# Identification of Intensity Ratio Break Points from Photon Arrival Trajectories in Ratiometric Single Molecule Spectroscopy

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{||}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].

_{||}+ I

_{⊥}without affecting the intensity ratio, I

_{||}/I

_{⊥}, are eliminated from the monitored quantity, I

_{d}, 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].

_{d}) 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.

## 2. Results and Discussion

#### 2.1. Threshold Values for Significance

_{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:

_{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].

_{excl}= 10 points at the beginning and the end of the investigated sequence of intensity ratio points from the analysis.

#### 2.2. Accuracy

_{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, $\mathcal{L}$(k′), calculated at each trial break point, k′, for this example is displayed in panel (b) of Figure 4. The maximum likelihood value, $\mathcal{L}$

_{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, $\mathcal{L}$

_{max}= L(k̂), as the best estimate, k̂, for the location of the actual break point, k.

#### 2.2.1. Distribution of Location Error

_{i}/ρ

_{i}

_{+1}. The probability P(Δ k̂) falls off approximately exponentially from a maximum probability at zero error, Δ k̂ = 0 and only depends on the change in the intensity ratio, ρ

_{i}/ρ

_{i}

_{+1}, at the break point, but not on the length of the sample, L, or on the relative location, k/L, of the break point within the sample. For sequences similar to the example pictured in Figure 4 with ρ

_{i}/ρ

_{i}

_{+1}= 1.30 and 〈N

_{photon}〉 = 25 photons per bin, over 20% of all break points are identified without error, Δk̂ = 0. The average deviation of all identified break points is 〈|Δk̂|〉 = 3.7 bins. The average deviation, 〈|Δk̂|〉 is shown in Figure 6 as a function of the relative change in intensity ratio, ρ

_{i}/ρ

_{i}

_{+1}, for a bin size of 〈N

_{photon}〉 = 25 photons per bin. Also indicated in Figure 6 is the percentage of correctly identified break point locations, P(Δk̂ = 0).

_{photon}〉, determines the photon counting and single molecule blinking noise of the binned intensity ratios, the number of intensity ratio points to be analyzed between two break points, and the time resolution of the analysis. The distribution of the error of the estimated location, however, if measured in terms of the absolute deviation in time instead of as a number of bins, proved to be independent of the bin size chosen for the analysis of a given photon sequence. Binning fewer photons per bin might increase the time resolution per bin, but due to the increased photon counting noise per bin will not change the timing error in the estimated break point locations. The only way to reduce the timing error for break points even further is to record the photon trajectory at a higher intensity, thus increasing the number of photons available for analysis for the same number of break points.

#### 2.2.2. Estimation of Location Error

_{1−}

_{α}, which defines the confidence interval for k̂, depends strongly on the length, L, of the sequence tested, the average number of photons per bin, 〈N

_{photon}〉, and the size of the step at the break point, ρ

_{i}/ρ

_{i}

_{+1}. However, for any combination of these three parameters investigated we find that the threshold value τ′

_{1−}

_{α}falls within 20% of a common upper bound, if described as a function of the variable $\mathcal{L}$

_{max}/L, where $\mathcal{L}$

_{max}is the likelihood of the estimated break point for a particular sequence of length, L. We describe this upper bound empirically through a power law that approaches a constant level for small values of $\mathcal{L}$

_{max}/L.

_{max}/L, τ′

_{0}is a constant threshold for small values of $\mathcal{L}$

_{max}/L, A signifies an amplitude, and λ is a power-law exponent. Table 2 lists the resulting parameters describing this upper bound for various levels of confidence, 1 − α. In tests on simulated photon trajectories of various lengths and step sizes we find that the fraction of the actual break points that lie within the thus estimated confidence interval is close to the expected value 1 − α for most parameter combinations. Only for very short sequences or few photons per bin do we observe the fraction of break points within the confidence interval to drop below 1 − α.

#### 2.3. Sensitivity

_{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.

_{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.

#### 2.4. Comparison to Existing Methods

_{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.

## 3. Single Molecule Experiment

_{||}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

_{photon}〉 ~ 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,

_{photon}〉 = 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, I

_{d}, (Equation 1) which is traditionally used in fluorescence microscopy.

#### 4.1. False Positives

_{photon}〉 = 25 photons on average.

_{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.

#### 4.2. Location of Break Points

_{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, $\mathcal{L}$

_{max}= $\mathcal{L}$(k′

_{max}), as our maximum likelihood estimate, k̂, if $\mathcal{L}$

_{max}exceeds the threshold value τ

_{1−}

_{α}for a sequence of length L at the confidence level 1 − α.

_{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

_{min}Δt = 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.

#### 4.3. False Negatives and Error of Location Estimate

_{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.

_{max}= $\mathcal{L}$

_{max}−$\mathcal{L}$(k) between the likelihood at the maximum, $\mathcal{L}$

_{max}= $\mathcal{L}$(k̂) and the likelihood for a break at the actual break point, $\mathcal{L}$(k). From the distribution of this likelihood difference, Δ$\mathcal{L}$

_{max}, we find a threshold value τ′

_{1−}

_{α}such that for a fraction 1−α of the N trials Δ$\mathcal{L}$

_{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 $\mathcal{L}$(k

^{−}) = $\mathcal{L}$(k

^{+}) = $\mathcal{L}$

_{max}− τ′

_{1−}

_{α}(see Figure 4).

#### 4.4. Algorithm

_{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 {k̂

_{0}. . . 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.

_{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:

_{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 x

_{2}for the entire tested sequence, {ρ

_{0}, . . ., 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.

#### 4.5. Simulation of Photon Sequences With Multiple Break Points

## 5. Conclusions

## Acknowledgements

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**Figure 1.**Waiting times between two consecutive photons recorded in two perpendicular polarization directions as a function of photon arrival time for the fluorescence of a single rhodamine B molecule in a solid polymer matrix. “Blinking” leads to frequent gaps (“dark” periods) in the stream of photons with durations on the order of a millisecond. Line: Result of the identification of intensity change points in each detection channel separately with a maximum likelihood method [29].

**Figure 2.**Threshold values, τ

_{1−}

_{α}, for the likelihood measure $\mathcal{L}$(k′) corresponding to statistically significant intensity ratio break points at a level of confidence of 1 − α as indicated. Points: results of N = 100,000 random number simulations with 〈N

_{photon}〉 = 25 photons per bin and 〈ρ〉 = 1. Lines: smooth fits with an empirical function (Equation 2).

**Figure 3.**Probability for false positive break points as a function of the trial location k′ in simulated intensity ratio trajectories without a break. The ideal, uniform, distribution at a level of confidence of 1−α = 99% is indicated by the dotted line. The first and last N

_{excl}= 10 points, whose probability for false positives approaches or exceeds the ideal target value α, are excluded in the analysis.

**Figure 4.**Identification of the most likely break point in a simulated example (

**a**) Logarithm of the intensity ratio, log(ρ), with 〈ρ〉 = 1, for 1000 simulated points (red) with 〈N

_{photon}〉 = 25 photons per bin and a 30% change in the intensity ratio at the center point (thin vertical line). The average intensity is indicated with a thick line (blue); (

**b**) Likelihood measure, $\mathcal{L}$(k′), for a break at all possible trial positions, k′, for the trajectory in panel (

**a**). The threshold value, τ

_{1−}

_{α}, for a level of confidence 1 − α = 99% is indicated with the dashed line. The position, k̂, of the maximum likelihood, $\mathcal{L}$

_{max}= $\mathcal{L}$(k̂), is taken as the best estimate for the break location. Inset: Expanded view of the likelihood for a break around the maximum, illustrating the determination of the confidence interval k

^{−}. . . k

^{+}, around k̂ using the threshold value τ′

_{1−}

_{α}(blue).

**Figure 5.**Distribution of the deviation between actual and estimated location of the break point, Δk̂ = k̂ − k, for various changes in the intensity ratio at the break point, ρ

_{i}/ρ

_{i}

_{+1}, ranging from 1.2 to 2.0 in steps of 0.1, for an average number of 〈N

_{photon}〉 = 25 photons per bin.

**Figure 6.**Solid line: Average error, 〈|Δk̂|〉, for the estimated location, k̂, of the break point, k, as a function of the change in the intensity ratio at the break point ρ

_{i}/ρ

_{i}

_{+1}for 〈N

_{photon}〉 = 25 photons per bin. Dashed line: Percentage of correctly identified break point locations, P(Δ k̂ = 0). We find no statistically significant dependence of the error on the length of the sample or on the position of the break point within the sample.

**Figure 7.**Probability of undetected break points (false negatives), P, as a function of the magnitude of the change in intensity ratio at the break point, ρ

_{i}/ρ

_{i}

_{+1}, and length of the test sequences, L, for sequences containing one break point in the center. 〈N

_{photon}〉 = 25 photons per bin and 1 − α = 99%. Contour lines are spaced in 10% intervals.

**Figure 8.**Fraction of undetected break points (false negatives) for various changes in the intensity ratio, ρ

_{i}/ρ

_{i}

_{+1}, as indicated, in sequences of photons with 2000 photons per break point as a function of the average bin size, 〈N

_{photon}〉, for 1 − α = 99% using the analysis algorithm invoking the additional safeguards discussed in the text.

**Figure 9.**Ratio of the intensity of the two polarization components of the fluorescence from a single molecule embedded in a rigid polymer matrix, binned at 2 ms (red points). Result from the statistical analysis described in this paper (black line) indicating the expected lack of any reorientations of the probe molecule. Results from the separate identification of intensity break points in each polarization channel (blue line), frequently misidentifying blinking events as single molecule reorientations.

**Figure 10.**(

**a**) Intensity of the two polarization components, I

_{||}and I

_{⊥}, of the fluorescence emitted by a single rhodamine B molecule embedded in poly(vinyl acetate) at the glass transition temperature; (

**b**) Ratio of the intensity of the two polarization components, ρ = I

_{||}/I

_{⊥}, (green) and sequence of single molecule angular jumps (black), reconstructed using the analysis method described in this paper.

**Figure 11.**Distribution of the logarithm of the intensity ratio, ρ = I

_{||}/I

_{⊥}, for a simulated photon arrival trajectory, binned with 〈N

_{photon}〉 = 25 photons per bin (red), with fit to a Gaussian distribution (blue).

**Figure 12.**Schematic representation of the algorithm employed to systematically identify a sequence of most likely break points in an experimental trajectory in a self-consistent manner.

**Table 1.**Best-fit parameters for the empirical fit of Equation 2 to the threshold, τ

_{1−}

_{α}, for statistically significant differences in two sections of intensity ratio change points as determined in random number simulations. α: probability of false positive identification, α

_{eff}: probability of false positive identification if the additional safeguard is used that N

_{min}= 10 consecutive points have to surpass the threshold value, τ

_{1−α}.

Nominal Confidence, 1 − α | Effective Confidence, 1 − α_{eff} | Amplitude, A | scale, s | Exponent, λ |
---|---|---|---|---|

90% | 98% | 3.62 | 0.565 | 0.285 |

95% | 99% | 4.15 | 0.567 | 0.249 |

98% | 99.8% | 4.98 | 0.580 | 0.227 |

99% | 99.95% | 5.79 | 0.608 | 0.231 |

99.5% | 99.98% | 6.80 | 0.655 | 0.253 |

**Table 2.**Parameters for the empirical description of the upper bound for the threshold τ′

_{1−}

_{α}, according to Equation 3, to estimate the confidence interval k

^{−}. . . k

^{+}, for the estimated break point location, k̂, at a level of confidence of 1 − α.

confidence, 1 − α | small $\mathcal{L}$_{max}/L constant threshold, τ′_{0} | amplitude, A | exponent, λ |
---|---|---|---|

68% | 1.10 | 27 | 1.15 |

90% | 1.95 | 34 | 1.12 |

95% | 2.65 | 37 | 1.15 |

98% | 3.40 | 41 | 1.15 |

99% | 4.20 | 43 | 1.15 |

99.5% | 5.20 | 45 | 1.18 |

© 2012 by the authors; licensee Molecular Diversity Preservation International, Basel, Switzerland. This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Bingemann, D.; Allen, R.M. Identification of Intensity Ratio Break Points from Photon Arrival Trajectories in Ratiometric Single Molecule Spectroscopy. *Int. J. Mol. Sci.* **2012**, *13*, 7445-7465.
https://doi.org/10.3390/ijms13067445

**AMA Style**

Bingemann D, Allen RM. Identification of Intensity Ratio Break Points from Photon Arrival Trajectories in Ratiometric Single Molecule Spectroscopy. *International Journal of Molecular Sciences*. 2012; 13(6):7445-7465.
https://doi.org/10.3390/ijms13067445

**Chicago/Turabian Style**

Bingemann, Dieter, and Rachel M. Allen. 2012. "Identification of Intensity Ratio Break Points from Photon Arrival Trajectories in Ratiometric Single Molecule Spectroscopy" *International Journal of Molecular Sciences* 13, no. 6: 7445-7465.
https://doi.org/10.3390/ijms13067445