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Logo of nihpaAbout Author manuscriptsSubmit a manuscriptHHS Public Access; Author Manuscript; Accepted for publication in peer reviewed journal;
 
Eur Biophys J. Author manuscript; available in PMC 2017 April 1.
Published in final edited form as:
PMCID: PMC5125805
NIHMSID: NIHMS830238

Dividing organelle tracks into Brownian and motor-driven intervals by variational maximization of the Bayesian evidence

Abstract

Many organelles and vesicles in live cells move in a start–stop manner when observed for ~10 s by optical microscopy. Changes in velocity and directional persistence of such particles are a potentially rich source of insight into the mechanisms leading to the start and stop states. Unbiased assessment of the most probable number of states, the properties of each state, and the most probable state for the particle at each moment can be accomplished by variational Bayesian methods combined with a hidden Markov model and a Gaussian mixture model. Our track analysis method, “vbTRACK”, applied this combination of methods to particle velocity v or changes in the direction of travel evaluated from simulated tracks and from tracks of peroxisomes in live cells. When tested with numerical data, vbTRACK reliably determined the number of states, the mean and variance of the velocity or the direction of travel for each state, and the most probable state during each frame. When applied to the tracks of peroxisomes in live cells, some tracks separated into two states, one with high velocity and directionality, the other approximately Brownian. Other tracks of particles in live cells separated into several diffusive states with distinct diffusion constants.

Keywords: Particle tracking, Mean-squared-displacement, Variational Bayes, Hidden Markov model, Gaussian mixture model

Introduction

When observed in a microscope at 60× magnification, vesicles in live cells start, stop, slow down, and change direction frequently (Allen et al. 1985; Ross et al. 2008; Zajac et al. 2013). The eye cannot give a quantitative characterization of the states, nor easily discern if there are other, less obvious states than “stop” and “go”. The hypothesis of this paper is that this phenomenon can be attributed to two or more states of vesicle transportation which operate at different times, and the goal of this study is to find an algorithm that can test whether a small number of distinct states exists, enumerate them, and distinguish them within the time series for further characterization.

A number of algorithms have been proposed to extract physical mechanisms from single vesicle tracks. Several of these methods use the mean squared displacement (MSD) of the particle or the turning angle to distinguish between driven and Brownian segments (Saxton 2007; Arcizet et al. 2008; Bruno et al. 2011; Weihs et al. 2012). The boundaries between driven and Brownian segments of tracks are defined by a user-selected threshold in these studies.

Probabilistic methods based on Bayesian statistics make better use of all the information inherent in single particle tracks and reduce the need for such user-determined thresholds. For example, free diffusion, anomalous diffusion, confined diffusion, and directed motion can be distinguished by Bayesian analysis of the MSDs of time-invariant tracks (Monnier et al. 2012). The confining potential of a membrane-bound receptor for a pore-forming toxin within the surface of a cell is measurable with a Bayesian inference scheme (Türkcan et al. 2012). Transition rates between diffusive states of single protein molecules in live cells can be determined by Bayesian analysis (Persson et al. 2013). Individual frames within tracks of particles in A7 cells can be assigned to either microtubule-based transport or myosin-activated transport by analyzing directional persistence within a hidden Markov model (Rőding et al. 2014).

The goal of our paper was to test whether a program combining variational Bayes analysis, a hidden Markov model, and a Gaussian mixture model (MacKay 1992; Attias 1999; Bishop 2006; Bronson et al. 2009) could separate the time-series of individual tracks into motor-driven and diffusive intervals. We call this combination vB-hmm-gmm. The variational Bayesian method provides a novel scoring metric which determines the optimum mixture distribution efficiently (Bishop 2006). Expectation maximization adjusts the amplitude, mean, and standard deviation of each Gaussian so as to maximize the evidence in Bayes’ Rule. The hidden Markov model (hmm) uses the fact that the (x, y)n pairs fall into a specific time sequence. The first-order Markov model assumes that the probability that the system is in state Sn at time tn is dependent on the value S(tn−1) of the preceding frame and conditionally independent of earlier states. The program learns the amplitude of the influence of S(tn−1) on the probability (S|tn) from the dataset. A Gaussian mixture model (gmm) is expected to be a good fit to the experimental probability distribution for velocities. vB-hmm-gmm uses the Viterbi algorithm (Viterbi 1967) to determine the most probable sequence of hidden states. Equations combining these features are derived in Bishop’s textbook (Bishop 2006).

Bronson et al. have used the vB-hmm-gmm equations to analyze time-series FRET data (Bronson et al. 2009). Their code library, named “vbFRET”, has been deposited on SourceForge.net. We use this library, supplementing it with additional software which we have written, to apply the vB-hmm-gmm method to particle tracks, calling our code library “vbTRACK”. Our code is available open source via http://vbTRACK.sourceforge.net.

Evaluation of velocity and turning angle from tracks

Both velocity v and directional change θ were calculated from the raw tracks and then used as input to vbTRACK. Velocity was defined operationally as displacement/elapsed time. The amplitude of the velocity was evaluated as Δx2+Δy2/Δt, with Δt = 0.01 s fixed by the frame rate. The sign of v was determined from the sign of Δx. This ensured that v¯=0 for purely diffusive processes. When v was evaluated from the tracks of peroxisomes in live cells, it was smoothed with a Savitzky–Golay filter.

The directional change θn at frame n was evaluated from the position of the vesicle in frames n − 1, n, and n + 1, as shown in Fig. 1.

Fig. 1
Definition of the turning angle θn between frames n and n + 1. The line connecting (x,y)n−1 to (x,y)n defines the reference direction for θ = 0

An additional step was applied to the directional data to accommodate the Gaussian mixture model employed by vbTRACK, because θ for Brownian motion had a uniform distribution over [−π, π], not a Gaussian distribution, and a single uniform distribution would be fitted by a large number of Gaussians, not one. To get around this problem, we applied the Fundamental Transformation Law for Probabilities, which shows that the inverse error function transforms a uniform distribution to a Gaussian (Press et al. 2007). This transformation was applied to the θ dataset, giving a new dataset designated by θ′. Supplemental File S1 demonstrates how this transformation works.

Using either v or θ′ from a single track as input data to vbTRACK, the program found the model which best explained the data.

Overview of the program

Figure 2 is a simplified flowchart of the program, showing initialization of the model and its parameters, adjustment of the parameters in an EM loop, and the key role of the variational lower bound L of the probability that the assumed model fits the data.

Fig. 2
Diagram of the main loops of the variational Bayes program. The program constructed k Gaussians and initialized the three parameters of each Gaussian by a random selection within their probability distributions. The parameters were optimized by iterating ...

Bayes’ theorem plays a central role in the definition of the variational lower bound function L. The theorem states:

p(θ|y,K)=p(y|θ,K)p(θ|K)p(y|K)
(1)

where p(θ|y, K) called the posterior, is the probability distribution of the model parameter θ, given the observed data y and the number of states K in the model; p(y|θ, K), called the likelihood, is the probability distribution of the data, given θ and K; p(θ|K), called the prior, is the probability distribution of θ, given K but not the data; and p(y|K), called the evidence, is the probability distribution of y given K only.

It might be thought that maximizing the likelihood would be the best route to maximizing the posterior. However, it has been shown that maximizing the evidence provides more robust and objective model selection (Attias 1999; Bishop 2006). It also does a better job at avoiding overfitting.

Rather than simply calculating the evidence and the posterior by brute force, the program (Fig. 2) minimized a functional F[q(z, θ)] where q(z, θ) is an analytic approximation to p(z, θ|y, K) (Bishop 2006). This maximized p(y|K) because

p(y|K)=F[q(z)]+DKL[q(z)p(z,|y,K)]
(2)

where DKL is the Kullback–Leibler divergence between q(z, θ) and p(z, θ|y, K); DKL is always positive.

The hidden Markov model accounted efficiently for the fact that the data {x,t} is not made up of independent samples from a probability distribution. Rather, each measurement xn was connected to the other measurements by the time tn at which it occurred. In a first-order Markov model, used here, p(S|tn) was influenced only by S(tn−1). The program “learned” the amplitude of the influence from the dataset. vB-hmm-gmm varied the model parameters to maximize p(y|K). This was done for K = 1, 2, 3,…Kmax. The value of K giving the global maximum of p(y|K) was the most probable model for the data. This achieved the first, key goal of vB-hmm-gmm, objective determination of the most probable model. It also achieved the second goal, determination of the mean and standard deviation of the velocity or angle and the standard deviation for that property for each state in the best model.

Having determined the best model, vB-hmm-gmm evaluated the probability that the vesicle was in state k at time t within a given track, using the Viterbi algorithm (Viterbi 1967). If the data was synthetic, this assignment of states could be scored against the known input state. For synthetic and real data, the track was divided into time segments (bouts) during which the vesicle remained in state one (driven) or state two (Brownian). To shed light on the physics of particle motion in state one, the displacements of all frames assigned to state one bouts were catenated, and the MSD for state one was computed. Similarly, from the catenated displacements for frames in state two, the MSD of state two was evaluated, etc.

Performance with simulated data

Simulated tracks were assembled by catenating bouts of directed motion with bouts of Brownian motion. To model directed bouts, each new point was placed at a distance from the previous point such that, at our frame rate of 100 frames/s, the amplitude of the velocity followed a Gaussian distribution with mean 800 nm/s and standard deviation 100 nm/s. The direction between adjacent displacements followed a Gaussian probability distribution constrained so that the persistence length of the driven bout matched the observed persistence length of filamentous actin, 18 μm (Gittes et al. 1993).

Vesicles undergoing Brownian motion in cells move in a random direction with zero mean velocity. This was modeled as two-dimensional diffusion, with the diffusion coefficient D given by the Stokes–Einstein relation D = kBT/3πηd. The diameter of the particle, d, was set at 100 nm, a typical diameter of vesicles in cells. The value of η was set at 1 Pa s, typical of measured cytoplasmic viscosity for vesicles (Luby-Phelps 2000; Hill et al. 2004). Noise was added to the driven bouts to simulate the effects of shot noise, stage vibrations, and other instrumental noise sources on vbTRACK. The added instrumental noise was scaled by the rms amplitude of the displacements during Brownian bouts to provide a sense of the magnitude of the added noise.

Figure 3 shows a simulated track for a 0.1 μm diameter bead undergoing three short bouts of driven motion separated by two long Brownian bouts. A small amount of noise (scaled value 0.01) was added to the driven bouts. At this noise level, the driven and Brownian bouts were easily distinguished by eye. However, if the noise was increased to 0.1, as shown in Fig. 3b, the beginnings and ends of the bouts were no longer so obvious to the eye. We will show that the Bayes program can assign states with high accuracy for both noise levels.

Fig. 3
An example of a simulated track for two different noise levels. In both cases, T = 310 K (37 °C), η = 1 Pa s, d = 0.1 μm. a Scaled noise = 0.01; b Scaled noise = 0.1. Each driven bout lasted ten frames, each Brownian bout 100 frames. ...

Although the values of x and y along such tracks are helpful visual cues for the state of the vesicle, they are not directly useful for the assignment of states. However, vesicle velocity and turning angle, which can be computed from the xy data, are expected to be much better discriminants of the vesicle state at each time t. Figure 4 shows the velocity and turning angle for the simulated track in Fig. 3a.

Fig. 4
The frame-to-frame velocity and change in direction computed from the xy track shown in Fig. 3a. a Velocity. The displacement in the x direction was used to assign a + or − sign to the velocity. b Change in direction. Note the uniform distribution ...

When the velocity or turning angle data set was used as input, vbTRACK first determined the probability that the data arose from 1, 2, 3, 4, or more Gaussian states. As an example, Fig. 5 shows the logarithm of the variational lower bound functional L [Eq. 10.3 in (Bishop 2006)] as the program iterated for a fixed value of k. After ten restarts for a given k, the program moved to the next value of k.

Fig. 5
Log L plotted against the number of iterations of the EM loop for k = 1, 2, 3, 4, and five Gaussian states, using the velocity data in Fig. 3a

The optimizations converged in less than 100 iterations of the EM loop for all values of k tested. Note that the highest value of log L was reached for two states, demonstrating that variational Bayes found the correct number of states and did not over fit the data. Additional noise (Fig. 3b) lengthened the convergence time and decreased the absolute value of log L, but the program still clearly identified k = 2 as the most probable number of states.

The next question we asked was how accurately frames were assigned to states after the convergence of variational Bayes and application of the Viterbi algorithm. For synthetic data, this question can be answered quantitatively. For the velocity data in Fig. 3a the assigned velocities are shown in Fig. 6.

Fig. 6
Scored velocity data for the numerical data shown in Fig. 3a. Red circles were used for frames generated by directed-motion physics. A small red circle means the frame was assigned correctly. A large red circle means that the frame was assigned incorrectly. ...

Figure 6 shows that vbTRACK scored 228 frames correctly and 1 frame incorrectly. Other numerical experiments might give different errors, but the error rate shown is typical of vbTRACK performance at this noise level.

Another way to present the results of the variational Bayes analysis is to label the tracks with colors corresponding to the state assignments provided by vbTRACK. Two such labeled tracks, generated with noise levels of 0.01 and 0.1, are shown in Fig. 7. Panels a and b show tracks for the two different noise levels. The colors of the symbols correspond to the assignment made by vbTRACK. When errors occurred, they were marked as red squares within a bout of blue circles, or vice versa. For noise level 0.01, 30 frames occupied state 1, a perfect score (panel a). For noise level 0.1, the program found 28 frames in state 1, 2 frames were assigned incorrectly (panel b). Both errors were at the edges of the bout.

Fig. 7
Performance of vbTRACK with synthetic data. a Synthetic track with noise level = 0.01 constructed with 10-frame driven bouts and 100-frame Brownian bouts. Symbols are colored according to the state assignments found by vbTRACK, with red squares assigned ...

Panel c shows log–log plots of the MSDs for the frames assigned by vbTRACK to state 1, to state 2, and for the unprocessed mixture for noise level 0.1. The slopes for states 1 and 2 were 1.8 and 1.03, close to the expected values 2.0 and 1.0 for directed and Brownian motion. The slope for “all states”, the raw data, was 1.2. Data with MSD slopes in this range are sometimes termed “superdiffusive”.

To determine whether the error rates became worse as the segments got shorter, the duration of the driven sections was allowed to vary between 3 and 100 while keeping the duration of Brownian sections constant at 100 frames. The fraction of correct positives and the fraction of false positives were determined for reduced noise levels of 0.01 and 0.1. The fractions of correct positives and false positives are shown in Fig. 8, for both velocity and θ as input, as a function of the duration of the driven sequence. The fraction of frames which were correct positives was evaluated as follows: Let s11 be the number of frames in a driven bout which were correctly assigned to state 1 and let s12 be the number of frames in the driven bouts which were incorrectly assigned to state 2. The fraction of correct positives is then s11/(s11 + s12). Similarly, let s22 be the number of frames in the Brownian bouts which were correctly assigned to state 2 and let s21 be the number of frames in Brownian bouts which were incorrectly assigned to state 1. The fraction of false positives is then s21/(s11 + s12).

Fig. 8
Scores for synthetic data using velocity or direction angle θ as input to the program. The duration of Brownian segments was 100 frames while the directed segments varied between 3 and 100. a The fraction of correct positives and false positives ...

As expected (Fig. 8a), the error rates became worse as the length of the driven segment decreased or the noise increased. For driven segments of length ten or more, the correct positive fraction was essentially 1.0 for both noise levels studied, and the fraction of false positives was small. For noise = 0.01 and four frames in the driven segments, the fraction of correct positives was essentially 1.0, and the fraction of false positives was still less than 0.2. However, with a tenfold increase in noise amplitude, the fraction of correct positives decreased to 0.63, while the fraction of false positives increased to 0.60, making results unreliable. With driven segments of length three or less, the correct positives dropped precipitously and the false positives rose steeply, showing that the results of a Bayes analysis using velocity as the input were completely unreliable for bouts of duration four or less. Because the input data were generated by statistical processes, the numerical experiments were carried out 100 times, scored, and the scores averaged for each choice of conditions.

A similar test with simulated data was carried out using frame-to-frame changes in the turning angle θ as the data provided to vbTrack. The results are shown in Fig. 8b. Overall, turning angle was slightly less accurate than velocity, but the difference was small.

Bayesian analysis of the movements of peroxisomes in live HME cells

Following validation with synthetic data, vbTRACK was applied to tracks of peroxisomes in cultures of live human mammary epithelial cells obtained from Lonza (Walkersville, MD, USA). The cells were cultured at 37 °C, 5 % CO2 in mammary epithelial growth medium (MEGM) supplemented with 0.4 % bovine pituitary extract, both from Lonza. Cells were grown in 35 mm glass-bottom dishes (WillCo Wells, Amsterdam) precoated with 100 μl of a 33 μg/ml collagen type IV, 67 μg/ml laminin to mimic the composition of a basement membrane. To fluorescently label the peroxisomes with GFP, 10 μl BacMam 2.0 peroxisome-GFP reagent (Life Technologies, Carlsbad, CA, USA) was added to each dish 1 day before imaging.

Images were acquired by a Nikon Eclipse Ti inverted epifluorescence microscope using a 60× NA 1.4 oil-immersion objective. Illumination was provided by an X-Cite 120 mercury arc lamp. Epifluorescence mode was provided by an FITC cube. A high-speed scientific CMOS camera (pco. edge, PCO, Kelheim, Germany, 6.5 × 6.5 μm pixel size) was used to digitize the images. Individual peroxisomes were tracked to subpixel precision, using Video Spot Tracker software (CISSM, University of North Carolina, Chapel Hill, NC, USA), as described previously (Smelser et al. 2015).

Most peroxisomes appeared to execute random motion. However, this motion is only partially thermal. The remainder is ATP-dependent and motor driven, but driven in random directions (Smelser et al. 2015). Occasionally, a peroxisome showed a mixture of random motion and fast, spatially directed, driven motion. Two tracks were selected to illustrate the performance of vbTRACK with real data. The first track, shown in Fig. 9, was selected because it shows track segments with rapid directed motion at some times but Brownian motion at other times. Such tracks were rarely observed. When directional change was used as the input to vbTRACK, a model with two states was most probable (Fig. 9c). The slopes of the log–log plot of the MSDs near tau = 0.5 s were 1.82 and 1.1 for states 1 and 2, indicating that state 1 was strongly motor driven, whereas state 2 was essentially random in direction (Fig. 9d). The slope of the MSD for the raw data was 1.56; this falls into the category of anomalous superdiffusion. To learn the velocities associated with states 1 and 2, velocity data for the same track were processed by vbTRACK, assuming the two-state model. The rms velocities found were1.49 μm/s for state 1 and 0.50 μm/s for state 2.

Fig. 9
Variational Bayes analysis of a peroxisome exhibiting directed in a live human mammary epithelial cell, with directional change θ transformed by the inverse error function as the input. a Raw track acquired at 100 frames/s. The duration of the ...

Figure 10 compares the results of vbTRACK for a more typical peroxisome track, with either velocity or directional angle θ was used as input. When velocity was analyzed, the probability was highest for the model with four Gaussian states. Their parameters (vrms ± SD) were 2.2 ± 0.49 μm/s (37 frames), 1.41 ± 0.22 μm/s (144 frames), 0.89 ± 0.19 μm/s (385 frames), and 0.40 ± 0.17 μm/s (433 frames). The MSDs for these four states are shown in Fig. 10d. Somewhat surprisingly, the slopes are 1.0 or less in all cases. However, when θ was used as input, the model with two states had the highest probability. In both cases, the MSDs are different from the results in Fig. 9, showing that the peroxisome in Fig. 10 followed four physical mechanisms at different times, but none was motor-driven at high velocity with directional persistence.

Fig. 10
Analysis of a typical peroxisome track in a live human epithelial cell, using either velocity (bd) or directional angle (eg) as input data. a Raw track acquired at 100 frames/s. b The same track after assignment of each frame to its ...

Discussion

The purpose of this paper was to test whether vbTRACK, a variational Bayes, hidden Markov, Gaussian mixture model program, could divide individual tracks of organelles in live cells into segments driven by distinct physical mechanisms. Using numerically generated tracks as input, vbTRACK was able to determine the most probable number of states with high accuracy, provided the segments were longer than about six frames. The edges of bouts were generally accurate to within ± 1 frame. This performance was better than methods which use a rolling window within an MSD curve.

Once the most probable number of states was established, the hidden Markov model within vbTRACK computed the probabilities p(t, k) that the system was in state k at time t. Finally, the Viterbi algorithm found the most probable sequence of states for the entire track; this was different from the sequence of most probable states.

The Gaussian mixture model used by vbTRACK permitted analytic simplifications which speeded up the processing (Bishop 2006). However, not all variables of interest for particle tracking were expected to have a Gaussian distribution. In particular, the turning angle θ, which seemed likely to be a good discriminator between motor-driven motion and random or Brownian motion, has a uniform distribution over the interval [−π, +π]. Fitting such a distribution with mixture of Gaussians required several Gaussians for this single physical process (Supplemental data). To circumvent this problem, θ was transformed to a new variable θ′ with the inverse error function. When vbTRACK was provided with θ′ data derived from a uniform distribution in θ, the program found that the most probable number of states was one (Supplemental data). This transformation thus allowed turning angles during a track to determine the number of states in synthetic tracks composed of mixtures of driven and Brownian segments.

Potential users of vbTRACK will be interested in the accuracy of the program. It was shown in Fig. 8 that the program became increasingly inaccurate as the driven sequence became shorter than ten frames. When velocity was the input variable, similar failures occurred when the velocity distributions of the states overlapped more. For a two dimensional diffusive process, the value of the velocity, when defined operationally as rms displacement/time interval, as in this paper, is 2D/t. for directed, motor-driven motion, by contrast, the time interval would have a small effect on the value of displacement/time interval. This suggests that longer time intervals might have advantages in separating diffusive states from driven states because the “velocity” of the diffusive state becomes smaller with longer t. We have not tested this.

When velocity and θ from the same synthetic track were used as simultaneous input data to vbTRACK, the scores were slightly worse than with velocity or θ separately. This was not the expected outcome and suggests that changes need to be made if vbTRACK is to work correctly with two dimensional input data.

Conclusions

A variational Bayes, hidden Markov, Gaussian mixture model was able to separate single tracks of moving vesicles in live cells into a mixture of driven, Brownian, and trapped Brownian bouts in an objective manner. The method was validated by tests with numerical data. Both instantaneous particle velocity and frame-to-frame turning angle were effective inputs for the program.

The Matlab source code for vbTRACK is available open source via http://vbtrack.sourceforge.net.

Supplementary Material

supplement

Acknowledgments

This material is based upon work supported by the National Science Foundation under Grant Number 1106105 (GH). MJM was supported by a Summer Research Grant from Wake Forest University. AMS was supported by National Institutes of Health (T32GM095440) and National Science Foundation graduate fellowship 0907738. We thank J. Bronson, J. Fei, J. Hofman, R. Gonzalez, C. Wiggins, E. Khan, K. Murphy, I. Nabney, and M. Beal for making their programs freely available. We also thank Jed Macosko and Keith Bonin for use of cell culture and video microscopy facilities.

Footnotes

Electronic supplementary material The online version of this article (doi:10.1007/s00249-015-1091-0) contains supplementary material, which is available to authorized users.

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