During a chemical or biological process, a molecule may transition through a series of states, many of which are rare or short-lived. Advances in technology have made it easier to detect these states by gathering large amounts of data on individual molecules. However, the increasing size of these datasets has put a strain on the algorithms and software used to identify different molecular states.

Now, White et al. have developed a new algorithm called DISC which overcomes this technical limitation. Unlike most other algorithms, DISC requires minimal input from the user and uses a new method to group the data into categories that represent distinct molecular states. Although this new approach produces a similar end-result, it reaches this conclusion much faster than more commonly used algorithms.

To test the effectiveness of the algorithm, White et al. studied how individual molecules of a chemical known as cAMP bind to parts of proteins called cyclic nucleotide binding domains (or CNDBs for short). A fluorescent tag was attached to single molecules of cAMP and data were collected on the behavior of each molecule. Previous evidence suggested that when four CNDBs join together to form a so-called tetramer complex, this affects the binding of cAMP. Using the DISC system, White et al. showed that individual cAMP molecules interact with all four domains in a similar way, suggesting that the binding of cAMP is not impacted by the formation of a tetramer complex.

Analyzing this data took DISC less than 20 minutes compared to existing algorithms which took anywhere between four hours and two weeks to complete. The enhanced speed of the DISC algorithm could make it easier to analyze much larger datasets from other techniques in addition to fluorescence. This means that a greater number of states can be sampled, providing a deeper insight into the inner workings of biological and chemical processes.

Single-molecule methods are powerful tools for providing insight into heterogeneous dynamics underlying chemical and biological processes otherwise obscured in bulk-averaged measurements (Moerner et al., 2015). Use of these techniques has expanded rapidly, with modalities spanning electrophysiology, fluorescence, and force spectroscopy to probe diverse physical phenomena. Generally, single-molecule data are obtained as a time trajectory where molecular behavior is observed as a series of transitions between a set of discrete states obscured by experimental noise. Following the growing realization that molecules involved in physiological and chemical processes exhibit complex kinetics and a diversity of behavior, there is an increasing demand for high-throughput technologies to adequately sample different sub-populations and rare but important events (Hill et al., 2017). As a result, there has been tremendous progress in improving both the number of single molecules that can be observed simultaneously and the total observation time of each molecule. For example, the observation window prior to photobleaching in conventional fluorescence paradigms such as single-molecule Förster resonance energy transfer (smFRET) or colocalization single-molecule spectroscopy (CoSMoS) can be dramatically extended with recently developed photostable dyes (Grimm et al., 2015; Altman et al., 2012). The current generation of metal-oxide semiconductor (sCMOS) detectors enables simultaneous imaging of 1 × 10⁴ molecules in a total internal fluorescence microscopy (TIRFM) configuration and can be coupled with nanofabricated zero-mode waveguides (ZMWs) to enable access to high concentrations (Levene et al., 2003; Chen et al., 2014; Juette et al., 2016). Non-fluorescence-based single-molecule experiments such as plasmon rulers, scattering, magnetic tweezers, and single-molecule centrifugation generate a tremendous amount of data through parallel measurement of hundreds of molecules with orders of magnitude longer recordings than a typical fluorescence experiment (Berghuis et al., 2016; Ye et al., 2018; Yang et al., 2016; Popa et al., 2016; Young and Kukura, 2019).

Unfortunately, despite these advances in generating statistically robust data sets, standard analysis algorithms impose a computational bottleneck at this scale of data generation (Hill et al., 2017; Juette et al., 2016). This is particularly true when the dynamics and physical states of a system are unknown.

Typical statistical modeling of single-molecule trajectories often adopts one of two approaches. The first is a probabilistic approach that models a molecule’s behavior as a Markov chain, wherein the molecule transitions between hidden discrete states whose outputs are measured experimentally (hidden Markov model, HMM). This involves estimating the transition probabilities between a small set of postulated states with defined outputs using methods to maximize the likelihood of the model given the observations or Bayesian inference to estimate model parameter distributions. Numerous software packages have been developed for implementing HMMs, such as QuB (Nicolai and Sachs, 2013; Qin et al., 2000), HaMMy (McKinney et al., 2006), SMART (Greenfeld et al., 2012), vbFRET (Bronson et al., 2009), ebFRET (van de Meent et al., 2014) and SPARTAN (Juette et al., 2016), each of which utilize a different HMM training method. For example, QuB implements the fast segmental k-means algorithm (SKM) which combines k-means clustering and the Viterbi algorithm to identify transitions between postulated states (Juang and Rabiner, 1990), whereas vbFRET adapts variational Bayesian inference for parameter estimation at faster speeds than traditional HMM training in both smFRET and single-particle tracking experiments (Blanco and Walter, 2010; Persson et al., 2013). Although powerful statistical tools are very useful for single-molecule analysis, HMMs have notable limitations, especially in the context of high-throughput analysis and unknown system dynamics. For example, HMMs are often used in a supervised manner where the user postulates model parameters such as the number of states, their measured outputs, and the allowed transitions between them. As this information is often not known a priori, it is desirable to test multiple models and rank them according to Bayesian probabilistic approaches or objective functions, such as the Bayesian Information Criterion (BIC). This process can dramatically increase the analysis time to ensure the parameters space has been sufficiently explored, which restrict their usefulness in high-throughput single-molecule analysis. Although variants such as infinite HMMs using Bayesian nonparametric inference try to naturally learn the trajectory of the states without the typical parametric model selection, these too are often computationally prohibitive for large data sets (Hines et al., 2015; Sgouralis and Pressé, 2017; Sgouralis et al., 2018).

The second class of single-molecule analysis approaches idealization as an unsupervised clustering problem from machine learning (Li and Yang, 2019). Typically, clusters of intensity values (e.g. states) are determined using bottom-up hierarchical agglomerative clustering (HAC) algorithms, which begin by treating each observation of N total observations as singleton clusters and perform N-1 iterations wherein pairs of clusters are merged until all data-points belong to a single cluster. For each number of possible clusters, an objective function can be minimized to find the optimal trade-off between the complexity and fit. This implementation results in time complexity of O(N²) owning to the need of computing a NxN similarity matrix to determine which clusters should be merged at each iteration. In practice, a separate algorithm called change-point (CP) detection precedes HAC to reduce the solution domain of the objective function by identifying statistically significant stepwise changes in signal over time. We denote this combination of algorithms as CP-HAC. At each identified CP, the data are divided into two segments, each described by the mean values of the data-points between sequential change-points. By using the segments as initial clusters rather than all N data-points, the HAC computation can be dramatically reduced. The pioneering application of CP-HAC to single-molecule data addressed CP detection and clustering in the presence of Poisson noise (Watkins and Yang, 2005). Variants of this framework such as STaSI use other merit functions for Gaussian noise, including the Student’s t-test for fast CP detection and minimum description length for state selection (Shuang et al., 2014). An advantage of CP-HAC methods is that they only require a confidence interval and/or an objective function for the analysis, unlike HMMs which require a model to fit. This makes them very attractive in situations where there is no prior knowledge about the different physical states. In common experimental modalities such as smFRET, CP-HAC methods offer superior computational speed over HMM approaches; however, their quadratic time-complexity renders them inefficient on long trajectories (Shuang et al., 2014). In addition, simulation studies have suggested CP-HAC algorithms yield lower event detection accuracy than HMM approaches (Hadzic et al., 2018).

Despite the utility of HMM and CP-HAC methods, there is an outstanding need for an analysis platform to provide accurate model-free idealization with sufficiently high computational performance to keep up with the increasing scale of data generation. Although advances in computing hardware can, to a degree, mitigate these issues (Smith et al., 2019; Song and Yang, 2017), there remains a pressing need for more computationally efficient algorithms. Here, we present a new algorithm for efficient and accurate idealization of large single-molecule datasets in a model-independent manner. Our method, DISC (DIvisive Segmentation and Clustering), enhances existing statistical learning methods and enables rapid state and event detection. The DISC algorithm draws inspiration from other model-free algorithms like CP-HAC and SKM that rely on unsupervised algorithms, such as k-means and hierarchical clustering. We advance these ideas by adapting divisive clustering algorithms from data mining and information theory to improve the rate and accuracy of identifying signal amplitude clusters (states) in a top-down process as opposed to the typical bottom-up clustering (Pelleg and Moore, 2000; Karypis et al., 2000; Hamerly and Elkan, 2003). We further couple our model-free state detection with the Viterbi algorithm to enable robust event detection on par with HMM methods (Juang and Rabiner, 1990; Qin, 2004; Viterbi, 1967). Overall, DISC is an unsupervised method that combines statistical learning approaches with the high event detection accuracy of HMMs at a fraction of the computational cost, enabling convenient application to large datasets.

Theory

Motivation

The goal of the DISC algorithm is time series idealization: the hard assignment of data points into discrete states. DISC approaches the problem of idealization as an unsupervised problem in machine learning wherein the number of significant states for a given single-molecule trajectory are not known a priori. This process of learning both the significant states and the transitions between them is accomplished in three phases: 1) divisive segmentation, 2) HAC, and 3) the Viterbi algorithm (Figure 1a, Figure 1—figure supplement 1). The first two phases use unsupervised statistical learning to identify the intensities of states following an appropriate user-specified objective function. The second phase uses the Viterbi algorithm to decode the most probable sequence of transitions between the identified states.

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The primary distinction of DISC vs other model-free single-molecule analysis algorithms is the use of divisive clustering. Rather than being limited by the quadratic time-complexity of bottom-up HAC, DISC adopts the exact opposite hierarchical clustering algorithm: top-down divisive clustering. Divisive clustering initializes all data-points to a single cluster and iteratively splits data into sub-clusters until each data point is its own singleton cluster. Given that there are 2 ^N−1-1 ways of spitting N data points into two sub-clusters, the complexity of top-down processes has led to their infrequent use. However, efficient implementations involving the use of sub-routines to determine how clusters should be split and whether the new clusters should be accepted has resulted in more efficient and accurate algorithms. For example, clusters can be sub-partitioned using k-means clustering and an objective function can determine if the fit of the data improves with an additional sub-cluster (Pelleg and Moore, 2000; Karypis et al., 2000; Hamerly and Elkan, 2003). Therefore, unlike HAC which makes a final decision at the end of the clustering, top-down processes make a binary decision during each iteration. For each iteration, if the objective function does not improve with the addition of new sub-clusters, the split is rejected. If the fit does improve, the split is accepted and each sub-cluster is again further partitioned into two new sub-clusters. The algorithm terminates once clusters cannot be split for further improvement of the objective function. Given this internal heuristic of cluster acceptance or rejection, the most probable number of states is typically identified within a few iterations, as opposed to the N-1 iterations necessary for HAC. In addition, when using k-means for partitioning data points into sub-clusters, the top-down algorithms can reach O(N) time complexity (Karypis et al., 2000). This simple implementation and accelerated computational speed make divisive clustering an attractive alternative that enables high-throughput single-molecule idealization.

Divisive segmentation

The first phase of DISC is divisive segmentation. Consider an observed single-molecule trajectory ${\displaystyle x=\left\{x_1,\dots x_N\right\}}$ where each $x_n$ is the observed intensity value $x$ at time-stamp n for N total observations contaminated by Gaussian noise. Like CP-HAC, the goal of the divisive segmentation is to identify and allocate each data-point into the optimal number of idealized states denoted by K. Following our Gaussian assumption, each state ${\varphi }_j\in \{{\varphi }_1,\dots ,{\varphi }_K\}$ is described by the mean (µ) and standard deviation ($\sigma$) of data points allocated to the state ${\varphi }_j=\left({\mu }_j,{\sigma }_j\right)$. We denote a series of transitions between states as ${\displaystyle y=\left\{y_1,\dots y_N\right\}}$ where $y_n$ ∈ ${\displaystyle \left\{{\varphi }_1,\dots ,{\varphi }_K\right\}}$ and 1 ≤ K ≤ N. Divisive segmentation aims to iteratively yield $x$ by determining whether data-points in a given cluster are better described by one or two states. At the onset of divisive segmentation, it is assumed that $x$ is described by a single idealized state. We will denote this initial fit as ${\displaystyle y_0=\left\{y_{0_1},\dots y_{0_N}\right\}}$ where $y_{0_i}$ ∈ $\left\{{\varphi }_0\right\}$ and ${\varphi }_0=\left({\mu }_0,{\sigma }_0\right)$.

Allocating each data-point into two unique states is accomplished in two sequential phases: CP detection and k-means clustering. As opposed to standard divisive algorithms that allocate data points via k-means clustering only, we find CP identification prior to clustering advantageous. Not only does it reduce the solution domain of the objective function, but it also speeds up subsequent clustering while providing a reasonable estimate of state transitions. CP detection is performed with the popular recursive binary segmentation algorithm (Watkins and Yang, 2005; Scott and Knott, 1974). For each time stamp n in $x$, a hypothesis test is conducted to evaluate the probability that a CP occurred at position n via

• H₀: a CP did not occur at position n

• H₁: a CP did occur as position n

In the context of single-molecule idealization, a CP is the location indicating a significant difference in mean intensity values between the data segments separated at location n, where the mean values of each segment are computed by

(1) ${\displaystyle {\mu }_1=\frac{1}{n}\sum _{i=1}^n{x}_i{\mu }_2=\frac{1}{N-n}\sum _{i=n+1}^N{x}_i}$

To determine whether there is a statistically significant difference between the two segments, we use a two-way Student’s t-test of unequal sample size but uniform variance to evaluate the differences in mean. This is the same approach used in STaSI (Shuang et al., 2014). Specifically, a t-value is computed for each position n by

(2) ${\displaystyle t_k=\frac{\left|{\mu }_1-{\mu }_2\right|}{\sigma \sqrt{\frac{1}{n}+\frac{1}{N-n}}}}$

where σ is the estimated standard deviation of uniform noise (Shuang et al., 2014). The most probable CP location c corresponds to the maximum t-value t_max given by

(3) ${\displaystyle c=\underset{k}{\mathit{a}\mathit{r}\mathit{g}\mathit{m}\mathit{a}\mathit{x}}k(t)}$

(4) ${\displaystyle t_{max}=\underset{k}{\mathit{m}\mathit{a}\mathit{x}}k(t)}$

For a user specified confidence interval, a critical value is used to determine whether to accept the change-point. If t_max > critical-value, we reject H₀ and the CP is accepted. This in turn segments the data at position c. As there are likely multiple CPs in $x$, the algorithm continues in a recursive manner by searching within each new segment ${\displaystyle s_1=\left\{x_1,\dots x_c\right\}}$ and ${\displaystyle s_2=\left\{x_{c+1},\dots x_N\right\}}$. This process terminates when no significant changes in mean intensity are found within any segment. Importantly, the confidence interval set by the user plays a crucial role by acting as a hard threshold for false positive rate.

Following the completion of CP detection, times-series $x$ can be described as a series of C+1 intensity segments where C is the total number of CP identified given by an idealized state trajectory where $y_n$ ∈ $\{{\varphi }_1,\dots ,{\varphi }_K\}$ and 1 ≤ K ≤ C+1. Like CP-HAC, the next goal is to discover the optimal number of states K into which to cluster the C+1 intensity segments generated by CP detection. Rather than iteratively merging each segment like bottom-up algorithms, we use divisive segmentation to cluster the data points in a top-down fashion. For divisive segmentation, all identified segments are partitioned into two unique clusters using the k-means algorithm, where the center of each cluster is described by the mean values of the CP-idealized data points within the cluster. For efficiency, DISC uses a modified k-means algorithm that is both deterministic and faster than standard implementations through use of triangle inequality for computational reduction (Elkan, 2003). Overall, this results in a series of transitions between two states ${\displaystyle y_1=\left\{y_{1_1},\dots y_{1_N}\right\}}$, where ${\displaystyle y_{1_n}}$ ∈ {$\varphi$₁, $\varphi$₂} and ${\varphi }_j=\left({\mu }_j,{\sigma }_j\right)$ the corresponds to a state assignment for each observation in $x$.

Now that the data-points are allocated to two separate clusters with identified transitions, the goal is to determine if $x$ is better fit with one or two states ($y_0$ vs $y_1$). Like CP detection, this decision follows a hypothesis test where

• H₀: the data corresponds to one unique state

• H₁: the data corresponds to two unique states

To determine whether one or two states provides a better fit, we use the Bayesian Information Criterion (BIC) which is defined in a general form as

(5) ${\displaystyle BIC=-2\ln (\widehat{\mathcal{ℒ}})+Mln(N)}$

where ${\displaystyle \widehat{\mathcal{ℒ}}}$ is the likelihood for the estimated model with M free parameters (Schwarz, 1978). The likelihood that the observations $x$ arose from a single state ($y_0$) is simply the product of the probability densities of a Gaussian distribution evaluated for each $x_i$. For the multi-state fit of $y_1$, the model extends to a mixture of 1D Gaussians whereby ${\displaystyle \widehat{\mathcal{ℒ}}}$ is computed as a linear combination of each K Gaussian components, corresponding to each state ${\varphi }_j\in \{{\varphi }_1,\dots ,{\varphi }_K\}$ with ${\varphi }_j=\left({\mu }_j,{\sigma }_j\right)$ weighted by a mixing coefficient (${\pi }_j$) (Bishop, 2006).

(6) ${\displaystyle \mathcal{𝒩}\left(x\right|\mu ,\sigma )=\frac{1}{\sigma \sqrt{2\pi }}\mathrm{e}\mathrm{x}\mathrm{p}\left(\frac{-{\left(x-\mu \right)}^2}{{\sigma }^2}\right)}$

(7) ${\displaystyle \widehat{\mathcal{ℒ}}=\prod _{i=1}^N\sum _{j=1}^K{\pi }_j\ast \mathcal{𝒩}\left(x_i\right|{\mu }_j,{\sigma }_j)}$

To test the null hypothesis that $x$ is described by one state instead of two, BIC values are computed for ${\displaystyle x}$ with a fit of a single-state (BIC₁) and fit with two-states from divisive segmentation (BIC₂). If BIC₂ > BIC₁, H₀ is accepted and we believe $x$ is sufficiently described by a single-state. If the BIC₂ ≤ BIC₁, the H₀ is rejected and $x$ is split into two states. Assuming two-states are identified on the first iteration, the sequential process of CP detection and bi-partitioning with k-means clustering continues in a recursive fashion within data points belonging to each of the newly identified states (Figure 1b; Pelleg and Moore, 2000). Divisive segmentation continues to identify sub-clusters within each identified cluster until no cluster can be further partitioned. Overall, the recursive bi-partitioning and binary decision making in both CP detection and divisive clustering result in a reduced time complexity on the order of O(Nlog(N)).

Agglomerative clustering

Self-termination of divisive segmentation results in a series of estimated states and transitions within the trajectory. While this algorithm is exceptionally fast owing to its top-down greedy design, the reliance on local choices for state assignments rather than evaluation of the entire trajectory for an optimal decision can produce sub-optimal results. This error often surfaces as an over-sampling of the number of states and an under-sampling of the kinetic transitions. Over-fitting the number of states results from a downward dissemination of error from early splits: if bisecting a given cluster is suboptimal, two different parent clusters may each produce highly similar and redundant sub-clusters thereby overfitting the number of states. Fortunately, this over-estimate of the number of states can be corrected using bottom-up clustering. Therefore, the second phase of DISC uses HAC to compute the similarities between all identified states at a global level and assess the fit of the whole trajectory rather than segmented portions. Like CP-HAC schemes, an objective function is used to determine the overall fit and number of states in the trace by merging highly similar clusters whose separation may arise during divisive segmentation (Figure 1b). For a general application, we continue to use BIC for evaluating fit vs complexity. The similarity between neighboring states $\varphi$_i and $\varphi$_j is computed using Ward’s minimum variance method (Ward, 1963), which considers the number of data points in each state (n) and the Euclidean distance between the means of the states,

(8) ${\displaystyle d({\varphi }_i,{\varphi }_j)=\sqrt{\frac{2n_{\varphi i}{n}_{\varphi j}}{n_{\varphi i}+n_{\varphi j}}}{\Vert {\mu }_{\varphi i}-{\mu }_{\varphi j}\Vert }_2}$

The improvement of HAC on divisive segmentation for state detection is shown in Figure 2. Although divisive segmentation alone tends to slightly over-estimate the number of states, it provides a more reasonable estimate than CP detection alone (Figure 2a). The comparative performances in terms of speed and accuracy of these algorithms is further explored in Figure 2b. While CP detection alone (Figure 2b, blue) is very fast, it consistently yields a higher number of total states as compared to the ground truth. As CP-HAC frameworks must explore this large state space in its entirety, they can achieve higher accuracy than CP detection alone, but they are much slower algorithms (Figure 2b, green). In contrast, the use of top-down clustering in divisive segmentation dramatically reduces the total state-space for exploration, resulting in a faster algorithm than CP-HAC with much higher accuracy than CP detection alone (Figure 2b, purple). Finally, the sequential combination of divisive segmentation and HAC used in DISC lead to the highest state detection accuracy with minimal computational cost (Figure 2b, orange).

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Simulated data of varying dynamics.

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Viterbi algorithm

Following state refinement with HAC, the trajectory is again described as a series of temporal transitions between identified intensity states. Although the overall states are well estimated at this point, fast transitions are often missed during CP analysis of single-molecule trajectories (Hadzic et al., 2018). To ensure events are accurately detected, the final phase of DISC applies the Viterbi algorithm (Viterbi, 1967).

The goal of the Viterbi algorithm is to identify the most probable sequence of hidden states through a series of observations. In our scenario, we have K total states and N total observations in our trajectory . In a naïve manner, determining the most probable sequence of hidden states y could be accomplished by evaluating the likelihood of every possible hidden state sequence and choosing the most probable. However, as there are K^N possible paths though the trajectory, this quickly becomes computationally intractable. A solution to this problem is the Viterbi algorithm, which makes use of dynamics programming to store only the most optimal state sequencing leading up to a given time point. In general, the Viterbi algorithm uses the observation that that the most probable state sequence leading up to data point n can be deduced by examining the most probable path leading up to the previous time point, n-1. Dynamic programming is used to keep track of all the optimal state sequences leading to all possible states for a given time point n-1 which reduces the amount of required computations. Since there are there are K states at time step n-1, the Viterbi algorithm stores K possible state sequences leading up the previous time point n-1. At time point n, there are now K² paths to consider, given K possible paths leading out of K states. By examining the optimal sequence up to time point n and considering the probability of state transition between time points n-1 and n, the most optimal state sequence up to time point n can be constructed. The state assignment of the first data point in the sequence can be determined by a provided initial probability of observing each state. Therefore, the time complexity of idealization with Viterbi is quadratic in the number of states K and linear with the number of observations N, O(K²N), which is dramatically lower than an exhaustive search.

Formally, the Viterbi algorithm is described with a K x N trellis for states $j\in K$ and observations $n\in N$ (Figure 1a). Each cell of trellis $v_n\left(j\right)$ represents the probability of being in state j after seeing the first n observations and passing through the most probable state sequence for the given model parameters, $\lambda$. The value $v_n\left(j\right)$ is computed by recursively taking the most probable path up to this cell by

(9) ${\displaystyle v_n\left(j\right)=\underset{y_1,\dots y_{n-1}}{max}p(y_1\cdot \cdot \cdot y_{n-1},x_1\cdot \cdot \cdot x_n,y_n=j|\lambda )}$

where $\lambda$ is first order Markov process of $\lambda =(\pi ,a,b)$. The primary components of a hidden Markov model include the initial probability of observing each state $\varphi$_j given by $\pi$ where $\sum _{j=1}^K{\pi }_j=1$; a transition probability matrix $a$ of size K x K where each element $a_{ij}$ is the probability of moving from $\varphi$_i to $\varphi$_j, each element $a_{ii}$ is the probability of staying in $\varphi$_i and $\sum _{j=1}^K{a}_{ij}=1$; and an emission probability matrix $b$ of size K x N where each element $b_j\left(x_n\right)$ is the probability of an observation $x_n$ arising from $\varphi$_j. The values of each component are computed for each trajectory using the fits obtained from sequential steps of divisive segmentation and HAC. Using these parameters, we can compute the most probable path for arriving in $\varphi$_j at time points n by the following recursion

(10) ${\displaystyle v_n\left(j\right)=\underset{\mathit{i}\in \mathit{N}}{max}\left\{v_{n-1}(i)a_{ij}{b}_j(x_n)\right\}}$

(11) ${\displaystyle {\psi }_j\left(n\right)=\underset{\mathit{i}\in \mathit{N}}{\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{a}\mathrm{x}}\left\{v_{n-1}(i)a_{ij}{b}_j(x_n)\right\}}$

where ${\psi }_j\left(n\right)$ is a helper function to store the $n-1$ state index $i$ on the highest probability path.

Upon termination, the forward likelihood of the entire state sequence y up to time point N+1 having been produced by the given observations and HMM parameters is 

(12) ${\displaystyle P\left(y|x,\lambda \right)=\underset{\mathit{i}\in \mathit{N}}{max}\left\{v_T(i)\right\}}$

Deducing the most probable hidden state sequence $y$ through observations $x$ can be accomplished in a backtracking step by

(13) ${\displaystyle y_n={\psi }_n(y_n+1)N\ge n\ge 2}$

To assess the improvement of the Viterbi algorithm for event detection, we simulated a two-state system with a constant k_on and varying k_off rate (Figure 2c). As shown previously (Hadzic et al., 2018), we found that results from CP detection alone were accurate for slower events, but often failed to identify faster transitions (Figure 2d). By refining the results of unsupervised clustering with the Viterbi algorithm, we found that event detection accuracy was significantly improved over CP detection and clustering alone across two-orders of magnitude of varying k_off (Figure 2d). Notably, as the changes in rates also affect the change in state occupancy, the high accuracy values returned after Viterbi refinement further highlight the power of DISC for resolving short-lived and rare transitions (Figure 2—figure supplement 1). In general, this improvement was anticipated since we are not the first to apply the Viterbi algorithm to the problem of idealization using unsupervised clustering. Although commonly used in the application of HMMs for hard assignment of data-points into K discrete states, the SKM algorithm has shown that the Viterbi algorithm can successfully decode a path sequence following state clustering using k-means as opposed to more rigorous HMM training procedures (Juang and Rabiner, 1990; Qin, 2004). Therefore, both SKM and DISC can yield the event detection power of standard HMM approaches without the need of rigorous model training. However, unlike SKM, DISC has the added benefit of identifying the states naturally without the need for any user supervision such as initial state specification. This makes DISC a powerful alternative as a computationally efficient unsupervised single-molecule analysis algorithm.

Modular nature of DISC

While the parameters used above are valid for trajectories with Gaussian noise, we do not claim they are optimal for all experimental modalities. We have intentionally developed DISC as a flexible framework for adaptation to different types of data. Although we used the Student’s t-test for CP detection in the presence of Gaussian noise, additional merit functions may be more appropriate in different situations (Song and Yang, 2017; Watkins and Yang, 2005; Li and Yang, 2019). The same holds true for the use of BIC for state selection as other objective function can be substituted as needed. For example, the harsh penalty for parameters in BIC can lead to underfitting in certain cases; therefore, less stringent Akaike information criterion (AIC) or Hannan-Quinn information criterion (HQC) may be more appropriate depending on the separation of states and noise (Akaike, 1974; Hannan and Quinn, 1979). It is important to note that while DISC performs idealization through unsupervised clustering, obtaining accurate results does require the user to determine the appropriate information criterion and CP detection methods as idealization results heavily depend on these variables. Critically, the central innovation of DISC is to take advantage of the best features of both top-down and bottom-up forms of cluster identification that leads to both fast and accurate state detection.

We developed a new algorithm for rapid and accurate unsupervised idealization of single-molecule trajectories. Our approach combines unsupervised statistical learning of discrete states with the event detection power of the Viterbi algorithm to quickly identify both significant states and transitions in a model-independent manner. Software implementing the DISC algorithm that includes a graphical user interface is available at https://github.com/ChandaLab/DISC (Chanda et al., 2019; copy archived at https://github.com/elifesciences-publications/DISC).

Like CP-HAC methods, DISC is not a fully probabilistic approach. While fully probabilistic HMM training approaches are beneficial for providing unbiased estimates of the parameter distributions, their high accuracy comes at the cost of significantly increased computational time. This cost is especially apparent for the recently developed infinite HMM approaches that aim to learn the true number of states from a potentially infinite number of possibilities. However, accomplishing this task costs hundreds of iterations per trace to provide a reproducible fit with some approaches taking days to analyze single trajectories (Hines et al., 2015; Sgouralis and Pressé, 2017; Sgouralis et al., 2018). Thus, while exhaustive search algorithms may be desirable in other contexts, they are clearly not suited for large datasets associated with high-throughput experiments. In contrast, by simulating data closely resembling the binding of fcAMP to pacemaker channels, we find that DISC surpasses the accuracy of common CP-HAC and HMM algorithms with a dramatic improvement in computational speed. Therefore, DISC satisfies the need for accuracy and speed in high-throughput analysis. In this regard, DISC is like the SKM algorithm for estimating the parameters of an HMM without direct HMM training. However, unlike SKM which relies on user-specified states, DISC uses unsupervised statistics to learn the states. Therefore, DISC offers the idealization power of SKM with the state-learning capabilities of CP-HAC.

We used DISC to analyze a large dataset obtained from ZMWs to evaluate cooperativity between CNBDs from pacemaker ion channels upon ligand binding. The rapid and robust idealization provided by DISC enabled stringent trace selection to ensure only reliable trajectories were analyzed. These data show that, in contrast to model inferences from ensemble measurements of HCN2 channels, CNBD tetramers do not exhibit cooperative ligand binding. This result suggests that allosteric interactions between binding sites may be coordinated by the channel’s transmembrane domains.

Although we have demonstrated the use of DISC on single-molecule fluorescence data, the framework can be easily extended to other data paradigms due to its modular nature. For example, the use of BIC for state determination or the Student’s t-test for change-point analysis could be interchanged with other information theoretic approaches or merit functions where appropriate. To allow for easy comparison of a given data set, the provided software and graphical user interface (GUI) allows the user to select from several options the desired parameters such as choice of information criteria. This flexibility makes DISC suitable for a wide array of experimental data provided it can be described as a series of transitions between discrete states, including, for example, single-channel current recordings, force spectroscopy and smFRET. However, there is no inherent knowledge within DISC to consider various sources of experimental noise, such as photo-blinking or baseline drift; therefore, correcting for these noise sources prior to DISC analysis will likely improve idealization accuracy.

Finally, our results show that DISC provides a dramatic improvement in computational speed over current state-of-the-art approaches while either improving or maintaining high accuracy for both state determination and event detection. This increase in speed is directly applicable to analyzing the growing datasets obtained in single-molecule fluorescence paradigms to adequately sample population dynamics. For example, the use of sCMOS camera enables smFRET measurements of tRNA conformational changes during protein translations across thousands of molecules simultaneously with millisecond resolution (Juette et al., 2016). Additionally, magnetic tweezers have enabled week-long mechanical measurements of single-protein folding and unfolding, shifting observable dynamics to pathological time-scales and allowing the detection of rare events (Popa et al., 2016). Thus, highly computationally efficient and robust algorithms such as DISC may be well suited for analysis of a wide variety of single molecule datasets beyond the standard smFRET data.

Simulated and raw data in addition to analysis scripts are available at https://doi.org/10.5281/zenodo.3727917.

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Author details

1. David S White

   1. Department of Neuroscience, University of Wisconsin-Madison, Madison, United States
   2. Department of Chemistry, University of Wisconsin-Madison, Madison, United States

   Contribution

   Conceptualization, Resources, Data curation, Software, Formal analysis, Investigation, Methodology

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0003-0164-0125

2. Marcel P Goldschen-Ohm

   Department of Neuroscience, University of Texas at Austin, Austin, United States

   Contribution

   Conceptualization, Software, Formal analysis, Investigation, Methodology

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0003-1466-9808

3. Randall H Goldsmith

   Department of Chemistry, University of Wisconsin-Madison, Madison, United States

   Contribution

   Conceptualization, Supervision, Investigation, Methodology, Project administration

   For correspondence

   rhg@chem.wisc.edu

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0001-9083-8592

4. Baron Chanda

   1. Department of Neuroscience, University of Wisconsin-Madison, Madison, United States
   2. Department of Biomolecular Chemistry University of Wisconsin-Madison, Madison, United States

   Present address

   Department of Anesthesiology, Washington University School of Medicine, St. Louis, United States

   Contribution

   Conceptualization, Supervision, Funding acquisition, Investigation, Methodology, Project administration

   For correspondence

   chanda@wisc.edu

   Competing interests

   Reviewing editor, eLife

   "This ORCID iD identifies the author of this article:" 0000-0003-4954-7034

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