PMC

Values of the threshold ρ for 5% false positive rate (error fraction) for the uncorrected (A) and corrected (B) log likelihood function, as a function of the number N of photons in the interval. The correction procedure is discussed in Section 3.2. The curves for n_p > 1 polarization channels are discussed in Section 3.3.

MCCP correction factors for (A) the expected value E[ ] and (B) the standard deviation σ_x of the log likelihood function () for N = {50, 100, 500, 1000, 5000, 50,000} and n_p = 8. The horizontal axis x = m/N indicates the position of the mth photon across the interval normalized to the total number of photons. Only half the distribution is shown; the correction factors are symmetric about x = 0.5. These functions are needed to evaluate the correction given in .

The power of the MCCP algorithm to detect short-duration (transient) states in the myosin V ATPase cycle, specifically, the short-lived detached state after the motor head releases from actin but before it steps and rebinds, is determined from photon emission rates simulated using the angles in . Simulations with n_p = 8 (left) and n_p = 16 polarization channels (right) indicate one (top), two (middle), or three (bottom) detected changepoints as the number of photons in the transient state is increased from 0 to 1200 for various SBRs. Requiring that the interval be detected with at least 90% accuracy (dashed curves, middle panels) significantly increases the number of photons needed to identify the state reliably (see text).

Power of the MCCP algorithm to detect changepoints of different magnitudes, as a function of the number of polarization channels n_p = 8 (A and C) and n_p = 16 (B and D) and the number of photons in the interval. Top row, solid lines, and symbols: The fraction of changepoints detected versus an arbitrary relative photon rate change of χ and various N. Dotted lines: The fraction of changepoints that were detected and assigned a time lying within the confidence interval of the true time. The high fraction meeting this condition indicates that these confidence intervals are conservative. Bottom row: The fraction of changepoints detected for an angle change corresponding to the tilting motion of a probe attached to the myosin V lever as it steps (see ), as a function of signal-to-background ratios (SBR) for various N.

The solid curve shows the distribution of false positives for n_p = 16 polarization channels across the interval for uncorrected log likelihoods ; it is strongly peaked near the edge of the interval, then decays slowly to a minimum at the center. The distribution becomes increasingly peaked as N is increased from N = 1000 (panel A) to 10,000 (panel B). The fraction of the total probability lying within the first and last 5% of each interval is 30% and 60% (instead of 10%) for N = 1000 and 10,000, respectively. Applying the correction factors (see ) to the log likelihood and excluding 2.5% of the photons from near the edges (vertical-dashed lines, see Section 4.1.1) result in a nearly uniform distribution of false positives (dotted curve). For comparison, a uniform distribution with total false positive rate 5% would look like the horizontal-dashed line.

Application of changepoint analysis to experimental data on the motions of the molecular motor myosin V. Dots show polar (θ) and azimuthal (φ) angles of a fluorescent probe attached to one lever arm of the motor inferred from photon rates obtained by the traditional time-binned method. The angles are defined in a system whose polar axis is the optical axis of the microscope. There are many outlier points, in part reflecting transitions that occur in the middle of a time bin. Solid lines show those same angles inferred from all the photons that lie between successive changepoints (dashed lines), indicating a clear alternating stride between well-defined values of φ. For each state, five lines are drawn to indicate the uncertainty in the fit angles, as described in Section 3.4. Generally, these lines are too close to distinguish.

(A–C) Illustration of changepoint detection methods on simulated data with N = 200 photons and a ratio between the high and low rates of χ = 3. (A) The photons are binned into 20 constant-width temporal bins. Examining the graph by eye, we may guess that there is a change in photon rate somewhere around the vertical-dashed line, but neither this change time nor the initial and final rates (I₁ and I₂), nor even the existence of a changepoint, are clear. (B) As described in Section 1.3, the kink in the cumulative distribution of photon arrival times gives a much clearer indication of changepoint time, and the two slopes flanking that point yield the corresponding photon rates. Because these are simulated data, we can compare the actual (triangle) and inferred (dashed line) changepoint times. This chapter describes a quantitative implementation of this simple observation. (C) The peak of the log-likelihood surface occurs at photon sequence number m = 99 (vertical-dotted line), and the 95% confidence intervals at m = 98 and 102 enclose the actual changepoint (inset, vertical lines). (D–F) Illustration on real experimental data. A bifunctional fluorescent dye molecule was attached to one of the two lever arms of a myosin-V molecular motor. The motor bound to an immobilized actin filament and began its mechanochemical cycle in the presence of 10-μm ATP. The dye was excited by polTIRF in each of the several incident polarizations (see Section 2.1), and individual emitted photons were detected after passing through a polarization splitter. (Time-stamped data also arise in FRET measurements.) (D) shows the photon counts in a set of 20-time bins (total of N = 1280 photons recorded). No changepoint is visible to the eye. (E) separates the total counts into the several “flavors,” or tagged subpopulations, of emitted photons. Of these, two have been selected for display as solid and dashed curves. A changepoint is visible, but its time cannot be established to greater accuracy than about two time bins. (F) shows the cumulative distribution described in Section 1.3. Each photon time series displays a sharp kink, and moreover, the two curves’ kinks occur at the same time (vertical position).