1. Introduction
Single molecule experiments can probe structure and dynamics on a molecular scale, revealing details that in traditional bulk experiments remain hidden behind ensemble averages [
1–
8]. In particular, the technique allows the observation of individual events of molecular dynamics, yielding distributions of and correlations between different dynamical properties. Unfortunately, as the method is still young [
5], few analysis methods are available to harvest the vast pool of information from the original fluorescence intensity or photon arrival trajectories. Single molecule experiments are therefore often analyzed with correlation functions building on the common practice for bulk experiments. Even though differences between individual molecules observed in the sample are preserved, many details embedded in the temporal sequences of events are lost in the correlation procedure, which time-averages over the trajectory. Furthermore, properly averaged autocorrelation functions require a trajectory length of at least 100-times the correlation time [
9], while experimental trajectories are often much shorter than this minimum due to the limited photochemical lifetime of the probe molecules.
Recently, methods were introduced that permit the correlation analysis of photon arrival times directly without binning [
10,
11], which, while increasing the time resolution achievable, still suffer from the time-averaging intrinsic in correlation analysis. Starting with the analysis of quantum dot and single molecule “blinking”, a drop of the total intensity to zero [
1,
12–
19], often associated with excursions to the triplet state of the single molecule [
20,
21], the advantage of the identification of individual events in the single molecule trajectory became obvious. Numerous methods that are based on model dynamics were subsequently introduced, such as the analysis of hidden Markov chains, for example through Bayesian analysis [
22–
24], photon counting histograms [
25], or maximum likelihood analysis [
26,
27]. These approaches are appropriate for single molecule dynamics with a well-known number of accessible states, for example in the case of blinking or enzymatic turnovers, which can be approximated as “on” and “off” states, but even this simplification is debated [
28].
Several model-independent methods are described in the literature, which are capable of detecting individual intensity change points directly from a single-channel photon arrival trajectory with Bayesian or maximum likelihood approaches [
29–
31], yielding the times of sudden changes in a piece-wise constant fluorescence intensity trace. These methods do not require the assumption of an underlying mechanism or a limited number of accessible states, and can therefore be applied more general. Typical applications are the identification of enzymatic turnovers [
32], motor protein movements [
33], or nanoparticle blinking [
18,
19,
34], all leading to large fluctuations in the fluorescence intensity associated with the dynamical event of interest. Those sudden changes in the fluorescence are common in single molecule spectroscopy, caused by jumps between states with different emission characteristics as opposed to the continuous changes seen in bulk.
In addition to the single-channel intensity detection employed in the examples above, ratiometric measurements, that is the simultaneous recording of two intensity channels, is another widely used technique in single molecule spectroscopy [
2,
35,
36]. Examples are the detection of two different polarization directions of the emitted fluorescence,
I|| and
I⊥, to observe the angular reorientation of a single probe molecule reporting on polymer [
37–
39] or protein dynamics [
40] as well as the detection of two different emission wavelengths, either to observe shifts in an emission spectrum [
41] or to determine the distance between a pair of single molecules showing Förster resonance energy transfer (
sp-FRET) [
42–
44].
Instead of investigating the intensities directly, a ratiometric analysis focuses on a normalized intensity ratio, for example, in the case of single molecule orientational motion, the reduced linear dichroism [
45] (
Equation 1).
Here, effects of the photodynamics of the probe molecule, which might change the total intensity,
I||+
I⊥ without affecting the intensity ratio,
I||/I⊥, are eliminated from the monitored quantity,
Id, and only changes in the polarization direction of the emission, caused by single molecule reorientation, are recorded. It would be desirable to detect sudden changes in the ratiometric measure similar to their detection in the single-channel analysis methods discussed above. A two-state Markov-chain approach has for example been used to analyze
sp-FRET experiments [
44,
46].
This intensity ratio is central to any of the two-channel experimental methods mentioned above. As an example for its usefulness we will here discuss one application, monitoring the fluorescence polarization direction of the single molecule emission. However, the method described here is general and can easily be adapted to any ratiometric single molecule technique.
To construct a model-free approach for the detection of change points in a ratiometric variable one might imagine analyzing both intensity channels separately using one of the single-channel model-free methods [
29–
31] and combining the results. However, single probe molecule “blinking” that leaves the ratiometric measure of interest (such as the reduced linear dichroism,
Id) undefined during any “dark” periods interferes with this approach. An example illustrating the high frequency and short durations of these blinking events is shown in
Figure 1 for the two polarization directions of the fluorescence from a single rhodamine B molecule immobilized in a solid polymer matrix. Short gaps in the continuous stream of photons indicate frequent blinking events with typical durations on the order of a millisecond. Also shown are the results of a statistical analysis routine [
29] analyzing the photon arrival times in each detection channel separately. This routine, in combination with a subsequent coincidence analysis [
47] would need to identify all of these blinking periods to avoid false positive ratiometric change points, the quantity of interest in the applications mentioned above. As can be seen from
Figure 1 this is clearly not the case, justifying the search for a new statistical analysis method dedicated to the identification of ratiometric change points, as presented in this paper.
3. Single Molecule Experiment
In single molecule spectroscopy, a fluorescent probe is embedded in the matrix of interest at a very low concentration [
3–
5]. We use rhodamine B as a probe molecule in the polymer poly(vinyl acetate) in the vicinity of the glass transition temperature of the matrix. Details of the experiment are published elsewhere [
38,
48]. Very briefly, a high-
NA microscope objective focuses a
cw He-Ne laser onto the sample and collects the fluorescence from the probe molecules. After spectral filtering, a dielectric polarization cube splits the emission into two perpendicular polarization directions, which are detected on separate single-photon-counting photodiodes. For later statistical analysis, the arrival timestamps of every detected photon in each channel are continuously recorded.
The recorded fluorescence intensity in both polarization directions,
I|| and
I⊥, as well as the ratio of these two fluorescence intensities,
ρ =
I||/I⊥, exhibit sudden changes, as shown in
Figure 10. While changes in the total fluorescence intensity,
I =
I|| +
I⊥, could be caused by the probe molecule’s photodynamics, such as excursions to the triplet state [
1,
13,
32], or fluctuations in the fluorescence lifetime due to changes in the probe environment [
49], changes in the ratio of the fluorescence intensity in the two polarization directions,
ρ =
I||/I⊥, indicate reorientations of the probe molecule.
4. Analysis Method
The analysis method described here identifies statistically significant changes in the expectation value of the observed intensity ratio. We bin photons from the original photon arrival trajectory with a short bin width of Δ
t = 5 ms, which is larger than the lengths of typical blinking periods for rhodamine [
12,
50], corresponding to an average number of about
〈Nphoton〉 ~ 25 photons per bin at typical intensities in our experiments. For each bin we calculate the logarithm of the ratio of the intensity in the two detection channels,
Stochastic photon counting noise leads to a near-normal distribution [
51,
52] of log(
ρ) as illustrated in
Figure 11 for simulated photon arrival times in two channels with a constant average intensity ratio of
〈ρ〉 = 1 and
〈Nphoton〉 = 25 photons per bin. At these bin sizes bins without photons are extraordinarily rare and do not require special treatment. The near-normality of the distribution for log(
ρ), combined with the numerous statistical tools available for the normal distribution, is the reason why we analyze the logarithm of the intensity ratio (
Equation 4) instead of the linear dichroism,
Id, (
Equation 1) which is traditionally used in fluorescence microscopy.
4.1. False Positives
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
〈Nphoton〉 = 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,
pmax, found in each of the
N trials of a given length,
L. We exclude the
p-values for the first and last
Nexcl = 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
τ1−α, such that for a fraction 1−
α of the
N trials the maximum
p-value,
pmax, falls below the threshold
τ1−α, where 1 −
α is the level of confidence.
4.2. Location of Break Points
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,
(
k′) = −log(
p), for a break at this trial position,
k′. We repeat the test for all trial break points,
k′ =
Nexcl . . . L −
Nexcl in the sequence (again excluding the first and last
Nexcl = 10 points). We accept the break point,
k′
max, with the maximum likelihood,
max =
(
k′
max), as our maximum likelihood estimate,
k̂, if
max exceeds the threshold value
τ1−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 Nmin = 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, tw, to about tw,min ~ NminΔt = 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, ρi/ρi+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.
4.3. False Negatives and Error of Location Estimate
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, Δ
ρ/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
Nmin = 10 points above the threshold,
τ1−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
ρi/ρi+1 = 1.1 to
ρi/ρi+1 = 2.0, with an average number of
〈Nphoton〉 = 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, Δ
k̂ =
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,
ρi/ρi+1, at the break point,
k.
Furthermore, we determine the difference in the likelihood measure Δ
max =
max−
(
k) between the likelihood at the maximum,
max =
(
k̂) and the likelihood for a break at the actual break point,
(
k). From the distribution of this likelihood difference, Δ
max, we find a threshold value
τ′
1−α such that for a fraction 1−
α of the
N trials Δ
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
(
k−) =
(
k+) =
max −
τ′
1−α (see
Figure 4).
4.4. Algorithm
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, {
k̂i}, 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,
k̂i−1, lengthening the sequence under test by
Nstep = 5 bins at a time. As an additional safeguard against spurious break points we also require that any probable new break point,
k̂i, is confirmed
Nrepeat = 2 more times in additional sequences that start with the same previously identified break point,
k̂i−1, but are lengthened by an additional
Nstep bins each time. If the break point is reproducible, we subsequently double-check the previously identified change point
k̂i−1 in the sequence bracketed by the two adjacent break points
k̂i−2 and (the newly identified)
k̂i. If
k̂i−1 is confirmed as the most likely break point location between
k̂i−2 and
k̂i, the process continues from break point
k̂i to search for a new break point
k̂i+1. However, if the previously identified most likely break point location,
k̂i−1, differs from the new location of the break point between
k̂i−2 and
k̂i, or if break point
k̂i−1 is no longer statistically significant given the new sequence limit
k̂i, break point
k̂i−1 is modified accordingly (or eliminated all together) and the confirmation check continues backwards until the sequence {
k̂0 . . . k̂i} 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, {
k̂i}, with
i = 0
. . . n.
We calculate the corresponding intensity ratios, ρi, between two identified change points, k̂i and k̂i+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 χ̄i and si 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, . . ., ρL} 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.
4.5. Simulation of Photon Sequences With Multiple Break Points
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).