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We describe a statistical method to analyze dual-channel photon arrival trajectories from single molecule spectroscopy model-free to identify break points in the intensity ratio. Photons are binned with a short bin size to calculate the logarithm of the intensity ratio for each bin. Stochastic photon counting noise leads to a near-normal distribution of this logarithm and the standard student

Single molecule experiments can probe structure and dynamics on a molecular scale, revealing details that in traditional bulk experiments remain hidden behind ensemble averages [

Recently, methods were introduced that permit the correlation analysis of photon arrival times directly without binning [

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 [

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 [_{||} and _{⊥}, to observe the angular reorientation of a single probe molecule reporting on polymer [

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 [

Here, effects of the photodynamics of the probe molecule, which might change the total intensity, _{||}+ _{⊥} without affecting the intensity ratio, _{||}_{⊥}, are eliminated from the monitored quantity, _{d}

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 [_{d}

The threshold value, _{1−}_{α}_{1−}_{α}_{photon}_{photon}_{1−}_{α}

where

If the analysis is performed with the additional safeguard that _{min}_{1−}_{α}_{eff}_{eff}_{i}_{i}_{+1}, and the waiting times, _{w}_{i}_{i}_{+1}, or a short waiting time, _{w}

Trial break points, _{excl}

To illustrate the strength of the analysis method, _{photon}_{max}_{1−}_{α}_{max}

The distribution of the deviation, Δ_{i}_{i}_{+1}. The probability _{i}_{i}_{+1}, at the break point, but not on the length of the sample, _{i}_{i}_{+1} = 1.30 and _{photon}_{i}_{i}_{+1}, for a bin size of _{photon}

The selection of the bin size, _{photon}

From _{1−}_{α}_{photon}_{i}_{i}_{+1}. However, for any combination of these three parameters investigated we find that the threshold value _{1−}_{α}_{max}_{max}_{max}

where _{max}_{0} is a constant threshold for small values of _{max}

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 _{jumps}_{i}_{i}_{+1}, the average number of photons per bin, _{photon}_{photon}

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

For a given sequence of photons, smaller bin sizes, _{photon}

Returning to the challenge of analyzing single molecule photon arrival trajectories containing frequent blinking events (

As an example of the application of the proposed method to a single molecule trajectory with reorientations, _{i}_{i}_{+1}

In single molecule spectroscopy, a fluorescent probe is embedded in the matrix of interest at a very low concentration [

The recorded fluorescence intensity in both polarization directions, _{||} and _{⊥}_{||}_{⊥}_{||} + _{⊥}_{||}_{⊥}

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_{photon}

Stochastic photon counting noise leads to a near-normal distribution [_{photon}_{d}

For normally distributed populations the student _{photon}

At every point, _{max}_{excl}_{1−}_{α}_{max}_{1−}_{α}

The simulated example trajectory (with a break point) shown in _{excl}_{excl}_{excl}_{max}_{max}_{max}_{max}_{1−}_{α}

For the analysis of our experimental data we choose a conservative level of confidence of 1−_{min}_{w}_{w,min}_{min}_{i}_{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, _{min}_{1−}_{α}_{i}_{i}_{+1} = 1.1 to _{i}_{i}_{+1} = 2.0, with an average number of _{photon}_{i}_{i}_{+1}, at the break point,

Furthermore, we determine the difference in the likelihood measure Δ_{max}_{max}_{max}_{max}_{1−}_{α}_{max}_{1−}_{α}_{1−}_{α}^{−} ^{+} for the estimated break point ^{−} and ^{+} are the two points to the right and left of ^{−}) = ^{+}) = _{max}_{1−}_{α}

The application of the _{i}_{i}_{−1}, lengthening the sequence under test by _{step}_{i}_{repeat}_{i}_{−1}, but are lengthened by an additional _{step}_{i}_{−1} in the sequence bracketed by the two adjacent break points _{i}_{−2} and (the newly identified) _{i}_{i}_{−1} is confirmed as the most likely break point location between _{i}_{−2} and _{i}_{i}_{i}_{+1}. However, if the previously identified most likely break point location, _{i}_{−1}, differs from the new location of the break point between _{i}_{−2} and _{i}_{i}_{−1} is no longer statistically significant given the new sequence limit _{i}_{i}_{−1} is modified accordingly (or eliminated all together) and the confirmation check continues backwards until the sequence {_{0} _{i}_{i}

We calculate the corresponding intensity ratios, _{i}_{i}_{i}_{+1}, directly from the number of photons recorded between these two times in each detection channel. To accelerate the calculation we approximate the

with

where _{i}_{i}_{2} for the entire tested sequence, {_{0}_{L}

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 [

Single molecule spectroscopy is a new and very powerful experimental technique, calling for new analysis methods. The statistical method described in this paper identifies sudden changes in a ratiometric variable, the ratio between two fluorescence intensities, indicating the times of individual dynamical events of the single probe molecule, from the recorded photon arrival times in the two detection channels. This model-free analysis approach provides quantifiable error estimates for the jump times at a chosen level of confidence. Tests on simulated photon arrival trajectories indicate that the analysis method locates all events of significant magnitude with little or no error, and still recovers a large fraction of jumps with small amplitudes. The approach described in this paper is general and can easily be applied to different functional forms of the ratiometric variables to analyze other single molecule experiments, such as

This work was made possible through NSF grant CHE-0749863

_{n}

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 [

Threshold values, _{1−}_{α}_{photon}

Probability for false positive break points as a function of the trial location _{excl}

Identification of the most likely break point in a simulated example (_{photon}_{1−}_{α}_{max}^{−} ^{+}, around _{1−}_{α}

Distribution of the deviation between actual and estimated location of the break point, Δ_{i}_{i}_{+1}, ranging from 1.2 to 2.0 in steps of 0.1, for an average number of _{photon}

Solid line: Average error, _{i}_{i}_{+1} for _{photon}

Probability of undetected break points (false negatives), _{i}_{i}_{+1}, and length of the test sequences, _{photon}

Fraction of undetected break points (false negatives) for various changes in the intensity ratio, _{i}_{i}_{+1}, as indicated, in sequences of photons with 2000 photons per break point as a function of the average bin size, _{photon}

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.

(_{||} and _{⊥}_{||}_{⊥}

Distribution of the logarithm of the intensity ratio, _{||}_{⊥}_{photon}

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.

Best-fit parameters for the empirical fit of _{1−}_{α}_{eff}_{min} = 10 consecutive points have to surpass the threshold value, _{1−α}.

Nominal Confidence, 1 − |
Effective Confidence, 1 − _{eff} |
Amplitude, |
scale, |
Exponent, |
---|---|---|---|---|

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 |

Parameters for the empirical description of the upper bound for the threshold _{1−}_{α}^{−} ^{+}, for the estimated break point location,

confidence, 1 − |
small ℒ_{max}_{0} |
amplitude, |
exponent, |
---|---|---|---|

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 |