Quantifying collective motion patterns in mesenchymal cell populations using topological data analysis and agent-based modeling

Abstract

Fibroblasts in a confluent monolayer are known to adopt elongated morphologies in which cells are oriented parallel to their neighbors. We collected and analyzed new microscopy movies to show that confluent fibroblasts are motile and that neighboring cells often move in anti-parallel directions in a collective motion phenomenon we refer to as “fluidization” of the cell population. We used machine learning to perform cell tracking for each movie and then leveraged topological data analysis (TDA) to show that time-varying point-clouds generated by the tracks contain significant topological information content that is driven by fluidization, i.e., the anti-parallel movement of individual neighboring cells and neighboring groups of cells over long distances. We then utilized the TDA summaries extracted from each movie to perform Bayesian parameter estimation for the D’Orsgona model, an agent-based model (ABM) known to produce a wide array of different patterns, including patterns that are qualitatively similar to fluidization. Although the D’Orsgona ABM is a phenomenological model that only describes inter-cellular attraction and repulsion, the estimated region of D’Orsogna model parameter space was consistent across all movies, suggesting that a specific level of inter-cellular repulsion force at close range may be a mechanism that helps drive fluidization patterns in confluent mesenchymal cell populations.

1. Introduction

In the US alone, chronic wounds affect about 6.5 million patients and impose a > $25 billion annual burden on the healthcare system. Each year, about 70 million inpatient and outpatient surgeries add to the national need for better wound care [42]. Central to the robustness of wound healing is the orchestrated migration of several cell types directed by multiple types of spatial cues. One critical cell type that must migrate into the wound bed are fibroblasts, prototypical mesenchymal cells that sense and respond to chemotactic (gradients of soluble ligands), haptotactic (gradients of immobilized ligands), and durotactic (gradients of mechanical stiffness) cues. As they invade the wound, which occurs over several days or weeks, fibroblasts are most responsible for rebuilding the tissue, by restoring collagen-rich extracellular matrix (ECM) and mechanically closing the wound. Directed migration of mesenchymal cells is also critical during embryonic development and has been implicated in cardiovascular and fibrotic diseases [3, 18, 31]. Given the central role that fibroblasts and other mesenchymal cells play in physiological and pathological settings, there is a pressing need for a rigorous understanding of mechanisms governing the collective migration of these cells [8].

With few exceptions, it is often the norm that cell migration studies are conducted at low cell density, with isolated cells, whereas high density is the norm in tissues [13, 15]. Cell-cell contact is systematically avoided in experimental studies because of a phenomenon known as contact inhibition of locomotion (CIL), which was first described approximately 60 years ago [1]; when fibroblasts and certain other cell types collide with one another, protrusion ceases at the site of contact, and cell movement is biased towards another direction. Although CIL is confounding for studies attempting to isolate the influence of environmental cues, it is relevant for cell movements in tissues [43]. Recent work has led to the specification of molecular pathways that are required for chemotactic or haptotactic migration [4, 28, 30, 51, 53], but several critical gaps remain. Imaging of cells during wound healing clearly shows that an invading and actively proliferating population of fibroblasts achieves a high cell density [13, 34, 40]. Yet, how non-adhesive cell-cell contacts shape migration in densely packed mesenchymal cell populations is still under investigation.

It has been established that fibroblasts in a confluent monolayer adopt morphologies that differ from those of isolated cells. And it has been reported that confluent fibroblasts, though completely surrounded by neighboring cells, are motile [12]. Analysis of the data collected for our current study reveals that cell crowding drastically alters cell migration behavior in that the movements of crowded cells, constrained as they are by CIL, are correlated and far more coherent than those of isolated cells. We refer to this collective motion phenomenon as “fluidization” of the cell population. As previously reported [20], time lapse microscopy showed that confluent fibroblast cells spontaneously arrange themselves into a nematic order. We performed a close analysis of the movement of individual cells within this order, made possible by our custom designed tracking algorithm for single cells in time-lapse microscopy recordings. These revealed that neighboring cells oriented parallel to each other were often moving in opposite directions. In light of this observation, new analytical tools that can track the direction of single cell movements are necessary to fully understand fluidization in fibroblast cells.

One type of modeling paradigm that could be used to capture fluidization patterns is agent-based modeling, however it is currently challenging to estimate parameters for ABMs from experimental data. Agent-based models (ABMs) have been used in a wide range of biological applications to study how the interaction between neighboring individuals in a population can produce emergent collective behaviors [21, 25, 26, 49]. For example, the Viscek and D’Orsogna ABMs are seminal models exemplifying how parameters that control a few simple interaction rules, e.g., the strength of attraction, repulsion, or alignment with neighbors, can lead to strikingly different population-level spatial patterns such as rotating rings and vortices that appear similar to behaviors observed in bird flocks or fish schools [21, 49]. Instead of simulating an ABM at different parameter values and observing the resulting model patterns, we are interested in performing the inverse problem, in which we observe data exhibiting a certain pattern and use an ABM to infer the model parameters that generated the observed behavior. This parameter estimation procedure is performed to shed light on the biophysical rules and parameters that lead to fluidization. Parameter estimation for ABMs is a challenging task, however, due to these models’ computationally intensive nature and a lack of established methods to quantify their output patterns [2, 10]. At present, the most commonly-used form of ABM parameter estimation involves model coarse-graining, where ABM rules are converted into a continuous differential equation model describing how the population-level agent density changes with time. Model coarse-graining is advantageous because there are many well established techniques for performing parameter estimation with differential equation models, however, the resulting differential equation models often fail to capture the ABM’s individual-level details and can lead to inaccurate parameter estimates [5, 22, 23, 35, 52]. There is thus a need for the development and establishment of methods capable of performing accurate parameter estimation for ABMs.

To achieve the goal of estimating parameters for an ABM that captures fluidization patterns, we propose to build upon previous work showing that techniques from topological data analysis (TDA) can be used to extract information from ABM simulations to accurately classify global behavior of collective movement patterns [11]. Recent biological studies have used TDA to quantify patterns on insect swarming, retinal vasculature, pulmonary disease, and zebrafish stripes. Persistent homology (PH) is a computational tool from TDA that quantifies the presence of topological features (e.g., connected components and loops) over some underlying scale parameter. PH is classically applied to a static point clouds of data, but the output data from ABM simulations consists of time-varying point clouds, where each point represents the location of an agent over time. Extensions have been developed to apply PH methods to time-varying point clouds. Specifically, the “Contour Realization Of Computed K-dimensional hole Evolution in the Rips complex” (Crocker) can be used to compute topological summaries that are functions of time and spatial scale [45]. Previous studies have shown that Crockers provide significant information for performing agent-based parameter classification [11]. Here, we investigate the approach of leveraging Crockers and agent-based models to perform parameter estimation from experimental data collected from mesenchymal cell populations. These estimated parameter values allow us to quantify patterns of fluidization in the data and understand the individual interactions that lead to this behavior.

To accomplish the goal of quantifying patterns of fluidization, we developed a quantitative framework for analyzing time-lapse microscopy movies using machine learning, topological data analysis, agent-based modeling, and Bayesian inference (Figure 1). We developed a custom single-cell tracking algorithm to estimate the angle and position of cells in movies of confluent populations where fluidization was observed. Single cell tracking reveals a strong bias towards parallel and anti-parallel movement of neighboring cells, indicating the presence of fluidization (Section 2.1). We then converted single cell tracks into time-varying point clouds and extracted topological information using Crocker matrices [45]. We used a statistical test on Crocker matrices to show that they contain significant topological information content, i.e., that they contain non-random topological patterns. We created visualizations showing that the information Crocker matrices capture are the anti-parallel movement of neighboring groups of cells, thereby exemplifying that TDA can be used to bridge the gap between quantifying individual cell behavior and patterns of collective cell movement in experimental data (Section 2.2). We then utilized the D’orosgna agent based model (ABM) as a simple biophysical description for cellular attraction and repulsion in order to investigate the mechanisms through which individual cell interactions generate the topological signatures of fluidization. In agreement with previous studies, we showed that the entire stability diagram of the D’orosgna ABM can be summarized with Crocker matrices, strongly suggesting that TDA summaries provide enough information to perform parameter estimation with experimental data (Section 2.3). We then used simulated data to perform a computational study showing that parameters can be accurately recovered for the D’Orsogna ABM using Crocker matrices through Bayesian estimation with Approximate Bayesian Computation. Finally, we performed Bayesian estimation with Crocker matrices on experimental data. Our parameter estimation results on experimental data suggest that fluidization patterns may, at least in part, be generated by a simple mechanism in which there are specific values for (i) the ratio between the range of the repulsive and attractive forces between cells, and (ii) the ratio between the magnitude of the repulsive and attractive forces between cells.

Figure 1:

Outline of our single-cell tracking and analysis framework with a results summary at each step. (a) We start by collecting a set of images of nuclei in a confluent population of moving cells using time-lapse fluorescence microscopy. (b) We developed a custom machine learning-based algorithm to generate single-cell tracking data for the position and angle of each detected cell in all movie frames. (c) Topological data analysis is performed on single-cell trajectories by treating them as time-varying point clouds and computing Crocker matrices. (d) Crocker matrices are used for Bayesian parameter inference with the D’orsogna agent-based model. Parameter estimation was tested on simulated data and then applied to experimental data for six replicates.

2. Results

2.1. Single cell tracking reveals fluidization patterns as a bias towards parallel or anti-parallel movement of neighboring cells

2.1.1. Single-cell tracking resolves anti-parallel cell movement

We collected six movies of an actively moving confluent population of cells; each movie consists of 100 time lapse images taken at 10 minute intervals of NIH 3T3 fibroblast cells with fluorescently marked nuclei (Supplementary Movie 1, see “Data” Section 4.1). The center of mass of each nuclei is used as a proxy for each cell center. Initial cell densities in each movie range from 800–2000 cells per frame, Movies 1-3 were initialized at higher densities than Movies 4–6. We developed a custom single cell tracking pipeline (see “Methods” Section 4.2.1) to localize and track the center of each cell through time. Our custom single cell tracking pipeline was able to resolve anti-parallel flow of adjacent cells, as well as anti-parallel flow of adjacent groups of cells (Figure 2a,b,c)). In contrast to previous studies that used particle image velocimetry (PIV) to quantify nematic order of confluent cell populations [20], we found that single cell tracking was necessary for capturing fluidization patterns manifested as a bias towards parallel or anti-parallel movement 146 of neighboring cells (Supplementary Figure S2). Note that we refer to the angle of the cell’s velocity vector as “cell angle”.

Figure 2:

Fluidization is detectable in single cell tracking data. (a) The angle of cell movement overlaid for an image of cells with a nuclear marker (frame 75 of Movie 1). Arrowheads are colored by angle from 0 to 360 degrees. Angles are overlaid only for cells with a velocity above the 25th percentile among all cells in the video. The black colored regions at the edge of the image are present because the frame was shifted to remove jitter. (b),(c) Magnifications of the rectangular regions outlined in (a) exemplifying that fluidization is evident as anti-parallel movement among groups of adjacent cells. (d) A diagram illustrating the concept of fluidization in which groups of cells stream past each other in opposite directions.

2.1.2. The distribution of neighboring cell angles is bimodally distributed

To quantify the prevalence of parallel or anti-parallel movement, we calculated the relative angle between each pair of neighboring cells. The cutoff distance for neighboring cells was defined to be approximately 4 cell widths, which appeared visually to be a biologically relevant distance at the experimental cell densities we used to compare the movement of neighboring cells. We found that the distribution of the difference in angle between neighboring cells is bimodally distributed, with peaks at 0 degrees and ±180 degrees (Figure 3). The presence of bimodality indicates a strong bias toward parallel and anti-parallel alignment among the movement of neighboring cells. To test whether bimodality in the distribution of neighboring cell angles was influenced by the cell angles and not due to the spatial distribution of cell locations, we randomly shuffled the angles for all cells in all frames of each movie. In other words, we compare the distributions of neighboring cell angles generated from the fluidization movement and random movement. Specifically, we created an artificial “shuffled” data set, where the location of each cell over time was its measured cell tracking location, but each angle of movement at each time point was randomly sampled (without replacement) from all computed cell angles. Recomputing the distribution of neighboring cell angles after shuffling erased bimodality and resulted instead in a uniform distribution (Supplementary Figure S3).

Figure 3:

Distribution of difference in angle between neighboring cells for all frames in Movies 1–6, corresponding to figures (a)-(f), respectively. Neighboring cells were defined as being closer than 4 cell widths.

2.2. Topological data analysis extracts information about the collective cell motion patterns of fluidization in experimental data

While the analysis of neighboring cells reveals the bias in movement toward parallel and anti-parallel directions, we also observed that these individual cell interactions appear to align to nearby groups of cells that move in concert in parallel and anti-parallel directions (Supplementary Movie 1). We hypothesized that computational techniques from topological data analysis (TDA) could be used to bridge the gap between analyzing the behavior of individual cells to the collective behavior of groups of cells. This hypothesis builds upon previous findings demonstrating the effectiveness of TDA in accurately predicting parameters that generate collective motion behavior in agent-based model simulations [11]. Here, we briefly describe the computation of Crocker plots, a TDA technique used to summarize topological patterns in time-varying point clouds. Additional details are described in the Methods Section 4.2.2.

2.2.1. Calculating topological summaries of time-varying point clouds derived from single-cell tracking

For application of TDA to Movies 1-6, we used 3d point cloud data consisting of the location and angle of movement of each tracked cell in each movie frame, i.e., each cell corresponds to a point $(x,y,\theta )$, where the angle $\theta \in [0,360]$ is in degrees. A graph-like object called a “simplicial complex” is constructed from the point cloud data in which the nodes of the graph correspond to each cell in the 3d point cloud and the presence of an edge in the complex depends on whether the two data points it joins are within a given distance or “proximity” $\left(ϵ\right)$ from each other (see Section 4.2.2 for our choice of distance metric). We calculate the $n$-th Betti curve (Betti-$n$), which contains information about the number of $n$−cycles in a simplicial complex as a function of $ϵ$. For simplicity, we focus only on Betti-0 in this results section since we found that it was sufficient to detect fluidization. Moreoever, the Betti-0 curve is straightforward to interpret, it is simply the number of connected components as a function of the proximity parameter $ϵ$. To summarize the topology of a movie, Crocker matrices are constructed by concatenating Betti curves computed at all time points, generating a matrix where the columns and rows correspond to time points and $ϵ$, respectively. Crocker matrices can be visualized as 2-d contour plots in which the color denotes the contour level for the number of $n$-cycles at each proximity value and frame number, where one Crocker matrix corresponds to a specific value of $n$; counts above 250 are displayed as white. The left columns of Figure 4 and 5 are the Crocker plots for Movies 1-3 and Movies 4–6, respectively.

Figure 4.

Crocker plots summarizing the topology of the high density videos. Only cells for which the speed was above the 25th percentile in the movie were used. Cell location (x,y) and the angle of velocity (between 0–360 degrees) were to make a 3d feature vector for each cell, from which the persistent homology was calculated. The color legend denotes the contour levels for the number of simplicial complexes at each proximity value and frame number; counts above 250 are displayed as white for visualization purposes. Left: Crocker plots for Movies 1-3. Middle: Crocker plots for Movies 1-3 calculated by randomly shuffling cell angles. Right: Difference in Betti-0 between the non-shuffled and shuffled videos, calculated over all frames and Betti 0 values greater than 200. Significant differences between unshuffled and shuffled Betti 0 plots were found for all videos (P< 2e-16, Paired t-test).

Figure 5.

Crocker plots summarizing the topology of the high density videos. Only cells for which the speed was above the 25th percentile in the movie were used. Cell location (x,y) and the angle of velocity (between 0–360 degrees) were to make a 3d feature vector for each cell, from which the persistent homology was calculated. The color legend denotes the contour levels for the number of simplicial complexes at each proximity value and frame number; counts above 250 are displayed as white for visualization purposes. Left: Crocker plots for Movies 4–6. Middle: Crocker plots for Movies 4–6 calculated by randomly shuffling cell angles. Right: Difference in Betti-0 between the non-shuffled and shuffled videos, calculated over all frames and Betti 0 values greater than 200. Significant differences between unshuffled and shuffled Betti 0 plots were found for all videos (P< 2e-16, Paired t-test).

2.2.2. Crocker matrices generated by single-cell tracking data contain significant topological information content

We first sought to determine what topological information the computed Crocker matrices for Movies 1-6 encode about cell movement. Because we do not have a prior expectation about how movement of a population of cells collectively generates an observed pattern in a Crocker matrix, we created an artificial point cloud for each movie frame to compare the experimental data against. In particular, we randomly shuffled cell angles by reassigning a new angle to each cell location. The new angles were sampled without replacement from a random shuffling of the recorded angle values from all frames. By not shuffling the cell locations, and only shuffling cell angles, we could isolate the importance of cell angle in generating significant topological differences in the movement patterns encoded in the Crocker matrices. Specifically, we computed the difference in Betti-0 between the Crocker matrices computed from the non-shuffled and shuffled data and aggregated the results into a distribution for each of the Movies 1-6. The Crocker matrices in Figures 4 and 5 visualize the Betti-0 curves at each time point $t_i$ where $i=1,\dots ,100$ denotes the frame number and the proximity parameter $ϵ$ is evaluated over a linear mesh ${ϵ}_j$, where $j=1\dots ,110$ and the mesh spans [0.08,0.15]. We compared the Betti-0 distributions for the non-shuffled and shuffled Crocker matrices, where each distribution consists of all Betti-0 values at each $\left(t_i,{ϵ}_j\right)$ pair, i.e., 100×110 values for each distribution. When testing for statistically significant differences, we also removed any Betti-0 values in both the non-shuffled and shuffled distributions when Betti-0 was less than 200 in the non-shuffled case. Filtering Betti-0 in this way focuses on differences that happen during the transition from a completely separated point cloud in which each cell is its own connected component to a point cloud where most cells have aggregated into a connected component with neighboring cells. We chose to filter based on Betti-0 instead of implementing a proximity parameter cutoff because the $ϵ$ value at which transitions between separated and aggregated point clouds occur is variable between movies. We found that the Betti-0 distributions were significantly different between the non-shuffled and shuffled Crocker matrices for all six movies (Figures 4 and 5, P< 2e-16, Paired t-test). As described in the “Data” Section 4.1, Movies 1-3 were collected from over-confluent cell populations initialized at a higher initial density than the confluent cell populations in Movies 4–6. We found that the mean difference between the non-shuffled and shuffled Crocker matrices was greater in the over-confluent movies, where the mean difference was at least −100 (Figures 4), compared to the confluent movies where the mean difference was at most −50 (Figure 5). These results suggest that the topological information computed for time-varying points clouds arising from single cell tracking can be used to quantify collective patterns of cell movement.

2.2.3. Intepreting topological information encoded in Crocker matrices through visualization

We then created a visualization to intepret topological differences that are encoded in the Crocker matrices generated from experimental data. We hypothesized that the Crocker matrices encode information about group level behavior in which groups of cells move in anti-parallel directions relative to other nearby groups.

This group behavior is analogous to the bias in individual cell movement towards parallel and anti-parallel alignment with neighbors shown in Figure 3. For each frame of a movie and at each proximity value, the Crocker matrices consist of a Betti-0 value representing the number of connected components present in 3d $(x,y,\theta )$ space. We visualized these components as connected graphs in which edges join any two cells that are within an $ϵ$ distance from each other in this 3d space. The graphs displayed by increasing the proximity parameter $ϵ$ from 0 to 0.12 revealed topological differences between the non-shuffled and shuffled data (Figure 6). Visualizing the non-shuffled data shows a more rapid increase in the size of each group of cells as the proximity parameter increases as compared to the shuffled data. This observation reflects the significant difference in Betti-0 distributions seen in Figures 4 and 5 in which the non-shuffled data start to form components at lower proximity values than the shuffled data. The visualization in Figure 6 also shows that different connected components that overlap move in opposite directions. We note that this result is expected since any overlapping connected components moving in the same direction would merge into a single component in the visualization. However, the useful information conveyed in Figure 6 is the difference in the rate at which overlapping connected components occurs between the non-shuffled and shuffled data, since connected components can also overlap by random chance. While connected components do overlap in the shuffled data, they tend to do so at higher proximity values, and occur much less often and with smaller component sizes. We made similar observations from visualizing the topology for the other 5 movies (Supplementary Figures S4–S8).

Figure 6:

Visualization of connected components using proximity parameter values of ϵ = 0.08, 0.1, 0.12 for frame 75 of Movie 2, calculated using non-shuffled (a),(b),(c) and shuffled (d),(e),(f) angle data. The angle of cell movement is overlaid with cell nuclei, arrowheads are colored by angle from 0 to 360 degrees. Edges in connected components are similarly colored by the average angle of all cells in each component. Note that the cells are connected based on both cell location (x, y) and the cell angle of velocity, highlighting the significance of the angle dimension in determining connectivity.

2.3. Parameter estimation using topological data analysis identifies D’Orsogna model parameters that recapitulate fluidization patterns in experimental data

Our motivation for investigating whether Crocker matrices computed from experimental data contained significant topological information to summarize fluidization patterns was based on previous work by Bhaskar et al. showing that TDA could be used to classify emergent collective motion data simulated from agent-based models (ABMs) and to accurately recover model parameters from simulated data [11]. We note that the results presented in this work is distinct from this prevous study since we perform parameter estimation on both experimental and simulated data, whereas Bhaskar et al. perform parameter classification on simulated data. To investigate the biophysical mechanisms underlying the fluidization patterns described in the previous sections, we next performed parameter estimation for a simple ABM by utilizing TDA to formulate the cost function. Specifically, we leveraged Approximate Bayesian Computation to estimate parameters by minimizing the distance between Crocker matrices from an ABM and experimental data.

Based upon a previous study by Bhaskar et al. [11], we used the most parsimonious ABM in the literature that was capable of achieving collective motion patterns similar to the fluidiziation we observed in experimental data. Specifically, we chose to use the D’Orsogna model [21], which assumes that the velocity of individual agents are subject to either attractive or repulsive forces of neighboring agents (see Section 4.2.3 for a description of the D’Orsogna ABM). While the D’Orsogna model is a minimal, phenomenological model, it can be used to simulate a wide ranges of global behaviors including single mill, double mill, collective swarming, flocking, double ring, and escape, with flocking behavior being most visually similar to our description of fluidization (see Fig. 7e for flocking behavior).

Figure 7:

Visualization of the D’Orsogna model outputs for different emergent behaviors of collective motion. (a)-(b) Betti-0 and Betti-1 colormap produced by t-SNE and PCA, respectively, that can be used to visually cluster different behaviors. First columns of (c)-(h) show example snapshots for different emergent behavior of collective motion: (c) $(C,L)=(0.2,1.5)$; (d) $(C,L)=(0.7,2.5)$; (e) $(C,L)=(1.5,0.7);$ (f) $(C,L)=(1.8,0.4)$; (g) $(C,L)=(2.0,0.1))$; (h) $(C,L)=(1.4,1.5)$; and (h) $(C,L)=(2.5,2.5)$. Second and third columns of (c)-(h) show the corresponding Betti-0 and Betti-1 Crocker matrices.

2.3.1. Crocker matrices summarize the global behavior of the D’Orsogna model

Before proceeding with parameter estimation to identify parameters that replicate the fluidization seen in the experimental data, we first investigated whether parameter space for the D’Orsogna model could be accurately summarized by the topological information encoded in Crocker matrices. Our work builds on a previous study by Bhaskar et al. which showed that parameter classification was possible using Crocker matrices [11] but did not investigate the problem of parameter regression which is the typical framework for estimating model parameters. We followed the procedure used by Bhaskar et al. of non-dimensionalizing the D’Orsogna model (Eqs. (2)–(4), Section 4.2.3). This process reduces the number of parameters from 7 to 4, enabling the usage of unit-free time, position, and velocity variables, while preserving the original dynamics and structure. The four dimensionless parameters are defined as: $\alpha ,\beta ,C=C_r/C_a$, and $L=l_r/l_a$. We used the non-dimensionalized model to test for parameter identifiability. In addition, the usage of dimensionless parameters allows the interpretability. For example, $C$ represents the ratio between the magnitude of the repulsive and attractive forces between cells and $L$ represents the ratio between the range of the repulsive and attractive forces between cells. From a biological standpoint, long-range attraction and short-range repulsion are most biologically relevant [32]. It is also important to note that only the non-dimensional parameters $C$ and $L$ are known to govern global collective behaviors for the D’Orsogna model [21]. For our analysis, we fixed $\alpha =1.0$ and $\beta =0.5$ to be consistent with previous studies [11, 21].

We first tested whether Crocker matrices contained enough information to encode the global structure of $C,L$ parameter space for the D’Orsogna model. We simulated the D’Orsogna model for 30 values of $C$ and $L$ equally-spaced ranging from 0.1 to 3.0, resulting in 900 total combinations. We then computed Crocker matrices for Betti-0 and Betti-1 for each of the 900 simulations. We leveraged a nonlinear dimensional reduction method, t-distributed stochastic neighbor embedding (t-SNE) [48] to reduce the combined pair of Crocker matrices for each $C,L$ pair down to 3 dimensions, in which one 3×1 vector corresponds to one $C,L$ pair. (Figure 7a). In Figure 7a (Left), these vectors are displayed as scatter points in a 3D space, and each point is colored by using the points’ 3D t-SNE coordinates as an RGB value. We then used these colors to paint a plot of the 30 × 30 matrix for all 900 $C,L$ values, i.e., the color of each point in the t-SNE scatter plot is mapped back to its corresponding $C,L$ pair that generated the corresponding simulation (Figure 7a (Right)). We found that the boundaries between the different colors in the $C,L$ parameter clearly demarcate the global bifurcation diagram previously reported for the D’Orsogna model [21]. Similar results using a linear dimensionality reduction technique, namely principal component analysis (PCA), are shown in Figure 7b. In Figure 7c–h, we display representative snapshots of a D’Orsogna model simulation for different regions of parameter space corresponding to different collective motion behaviors.

2.3.2. D’Orsogna model parameters are identifiable by using Crocker matrices with Bayesian estimation

Because our dimensionality reduction visualization of Crocker matrices showed that they contained enough information to recapitulate the global bifurcation diagram for the D’Orsogna model, we next used simulated data to investigate which regions of D’Orsogna model parameter space are identifiable with Bayesian parameter estimation. In particular, we performed a computational practical identifiability study by simulating the D’Orsogna model at several different parameter combinations of $(C,L)$ and showed that we could recover those parameters from Crocker matrices. We note that, while a good prior can help address identifiability in Bayesian inference frameworks, our computational study was conservative in that we assumed uniform priors. Moreover, we chose parameter combinations for our simulation study that were difficult to recover since they lie on the boundary between regions of parameter space that produce qualitatively different collective motion patterns. Since it is infeasible to directly evaluate the likelihood function for Bayesian inference using the D’Orsogna model, we leveraged Approximate Bayesian Computation (ABC), an alternative approach to conventional Bayesian estimation methods. ABC approximates the posterior distribution by accepting parameters that produce simulated data similar to the observed data, eliminating the need for formulating the likelihood. While several ABC techniques have been developed and validated in previous studies [9, 33, 39, 44] we used the prototype rejection-ABC algorithm [39]. We applied this ABC algorithm by generating potential parameter sets from a prior distribution and simulating data for each parameter, $\mathit{q}:=(C,L)$, where $C$ and $L$ are the non-dimensional ABM parameters. We used a uniform prior distribution over $(C,L)\in [0.1,3]\times [0.1,3]$. The parameter set, $\mathit{q}$, is accepted if

$$SSE\left({\stackrel{\mathbf{ˆ}}{\mathbf{C}}}_{0,1}(\mathit{t},\epsilon ,\mathit{q});{\mathbf{C}}_{\mathrm{0,1}}^o(\mathit{t},\epsilon )\right)\le \tau$$ (1)

where $SSE\left({\stackrel{\mathbf{ˆ}}{\mathbf{C}}}_{0,1}(\mathit{t},\mathit{\epsilon },\mathit{q});{\mathbf{C}}_{0,1}^o(\mathit{t},\mathit{\epsilon })\right)$ is the sum of squares error between two Crocker matrices. The notation ${\stackrel{\mathbf{ˆ}}{\mathbf{C}}}_{0,1}(\mathit{t},\mathit{\epsilon },\mathit{q})$ denotes the concatenated pair of Crocker matrices for Betti-0 and Betti-1, calculated at time points in the vector $t$ over proximity values in the vector $\epsilon$, at some $C,L$ parameter combination represented by $\mathit{q}$. Similarly, the notation ${\mathrm{C}}_{0,1}^o(\mathit{t},\mathit{\epsilon })$ denotes the Crocker matrices computed from observed data. The ABC algorithm requires the user to choose a hyperparameter, termed the “tolerance”, denoted above by $\tau$. The pseudocode and additional details for our application of the prototype rejection-ABC algorithm can be found in the Supplementary Materials Section 2.2. To find regions of $C,L$ parameter space that were identifiable, we applied ABC to simulated data and tested whether we could recover the known parameters. We chose six parameter values to simulate data that located in six regions of parameter space with distinct collective motion behavior; the chosen parameter values are identical to those used in Figure 7. In Figure 8, we display the ABC posterior densities for three sets of parameter values, and comparisons between Crocker matrices from data simulated using the observed (ground-truth parameters) and ABC predictive (estimated parameters). In all three cases, the median of the ABC posterior density almost exactly matches the ground-truth parameter set. Moreover, the corresponding Crocker matrices from the ground-truth and estimated parameters are nearly indistinguishable. Similar results were found for the other three parameter sets (Supplementary Figure S9). We want to note that only using Betti-0 Crocker matrix might not be sufficient to distinguish the parameter regions that contain two cases in Supplementary Figures S9a and S9b. Therefore, we compute Crocker matrices for both Betti-0 and Betti-1. Comparisons between snapshots of simulations using the ground-truth and estimated parameters reveals that, while the exact location of all agents is not preserved, the topological structure is identical (Supplementary Figures S10 and S11).

Figure 8:

The inferred parameters for the D’Orsogna model using simulated data. Posterior densities from ABC. The left column shows the inferred ABC-posterior density, the median of the posterior (ABC-median, black dot), and the ground-truth parameter value (True, white star). The middle and right columns display the Crocker matrices for the data simulated from the “True” parameter value (Observed) and the inferred parameter value (ABC predictive), respectively. Ground-truth parameter values used for simulation are (a) $(C,L)=(0.2,1.5)$ (b) $(C,L)=(0.7,2.5)$, and (c) $(C,L)=(1.5,0.7)$.

2.3.3. Parameter estimation for the D’Orsogna model using experimental data

After validating our ABC parameter estimation approach using Crocker matrices to summarize information for simulated data, we then applied ABC to experimental data. Specifically, single-cell tracks generated by Movies 1-6 by using the single-cell tracks were used to compute Crocker matrices to perform parameter estimation with the D’Orsogna model. However, due to the inconsistency in the number of detected cells in each frame, we subsampled the particle trajectories to keep the number of cells consistent accross all frames for simulation (see Supplementary Section 2.3). As mentioned above, only the non-dimensional parameters $C$ and $L$ are known to govern the collective behaviors. Therefore, we ran parameter estimation on the experimental data using the non-dimensionalized model to ensure the parameter identifiability. Figure 9 displays the results for Movies 1-3 in a format similar to those used for the simulated data, namely, a plot of the inferred posterior densities and comparisons between the Crocker matrices. Results for Movies 4–6 are shown in Supplementary Figure S12. Comparisons of snapshots between the experimental data and simulated data are shown in Supplementary Figures S13 and S14 for Movies 1-3 and Movies 4–6, respectively. While we do not have a ground truth for evaluating our results, our results from simulated data and the consistency of results across experimental data sets strongly suggest that fluidization behavior can at least partially be explained by biophysical mechanisms that map at least phenomenologically to the parameters $C$ and $L$ of the D’Orsogna model. In particular, the estimated value of $C$ had a mean of 1.23 (SD=0.11), and the estimated value of $L$ had a mean of 0.73 (SD=0.03) for Movies 1-6. The similarity between the Betti-0 and Betti-1 Crocker matrices from the experimental data and D’Orsogna model strongly indicate our ability to infer ABM parameters that can generate simulations closely matching the topology of real data.

Figure 9:

The inferred parameters for the D’Orsogna model using experimental data. The left column shows the inferred ABC-posterior density and the median of the posterior (ABC-median, black dot). The middle and right columns display the Crocker matrices for the experimental data (Observed) and the D’Orsogna model simulated at the inferred parameter value (ABC predictive), respectively. Inferred parameter values (ABC-Median) for Movies 1,2, and 3 with results in (a), (b), and (c), respectively are $(C,L)=(1.10,0.79)$ (b) $(C,L)=(1.18,0.73),$ and (c) $(C,L)=(1.46,0.69).$

3. Discussion

This study used a combination of video microscopy, machine learning, topological data analysis (TDA), agent-based models (ABMs), and Bayesian inference to quantify fluidization patterns that arise from the collective interaction of mesenchymal cells within a confluent monolayer. Although detailed studies of directed cell migration are currently being conducted with isolated cells, they ignore the emergent dynamics of cell populations at the high densities observed in tissues. Our work produced novel computational tools and biological insight into how information from multiple scales and spatial cues are integrated in the collective migration of mesenchymal cells in dense tissues relevant to physiological processes such as embryonic development and wound repair.

Our results and visualizations involving Crocker matrices to summarize the topology of collective cell movement strongly suggest that not only do neighboring cells align parallel and anti-parallel over time, but neighboring groups of cells also have this behavior and that this group movement pattern is coherent over long distances. Our findings suggest that TDA can be used to bridge the gap between local behavior and group behavior, and that Crocker matrices contain a statistically significant amount of information for accomplishing this task. Determining whether TDA encodes topological signatures of fluidization is an important question for developing new mathematical models of collective cell migration since the simple assumption of random diffusion is often the most common assumption made in deriving such models [12, 22, 29, 36, 50].

We investigated whether TDA could provide enough information for performing parameter estimation with ABMs and thereby provide insight into the role of cell-cell interactions in governing collective motility in dense cell populations. Our results showed that the information contained in Crocker matrices was sufficient to recapitulate how parameter space for the D’Orsogna ABM is organized according to different emergent behaviors the model is capable of producing, i.e., flocking, double mill, escape, etc. Building upon this finding, we used simulated data to show that D’Orsogna model parameters were identifiable using Bayesian estimation with Crocker matrices to summarize topological information of the ABM simulation. We then performed the same Bayesian inference procedure with experimental data and showed that estimated parameters for the D’Orsogna ABM across six different movies were highly similar and located in the region of D’Orsogna model parameter space corresponding to fluidization. Moreover, these results were robust across experiments with different confluencies, in which Movies 1-3 were collected from cell populations initialized at a higher density than Movies 4–6. The high level of consistency in parameter estimation results (CV=0.09 for $C$ and CV=0.04 for $L$) strongly suggests that one of the key mechanisms driving fluidization patterns in dense populations can be modeled as a balance between (i) the strength of attractive and repulsive intercellular forces and (ii) the length scale of those interactions. One plausible intepretation for the estimate of $L$ being less than 1 and C being greater than 1 is that the D’Orsogna model in this parameter range could be used to describe contact inhibition of locomotion (CIL). Specifically, these parameters correspond to a mechanism in which the dominant inter-cellular force at close range is repulsion, which is similar to CIL in which cells change their direction when coming into contact with one another. In this study, since the ABC-posterior densities are non-Gaussian distributions, we choose to use the median for $C$ and $L$ as the representation for the estimated parameters in this study. For future work, unsupervised machine learning approaches such as $k$-means clustering could potentially be used to find the cluster centers of the ABC-posterior densities as representation for estimated parameter locations.

We note that even though the overall pattern encoded in the Crocker matrices was visually similar between the experimental data and the D’Orsogna ABM simulated at the estimated parameters, there still exists some discrepancy between the patterns which might have arisen from model bias. In particular, the D’Orosgna model does not capture the slight downward trend of the contours in Betti-0 and the precise values at which Betti-1 is greater than 250 for some of the experiments, i.e., Movies 1-6 (Figure 9 and Supplementary Figure S12). The ABM we considered in this work was chosen because it was the simplest model that has been shown to have the capacity to generate patterns similar to fluidization. Specifically, the D’Orosgna ABM can exhibit flocking behavior in which long lines of agents moves as a group in approximately the same direction while being adjacent to other groups that move in a different direction, albeit not exactly anti-parallel. In future work, we will investigate whether models that more accurately reflect the underlying biophysics of mesenchymal cell movement and inter-cellular interactions could produce patterns which more accurately capture the parallel and anti-parallel movement that is characteristic of fluidization. For example, the study by Haraiwa [25] investigated the behavior of an agent-based model of migrating cells that incorporates inter-cellular contact communication through contact following (CF) and contact inhibition and attraction of locomotion (CIL and CAL). By varying the parameters describing the strength of CF, CIL, and CAL, Haraiwa found that the ABM is capable of producing many distinct dynamic self organization patterns, such as rotating aggregates, spirals, rings, and a new pattern called snakelike dynamic assembly that enables high directional accuracy towards a faint external signal in a collectively migrating cell population. In a later work, Haraiwa extended this ABM to include more specific dynamics involving modeling of the basal- and apical-side of cells and incorporating cell shape [26]. With these additional details in the ABM, Haraiwa was able to find parameter regimes that lead to highly motile dynamics, including spontaneous swirling. The TDA-based analysis pipeline presented here lays the foundation for future model selection and validation studies of ABMs to quantitatively test whether including CIL mechanisms can improve the ability of such models to match topological information from time-lapse microscopy recordings of collective motion in cell populations. Determining the optimal choice of topological filtration that can provide the maximum amount of information for parameter estimation from a data set remains an open area of research [38]. In this work, the Vietoris-Rips complexes that form the filtration were computed using a simple distance metric for the symmetric difference between two angles. We note that we did not consider other distance metrics and leave it to future work to investigate the impact of these metrics and different choices of topological filtration on parameter estimation accuracy.

4. Data and Methods

4.1. Data

A set of time-lapse images was collected using widefield epifluorescence microscopy to image NIH 3T3 fibroblast cells labeled with a far-red DNA-binding fluorescent marker which allows us to see the cell nuclei [17]. Within an hour of the images being collected, the cells are treated with platelet-derived growth factor subunit B (PDGF-BB). The data were obtained from six non-overlapping locations inside a single dish containing two different densities of cells at the initial time point in the experiment. The cell density is measured via the confluency—a common metric of cell density for assay experiments that describes the percentage of the vessel or dish that is covered by cellular material [19]. We refer to the images that are 100% confluent as the “confluent” images, and the images which are more densely-packed as “over-confluent”. In other words, while both confluent and over-confluent images refer to images with 0 spaces between the cells, i.e., the surface of the dish is fully covered by the cells, the over-confluent images have higher cell density. To achieve the higher cell density, over-confluent cells are forced to be less spread and to adopt different morphologies. Three of the six non-overlapping locations where time-lapse images were taken show portions where the cell density is over-confluent (Movies 1-3); the other three locations contain confluent cells (Movies 4–6). One hundred images were collected at each of these locations in 10 minute intervals, resulting in time-lapse movies lasting approximately 17 hours and a total of 600 individual images. Images were acquired as single-channel, 16-bit TIFF images with a resolution of 1344 pixels by 1024 pixels and a spatial resolution of 1.157 pixels per micrometer (μm). Supplementary Movie 1 contains all six movies. Note that some frames contain regions of black pixels near the boundary because each frame was automatically shifted to remove jitter.

4.2. Methods

4.2.1. Single-cell tracking

Cell localization and trajectory mapping from time-lapse imaging.

We developed an automated cell tracking pipeline combining (1) a custom deep learning method for cell localization and (2) the Crocker–Grier particle tracking algorithm [16] [Crocker–Grier ref] as implemented in the Trackpy Python package [46] (Supplementary Figure S15). For step (1), we developed a convolutional neural network (CNN) that takes as input a grayscale image of the cell nuclei (a frame of a movie) and outputs an image called a p-map consisting of small Gaussian peaks which represent the probability or proximity to nuclei centers of mass, which we use as a proxy for a point location of each cell. This p-map approach was shown to be particularly effective at detecting cell nuclei in microscopy data [27]. We based our p-map generating CNN architecture on U-Net [41]; additional details about the CNN architecture, loss function, training, and data augmentation can be found in Supplementary Materials Section 2.1. For step (2), we used the sequences of p-maps for each movie as inputs to the Crocker–Grier tracking algorithm, which locates the signal peaks based on a particle diameter setting and then links these ”particles” over time using nearest-neighbor search. We accomplished this with the Trackpy software using a particle diameter of 11 pixels, a nearest-neighbor search range of 10 pixels, and a tracking memory of 5 frames.

Post-processing.

We post-process the resulting trajectories by removing image jitter, interpolating trajectories with temporary gaps, computing velocities using finite differences, and removing trajectories which are too short to be useful (trajectory length < 10 frames). We calculated the speed of each detected cell in all frames and found that the cell speed is lognormally distributed for each of the six movies (Supplementary Figure S1). We removed slow moving cells with speed below the 25th percentile of the lognormal distribution of speeds for each movie from downstream analysis since faster moving cells likely contain most of the information about how cell movement is flowing in the population.

4.2.2. Topological data analysis

Homology is a fundamental topological concept that formalizes the congruence of manifolds based on the number of n-dimensional holes they enclose. Persistent homology (PH) provides a multi-scale summary of the shape of data by calculating Betti numbers. Here, we summarize the PH calculations used in our analysis and refer the reader to [45] for a more thorough explanation of PH and its application to collective motion in biology.

Simplicial complexes and filtration.

We computed the PH of point clouds corresponding to the time-varying locations and angles of a population of cells at each frame of all movies by computing Betti curves and concatenating them into crocker plots. We denote $X_f={\left\{\left(x_i,y_i,{\theta }_i\right)\right\}}_{i=1}^{N_f}$ as the point cloud for a specific frame $f$ from one of the six movies that contains $N_f$ cells. In topology, points are known as 0-simplices.

We implement the Vietoris-Rips filtration to analyze the topology of each point cloud $X_f$. This is performed by first generating higher-dimensional $k$-simplices from the 0-simplices; the collection of all such simplices is commonly referred to as a simplicial complex. We construct each simplicial complex according to the following rules: for a proximity parameter $\epsilon >0$, each $k$-simplex is a line, triangle, or higher-dimensional shape connecting any $k+1$ points that are all pairwise within distance $\epsilon$ of each other. The distance between 2 points is defined as the sum of the euclidean distance in the $x$ and $y$ dimensions plus the distance in the $\theta$ dimension. For the $\theta$ dimension, we assume that the difference between two angles is symmetric with a maximum of 180 degrees, i.e., ${\theta }_1-{\theta }_2=\mathrm{m}\mathrm{i}\mathrm{n}\left\{{\theta }_1,{\theta }_2\right\}-\mathrm{m}\mathrm{a}\mathrm{x}\left\{{\theta }_1,{\theta }_2\right\}+360$. By taking an increasing sequence of proximity parameters $T=\left({\epsilon }_0,{\epsilon }_1,\dots ,{\epsilon }_m\right)$, we create an entire nested family of simplicial complexes, also known as a filtration, given by ${\left(V{R}_{\epsilon }\left(X_f\right)\right)}_{\epsilon \in T}$ [14]. We finally compute the kth Betti number by quantify the number of $k$-dimensional holes in each $V{R}_{\epsilon }\left(X_f\right)$. Specifically, the kth Betti number is defined as the rank of the kth homology group of $V{R}_{\epsilon }\left(X_f\right)$, which quantifies the number of $k$-dimensional holes enclosed by $V{R}_{\epsilon }\left(X_f\right)$ [6]. The zeroth and first Betti numbers are often referred to as connected components and loops, respectively.

Crocker plots.

Given a computed filtration, we calculate the kth Betti at each value of $\epsilon$. The changes in Betti numbers as a function of $\epsilon$ are what encodes the topology of multiple spatial scales in each point cloud. The complete persistent homology of the filtration can be summarized by considering the Betti number of each complex in the filtration as a function of the proximity parameter $\left(\epsilon \right)$ to form a Betti curve [24]. Here we utilize a previously developed approach for analyzing time-varying or dynamic point cloud data sets (such as cell trajectories). Specifically, we compute Betti curves for each discrete time step, and then concatenate them into a single matrix which summarizes the homology of the entire time-series. This matrix is known as the “Contour Realization Of Computed K-dimensional hole Evolution in the Rips complex” or crocker plot [45]. Crocker plots characterize changes in Betti numbers over time by plotting the level sets or contours of the concatenated Betti curves and has previously been used to visualize the homology of dynamic point cloud data for the sake of understanding the topology of biological models [47].

Implementation details.

We used the Ripser package to calculate Betti curves that comprise the Crocker plots used in our analysis. Ripser is a C++ library that utilizes many state-of-the-art optimizations for calculating persistent homology to drastically reduce both memory demands and computation time [7]. We utilized the Python port of Ripser in the Scikit-TDA library or all persistent homology computations in the 516 ensuing analysis.

4.2.3. D’Orsogna agent-based model

The D’Orsogna model is a system of ODEs with that describe the interactions between $N$ identical particles with equal mass $m$ and indexing by $i\in 1,2,3,\dots ,N$. The system consists of $2Nk$ coupled ODEs where $k$ is the dimensionality of the model. We utilize two-dimensional particle motion datasets, and thus consider only the model where $k=2$, but note that the model generalizes to 1 or 3 dimensions without complication. The model is given in [21] by the pair of equations:

$$\frac{d{\mathit{x}}_i}{dt}={\mathit{v}}_i,$$ (2)

$$m\frac{d{\mathit{v}}_i}{dt}=\left(\alpha -\beta {\left|{\mathit{v}}_i\right|}^2\right){\mathit{v}}_i-{\nabla }_{i}U\left({\mathit{x}}_i\right)$$ (3)

where $U\left({\mathit{x}}_i\right)$ is a generalized form of the Morse potential [37] suited for cellular attraction and repulsion:

$$U\left({\mathit{x}}_i\right)=\sum _{j\ne i}{C}_r{e}^{-\left|{\mathit{x}}_i-{\mathit{x}}_j\right|/{\ell }_r}-C_a{e}^{-\left|{\mathit{x}}_i-{\mathit{x}}_j\right|/{\ell }_a}.$$ (4)

The dynamics of a particle depend only on its own position $x_i$, its own velocity $v_i$, and the positions of other particles ${\mathit{x}}_j,j\ne i.\alpha$ and $\beta$ are the self-propulsion coefficient and the friction coefficient respectively, which govern how each particle will accelerate to an asymptotic velocity if no other forces are acting on it. All other parameters determine how the particles experience these two-body interactions, with ${\ell }_a$ and ${\ell }_r$ representing the attractive and repulsive potential ranges respectively, and $C_a$ and $C_r$ providing the magnitude of these forces. Following previous work [11], after nondimensionalizing Eqs. (2)–(4), we obtain the nondimensionalized version of the D’Orsogna model, which effectively reduces the number of parameters from 7 to 4 and allows the use of unit-less time, position, and velocity variables, yet maintains the same dynamics and structure. The four dimensionless parameters are: $\alpha ,\beta ,C=C_r/C_a$, and $l=l_r/l_a$.

• Single cell tracking unveils anti-parallel movement in mesenchymal cells population.

• Topological analysis quantifies collective motion in experimental data.

• Topological information can be used to estimate parameters for an agent-based model.

• An agent-based model recapitulates experimental collective motion.

• A specific balance of inter-cellular forces drives the motion in experimental data.

6. Code and data availability statement

The code and data are publicly available at: https://github.com/kcnguyen3191/Fluidization-TDA-ABM-ABC