20 to L = 5000 points, without a break in the intensity ratio, ρ, (testing for false positives). For these simulated photon arrival trajectories we calculate the logarithm of the intensity ratio, log(ρ), for bins with 〈N_photon〉 = 25 photons on average.

At every point, k′, with k′ = 0. . . L, in these intensity ratio sequences we determine the Student’s t-test’s p-value using standard statistical procedures [53] to test for statistical differences between the sequences before and after the trial point k′, {0, . . ., k′− 1} and {k′, . . ., L}, as if k′ were an actual break point in the intensity ratio sequence. As the sequences simulated to test for false positives do not contain an actual break point, we save the maximum p-value, p_max, found in each of the N trials of a given length, L. We exclude the p-values for the first and last N_excl = 10 potential break points, as those show very large fluctuations due to the small size of one of the two sections tested (Section 2.1). For each length, L, we find a threshold value τ_1−α, such that for a fraction 1−α of the N trials the maximum p-value, p_max, falls below the threshold τ_1−α, where 1 − α is the level of confidence.

The simulated example trajectory (with a break point) shown in Figure 4 illustrates the approach used for the analysis of our experimental results. We split a sequence of intensity ratio points into two sections at a trial break point, k′, calculate the p-value, p, for the statistical significance of different means to determine the likelihood, ℒ(k′) = −log(p), for a break at this trial position, k′. We repeat the test for all trial break points, k′ = N_excl . . . L − N_excl in the sequence (again excluding the first and last N_excl = 10 points). We accept the break point, k′_max, with the maximum likelihood, ℒ_max = ℒ(k′_max), as our maximum likelihood estimate, k̂, if ℒ_max exceeds the threshold value τ_1−α for a sequence of length L at the confidence level 1 − α.

For the analysis of our experimental data we choose a conservative level of confidence of 1−α = 99% that in addition has to be surpassed by at least N_min = 10 consecutive trial break points around k̂ to further guard against false positive identifications caused by additional (non-photon counting) noise in the experiment, for example through frequent blinking of the probe molecule. This safeguard limits the shortest detectable distance between break points, t_w, to about t_w,min ~ N_minΔt = 50 ms and raises the detection threshold for break points significantly. The analysis of our experimental data therefore rejects some fraction of the break points with the smallest change in intensity ratio, ρ_i/ρ_i+1. However, for our particular application these rejections are much less of a concern than the inclusion of just a few false positive break points.

In separate stochastic simulations of photon streams that feature one intensity ratio break in the center, k = L/2, as illustrated in Figure 4, we determine the fraction of missed break points (false negatives) as a function of the change in intensity ratio, Δρ/ρ, and the length, L, of the sequence probed. We choose a level of confidence of 1 − α = 99% plus an additional safeguard of a minimum of N_min = 10 points above the threshold, τ_1−α, to test the routine under the same conditions as in the analysis of our experimental trajectories. We perform N = 10,000 trials for intensity changes of ρ_i/ρ_i+1 = 1.1 to ρ_i/ρ_i+1 = 2.0, with an average number of 〈N_photon〉 = 25 photons per bin with equal intensity in both channels when averaged over the entire trajectory. For successfully identified break points we determine the distribution of the error, Δk̂ = k̂ − k, for the estimated location of the break point, k̂, as a function of the length of the sequence, L, and the change in the intensity ratio, ρ_i/ρ_i+1, at the break point, k.

Furthermore, we determine the difference in the likelihood measure Δℒ_max = ℒ_max−ℒ(k) between the likelihood at the maximum, ℒ_max = ℒ(k̂) and the likelihood for a break at the actual break point, ℒ(k). From the distribution of this likelihood difference, Δℒ_max, we find a threshold value τ′_1−α such that for a fraction 1−α of the N trials Δℒ_max falls below τ′_1−α where 1−α signifies the level of confidence for the location error estimate. We use this threshold value τ′_1−α to find the confidence interval k⁻ . . . k⁺ for the estimated break point k̂, where k⁻ and k⁺ are the two points to the right and left of k̂, respectively, such that ℒ(k⁻) = ℒ(k⁺) = ℒ_max − τ′_1−α (see Figure 4).

The application of the t-test as described above can only locate one most probable change point, k̂, in a given sequence of points. To find all change points, {k̂_i}, in an experimental trajectory, we systematically test for potential break points in a slowly growing section of the trajectory, starting at the last identified break point, k̂_i−1, lengthening the sequence under test by N_step = 5 bins at a time. As an additional safeguard against spurious break points we also require that any probable new break point, k̂_i, is confirmed N_repeat = 2 more times in additional sequences that start with the same previously identified break point, k̂_i−1, but are lengthened by an additional N_step bins each time. If the break point is reproducible, we subsequently double-check the previously identified change point k̂_i−1 in the sequence bracketed by the two adjacent break points k̂_i−2 and (the newly identified) k̂_i. If k̂_i−1 is confirmed as the most likely break point location between k̂_i−2 and k̂_i, the process continues from break point k̂_i to search for a new break point k̂_i+1. However, if the previously identified most likely break point location, k̂_i−1, differs from the new location of the break point between k̂_i−2 and k̂_i, or if break point k̂_i−1 is no longer statistically significant given the new sequence limit k̂_i, break point k̂_i−1 is modified accordingly (or eliminated all together) and the confirmation check continues backwards until the sequence {k̂₀ . . . k̂_i} is self-consistent. The algorithm is represented graphically in Figure 12. This approach eventually yields a time sequence of n most likely intensity ratio change points, {k̂_i}, with i = 0. . . n.

We calculate the corresponding intensity ratios, ρ_i, between two identified change points, k̂_i and k̂_i+1, directly from the number of photons recorded between these two times in each detection channel. To accelerate the calculation we approximate the p-value, p, through the following equation that we determined empirically:

with

where x = log(ρ), while χ̄_i and s_i are the maximum likelihood estimators for mean and standard deviation, determined for the intensity ratio sequence sections before (i = 1) and after (i = 2) the trial break point, k′. To further improve the speed of the calculation, we pre-calculate cumulative sums for x and x₂ for the entire tested sequence, {ρ₀, . . ., ρ_L} to quickly determine averages and standard deviations for the two samples on either side of all possible trial break points k′ from differences between the corresponding two elements of the cumulative sums.

To test the performance, sensitivity, and reliability of the analysis routine algorithm, we simulate the following three types of trajectories with multiple break points: (a) square wave intensities with constant waiting times and constant intensity jumps, varying both parameters independently in separate runs; (b) trajectories with constant waiting time but intensity jumps of random amplitude, varying only the constant waiting time in separate simulation runs; as well as (c) photon sequences that are comparable to experimentally recorded trajectories. The simulation of realistic experimental single molecule trajectories is based on a recently proposed model for the dynamics of glasses [38], which allows us to simulate the waiting times between changes in the fluorescence polarization recorded for the single probe molecule. We simulate the changes in the intensity ratio, ρ, at these break points from angular jump trajectories of the single molecule that stem from random walks on a sphere with isotropic exponential jump size distribution. Accounting for the numerical aperture of the microscope objective [54], we subsequently calculate the intensities in the two polarization directions for each orientation, randomly pick photon arrival times with an exponential waiting time distribution, consistent with these intensities in the two detection channels and finally bin the photons as done in the experiment. In these simulations we assume that waiting times and jump sizes are uncorrelated, which is consistent with our experimental results. The purpose of these different simulations is to determine the percentage of identified jumps (sensitivity), the average error in the estimated location of break points (accuracy) of the utilized algorithm (Section 4.4).