PLANT: A Method for Detecting Changes of Slope in Noisy Trajectories

Abstract

Time traces obtained from a variety of biophysical experiments contain valuable information on underlying processes occurring at the molecular level. Accurate quantification of these data can help explain the details of the complex dynamics of biological systems. Here, we describe PLANT (Piecewise Linear Approximation of Noisy Trajectories), a segmentation algorithm that allows the reconstruction of time-trace data with constant noise as consecutive straight lines, from which changes of slopes and their respective durations can be extracted. We present a general description of the algorithm and perform extensive simulations to characterize its strengths and limitations, providing a rationale for the performance of the algorithm in the different conditions tested. We further apply the algorithm to experimental data obtained from tracking the centroid position of lymphocytes migrating under the effect of a laminar flow and from single myosin molecules interacting with actin in a dual-trap force-clamp configuration.

Introduction

The output of several biophysics experiments, such as DNA elongation via magnetic and optical tweezers (1, 2, 3, 4), myosin motor motility (5), and cell tracking under shear flow (6, 7), consists of time-trace curves. Many of these curves show alternate regions of pauses and linear increments usually associated with different events taking place at the molecular level (6, 7). The identification of such regions and the associated physical parameters, such as segment duration and slope, are essential to better understanding the underlying dynamics of the ongoing processes. However, the noise produced by thermal fluctuations affects these data and limits their accurate quantification. Thus, the automatic detection of the different features of the time-trace curves remains a challenging computational problem.

Several approaches have been developed for the automatic segmentation of time traces (8, 9, 10, 11), mainly for the dimensionality reduction in data mining (12). One of the most common approaches is the approximation of the original data by different straight lines, usually referred to as piecewise linear approximation (PLA). Given a time series, this approach consists of constructing a piecewise linear function that produces the best representation, either using a limited number of segments or not exceeding a user-specified threshold for the error (8). Generally speaking, the segmentation algorithms based on the PLA can be grouped into three different categories: sliding window, top-down, and bottom-up algorithms (8). In the sliding window algorithms, a segment is grown until the error after adding a new point to the segment exceeds some specified threshold. Although this type of approach is ideal to analyze the data while being generated (i.e., online segmentation), it tends to overestimate the number of segments (13). On the other hand, the top-down type of algorithms works by recursively splitting the entire trace until a certain error threshold is reached. Last, bottom-up algorithms start by arbitrarily dividing the trace into multiple segments and recursively merging the two more similar adjacent segments. The process is stopped when some criterion fixed by the user is met. Although, in general, top-down and bottom-up algorithms have been shown to perform better than the sliding window algorithm (8), there are several variations of these three methods, each of them usually adapted to perform better on specific types of data (14, 15, 16, 17).

A crucial element of the PLA consists of determining the breakpoints between segments of different slopes. This is related to a more general problem in statistics concerning the determination of the actual states of a system whose observables are corrupted by noise (18). This problem can be approached by so-called change-point analysis (19), which was first introduced by Page (20, 21) and assesses whether a statistically significant change in a given observable occurs at some point within the time trace (22, 23). Algorithms based on change-point analysis have been developed to segment time traces originated by a variety of experiments, including biophysical and single-particle experiments (22, 24, 25, 26, 27, 28, 29).

Here, we describe a bottom-up type of PLA algorithm that uses change-point analysis to robustly reconstruct time traces with constant noise. Unlike conventional bottom-up algorithms, in the first step, we find the points at which there is potential variation of the slope, and then we recursively merge the adjacent segments. We follow a likelihood-based approach, and therefore, we do not impose any condition on the number of segments or on the residual error. Although most change-point algorithms developed for the segmentation of biophysical data were meant to detect abrupt changes of the mean value or the variance in time traces, our algorithm deals with the detection of change points in a multiple linear regression model (30). A similar approach was recently described and used to detect changes in DNA looping dynamics during replication (28) or changes in diffusion and/or velocity in single-particle trajectories (29). These works followed the binary segmentation scheme first described in (24) and used a maximal likelihood ratio test, whereas our method relies on a different segmentation approach to find putative change points and later applies a Fisher test to eliminate spurious identifications. This scheme allows us to evaluate all the points within a trace as potential change points and to further refine them according to a hierarchical statistical criterion. We test our algorithm on a variety of conditions, including different noises, slopes, and segment lengths, and provide reasonable explanations about the observed results. This analysis shows a performance equal or higher than that of other published methods. Although, strictly speaking, our method is less generally applicable than others because of the constraint of a nearly constant noise along the trace, this condition is met—at least locally—in several types of single-molecule experiments, such as those discussed in the last sections of this article. Finally, we validated our algorithm on time traces experimentally obtained from 1) tracking the centroids of lymphocytes moving on endothelial cells under the presence of a constant shear flow and 2) ultrafast force-clamp spectroscopy experiments aimed at quantifying force-dependent kinetics of myosin from skeletal muscle.

Materials and Methods

Algorithm description

The flow diagram of the algorithm is depicted in Fig. 1 A. We consider input data consisting of a trajectory in which an observable y is sampled at N discrete time points $t_i$. The working hypothesis is that the trajectory $\left(t_i,y\left(t_i\right)\right)$ is composed of a sequence of K-adjacent linear segments bounded by $K+1$ change points—including the trajectory boundaries—occurring for values of the index $i=c{p}_k$ and corrupted with Gaussian noise having constant variance ${\mathit{\sigma }}^2$ (Fig. 1 B). Given a minimal segment length $N_{MIN}$, an estimation of the noise ${\mathit{\sigma }}^2$, and a confidence level $1-\alpha$, the objective of the algorithm is to efficiently and precisely determine the change points $c{p}_k$ (Fig. 1 C) and thus reconstruct the original trace to finally be able to measure the segment duration and slope (Fig. 1 D). The algorithm is composed of two main sequential blocks, the first of which (WINLINFIT) recursively selects a set of candidate change points associated with a potential variation in the slope by means of least-square fitting and likelihood evaluation. The second (MERGING) refines the number of change points by hierarchically executing an hypothesis test on adjacent segments with a confidence level $1-\alpha$. Analysis of time traces of various lengths shows that the time complexity of the algorithm scales as $O\left(n^2\right)$, and it typically takes $\sim 0.5\text{s}$ to analyze a 100-point-long trace on a MacBook Pro with a 2.2 GHz Intel Core i7 processor and 16 Gb random access memory. For long data streams, segmentation of the trace in overlapping segments and sequential application of the algorithm can be used to reduce computational time. Alternatively, parallelization schemes, such as those described in (31, 32), can also be used to improve scalability in runtime and computational memory.

Figure 1.

The principles of the PLANT algorithm. (A) A flow chart of the algorithm is shown. (B) Representative simulated input trajectory is shown as a series of joint linear segments corrupted with Gaussian noise. (C) The WINLINFIT block detects points compatible with a change of slope (full and empty symbols), which are further refined by the MERGING block based on their statistical significance (full symbols). (D) Reconstructed trajectory, from which it is possible to calculate the slopes and durations of each segment as well as their errors, is shown.

WINLINFIT

The WINLINFIT module operates on the input trajectory to provide an initial estimation for the change points $c{p}_k$. Under the hypothesis that the trajectory is a concatenation of linear segments in the presence of Gaussian noise, performing a least-square fitting over each of these segments with a linear model $f\left(t_i\right)=a_k+b_k{t}_i$ allows us to calculate the estimators ${\hat{a}}_k$ and ${\hat{b}}_k$ (33). Therefore, by applying the principle of maximal likelihood, we obtain that

$$s_k^2=\frac{\stackrel{c{p}_{k+1}}{{\sum }_{i=c{p}_k}}{\left(a_k{t}_i+b_k-y_i\right)}^2}{c{p}_{k+1}-c{p}_k+1}.$$ (1)

The quantity $s_k^2$ can also be interpreted as the variance of a sample of length $N_k=c{p}_{k+1}-c{p}_k+1$, and its expectation value provides the (biased) estimator of the population variance ${\mathit{\sigma }}^2$. It has been demonstrated that $s_k^2$ has a Pearson type III probability distribution (34) as follows:

$$p\left(s_k^2,N_k{|\sigma }^2\right)=\frac{{\left(\frac{N_k}{2{\sigma }^2}\right)}^{\frac{N_k-1}{2}}{\left(s_k^2\right)}^{\frac{N_k-3}{2}}{e}^{-\frac{N_k{s}_k^2}{2{\sigma }^2}}}{\mathit{\Gamma }\left(\frac{N_k-1}{2}\right)},$$ (2)

where $\mathit{\Gamma }(\cdot )$ represents the gamma function.

Although for the trajectories under examination neither the number $K+1$ nor the location of the $c{p}_k$ are known, in many cases it is possible to obtain an estimation of ${\mathit{\sigma }}^2$ either from controlled experiments or from the trajectory itself. An example of such a procedure is provided in Supporting Materials and Methods. We can split the trajectory in all the possible combinations of segments having length $N_k\ge N_{MIN}$, with $N_{MIN}\ge 3$ imposed by the condition of having a number of points larger than the degrees of freedom for a linear fit. The total number of segments is calculated as follows:

$$\begin{array}{l}\stackrel{N}{\sum _{N_k=N_{MIN}}}\left(N_k-N_{MIN}+1\right)=\cdots \\ =\frac{1}{2}\left(N_{MIN}-N-2\right)\left(N_{MIN}-N-1\right).\end{array}$$ (3)

We thus perform a linear fit on each segment, building a matrix of variances. For the sake of simplicity, we will now label the segments with the couple of indexes corresponding to their bounding $cp$ values, such that $N_k=N_{ij}=j-i+1$. With this notation, we obtain the following:

$$s_{ij}^2=\frac{{\sum _{l=i}^j\left({\hat{a}}_{ij}{t}_l+{\hat{b}}_{ij}-y_l\right)}^2}{j-i+1},$$ (4)

and we evaluate the corresponding likelihood $p_{ij}=p\left(s_{ij}^2,N_{ij}{|\sigma }^2\right)$ by means of Eq. 2. The segments fully contained in a linear fragment of the trajectory will provide a higher value for the probability, whereas those spanning over two or more linear regions will display larger deviations from the expected variance and thus lower $p_{ij}$ values. We would like to point out that here we neglect the statistical multiplicity associated with change point selection, which would require a more complicated treatment. Although this simplification limits somewhat the change point localization precision, it largely simplifies the calculations. We then apply a recursive routine by means of which the segment with the maximal $p_{ij}$ value is selected and the coordinates of its endpoints ($t_i$ and $t_j$) are saved as potential change points. All the elements of the matrix $p_{ij}$ corresponding to segments with one end point falling within the interval $\left(t_i,t_j\right)$ are removed and excluded from the subsequent analysis. The process is repeated until the whole trajectory is segmented in a series of contiguous fragments with change points $cp$.

MERGING

Once the potential $cp$ values are obtained from the WINLINFIT step (empty and full circles in Fig. 1 C), the MERGING block statistically refines their occurrence by hierarchically applying a Fisher test with confidence level $1-\alpha$. The MERGING block determines whether some of the change points correspond to a false positive detection (empty circles in Fig. 1 C), the removal of which would allow the adjacent segments to merge. For each pair of adjacent segments $\left(i,j\right)$ and $\left(j,k\right)$, we test the null hypothesis $H_0$—which states that the segments belong to the same linear region $\left(i,k\right)$ and that there is no change point j between them—versus the alternative hypothesis $H_1$, which states that the segments do correspond to different linear behaviors. In the first case $\left(H_0\right)$, we perform a linear fit over the whole region $\left(i,k\right)$, whereas in the second case $\left(H_1\right)$, two different linear fits are executed. The significance level α represents a type I error, i.e., the probability of rejecting the null hypothesis when no change point is present. Then, the Fisher statistic is computed as follows (35):

$$\begin{array}{l}F=\frac{RS{S}_{H0}-RS{S}_{H1}}{RS{S}_{H1}}\left(\frac{N_{ik}-p_{H1}}{p_{H1}-p_{H0}}\right)=\cdots \\ =\left(\frac{\left(k-i+1\right)s_{ik}^2}{\left(j-i+1\right)s_{ij}^2+\left(k-j+1\right)s_{jk}^2}-1\right)\left(\frac{k-i-3}{2}\right),\end{array}$$ (5)

where $RS{S}_{Hi}$ and $p_{Hi}$ are the squared sum of residuals and the number of degrees of freedom associated with the two hypotheses $i=\mathrm{0,1}$, respectively. We then execute the hypothesis test on the pair of segments showing the minimal F value (higher probability of merging). The value of F is compared with the critical value of the Fisher-Snedecor distribution ${\mathcal{F}}_{\alpha }$ (35) with a level of confidence of $1-\alpha$. In the case that the hypothesis $H_0$ cannot be rejected, the segments are merged, and the Fisher statistic is recalculated for the updated list of $cp$ values. The procedure is repeated as long as the segment pair with the minimal F verifies the null hypothesis condition $\text{min}\left\{F\right\}\le {\mathcal{F}}_{\alpha }$.

For multidimensional trajectories, under the assumption that the noise level is the same in all the dimensions, the algorithm can be extended in a straightforward fashion by calculating the total $s_k^2$ and the squared sums of residuals as a sum over all the dimensions and adjusting the number of degrees of freedom accordingly.

Results and Discussion

General considerations

To first investigate the performance of our algorithm in identifying a change point, we used numerical simulations. For this, we used the scheme depicted in Fig. S1 A, which is made of trajectories composed of two different linear segments with lengths $L_1$ and $L_2$, slopes $m_1$ and $m_2$, and noise σ. For each set of parameters, 500 traces in total were simulated and analyzed with the PLANT (Piecewise Linear Approximation of Noisy Trajectories) algorithm using $\alpha =0.05$. The noise SD was automatically estimated by means of the algorithm routine described in Supporting Materials and Methods. To quantify the algorithm performance, we evaluated several figures of merit. An important parameter is the power of detection, i.e., the probability of detecting a change in slope when it is known that there is a change within the trajectory. In Fig. 2, A–C, we show that the power of detection depends in a sigmoidal fashion on the noise σ. Fig. 2 A reports the dependence on the segment length, showing that traces formed by segments of the same length with different slope values but the same slope difference $\left|m_2-m_1\right|$ display analogous behavior. In addition, the larger difference $\left|m_2-m_1\right|$ (Fig. 2 B) produced a higher power of detection. We also investigated traces composed by segments of different lengths, finding that the power of detection is actually a function of the segment length through the quantity $N^{\text{∗}}=2N_1{N}_2/\left(N_1+N_2\right)$ (Fig. 2 B). As discussed in Supporting Materials and Methods, the dependence on these parameters can be expressed in terms of a scaled variable ξ, defined in Eq. S6, for which all the curves collapse into the same one (Fig. 2 C).

Figure 2.

Algorithm performance for $cp$ detection calculated on the simulation scheme depicted in Fig. S1A. (A) The power of detection is shown as a function of the noise $\sigma$ for segment lengths $N_1=N_2=\mathrm{10,20,40,80}$ (from left to right, represented in different tones of gray) and slopes $m_1=0.2,0.5$ and $m_2=0.5,0.8$ ($●$ and $\mathit{△}$, respectively) with same slope increment $\left|m_2-m_1\right|=0.3$. (B) The power of detection is shown as a function of the noise $\sigma$ for segment slopes $m_1=0.2$ and $m_2=0.3,0.5,0.8,1.3$ (from left to right, represented in different tones of gray) and lengths $N_1=13,N_2=30$$(●)$, and $N_1=N_2=20$$\left(\mathit{△}\right)$ with similar $N^{\ast }$ ($N^{\ast }\simeq 18\text{and}{N}^{\ast }=20$, respectively). (C) The powers of detection for several segment lengths and slopes collapse onto each other once expressed in terms of the rescaled variable $\xi$, defined in Eq. S6. The lines in (A)–(C) correspond to the function given in Eq. S7. (D) Change point localization precision, normalized to the trace length for traces composed by: segments with slopes $m_1=0.2,m_2=0.3$ and lengths $N_1=20,N_2=40$$(○)$; segments with slopes $m_1=0.2,m_2=0.3$ and lengths $N_1=N_2=80$$(\square )$; segments with slopes $m_1=0.2,m_2=0.5$ and lengths $N_1=N_2=80$$\left(\mathit{△}\right)$; segments with slopes m₁ = 0.2, $m_2=1.1$ and lengths $N_1=N_2=80$$\left(\mathit{▽}\right)$. The lines correspond to Eq. S3. (E) The error in the determination of the segment slopes normalized to the modulus of slope difference is shown. Symbols have the same meaning as in (D). Empty or filled symbols refer to the two different segments of the simulated trace. The lines correspond to Eq. S5. (F) The false positive identification rate obtained for $\alpha =0.05$ on traces of constant slope is shown as a function of $\sigma$ for different parameter sets: $m=0.1,N=40$$(○)$, $m=0.1,N=80$$(\square )$, $m=0.1,N=150$$\left(\mathit{△}\right)$, and $m=1.1,N=150$$\left(\mathit{▽}\right)$. Each point in the plot was obtained from 500 simulated traces. Dashed lines correspond to the average values.

Next, we calculated the precision in the localization of the $cp$, which was defined as the SD of the distribution of the detected $cp$ position minus the “true” $cp$ position. Fig. 2 D shows that this quantity depends on the noise, the slope, and the difference between slopes. We also evaluated the error on the detected slopes, as displayed in Fig. 2 E. $\mathit{\Delta }{m}_i$ is found to be independent on the segment slopes and to scale linearly with the noise. Last, in Fig. 2 F, we report the false positive rate, i.e., the probability of detecting a change point when there actually is no change point. The data show no clear trend with segment slope and noise because the statistical tests performed by the algorithm explicitly take into account the presence of the noise and its value. The false positive rate shows a dependence on the trace length ($\sim N^{1/2}$, Fig. S2 A), with values lower than the false positive rate specified through the type I error level $\alpha =0.05$ ($\mathrm{\lesssim }0.1$ for $L\mathrm{\lesssim }500)$), ensuring high specificity and robustness against spurious detection caused by noise-induced fluctuation. To allow comparison with other works that report the false positive rate per point within a trajectory, we would like to point out that in Fig. 2 F, we plot the false positive rate per trajectory. When normalizing by the trajectory length N, we obtain much lower values for the false positive rate. For the cases shown in Fig. 2 F, the average false positive rates per point are of the order of ${10}^{-4}$ already for relatively short trajectories ($<200$ points); as also reported by others (29), this value further decreases as a function of the trajectory length.

The actual scenario to which the algorithm is meant to be applied is more complex than the simulation scheme depicted in Fig. S1 A. This is because each segment of the trace is embedded within two others with different slopes and durations. Moreover, both a missed change point and a false positive detection will affect the determination of the parameters of the neighboring segments in a complex way that ultimately depends on the distribution of slopes and the duration of the trace. Therefore, the specific performance of the algorithm must be evaluated in every specific case by means of simulations. To provide a flavor of the ability of the algorithm to correctly identify a segment in a more realistic scenario, we also used a more complex simulation scheme composed by three neighboring segments (Fig. S1 B). The results of these simulations are described in Fig. S2.

Application of the algorithm to the analysis of trajectories from cell-tracking experiments

We applied our algorithm to analyze trajectories describing the motion of T cells over endothelial cells under shear flow (Fig. S3). Typical time traces of the T cell centroid coordinates in the directions perpendicular (x) and parallel (y) to the flow are shown in Fig. 3 A. Lymphocytes display heterogeneous motions, which at the experimental time resolution appear composed of a concatenation of segments. Each segment has an approximately constant velocity, but we observe large velocity variations between different segments, including a reversal of the velocity by moving against the flow direction $\left(v_y<0\right)$. Therefore, we applied our algorithm to a set of experimentally collected time traces to automatically characterize the different features in the cellular motion. The result of the algorithm applied to the y coordinate of the cell centroid position is shown in Fig. 3 A, where the shaded gray regions correspond to the detected changes of slope, and the dashed line represents the piecewise reconstructed trace, which has been shifted up for clarity. This analysis was applied to 50 traces corresponding to different cells with durations of 400–600 s, and the distributions of segment lengths (durations) and slopes (velocities) obtained from the analysis are represented as histograms in Fig. 3, B and C.

Figure 3.

Analysis of representative time traces from cell-tracking experiments. (A) Time traces of the x (gray line) and y coordinates (black line) of the position of the cell represented in Fig. S3 are shown. Shaded gray regions correspond to regions where the algorithm detected changes of slope. The dashed line corresponds to the piecewise reconstructed trace of the trajectory along the y-direction that has been shifted upwards for clarity. (B) A histogram shows the duration of the detected segments, divided according to the direction of the y-component of the velocity along (light gray shaded area) or against (white area) the shear flow. (C) A histogram shows the y-component of the velocity of the detected segments. In both panels, the gray bars correspond to experimental data, and the black lines are the resulting distributions of time and velocity after applying the algorithm to the simulated data.

To verify whether the motion along or against the flow had different characteristics, the data corresponding to the duration of the segments (Fig. 3 B) were split based on the direction of the velocity component along the flow direction $v_y$, which is reported in Fig. 3 C. Both distributions show an asymmetric shape, reflecting the fact that the T cells preferentially move along the flow. Despite this preference, molecular interactions established by cells allow them to sustain the motion against the shear flow with nonnegligible speed. We attempted to obtain a more quantitative insight from these data to link the observed features to a molecular model of cell migration (36, 37). To this aim, we performed stochastic simulations in which we generated synthetic traces with a priori known values of velocities and durations to evaluate the impact of the algorithm on the determination of these parameters (see Supporting Materials and Methods for a full description). By means of this approach, we could find a set of parameters that could nicely reproduce our experimental results (black lines in Fig. 3, B and C). This allowed us to estimate the real parameters of the distributions before the limitations introduced by instrumental resolution and algorithm-limited performance. The results showed that T cells display distributions compatible with simulations in which the movement along the flow direction occurs at an average speed of $39\pm 6$ nm/s, which decreases to $29\pm 5$ nm/s when moving against the flow. The typical duration of a segment traveling at nearly constant speed is $35\pm 6$ s, independent of the direction, producing an average “step” of ∼1.4 μm in the positive direction and ∼1 μm in the negative direction. In addition, the cells spend 2/3 of the time traveling along the flow. These results provide us with a first test of the ability of our algorithm to extract valuable information from noisy trajectories obtained from cell-tracking experiments. The values of cell velocity and segment duration that can be obtained by means of our algorithm thus constitute relevant microscopic parameters, which can be used to validate molecular models of cell migration (36, 37) and thus better understand the role played by molecular interactions.

Application of the algorithm to the analysis of trajectories from ultrafast force-clamp spectroscopy

We also applied the PLANT algorithm to data from ultrafast force-clamp spectroscopy experiments on actin-myosin interactions. Here, an actin filament is stretched between two optically trapped beads and set in contact with a third bead fixed onto the coverslip, which is coated with myosin II at single-molecule concentration (Fig. S4). In these experiments, a constant force F is applied to the bead-actin-bead “dumbbell” and alternated back and forth, thus producing a sawtooth-like time trace with regions of positive (negative) slope interrupted by flat regions corresponding to binding events of myosin to actin (Fig. 4 A). The algorithm could efficiently locate the flat segments along the traces. In agreement with previous results (3), the velocity distribution returned by our algorithm clearly showed the existence of two populations (gray histogram in Fig. 4 B): one with a mean velocity close to zero and corresponding to the binding events and another peaked around the velocity caused by the application of the force on the dumbbell ($v=F/\gamma$, where γ is the viscous drag coefficient of the dumbbell; unbound state). However, probably because of the presence of correlated noise, the algorithm produced a large number of very short segments with slightly different slopes. To refine the analysis and better assign the events to these two states, we used the same criterion described in the original manuscript (3), i.e., a threshold velocity corresponding to the weighted average of the peak velocities (dashed black line in Fig. 4 B).

Figure 4.

Analysis of ultrafast force-clamp spectroscopy experiments. (A) Fragments of traces corresponding to actin-myosin binding events are shown with different durations under a positive force of 1 pN and [ATP] = 20 μM. Dark gray areas delimit the nearly flat regions found by the PLANT algorithm, corresponding to binding events. White lines represent the reconstructed segments. (B) Histograms show the velocity of the fragments before (gray bars) and after (open bars) threshold-based classification. The dashed black line corresponds to the threshold value. (C) Cumulative histograms show the duration of actin-myosin binding events (gray) and the corresponding fit using a two-exponential cumulative probability density function (black line).

This procedure groups together adjacent segments with slopes compatible with the occurrence of the same state, producing a clear narrowing of the velocity distribution (open bars of the histogram in Fig. 4 B). As shown in Fig. 4 A, this procedure produces a reconstruction of the flat regions with high fidelity and allows us to detect binding events as short as 100 μs, corresponding to 20 data points. The algorithm retrieved 126 binding events on the analyzed portion of the trace. The cumulative probability distribution of the duration of these events is displayed in Fig. 4 C. In agreement with a previous analysis obtained by means of a different method (3), the distribution of bound-state duration was found to be compatible with a two-exponential function. From the fitting, we obtained two different rates $k_1=103\pm 5$ s^–1 and $k_2=\left(4\pm 3\right)\times {10}^3$ s^–1. Although the faster rate has a rather large error because it corresponds to durations of the same order of magnitude as the algorithm dead time, the rate values are fully consistent with earlier results and provide higher time resolution compared with previous methods (3). As such, the algorithm provides an efficient, automated tool to extract rapid kinetics from time traces generated by this type of single-molecule experiment, including, for example, those measuring the sequence-dependent interaction of single transcription factors along DNA (38).

Conclusions

We introduced PLANT, a bottom-up type of algorithm for detecting changes of slope in time traces affected by Gaussian noise. Unlike other algorithms, PLANT does not impose any condition on the number of segments composing the trajectory and includes a routine for the estimation of unknown noise from the data. By means of extensive numerical simulations, we have shown that it robustly reconstructs noisy trajectories with the power of detection of change points, which depends on the length of the segment, the noise, and the slope difference. The algorithm is also provided as a flexible and easy-to-use graphic user interface. Finally, we applied the algorithm on a set of experimental data obtained by tracking the position of T cells and single myosin molecules interacting with actin under force. These results provide a first test of the ability of our algorithm to extract valuable information from noisy trajectories. The program is available on the GitHub repository hosting service at http://github.com/cmanzo/PLANT and includes the program files, documentation, and demo data.

Author Contributions

A.S.-C. and C.M. designed and developed the software. A.S.-C. carried out the simulations and analyzed the data. A.S.-C., I.K.P., and L.G. performed experiments. F.S.P., M.C., M.F.G.-P., and C.M. supervised the research. A.S.-C., M.F.G.-P., and C.M. wrote the article. All authors approved the final version of the manuscript.

Editor: Antoine van Oijen.

Supporting Citations

References (39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52) appear in the Supporting Material.