Heteromotility analysis approach

Heteromotility analyzes motility features in a set of provided motion paths, as obtained through timelapse imaging, image segmentation, and tracking (see Methods). From these paths, 79 motility features are calculated to comprise a “motility fingerprint” (S1 Fig). These features include simple metrics of speed, total and net distance traveled, the proportion of time a cell spends moving, and the speed characteristics during that period. The linearity of motion is assessed by linear regression through all points occupied by the cell in the time series, taking Pearson’s r² as a metric of fit. Monotonicity is also considered for the distribution of points, using Spearman’s ρ². In some instances, cells have been proposed to have a directional bias when making turns [46]. Turn direction and magnitude features are provided that consider turns on various time intervals.

Another class of features is concerned with the directionality and persistence of motion. There are several possible ways to characterize the persistence of motion, hence we have included several metrics that allow for reasonable consideration. A cell’s “progressivity,” is considered as the ratio of net distance to total distance traveled. This serves as a metric of directional persistence [47]. Mean squared displacement (MSD) is considered for each cell for a variable time lag τ, and the power law exponent α is taken as a quantification of the relationship MSD ∝ τ^α. Directed motion may therefore be expected to have a larger value than undirected random-walk motion or non-motion, indicating a superdiffusive behavior [48].

The distribution of a cell’s displacement steps is also informative. Motion that exhibits a displacement distribution with a heavy right tail, referred to as a Levy flight, has been demonstrated to optimize the success of a random search [49]. This property has led to the Levy flight foraging hypothesis, suggesting that biological systems may perform Levy flight-like motion when searching for resources [50–52]. To assess the Levy flight-like nature of a cell’s motility behavior, the Heteromotility software provides metrics of displacement kurtosis for displacement distributions on multiple time scales. Larger values of kurtosis indicate heavier distribution tails, such that higher kurtosis may indicate more Levy flight-like motion. The Heteromotility software also considers the non-Gaussian parameter α₂, for which larger values indicate a more heavily tailed, Levy flight-like distribution [53, 54]. The second and third moments of the displacement distribution are also provided as features.

If displacements are considered as a time series, the self-similarity and long range memory may provide insight into the coordination of motility behavior [39]. The Heteromotility software calculates the autocorrelation function for displacements with variable time lags τ as a metric of self-similarity. Fractal Brownian motion (fBm) describes a Gaussian process B_H(t) for a time t on the continuous interval [0, T] with successive displacements that are not necessarily independent. The Hurst parameter H describes the self-similarity of a fBm process, with the interval 0 < H < 0.5 describing a process with negatively correlated successive displacements (large displacements are more likely to follow small displacements), H = 0.5 describing non-correlated, independent successive displacements (Brownian motion), and 0.5 < H < 1 describing positively correlated successive displacements (large displacements are more likely to follow large displacements)(see Supp. Methods for additional detail). The Heteromotility software estimates the Hurst parameter as a metric of long range memory using Mandelbrot’s rescaled range method (see Supp. Methods) [55] [56]. As Brownian motion displays a Hurst parameter H = 0.5, deviations from this value may indicate long range memory of a cell’s displacement series and coordinated motility behavior. Each of the motility features described here concerns single cells, and none directly measure correlations in motion between cells as may be observed during collective migration. However, cells with similar behaviors, such as groups of “leader,” and “follower,” cells, may display correlations in single cell features.

Heteromotility features are sufficient to distinguish canonical models of motion

In principle one could devise an infinite number of motion descriptors. How can we determine whether a given set of motion features is sufficient to capture meaningful aspects of motion? To test whether the feature set outlined above is sufficient to distinguish biologically relevant models of motion, we simulated and analyzed cell paths generated by four distinct motion models: unbiased random walks, biased random walks, Levy flights, and fractal Brownian motion with long range memory (H = 0.9)(see Methods for implementations). These data were initiatially simulated with the same mean displacement size, such that the motion models are not trivially seperable based on a parameter unrelated to their defining characteristics. This problem therefore represents the most challenging setting for separating models of these types. Large simulated data sets (n = 20,000) were generated with a range of track lengths. Samples with a range of sizes (members per class) were redrawn from these large populations three times per sample size/track length parameterization. We then analyzed the simulated data to determine if the feature set is sufficient to place each model in a distinct region of a joint feature space.

Visualization of high-dimensional data sets, such as the Heteromotility feature set, presents a fundamental challenge. Here we employ t-Stochastic Neighbor Embedding (t-SNE), a visualization method that embeds high-dimensional data into a low-dimensional map, to generate 2D projections of our high dimensional feature space [57] (see Supplemental Methods for discussion of t-SNE parameter selection). When Heteromotility feature space is visualized using t-SNE (perplexity = 50), these models of motion occupy distinct regions of feature space for a number of sample sizes and track lenghts. A sample visualization of a simulation with 1000 members per class and track lengths of 100 units is shown (Fig 1A)(see further examples, S2 Fig). Unsupervised hierarchical clustering was performed using Ward’s method [58] (see Supplemental methods for discussion). Unsupervised clustering segregates the various models with high accuracy based on Heteromotility features for a range sample sizes as small as 100 members per class, and lengths as short as 50 time steps. Accuracy tends to increase with larger sample sizes and longer track lengths (Fig 1B and 1C). Clustering was also performed on simulations with varied fundamental parameters, and accuracy is likewise high (Fig 1F). Parameters for the bias magnitude of biased random walkers, random walker displacement means, the Levy flier exponent, and fractal Brownian motion Hurst parameter were all varied. 10 groupings of the four simulations with varying parameter settings were sampled and classified as above (see S2 Table for parameter settings).

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Fig 1. Simulated models of motion can be differentiated based on Heteromotility features.

(A) Representative t-SNE visualization of Heteromotility feature space determined for simulated motion models (1000 members/class, 100 time steps)(perplexity = 50). (B) Unsupervised clustering accuracy for simulated data across a range of sample sizes and (C) track lengths. (D) The Confusion Matrix of a representative Random Forest classifier trained to distinguish four simulated models of motion shown in (A), mean accuracy of 99.8% (5-fold CV). (E) Features ranked by importance for Random Forest classification, where importance is determined as the decrease in accuracy when the feature is removed across all 36 simulated sample populations. The number of times the feature appears in the top 10 most important across the 36 populations is reflected in the ‘Count’ column. (F) Accuracy of classifying simulated motion models with varying parameters using hierarchical clustering and a Random Forest classifier. 10 populations of the four simulated motion models with varied parameters were generated and classified.

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Applying a supervised Random Forest classifier [59] to each of the simulated populations, we are able to discriminate the four models of motion with high accuracy (5-fold cross validation score) across a range of sample sizes and track lengths (S2 Fig) (see Supp. Methods for discussion on classification model selection). We present the performance of a sample Random Forest classifier, trained on simulations of 100 time steps with 1000 members per class (shown in Fig 1A), in a confusion matrix. A confusion matrix describes the errors made during classification in a matrix by listing the true classes as rows and the predicted classes as columns with values of the matrix cells representing the number of observations for each true class: predicted class pair (Fig 1D) [60]. In this way, values along the matrix diagonal represent correct predictions, and values off the diagonal represent incorrect predictions. As seen in the matrix, there is little confusion between Levy flights, fBm, and the random walks, but some confusion between unbiased and biased random walks.

The features important for effective classification can be determined by measuring the decrease in accuracy when each feature is removed from classification. For each of the simulated populations generated, we found the top 10 most important features based on this metric. We present the 10 features that appear most often across classifiers, and the mean decrease in accuracy associated with removal of each feature (Fig 1E). We find that the non-Gaussian parameter and other metrics of the displacement distribution, as well as metrics of self-similarity, dominate this list. This supports the notion that the non-traditional features provided by Heteromotility can be useful for discriminating biologically relevant models of motion. These results demonstrate robust detection of heterogenous motility phenotypes using a rich space of motion features. We also find high accuracy Random Forest classification of simulations with varied parameters (Fig 1F). As a proof-of-concept, this analysis of simulated motion indicates that the Heteromotility feature set is sufficient to recognize different motility phenotypes where they exist.

Wild-type and transformed MEFs occupy a shared region of motility state-space

Utilizing the Heteromotility software, we are able to analyze behavior in live cells and map this behavior into a cell state space. We initially sought to determine if this cell behavior state space was continuous, with cell states existing along a spectrum, or discrete, with a limited set of states cells could adopt. To answer this question, we first employed the mouse embryonic fibroblast (MEF) culture system. Wild-type MEFs (WT MEFs) and MEFs transformed with c-Myc and HRas-V12 overexpression constructs (MycRas MEFs) [61], which serve as a cancer model, were timelapse imaged for 8 hours to capture motility behavior. Images were segmented and tracked (see Methods), with cell centroid coordinates used as the cell location for tracking. By considering cell centroids, process extensions and retractions are considered as motile behaviors, in addition to displacement behaviors that would not be captured by an analysis of morphology alone [39]. Analysis of cell paths with the Heteromotility software revealed that WT and MycRas MEFs differentially occupy different sections of a shared motility space, as visualized by t-SNE (Fig 2A). Strikingly, MycRas MEFs occupy a sub-region of wild-type motility space, rather than a unique region.

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Fig 2. Wild-type and Myc/Ras transformed MEFs show differential occupancy within a shared motility state-space.

(A) t-SNE visualization of wild-type (blue) and Myc/Ras transformed (red) MEFs in motility space (perplexity = 50). (B) Hierarchical clusters visualized with t-SNE (MANOVA, p < 0.001; Silhouette S_i = 0.21). (C) Proportion of wild-type vs. transformed cells occupying each cluster. (D) Comparison of a subset of normalized feature values between clusters (*: Holm-Bonferroni corrected p < 0.01 by ANOVA) and (E) wild-type and transformed MEFs (*: Holm-Bonferroni corrected p < 0.01 by t-test). n > 250 cells per condition (pooled) from six independent experiments.

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We applied hierarchical clustering (Ward’s method) [58] to the motility state space generated from pooled WT and MycRas MEFs to identify heterogenous motility phenotypes (Fig 2B), considering a set of cluster validation indices to select the optimal number of clusters [62](See Supp. Methods for complete details). Clustering was performed on the first 30 principal components of the motility feature set, as this dimensionality preserves >95% of variation for each system we study (S3 Fig). The Silhoutte value is a metric of cluster validity on the interval [-1, 1], taking into account the similarity of samples within a cluster and the difference between clusters [63]. Higher values indicate that samples within a cluster are similar and clusters are distinct. The cluster partition displays a positive Silhouette value, indicating an appropriate cluster structure. Feature mean values between clusters are also confirmed to be significantly different by multivariate analysis of variance (MANOVA) [64]. A complete set of cluster validation metrics is provided (S1 Table). This result suggests that the state space is continuous, but can be decomposed into a set of overlapping states with characteristic behavioral phenotypes. We define these motility states as heterogeneous based on significant differenences from one other in feature space. This definition of heterogeneity is also followed for the other cell systems we investigate in this work.

WT and MycRas MEFs are found to differentially occupy different clusters within the shared set, as may be expected from their initial distributions in the space (Fig 2C). MycRas MEFs preferentially occupy Cluster 1, characterized by the highest average speed, proportion of time spent moving, and distance traveled, indicating a motile state. Conversely, WT MEFs preferentially occupy Cluster 2, characterized by the lowest average speed and time spent moving, and progressivity (net distance / total distance), indicating a less motile state (Fig 2D). A proportion of both MycRas and WT MEFs occupy Cluster 3, a population characterized by high kurtosis and progressivity, indicating a Levy flight like motile state that performs “jumping” motions. Observed visually, Cluster 2 cells move relatively little, Cluster 1 cells progress smoothly, and Cluster 3 cells exhibit erratic motion (S1 and S2 Videos). The transformation state-dependent distribution of MEFs between behavioral clusters is reproducible across all “round-robin” groupings of 4 out of the 6 biological experiments we perform (S4 Fig)(Supp. Methods). The clusters are also identifiable using only a small subset of features, or even a single feature, and the relative distribution of WT and MycRas cells within them is maintained (S5 Fig).

To estimate whether local cell density played a role in determining a cell’s motility behavior, we estimate a “local cell density index” based on the sum of inverse squares of distances between a cell and its neighbors using our cell tracking data. Constructing linear regression models between this local cell density index and each of our measured motility features, we find no meaningful relationships (S3 Fig). However, we note that our tracking data purposefully does not track every cell, such that the local cell density index we estimate is an imperfect representation of local cell density. We also note that our experiments are designed to minimize cell-cell contact, such that our lack of detected cell density influence may be the simple result of sparse growth conditions failing to reach a quorum sufficient to influence motility behavior.

MycRas and WT MEF cluster preferences are statistically significant by Pearson’s χ² test of the transformation state × cluster contingency table (p < 0.0001). Considered as a population, the MycRas MEFs demonstrate higher progressivity, mean squared displacement, time moving, average speed, and self-similarity metrics than WT MEFs. This indicates that as a population they spend more time moving in a directed manner (Fig 2E).

These results suggest that high motility and low motility states exist in both wild-type and transformed MEFs. Oncogenic transformation by c-Myc and HRas-V12 may then be viewed as an input that leads a larger proportion of cells to adopt the high motility state, rather than introducing a novel phenotype unseen in WT cells. This observation is consistent with studies of motility behavior in the context of cancer, which have long suggested that increased motility is a phenotype of malignant cells [65, 66]. Functionally, the increased motility of neoplastic cells may be related to their potential for tissue invasion [67, 68].

Importantly, the motility behavior of a cancer cell population may be indicative of disease progression and outcomes [69]. We trained supervised classification models to predict if a cell was wild-type or transformed based on its motility behavior. Support vector machine (SVM) classifiers are a set of models that learn a decision boundary between classes. SVMs have shown efficacy in many problem domains [70]. SVM classifiers were trained on the top 62% features selected by ANOVA F-value in a round-robin fashion for cross-validation, training on four experiments and testing on the remaining two iteratively. On average, round-robin trained SVM classfiers are able to classify individual cells as wild-type or MycRas transformed with ~70% accuracy based on motility features alone (S3 Fig). Parameters for feature selection and the SVM classifier were identified by a grid search (see Supp. Methods for details). While clusters are identifiable with only a single feature, we find classification accuracy decreases if more than ~62% of features are omitted (S5 Fig). The most important features for classification were estimated based on the decrease in round robin classification accuracy when a feature was removed, with larger decrease considered more important. The average moving speed, time spent moving, kurtosis, and autocorrelation are the most important features by this metric (S5 Fig).

Myoblasts display distinct motility states robust to perturbation

Our long-term goal is to understand the cell as a unit in a dynamical system, which entails not just identifying states, but also determining how various inputs from the extracellular environment may drive transitions between states. Mouse myogenic progenitor cells provide a well-characterized system in which chemical signals alter the behaviors of a mechanically-active cell type [71–74], and dynamic activation and lineage commitment processes can be manipulated [75, 76]. In light of this long-term goal, we applied Heteromotility analysis to the myogenic system to ask how myogenic cells occupy state space.

Myoblasts are the transit amplifying progenitor of the skeletal muscle, produced as the daughters of muscle stem cells [75, 76]. Myoblast motility has direct functional relevance, as muscle progenitors translocate along the muscle fiber to sites of injury during muscle regeneration [73, 77] and transverse fibers during development [78]. Primary myoblasts were timelapse imaged for 8 hours, with and without stimulation by the growth factor FGF2. FGF2 is a known mitogen, inhibitor of differentiation, and possible chemoattractant in myogenic cells in culture [79–81]. In vivo, FGF2 is released during muscle injury [82, 83], promoting expansion of the myogenic progenitor pool and possibly migration to sites of injury. As FGF2 is known to elicit different functional behaviors in myogenic cells, introducing this perturbation allows us to evaluate the robustness of myoblast motility states under multiple growth conditions.

Applying hierarchical clustering, two motility clusters are detected (Fig 3A). This partitioning scheme has a positive Silhouette value [63], significantly different cluster means by MANOVA, and optimizes cluster validation metrics (S1 Table). Cluster 1 is a less motile state characterized by lower average moving speed, distance traveled, and a higher kurtosis. Conversely, Cluster 2 is characterized by a higher total distance, average speed, mean-squared displacement (MSD), and proportion of time spent moving. These characteristics indicate that myoblasts occupy a two state system with a consistently motile state (Cluster 2), and a less motile state that exhibits more Levy-flight like displacement behavior (Cluster 1). Observed in videos, cells in Cluster 1 are less motile, and display less directed motion than cells in Cluster 2 (S3 and S4 Videos). Clustergram visualizations of the myoblast clusters using multiple linkages are provided (S6 Fig).

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Fig 3. Myoblasts display distinct motility states, shared by both FGF2+ and FGF- conditions.

(A) t-SNE visualization of hierarchical clusters in myoblast motility space (perplexity = 30) (MANOVA, p < 0.001; Silhouette S_i = 0.31). (B) Normalized feature means for motility state clusters (*: p < 0.05, Holm-Bonferroni corrected t-test). (C) t-SNE visualization of FGF2 treated (blue) and untreated (red) myoblasts in motility space. n ≈ 150 cells per condition, taken from two separate animals, with total n = 308 cells.

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Both FGF2 stimulated (FGF2+) and unstimulated (FGF2-) cells co-occupy both states, with no notable preference in state space induced by either condition (Fig 3C). This is confirmed quantitatively as a lack of preferential cluster occupancy between FGF2 treated and untreated cells (χ² test p > 0.05 of FGF2-treatment × cluster contingency table). This negative result suggests that FGF2 does not induce a notable effect on myoblast motility under these conditions.

Muscle stem cells display motility states reflecting activation

Until this point, the cells analyzed have not been undergoing any dramatic phenotypic transitions. In contrast, stem cells are specifically designed to undergo dynamic activation and differentiation processes that radically reshape cell phenotype on an hours to days long timescale. Would it be possible to observe such phenotypic transitions through the lens of cell behavior changes? We apply Heteromotility analysis to the muscle stem cell (MuSC) system during activation from quiescence in an attempt to observe dynamic transitions in cell behavior. MuSCs undergo an exit from quiescence in cell culture conditions, entering an activated state over a roughly 48 hour window, providing a model of a dynamic process where behavior state transitions would be expected [75, 84]. Heterogeneity within the MuSC population is well appreciated [11], suggesting that motility behavior may also be heterogenous during activation. Motility behavior is also relevant to MuSC function due to the physiological motility behavior of muscle progenitors during regeneration, as noted above.

Primary MuSCs were isolated from limb muscles by FACS (PI^-/CD31^-/CD45^-/Sca1^-/VCAM⁺/α7-integrin⁺) [85] and seeded on sarcoma-derived ECM coated well plates. After 24 hours in culture, MuSCs were timelapse imaged for 8 hours in DIC. At this stage, MuSCs have begun to activate (MyoD⁺), but are not yet committed to differentiation (MyoG⁺) [84]. Visualizing hierarchical clusters (Ward’s method) of MuSC motility features with t-SNE, it is apparent that multiple motility subpopulations are present (Fig 4A). We identify three distinct clusters with notably different phenotypes. This partitioning scheme has a positive Silhouette value [63], significantly different cluster means by MANOVA, and optimizes cluster validation metrics, as above (S1 Table)(see Supp. Methods for complete discussion). As in myoblasts, the clusters appear to separate based on differences in total distance, average speed, and time moving. Additionally, clusters segregate based on the linearity and progressivity of motion.

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Fig 4. MuSC motility states reflect progressive myogenic activation.

(A) t-SNE visualization of distinct motility states, as detected by hierarchical clustering (colors)(MANOVA, p < 0.001; Silhouette S_i = 0.197)(perplexity = 70). (B) t-SNE visualization of MuSC motility states and myoblasts in a shared space suggesting progressive myogenic motility states. (C) Average transition rate vectors for MuSC motility states (arrows, scaled for presentation) demonstrating transitions toward the myoblast motility phenotype over time in a shared PCA space. (D) Subset of normalized feature means for each motility state (*: p < 0.05, Holm-Bonferroni corrected ANOVA). (E) The magnitude (+/- SEM) of the mean transition vector for all cells in a given state. All data represent analysis of n > 1800 cells per condition pooled from three animals, with total n = 4316 cells.

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Cluster 1 is characterized by the lowest total distance, average speed, time moving, progressivity, and MSD. Observed visually, cells in Cluster 1 are immotile and appear morphologically rounded, lacking any filopodia characteristic of myogenic activation and motility (S5 Video). Clusters 2 and 3 are characterized by increasing measures of total distance, average speed, and time spent moving. Cluster 3 exhibits the highest kurtosis and non-Gaussian parameter, suggesting a jumping, Levy-flight like state. Cluster 2 exhibits the highest time moving, moving speed, progressivity, linearity, net distance, and MSD, indicating a more directed motility state. Visually, Cluster 2 cells largely move in a single direction, while Cluster 3 cells more often turn back and retrace a portion of their previous path (S6 Video)(Fig 4D)(p < 0.001 by ANOVA for all features displayed).

MuSCs incubated with FGF2 display a similar distribution in motility space compared to untreated cells (S7 Fig). Quantitatively, FGF2 MuSCs display no motility state preference relative to untreated cells (S7 Fig)(p > 0.05, χ² test of FGF2-treatment x cluster contingency table). On a population level, FGF2 does not appear to significantly alter any of the motility features calculated by our software. This negative result suggests that FGF2 may not play a role in regulating motility in these conditions at this stage of myogenic activation.

Visualized in a shared t-SNE space with myoblasts, the order of MuSC motility states suggests a progressive motion toward the myoblast state space as total distance, time moving, and speed metrics increase (Fig 4B). This is confirmed in a linear state space using principal component analysis (PCA), in which MuSC states and myoblasts segregate primarily along the first principal component (PC1)(Fig 4C). The top 20 features contributing to PC1 in this shared space are metrics of moving speed and time spent moving, suggesting that the features which vary most between quiescent MuSCs and activated myoblasts are related to the speed and frequency of movement. These quantitative results are in line with qualitative observations, in which quiescent cells begin in a relatively immotile state and gradually increase the speed of their motility as they activate in culture. These phenotypic changes also align well with the in vivo role of activating MuSCs. Upon injury, MuSCs must activate from quiescence and translocate along the fiber to the site of tissue damage, necessitating more motile behavior [72, 77]. This result indicates that MuSC motility states reflect progressive states of activation toward the eventual myoblast motility state space.

We explored this possibility with pseudotime analysis, which attempts to reconstruct a dynamic cellular process from high-dimensional single cell data under the assumption that the process is ergodic and cells move in a continuous manner through feature space over time [18, 86]. Analytically, this is most commonly achieved by fitting a minimum spanning tree through the data in reduced dimensional space, interpreting the tree’s longest axis as a temporal axis. Pseudotime analysis in the MuSC motility state space reflects the view that MuSC states are ordered in a progressive series (S7 Fig).

To further confirm that the MuSC motility states reflect states of activation, we quantified state transition rates and directions for cells in each MuSC state and myoblasts. Cell paths were divided into three equal length tracks (τ = 20 frames) and Heteromotility features were extracted for each of these subpaths. The state of a cell during each time interval τ was defined in two dimensions as the cell’s location along the first two principal components of a shared MuSC and myoblast PCA space. Transitions for each cell were calculated as the vector between sequential 2D state locations. The mean transition vector for a given cluster was calculated as the mean of all transitions made by all cells assigned to that cluster.

Visualizing transition rates as vectors originating from each cluster centroid in PCA space, it is evident that cells in each MuSC state progress toward the next state in the sequence over time (Fig 4C). The myoblast motility states are present at the end of the sequence, suggesting that MuSC states represent different stages of myogenic activation. Notably, the transition rates increase as a function of state progression, suggesting the kinetics of activation are non-linear (Fig 4F). In this way, we provide quantitative measurements of transitions between states of myogenic activation in single cells for the first time.

MuSC motility states are more dynamic than MEF and myoblast motility states

We next sought to investigate the dynamics of our motility state systems. A key benefit of using behavior features to define states is that a single cell can be subjected to a state assay at multiple time points, allowing state transitions to be detected. How long does a cell reside in a particular state? Can every state transition to every other state, or are transitions restricted to exist between particular states, as would be the case in a state automaton? The answers to such questions would help clarify the computational logic underlying cell behavior.

To visualize and quantify the dynamics of the motility state systems, we applied coarse-grained probability flux analysis (cgPFA), as implemented by Battle et. al. [87] (Supp. Methods). In this method, the first (PC1) and second (PC2) principal components are segmented into k bins, and we define each unique combination of bins on PC1 and PC2 as a unique state. To define cell state at multiple points using motility features, cell paths are segmented into τ length subpaths and motility features are extracted from each subpath. Cell state is defined for each subpath based on its position in coarse-grained PCA space, where each binned coordinate is treated as a state. Cell states are compared from one subpath to the next to quantify the dynamics of the motility state system (see Supp. Methods).

As a validation of our implementation, we performed cgPFA on simulated cell paths that varied their model of motility on a defined interval (τ = 20 time points) and compared them to invariant simulations, using subpaths of the same length as the variable model’s states (τ = 20 time points). For each bin, a state transition rate is calculated as the vector mean of transitions from that state bin into neighboring states in the von Neumann neighborhood. These transition rates are visualized as arrows atop each state bin. Arrow direction represents the direction of transition rate and arrow length represents the rate magnitude. State bins with high transition consensus therefore display longer arrows, indicating that most cells transition in the same direction. As a measure of state stability, the divergence of the vector field is displayed as a heatmap. States with negative divergence may be considered metastable, with more cells entering than exiting. A Levy flight model transitioning to a random walk displays a highly directed set of state transitions (Fig 5A), as compared to a simple random walk which displays minimal directionality (Fig 5B).

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Fig 5. Coarse-grained probability flux analysis (cgPFA).

cgPFA of (A) Levy flier to random walk simulations and (B) Random walk invariant simulations. cgPFA of (C) MycRas MEF, (D) WT MEF, (E) Myoblast FGF2+, and (F) MuSC FGF2+ motility states with subpaths of length τ = 20 time points (130 minutes). Each unique combination of bins between PC1 and PC2 is considered as a unique state. Arrows represent transition rate vectors, calculated for each state bin as the vector mean of transitions into the neighboring states in the von Neumann neighborhood. Arrow direction represents the direction of these transition rate vectors, and arrow length represents transition rate vector magnitude. Underlying colors represent the vector divergence from that state as a metric of state stability. Positive divergence indicates cells are more likely to leave a state, while negative divergence indicates cells are more likely to enter a state. 3D representations of (G) MEF WT, (H) Myoblast FGF2+, and (I) MuSC FGF2+ motility state divergence.

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Applying cgPFA to our biological systems with subpaths of length τ = 20 time points (130 minutes), both WT MEF and MycRas MEF systems display no obvious state flux (Fig 5C and 5D; S8 Fig). Topographically, MEF systems display a ‘basin’ of metastable states. This region has near zero divergence and low transition rates. States on the outer periphery of this metastable region have higher divergence and transition rates, indicating that these states are less stable (Fig 5C and 5D; S8 Fig). Visualizing state space divergence in three dimensions, the metastable states appear as a central valley, while the unstable states appear as peaks (Fig 5G). Myoblast motility state systems appear similar to the MEF systems by simple observation. The topology appears to be dominated by a metastable basin at the center, surrounded by unstable states on the edge of this basin. Topology is comparable between FGF2 treated and untreated conditions (Fig 5E, S8 Fig).

MuSC motility state systems appear more dynamic than MEF and myoblast systems. This transition directionality is apparent when viewing the transition vector fields, with many transition rates leading toward a metastable ‘valley’ and a metastable basin on the edge of state space (Fig 5F, S8 Fig). Visualized in three dimensions, these metastable regions are clearly visible as valleys in state space bordered by unstable ‘ridges’. We apply these cgPFA analyses with multiple temporal and course-grained resolutions, and find that the results are qualitatively similar for multiple state definition time scales and course-grained binning schemes (S9 Fig).

To determine how long MuSCs occupy a given state, we characterized the dwell times of each occupied state in the course-grained PCA space by exponential decay curve fitting, providing a time constant for each state. In these course-grained state spaces, dwell times appear exponentially distributed with time constants ranging from τ_TC ≈ 1 to τ_TC ≈ 3 (S10 Fig). The mean dwell times for state bins range from τ_dwell = 1 to τ_dwell ≈ 2.2, indicating that most cells transition at least once during the time course. Dwell times appear exponentially distributed, suggesting that the state transition process is memoryless on the timescales we observe. Time constants are positively correlated with the number of cells occupying a state, supporting the notion that states of high occupancy are metastable and therefore characterized by longer dwell times (S10 Fig). Topology, dwell times, and transition directionality are comparable between FGF2 treated and untreated conditions (Fig 5I, S8 and S10 Figs). These results suggest that the MuSC motility state system is more dynamic than the MEF or myoblast systems, consistent with the dynamic MuSC activation process.

MuSC motility states break detailed balance

A system with a discrete set of states at equilibrium should obey the law of detailed balance, such that each individual state transition A → B occurs at the same rate as the reverse transition B → A. Systems that break detailed balance transition between states in a directed manner, such that future behavior can be predicted by current state. Biological systems frequently break equilibrium when undergoing directed processes, but confirming that detailed balance is broken in a given scenario can prove challenging [87]. A system in detailed balance would be expected to exhibit equal and opposite transition vectors with no observable pattern of transitions. In our MuSC systems, visualizing transition rates by PFA suggests that detailed balance may be broken, as shown by the directed nature of transition rate vectors (Fig 5, S8 Fig).

To confirm statistically that detailed balance is broken in these systems, we defined N-dimensional state spaces based on the first N coarse-grained principal components and performed PFA in these spaces (ND-cgPFA)(see Methods). A 2N-dimensional matrix is generated representing every possible set of state transitions in a given N-dimensional state space. For example, a 1 dimensional state space is represented as a 2 dimensional matrix, each dimension coarse-grained into k bins (rows, columns). One dimension of this 2D space represents a cell’s initial state position at one time point, and the other represents the destination state position at the next time point (as in Fig 6A and 6B). This pattern is repeated for the construction of higher dimensional spaces. For instance, a 2D state space is represented as a 4D matrix, where the first two dimensions describe initial state, and the next two dimensions describe final state. The number of cells that exhibit each transition are recorded as the value of the corresponding position in the matrix (see Methods for further description).

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Fig 6. Analysis of state transition dynamics idicates MuSC motility states break detailed balance.

One-dimensional coarse-grained PFA of (A) Simulated Levy flier transition to a random walk, (B) simulated random walk. Transition pairs from N-dimensional cgPFA displayed as heatmaps for (C) WT MEFs (n = 312), (D) Myoblasts (FGF2+) (n = 150), (E) MuSCs (FGF2-) (n = 1838), and (F) MuSCs (FGF2+) (n = 2500). Heatmaps show the five most unbalanced transitions in a system, with colors and numerical insets indicating the number of cells that transitioned from a given state at t₀ to a given state at t₁. Significantly unbalanced transitions are outlined in red (p < 0.05, Benjamini-Hochberg corrected binomial test). A system in detailed balance would display no unbalanced transitions.

https://doi.org/10.1371/journal.pcbi.1005927.g006

A system in detailed balance would be expected to display symmetry for each pairwise set of transitions. To determine if each of our systems was in detailed balance, we performed ND-cgPFA at several levels of dimensional (N∈ {1, 2, 3, 4}) and coarse-grained (bins k ∈ {2, 3, 5, 7, 10, 15, 20}) resolution. At each level of resolution, we test each of the k^2N possible pairwise transitions for balance by the binomial test (H₀: p = 0.50).

To validate this method, we performed cgPFA on simulated cell paths generated using variable and invariant models of motion. A Levy flier that transitions to a random walk shows clearly unbalanced pairwise transitions (Fig 6A) at one dimension of resolution, while an invariant random walk displays symmetry about the diagonal and balanced pairwise transitions (Fig 6B). A simple binomial test for pairwise transitions finds multiple unbalanced transition pairs in the variant model, but not the invariant model (Fig 6A and 6B).

N-dimensional matrices cannot be visualized in their totality as in the 1D case. As noted above, we tested all k^2N possible transitions for balance at multiple dimensional and binning resolutions for each system. Here, the five transition pairs that are most unbalanced out of the possible set of k^2N transitions (by p value of the binomial test) in a given system and state space are presented as a 5-by-2 heatmap, where columns represent the initial and final states in a transition pair. We present the dimensional and coarse-grained resolution revealing the most asymmetry for each system, as noted above each heatmap. Visualizing ND-cgPFA pairwise transitions for MEF (Fig 6C), myoblast (Fig 6D), and MuSC state systems (Fig 6E and 6F) demonstrates that transitions with some degree of unbalance exist in each of the systems. By the binomial test (Benjamini-Hochberg corrected), only the MuSC system (FGF2+ and FGF2-) displays significantly unbalanced transitions in any of the course-graining schemes tested, confirming that detailed balance is broken. Our ND-PFA test is biased toward Type II error (false negatives) rather than Type I error (false positives) for multiple reasons, providing further confidence that the detailed balance breaking we identify is valid. It is therefore possible that detailed balance is also broken in the MEF and myoblast systems and our tools are simply not sufficient to detect this asymmetry.

We performed this ND-cgPFA analysis for multiple values of the timescale parameter τ and find that these results are consistent. At each value of τ, the MuSC system demonstrates broken detailed balance and the MEF and myoblast systems do not. On a very short timescale, it is unlikely that detailed balance is broken for a motility state system, even in a dynamic system like activating MuSCs. In line with this biological prior, we find that detailed balance is not broken in the MuSC system when we perform ND-cgPFA with the same number of τ = 20 length windows, but set them to overlap with a stride s = 1. In this overlapping scheme, only two time steps of difference are present between the first and final temporal window, accounting for only 13 minutes in real time in our experiments (S11 Fig).

As additional confirmation of MuSC motility state dynamism, we performed this symmetry breaking analysis using hierarchical clustering to define cell state over the whole motility feature space, rather than coarse-grained location along the principal components (S12 Fig). As with ND-cgPFA, our positive control variable model breaks detailed balance by the binomial test, while our negative control invariant models do not. MuSC systems again demonstrate greater pairwise asymmetry than MEF or myoblast systems in this test (S12 Fig).

An important question is raised when detailed balance is broken. Is the system stationary, with the same number of cells in each state over time, or non-stationary? We evaluated the stationary nature of the MuSC systems by forming a contingency table comparing state occupancy between time points for each set of length τ = 20 subpaths in the dataset and find a significant difference (p < 0.05, χ² test) for each scale where detailed balance breaking is detected. These results collectively demonstrate the more dynamic nature of the MuSC system and demonstrate that the MuSC motility state system is non-stationary and breaks detailed balance.

Conclusion

Our Heteromotility software provides a means of defining cell states and quantifying transitions between them by quantitative analysis of one aspect of cell behavior. We demonstrate motility behavior analysis is capable of identifying unique motility states in simulations and three biological systems. In our biological systems, these states appear to have robust definitions when perturbed, but each cell’s state identify appears dynamic. Real-time, quantitative assays of single cells such as Heteromotility analysis may reveal the dynamics of heterogenous cell systems, which cannot be accomplished by terminal molecular assays. We demonstrate this approach by showing that MuSCs transition through a progressive set of motility states toward the myoblast motility state in a predictable manner, breaking detailed balance.

Ethics statement

All mice were housed at the University of California San Francisco following UCSF Institutional Use and Care of Animals Committee guidelines. Adult male C57Bl/6 mice (2-4 m.o.) were used for myogenic cell experiments. Adult female C57Bl/6 mice (2-3 m.o.) were used as mothers to derive MEFs from E13.5 embryos [90].

Cell isolation and culture

Mononuclear cells were isolated from limb muscles of adult mice as described [91]. Myoblasts were isolated by negative-selection against CD31, CD45, and Sca1 using the EasySep Endothelial Selection kit and subsequent 5 minute preplating. Myoblasts were maintained in myogenic growth media (F10, 20% FBS, and [5 ng/mL] FGF2) on sarcoma-derived ECM coated dishes. All myoblasts used in experiments were passage 3-4. Wild-type muscle stem cells were isolated by FACS (PI-/CD31-/CD45-/Sca1-/VCAM+/α7-integrin+) and cultured as described [85]. Wild-type MEFs were isolated from E13.5 embryos as described [90]. Transformed MEFs were generated as described [61], generously donated by the authors. All MEFs were maintained in DMEM, 10% FBS, 1% Penicillin / Streptomycin.

Timelapse motility imaging

All images were performed in a temperature and CO2 controlled unit at 37°C and 5% CO₂. All experiments were performed with a 20X air objective, capturing images with a Hamamatsu C11440-22CU camera (pixel size 6.5 x 6.5 μm) using a Nikon Ti with automated XY stage. MuSCs were seeded at 500-1000 cells/well in 20 wells of an optically clear tissue culture treated, plastic bottomed 96 well plate coated with sarcoma-derived ECM (Sigma) immediately after FACS sorting. After 23 hours to allow for adaptation to culture, media was exchanged for the relevant experimental medium and MuSCs were incubated for 1 hour to adapt. MuSCs were imaged at 20X in DIC for 10 hours with a temporal resolution of 6.5 minutes / frame. Myoblasts were passaged and plated 24 hours prior to imaging at 300-500 cells/well. Media exchanges were performed 1 hr prior to imaging, in the same manner as MuSCs. Myoblast imaging was performed in the same manner as MuSCs. MEFs were similarly plated 16-20 hours prior to imaging at the same density and imaged in the same manner. For FGF2 perturbation experiments, 10 wells of an optically clear plastic bottomed 96 well plate were imaged in growth media with [5 ng/mL] FGF2, and the remaining 10 were imaged in growth media with [0 ng/mL] FGF2. Additional details are available in the Supplemental Methods.

Heteromotility analysis

DIC images were segmented using custom segmentation algorithms, optimized for each of the systems we investigated. Tracking was performed with a modified version of uTrack [92]. Code is available on the Heteromotility GitHub page. All code for the Heteromotility tool is available on the Heteromotility Github page (https://github.com/cellgeometry/heteromotility) and in the Python Package Index (PyPI). Detailed descriptions of the algorithms are provided in the Supplemental Methods.