Enhanced persistence and collective migration in cooperatively aligning cell clusters

Abstract

Most cells possess the capacity to locomote. Alone or collectively, this allows them to adapt, to rearrange, and to explore their surroundings. The biophysical characterization of such motile processes, in health and in disease, has so far focused mostly on two limiting cases: single-cell motility on the one hand and the dynamics of confluent tissues such as the epithelium on the other. The in-between regime of clusters, composed of relatively few cells moving as a coherent unit, has received less attention. Such small clusters are, however, deeply relevant in development but also in cancer metastasis. In this work, we use cellular Potts models and analytical active matter theory to understand how the motility of small cell clusters changes with N, the number of cells in the cluster. Modeling and theory reveal our two main findings: cluster persistence time increases with N, whereas the intrinsic diffusivity decreases with N. We discuss a number of settings in which the motile properties of more complex clusters can be analytically understood, revealing that the focusing effects of small-scale cooperation and cell-cell alignment can overcome the increased bulkiness and internal disorder of multicellular clusters to enhance overall migrational efficacy. We demonstrate this enhancement for small-cluster collective durotaxis, which is shown to proceed more effectively than for single cells. Our results may provide some novel, to our knowledge, insights into the connection between single-cell and large-scale collective motion and may point the way to the biophysical origins of the enhanced metastatic potential of small tumor cell clusters.

Significance

Small tumor cell clusters, typically 2–20 cells in size, show a dramatically increased metastatic potential (up to two orders of magnitude) compared with single tumor cells. However, in contrast to single-cell or collective tissue dynamics, little is known about the manner in which such small clusters migrate and why they are so much more effective at accomplishing metastasis. This work reveals that in the presence of only generic velocity alignment, clusters exhibit enhanced directional migration, an effect that increases with cluster size. Our work offers specific predictions for the scaling of the persistence time and the random motion of clusters of cells as a function of cluster size, effects that should be directly observable in experiments.

Introduction

Many cell types—even those that otherwise are largely stationary—possess an innate capacity to migrate, individually and autonomously, on two-dimensional (2D) substrates or in three-dimensional (3D) matrices. Properly regulated, cell migration contributes crucially to organismal functioning, as it drives vital processes such as morphogenesis, tissue formation, wound healing, and the inflammatory response. In pathology, cell migration likewise features prominently, and nowhere more so than in cancer metastasis. Cancer remains one of the leading causes of death in the developed world (1), and the vast majority of deaths due to cancer (∼90%) are a consequence of metastasis (2, 3, 4, 5). In metastasis, cells detach from a primary tumor and invade the surrounding extracellular matrix (ECM)—i.e., the three-dimensional cellularized material that provides structural support to tissue—migrating toward blood or lymphatic vessels. Once there, migratory cancer cells traverse the vessel wall (intravasation) and pass into the circulation system as circulating tumor cells (CTCs). Eventually, some of these CTCs may once again pass the vessel wall and navigate the local ECM to seed a secondary tumor (3,4,6).

The elimination of malignant tumors through early detection and timely resection, possibly combined with chemo-, radiation, and immune therapy, is the principal directive in treatment. Despite the seemingly straightforward sequence of metastatic events (often referred to as the “metastatic cascade”), this process remains poorly understood; effective countermeasures that directly interfere with metastasis itself are scarce (3). The process continues to hold surprises, too; although it was long generally held that distant metastases were mostly seeded by single tumor cells (7), recent experimental studies reveal significant contributions to metastasis from so-called CTC clusters: heterogeneous cell clusters consisting of ∼2–20 cells that have collectively detached from a single primary tumor and are collectively undertaking the entire metastatic cascade, invading, intravasating, and circulating as one conserved unit (2, 3, 4,6, 7, 8, 9, 10). These clusters are dangerously potent; a study of spontaneous breast cancer in mice revealed that over 97% of all observed metastases originated from CTC clusters rather than single CTCs (10). Other work highlights the importance of the cell-cell adhesion mediators such as E-cadherins in metastasis and likewise suggest that CTC clusters may possess a metastatic potential that is at least 50 times (and possibly over a hundred times) greater than for individual CTCs (2,6,8,10,11). CTC clusters are associated with lowered overall survival and lowered progression-free survival in a range of cancer types (3). It was convincingly shown that CTC clusters indeed remain a single unit throughout the journey from primary tumor to distant site (6,8); the pathway in which polyclonal CTC clusters would assemble from single CTCs at some point during metastasis is highly improbable (6). Finally, although collective metastasis is our main motivation, we note that collectively moving clusters also play a crucial role in many developmental processes; (12) and (13) emphasize their importance in, e.g., the neural crest, in mesoblasts in gastrulation, and in the extension of chick somites forming the sclerotome, among a number of other appearances in biology.

Overall, experimental findings have opened up a completely new field of study focusing on relatively small cell clusters in biology (12). To explain in particular the enhanced metastatic potential of CTC clusters, a number of hypotheses have been brought forward, including the cooperation of heterogeneous cell types within the CTC cluster, shielding from attacks by immune cells, a differential capacity for sensing and responding to chemical gradients (14,15), and the protection from pressures and shear forces while in the bloodstream (2,7,16). Yet much remains unknown about the genesis, transit, and the settlement of CTC clusters during metastasis (2).

The purpose of this article is to examine three physical-mechanical aspects of cluster motility. First, clusters are obviously larger than single cells. How do multiple erratic individual motile tendencies, with varying degrees of coordinated organization, add up to the collective motion of a small cluster of identical cells? Second, how does in-cluster heterogeneity (in intrinsic motility) affect motility at the cluster level? And third, how do these altered properties affect the ability of a cluster to perform durotaxis (17, 18, 19, 20), that is, to move directedly in the presence of a rigidity gradient? The latter has been shown to improve in large aggregates (17); how does it play out in smaller clusters?

To address these research questions, we combine coarse-grained simulations with analytical active matter theory. Specifically, we use the cellular Potts model (CPM) to simulate cell (cluster) motion. This model is augmented to capture two important features of collective motility: directional persistence and cell-cell alignment. Directional persistence captures the tendency of individual cells to persist directionally for some amount of time (20, 21, 22, 23). It is quantified by a persistence time, which corresponds to the average time it takes a cell to deviate significantly from an initial course. Cell-cell alignment refers to the tendency of densely packed motile cells to mutually inform the direction of their motion (13,24,25) and has been invoked to explain collective motility in dense systems (26). This may happen either by direct physical interactions such as volume exclusion and traction forces, by which cell-cell adhesions drag neighbors along, or in a more indirect fashion through contact inhibition of locomotion (27). Although the latter tends to cause cells to move away from each other, in dense systems this effect also suppresses convergent relative motion and is thus generally manifested as a parallel-aligning field. In this work, we model persistence and alignment using a Langevin and Vicsek-type (28) approach, respectively. More specifically, we implement persistent migration in two and three dimensions using a Langevin description for the stochastic rotational diffusion of the cells’ instantaneous direction of motion. Alignment is implemented in a Vicsek-like feedback mechanism and quantified by the relative weight assigned to neighbor velocities when updating the velocity of a given cell. To rationalize our CPM simulation results, we also develop an analytical model for finite-sized clusters composed of (aligning) active Brownian particles (ABPs), providing more theoretical insight into the cluster migration efficacy as a function of cluster size and cluster heterogeneity.

The study is organized as follows. We start with introducing the CPM and demonstrate how a persistent random walk and a Vicsek cell-cell alignment term are implemented in the CPM. We validate the implementation of persistence and alignment by analyzing the trajectories of single cells and cell clusters exploring homogeneous environments. We then discuss the theory of ABPs and use it to provide an analytical underpinning of the numerically observed behaviors. Finally, we relate the enhanced persistence of clusters to cell transport in a more complex, durotactic environment. We conclude by summarizing the main findings and provide some future directions and topics where our results may have an impact.

Methods

CPM

To simulate the motion of a CTC (cluster) through the ECM, we employ the so-called CPM (29,30). This model, credited for explicitly representing the cell shape, has been successfully applied to a wide variety of biological phenomena, including, for instance, blood vessel network formation, cancer cell invasion, and collective cell motion (25,31, 32, 33, 34, 35). The CPM is a variation on the classic Potts model (36) and consists of integer spins σ(x) ≥ 0 on a discrete square or cubic lattice (with a lattice constant a), whose sites are characterized by their position in space x. Biological cells are represented as (simply connected) domains of equal spin σ(x) > 0, whereas the medium or ECM is assumed to be homogeneous (which is, of course, a gross simplification but provides computational convenience and allows us to exclusively focus on cell-cell alignment) and depicted by σ = 0 (see Fig. 1).

Figure 1.

(A) Experimental observation of a CTC cluster taken from (10) is shown. (B) Visualization in 2D of CPM cells representing the experimentally observed CTC cluster is given. Different colors represent different spins σ with the medium (σ = 0) shown in white. (C) Example trajectories of an aligned four-cell CPM cluster are shown, with the black dots denoting the end point of each trajectory. Inset shows the respective cluster with the arrows denoting the polarity vector p_σ of each cell. To see this figure in color, go online.

Cell movement can then be imposed on the system via a modified Metropolis Monte Carlo algorithm (37, 38, 39); a candidate lattice site i is randomly chosen, and its spin value σ(x_i) is attempted to be changed to a randomly picked adjacent spin value σ(x_j). The attempt is (provided all cells remain simply connected) accepted with a Boltzmann probability, i.e., min(1, $e^{-\text{Δ}\mathcal{H}/T}$), where T parameterizes the energy associated with membrane fluctuations (29). The parameter is suggestively called T to emphasize the temperature-like role it plays in tuning dynamics from quiescent to actively disordered, but we stress that it is not an actual temperature. $\text{Δ}\mathcal{H}$ is the resulting change in a phenomenological Hamiltonian $\mathcal{H}$; the latter accounts for all physically relevant terms, which in the original model are (approximately) constant cell volume and a finite cell-cell interfacial tension yielding a Hamiltonian (29,30)

$$\mathcal{H}={\mathcal{H}}_{\text{volume}}+{\mathcal{H}}_{\text{adhesion}}=\lambda \sum _{\sigma }{\left(V_{\sigma }-V_{\sigma ,0}\right)}^2+\sum _{i,j}\frac{J_{\sigma \left({\mathbf{x}}_i\right),\sigma \left({\mathbf{x}}_j\right)}\left(1-{\delta }_{\sigma \left({\mathbf{x}}_i\right),\sigma \left({\mathbf{x}}_j\right)}\right)}{|{\mathbf{x}}_i-{\mathbf{x}}_j|}$$ (1)

Here, V_σ denotes the volume (in 3D) or area (in 2D) of cell σ, i.e., the number of lattice sites with σ(x) = σ; V_{σ, 0} is the preferred volume or area of the corresponding cell. The parameter λ represents the strength of the volume constraint, and the first sum is taken over all cell spins σ > 0. For the adhesion term, the sum is taken over nearest (contact adjacent) and, to enhance cell roundness, next-nearest (diagonally adjacent) neighbors i, j with J_{σ, σ′} (= J_{σ′, σ}) denoting the adhesion coefficient between cell σ and cell σ′ (or the medium σ′ = 0), |x_i − x_j| the distance between the neighboring sites (which ensures stronger energetic contributions from the nearest neighbors in comparison to the next-nearest ones), and δ_{σ, σ′} the Kronecker delta, which ensures that only lattice site pairs of different cells contribute to the surface energy. Note that generally, J_{σ, σ′} > 0, and that by choosing different values for the coefficient J between two cells and between a cell and the medium, we may implement preferential cell-cell adhesion. To quantify the evolution of the system within the CPM, we introduce the Monte Carlo step (MCS) as a time measure (29,30,38). The MCS is defined as N_l attempts to change a spin value, with N_l the total number of sites in the lattice; it ensures that, on average, each lattice site has one attempt every MCS, thereby decoupling the time step from the actual system size (38).

Activity and persistence

In its original formulation, i.e., Eq. 1, cell dynamics in the CPM arises solely from fluctuations in the cell volume and interfacial area (or cell area and interfacial length in 2D systems). As a result, the cells do not experience any directional bias. In real life, however, cells migrate actively and may exhibit biased, directional motion. This may be due to external guiding cues such as the local organization of the extracellular matrix and more generally in response to gradients of some kind. In such cases, the motion is called a “taxis.” The most well-known of these tactic motions is chemotaxis, in which cells move upstream in gradients of beneficial compounds such as nutrients or oxygen. To implement such directed motion, which we assume in one form or other to feature in CTCs migrating through the ECM, an additional energy bias $\text{Δ}{\mathcal{H}}_a$ is incorporated in the change of the Hamiltonian $\text{Δ}\mathcal{H}$. This bias promotes attempts that move the cell along a preferred direction that we shall call the polarization and is given by (25,39,40)

$$\text{Δ}{\mathcal{H}}_a=-\sum _{\sigma =\sigma \left({\mathbf{x}}_i\right),\sigma \left({\mathbf{x}}_j\right)}{\kappa }_{\sigma }\text{Δ}{\mathbf{R}}_{\sigma }\left(\sigma \left({\mathbf{x}}_i\right)\to \sigma \left({\mathbf{x}}_j\right)\right)\cdot {\mathbf{p}}_{\sigma }$$ (2)

Here, p_σ denotes the (unit) polarization vector of cell σ, i.e., the direction in which the cell is currently moving. ΔR_σ(σ(x_i) → σ(x_j)) is the center-of-mass displacement of cell σ that would result if the proposed move were accepted, and κ_σ > 0 measures the relative strength of active motion; this parameter controls the speed of cell σ.

Isolated cells in experiments generally exhibit persistent motion. That is, the direction of motion drifts on some characteristic timescale. Indeed, it has been shown that single-cell motility in 2D can, to first order, be accurately described by a persistent random walk (PRW) (21,22) (although more complex heterogeneous random walks have also been reported (41,42)) and that consequently, its mean-square displacement (MSD) is given by (23,43, 44, 45)

$$\langle {\left(\mathbf{r}\left(t\right)-\mathbf{r}\left(0\right)\right)}^2\rangle =4D_a\tau \left(e^{-t/\tau }+t/\tau -1\right)+4D_{t}t,$$ (3)

where r(t) is the position of a cell at time t. The displacement of the cell is composed of two parts: a purely diffusive part characterized by a passive diffusion coefficient D_t and a persistent contribution quantified by a persistence time τ and an active diffusion coefficient D_a $\equiv v_0^2\tau$/2 (with v₀ the active cell speed). At very short times t ≪ τ, the resultant motion is diffusive (MSD ∝ t) with diffusion coefficient D_t, ballistic (MSD ∝ t²) at intermediate timescales t ≈ τ, and diffusive again with an enhanced diffusion coefficient D_t + D_a at long times t ≫ τ.

Although the PRW accurately describes cell motility in 2D, the correct description in 3D involves an anisotropic persistent random walk model in which two persistent random walks for a primary and nonprimary direction of motion of the environment are combined (22). To make our general point and to facilitate comparison with an analytical model we will present later on in this work, we mostly restrict our simulations to 2D cell (cluster) motion. Nonetheless, we note that an extension to 3D leads to similar results (see Appendix A). Interpreting our polarity vector as the instantaneous direction of motion of our active cell, we impose a PRW by letting p_σ undergo rotational diffusion. This is implemented by expressing it in terms of its polar angle ϕ_σ, p_σ = [cos(ϕ_σ), sin(ϕ_σ)], and having it evolve in time according to a discretized angular Langevin dynamics process (23,43, 44, 45):

$${\varphi }_{\sigma }\left(t+\text{Δ}t\right)={\varphi }_{\sigma }\left(t\right)+\sqrt{\frac{2}{{\tau }_{\sigma }}}\text{Γ}\left(\text{Δ}t\right)$$ (4)

Here, Δt is the time step of the update that we set to 1 MCS, τ_σ is the implemented persistence time of cell σ (given in units of MCS), and Γ(Δt) is a stochastic white noise term with zero mean $\langle \text{Γ}\left(\text{Δ}t\right)\rangle$ = 0 and a variance equal to Δt: $\langle \text{Γ}\left(\text{Δ}t\right)\text{Γ}\left(\text{Δ}t\right)\rangle$ = Δt.

Vicsek alignment

With the update scheme given by Eq. 4, we have incorporated the persistent random walk into the CPM through reorientations of the polarity vector p_σ. This vector represents the currently preferred direction of motion and may be interpreted as an internal polarization of the motile machinery, i.e., the instantaneous polarization direction of cytoskeletal stress fibers, or, in a more pragmatic sense, as the orientation of the leading edge of the cell (25) (even though the direction of movement does not always line up perfectly with either of these two directions). For now, we treat p_σ as a proxy for some internal or external bias direction that guides the motion.

This brings us to a principal feature of this work: the effect of cell-cell alignment. As detailed in the Introduction, it is reasonable to assume that when cells are in contact with each other, they may influence each other’s direction of motion and thereby alter the polarity vector p_σ of nearby cells. Inspired by the capacity for neighbor-induced migration alignment in the context of, for instance, wound healing (26), we let this interaction between cells manifest itself as a tendency to align their respective polarities in parallel fashion. This is implemented numerically using an adaptation of the well-known Vicsek model (28,46), extending the update rule presented in Eq. 4 to

$${\varphi }_{\sigma }\left(t+\text{Δ}t\right)=\text{arg}\left(\gamma {\mathbf{p}}_{\sigma }\left(t\right)+\sum _{{\sigma }^{\prime }}{\mathbf{p}}_{{\sigma }^{\prime }}\left(t\right)\right)+\sqrt{\frac{2}{{\tau }_{\sigma }}}\text{Γ}\left(\text{Δ}t\right),$$ (5)

where γ is a weight factor that controls the degree of alignment, arg(a) denotes the angle of a vector a in polar coordinates, and the sum is taken over all cells σ′ that are in direct contact with cell σ. We shall call two cells in direct contact when there is at least one site with spin σ on the one cell that shares a boundary with the lattice site of a nearby cell σ′ (see Fig. 2). Note that this definition of neighborhood slightly deviates from the original formulation of the Vicsek model, which aligns all particles within an interaction radius. When γ → ∞, the alignment disappears (all direct contacts carry zero relative weight in the update scheme, recovering Eq. 4), whereas for γ = 1, the polarity vector p_σ instantaneously takes on the average direction of itself and its neighbors after each update and we have perfect local alignment. We will call this the fast-aligning regime, and it is also how the alignment in the original Vicsek model is implemented (28,46).

Figure 2.

Visualization of the polarity vector p_σ of a CPM cell σ (colored in yellow) and its direct neighbor polarities p_σ′ with which cell σ aligns according to a Vicsek-type model. Dashed arrows denote the polarity vectors of the cells in the cluster that are not in direct contact with cell σ. To see this figure in color, go online.

Simulation details

Each simulation starts with initiating a model CTC cluster by placing N_cells square cells—domains of equal size and each with a unique spin σ > 0—adjacent to each other on a square lattice with grid size a. The system is then equilibrated by running the CPM simulation for 500 MCS including only the original Hamiltonian (Eq. 1), without the active energy bias (Eq. 2). This is done to allow the cells to relax to a natural, smoothly convex shape. After this equilibration stage, we assign polarity vectors drawn from a uniform distribution of angles to each cell, set the cluster center of mass to R_c(t₀) $\equiv$ 0 (which defines the origin), and start the clock at t₀ = 0. We then run the actual simulation using the Hamiltonian (Eq. 1) including the active energy bias (Eq. 2). We proceed to track the cluster (or single-cell) center of mass R_c(t_n) at fixed time intervals Δt = t_{n + 1} − t_n = 1 MCS to generate the motile trajectory of the cluster (see Fig. 1 for example trajectories).

For now, we will assume all cell parameters to be spatially independent and each cell to be identical. That is, we set κ_σ = κ, τ_σ = τ, V_{σ, 0} = V₀, J_{σ, σ′} = J_cell-cell for σ, σ′ > 0, and J_{σ, σ′} = J_{cell-substrate} for σ $\vee$ σ′ = 0. Additionally, for all 2D simulations in this work, we have fixed the simulation pseudotemperature T = 1, the target area of the cells V₀ = 64 a², the area constraint strength λ = 1, the cell-cell line tension J_cell-cell = 0.5, and the cell-substrate line tension J_{cell-substrate} = 1. The positive difference between J_{cell-substrate} and J_cell-cell implies cells prefer boundaries with other cells over boundaries with the substrate. Furthermore, the fact that both values are individually positive implies that all boundaries experience a positive (contractile) line tension. Thus, effectively, this choice of J-values encodes both cell-cell adhesion and cortical tension. Combined with an active energy bias and cell persistence time of typically κ = 5 and τ = 500 MCS, respectively, these parameters ensure that cells tend to stick together, and by mapping a ∼1 μm and MCS ∼0.001 h, single cells have, consistent with experiment, a typical size of ∼10 μm, a speed of ∼50 μm/h, and a persistence time of ∼1 h (17,18,47).

Finally, to prevent unphysical disintegration of the cell shape as a consequence of strong cell-cell adhesion, we have included an additional shape-regulating contribution to the energy, $\text{Δ}{\mathcal{H}}_r$. This bias term forces the cells to prefer a circular shape, thus penalizing, e.g., fingering-type structures. In principle, this would be taken care of by the positive cortical tension, but for small systems, we find that lattice effects on the shape are non-negligible. The $\text{Δ}{\mathcal{H}}_r$ term may be physically interpreted as a bending rigidity of the cell cortex/perimeter, and we have verified that, despite the fact that it also suppresses cell protrusions at the exterior boundary, its precise value does not influence our main findings (see Appendix B for more details).

Results and discussion

Migration in uniform environments: phenomenology

Single-cell motion

Before proceeding to the effects of cell-cell alignment in clusters, we first validate our implementation of the PRW into the CPM. For this, we study the MSD of isolated cells for different values of τ and κ. To analyze the diffusive process, we plot the instant diffusion coefficients D_in = MSD/4t that follow from the calculated MSDs and fit the results with a PRW (Eq. 3). This is demonstrated in Fig. 3. The accurate fit confirms that indeed, the MSD follows a PRW (although note that because of poorer statistics, small deviations between simulation and theory persist at large times). By plotting the instant diffusion coefficient, we clearly recognize the transition from an initial “slow” diffusive process (constant D_in = D_t) via an intermediate ballistic regime (manifesting as D_in ∼t) to again a diffusive process with an increased diffusion coefficient D_in = D_t + D_a in the long time limit.

Figure 3.

(A and B) Plots of the instant diffusion coefficients D_in = MSD/4t (markers) are given, which have been calculated from 2D single-cell CPM simulations and are fitted with a PRW using Eq. 3 (lines). Results are obtained for (A) different implemented persistence times τ with a fixed value of κ = 5 and (B) different polarity strengths κ with a fixed value of τ = 500 MCSs. (C and D) Persistence time τ_p, active diffusion coefficient D_a, and thermal diffusion coefficient D_t were obtained from the fits shown in (A) and (B), respectively. Red dotted lines denote the predicted or fitted values corresponding to τ_p = τ, D_a ∝ κ²τ (assuming κ ∝ the active speed v₀), and a constant D_t. Data were obtained by time-ensemble averaging over 50 trajectories each consisting of 50,000 MCSs. To see this figure in color, go online.

We further characterize the cell motion from each fitted MSD by extracting the persistence time τ_p (we add the subscript to distinguish the fitted persistence time from the implemented one; we will do this throughout), the active diffusion coefficient D_a (or an average active cell speed v₀), and the “thermal” diffusion coefficient D_t. The resulting values of these parameters are shown as a function of both τ and κ in Fig. 3. These results confirm that the persistence time τ that we implement in the CPM is also the time observed in the simulation, i.e., by τ_p, and that it is independent of the polarity strength κ. Also, as anticipated, the active diffusion coefficient D_a scales linearly with τ and quadratically with κ. Taking into account that for a PRW, D_a ∝ $v_0^2\tau$, we conclude that τ_p = τ and v₀ ∝ κ. This implies that we dial in the persistence time and active speed of individual cells directly with τ and κ. The observed values for D_t remain constant upon changing both τ and κ. This, too, is as expected; the passive motion originates from the pseudothermal fluctuations in cell area and shape affected by the parameter T. This intrinsic randomness is completely independent of all other parameters.

Thus, consistent with earlier work in, e.g., (25) and (39), we have demonstrated that an active PRW can effectively be mapped onto the CPM. In a broader context than cell motility, we note that this implementation also provides a good framework for the study of a larger range of active (soft) materials and is not necessarily limited to a description of biological cells.

Aligned cell cluster motion

We now turn to the collective motion of CTC clusters, focusing specifically on the roles of cell-cell alignment and cluster size. Let us first consider the case of fast alignment (γ = 1) for a cluster of N_cells identical cells. The results for D_in, extracted from the corresponding MSDs, are shown in Fig. 4. The MSDs are well defined, indicating sufficiently large sample sizes, and are still accurately fitted with a PRW; we conclude that an aligning cluster, too, moves according to a PRW. The question, then, is how its parameters depend on cluster size and alignment.

Figure 4.

(A) Plots of the instant diffusion coefficient D_in = MSD/4t for fast-aligning (γ = 1) 2D cell clusters consisting of a variable amount of N_cells identical cells (markers) are given. The results have been fitted with a PRW using Eq. 3 (lines). Inset: average cell (cluster) velocity v₀ as a function of N_cells including a power-law fit, i.e., v₀ ∝ $N_{\text{cells}}^{0.1}$. (B) Cluster persistence time τ_p, active diffusion coefficient D_a, and thermal diffusion coefficient D_t were obtained from the fits shown in (A). Red dotted lines show a power-law fit for D_t and a comparison and fit of τ_p and D_a, respectively, to the derived results of the fast-aligning active Brownian particle theory, i.e., Eq. 13. Simulation parameters used were κ = 5 and τ = 500 MCSs. Data were obtained by time-ensemble averaging over 30 trajectories, each consisting of 50,000 MCSs. To see this figure in color, go online.

The resulting fit parameters, plotted as a function of N_cells in Fig. 4, allow us to extract these dependencies. We observe that the “thermal” diffusion coefficient decreases with the number of cells. This can be attributed to the increased size of the cluster, which results in weaker relative fluctuations in shape and size. A power-law fit yields D_t ∼$N_{\text{cells}}^{-0.8}$, showing that this decrease is roughly linear with the number of cells. However, note that these fits (and the subsequent ones) are only made over one decade in N_cells, and so the range of validity of the fitted powers is limited.

The persistence time of the cluster, on the other hand, is seen to increase linearly with the number of cells: τ_p ∼N_cellsτ. An intuitive explanation for this may be found in the fact that when the cells are strongly aligning (sufficiently small γ), (almost) all cells must simultaneously reorient toward the same direction to permit the entire cluster to change its course. This suggests that larger clusters of cells prone to alignment generally continue to move along the same direction for longer times, which corresponds to an increasing persistence time.

Similar to the persistence time, the active diffusion coefficient D_a also increases linearly with N_cells. We may understand this scaling by noting that for a PRW, the active diffusion coefficient is expected to scale linearly with the cluster persistence time D_a ∝ τ_p and thus, by extension, also with N_cells. This holds exactly when the cluster speed is independent of the cluster size and remains the same as that of a single isolated cell. We do see, however, that cell clusters actually have a slightly larger active speed than single cells, which we believe (even though a proper verification would require a continuous monitoring of the active forces) is a result of cells within the cluster actively pushing and pulling each other along the direction in which all of them, because of the fast alignment, want to migrate. This is demonstrated in the inset of Fig. 4, which shows the average absolute velocity v₀ as function of N_cells. However, a power-law fit yields v₀ ∝ $N_{\text{cells}}^{0.1}$; hence, there is only a weak dependence of the cell cluster speed on the number of cells that does not strongly influence the persistent motion of the cluster.

Interestingly, comparable results are obtained for 3D simulations (see Appendix A). This suggests that the effect of the Vicsek alignment of cell polarities on the cell cluster motion is the same in 3D as it is in 2D. In particular, it shows that within a more extended 3D model setup, the effect of alignment still results in a linear dependence of the persistence time on the number of cells in a cluster.

Up until this point, we have imposed fast neighbor alignment within the cluster by setting γ = 1. To assess the influence of the relative weight of the neighboring polarizations, we have calculated MSDs for homogeneous four-cell clusters experiencing weaker alignment (by increasing the value of γ). The results for D_in, extracted from the corresponding MSDs, are shown in Fig. 5 and, despite the weaker alignment, can still be accurately fitted with a PRW. The resulting fit parameters, plotted as a function of γ, are shown in Fig. 5. We notice that the “thermal” diffusion coefficient remains constant over the entire range, which is consistent with the fact that γ only influences the active motion of the cells. Interestingly, upon initial increase of γ (indicating more moderate degrees of alignment), the active diffusion coefficient and cluster persistence time also retain a constant value. Thus, within this regime of alignment, highly cooperative migration in the cluster is still produced. There is a finite bound on this effect, however—by increasing γ further (approximately beyond γ ∼τ), the system enters a regime in which the alignment is not sufficiently strong anymore. As a consequence, both parameters rapidly drop in value, and the cluster may even disintegrate, which is easily checked by counting the number of domains with σ > 0 on the lattice. Note that we have ensured that in the trajectories used for the MSDs, the cluster remained intact. In particular, the cluster persistence time decreases to the same value as that of a single cell, and the active diffusion coefficient becomes even smaller than its single-cell counterpart. Overall, this implies a complete loss of cooperative motion in the large γ limit and suggests that there exists a critical degree of alignment γ_c beyond which alignment is not sufficiently strong to have all polarities point in the same direction for all times. Physically, this marks the point at which cells within the cluster often want to travel in different directions (opposite polarity vectors) for increasingly long times. This allows them to slow down the cluster, decrease its persistence, or even actively pull themselves loose from the other adjacent cells, resulting in cluster disintegration.

Figure 5.

(A) Plots of the instant diffusion coefficient D_in = MSD/4t for 2D homogeneous four-cell clusters with different degrees of alignment γ (markers) are given. The results have been fitted with a PRW using Eq. 3 (lines). (B) Cluster persistence time τ_p and (C) active and thermal diffusion coefficients D_a and D_t, respectively, were obtained from the fits shown in (A). Red and blue dotted lines show the theoretically expected value for τ_p based on the derived results of the fast-aligning active Brownian particle theory, i.e., Eq. 13, and the implemented single-cell persistence time τ. Simulation parameters used were κ = 5 and τ = 500 MCSs. Data were obtained by time-ensemble averaging over 30 trajectories, each consisting of 30,000 MCSs. In this analysis, we have ensured that the cluster remained intact; trajectories in which the cluster fell apart, which for the largest γ happened in roughly half of the cases, have been discarded. To see this figure in color, go online.

Thus, we have demonstrated that fast alignment of the cells in the CPM will increase the persistence of the cluster, allowing it to move more directionally. This happens at the cost of a decrease in the translational diffusion coefficient. In the case we consider and that we assume to most closely represent actual cellular behavior, the overall motion is dominated by its active contribution (D_a ≫ D_t). As a result, the decrease in the “thermal” diffusion coefficient will hardly influence the overall motion. This implies that aligned clusters can, on average, cover more distance than single cells within a given timeframe, provided that v₀ is sufficiently large. In the context of CTC clusters, this suggests that when clusters experience an externally imposed polarity (through, e.g., tracks or anisotropy in the ECM), they are generally better able to follow such tracks collectively compared with single cells, enhancing the effects of contact guidance and possibly allowing them to reach targets such as blood vessels more easily.

Finally, we consider the effect of fast alignment for a heterogeneous cluster. In particular, we investigate how one less persistent cell influences the motion of an otherwise more persistent cluster. Indeed, individual cells typically show a variety of persistence times in experiments (22), and thus, CTC clusters will consist of a heterogeneous mixture of cells. To study this effect, we simulate the motion of a fast-aligning cluster consisting of N_cells = 4 cells, three of which have a “large” persistence time (denoted τ_large) of 1300 MCSs and one having a variable “small” persistence time (denoted τ_small). As before, we retrieve the cluster persistence time from fitting the collective MSD. The resulting values are plotted as a function of τ_small in Fig. 6. It shows that the collective benefit of alignment can become much smaller or even nonexistent (τ_p < τ_large) by adding a single cell with a small persistence to an existing cell cluster. We can understand this by realizing that a small persistence time corresponds to a rapid reorientation of the cell’s polarity. If the reorientation becomes too fast, the cell will drag along other cells toward this polarity as well, which results in a decrease of the cluster persistence. This demonstrates that to exhibit the additional directionality of aligning cluster motion, the spread in individual persistence times should not be too large.

Figure 6.

Plot of the cluster persistence time τ_p as a function of τ_small for a cluster consisting of one cell with an implemented “small” persistence time τ_small and three cells with a “large” persistence time τ_large. Circles denote the results obtained from PRW fits of the MSD, which in turn has been calculated from fast-aligning CPM cell cluster trajectories. Red dotted line represents the theoretical prediction of our fast-aligning active Brownian particle theory, i.e., Eq. 18. Simulation parameters used were N_cells = 4, γ = 1, κ = 5, and τ_large = 1300 MCSs. Data for the MSDs were obtained by time-ensemble averaging over 60 trajectories, each consisting of 50,000 MCSs. To see this figure in color, go online.

Active Brownian motion

Identical particles

To provide a more general framework for our results, we now seek to rationalize the observed benefits of cell-cell alignment in our CPM simulations using so-called active matter theory. One of the most widely used models in this field (23,43,44,48) is the ABP model. Such particles undergo Brownian motion with a “thermal” diffusion coefficient D_t while they simultaneously self-propel with an absolute speed v₀ along their orientation axis, called the director e_i(t). The index i here labels each of the N individual ABPs that together form a cluster in our theory. The evolution in time t of the position r_i(t) = [x_i(t), y_i(t)] of each particle i is captured by the stochastic differential Langevin equation (23,43, 44, 45,48,49)

$$\frac{d{\mathbf{r}}_i\left(t\right)}{dt}=v_0{\mathbf{e}}_i\left(t\right)+\sqrt{2D_t}{\mathit{\xi }}_i\left(t\right)$$ (6)

Here, ξ_i = $\left[{\xi }_{x_i},{\xi }_{y_i}\right]$ where ξ_α (α = x_i, y_i) represents an independent white noise stochastic process with zero mean, $\langle {\xi }_{\alpha }\left(t\right)\rangle$ = 0, and δ correlations $\langle {\xi }_{\alpha }\left(t^{\prime }\right){\xi }_{\beta }\left(t\right)\rangle$ = δ(t′ − t)δ_{α, β}.

Similar to the polarity vector of each CPM cell p_σ, the director of each ABP is parameterized by the polar angle ϕ_i(t) ∈ [0, 2π), i.e., e_i(t) = [cosϕ_i(t), sinϕ_i(t)]. In our aligning ABP model, we assume that ϕ_i evolves in time not only according to a stochastic rotational diffusion process but also because of a potential U that encodes velocity alignment,

$$\frac{d{\varphi }_i\left(t\right)}{dt}=-\eta \frac{\partial U}{\partial {\varphi }_i}+\sqrt{\frac{2}{\tau }}{\xi }_{{\varphi }_i}\left(t\right),$$ (7)

with η > 0 denoting a relaxation rate that controls how fast the alignment takes place, τ the single particle persistence time, and ${\xi }_{{\varphi }_i}$ another independent white noise stochastic process. For the aligning potential U, we write

$$U\left(\left\{{\mathbf{r}}_i\right\},\left\{{\varphi }_i\right\}\right)=-\sum _{|{\mathbf{r}}_i-{\mathbf{r}}_j|<r_c}{\mu }_{ij}\text{cos}\left({\varphi }_i-{\varphi }_j\right),$$ (8)

where {r_i}, {ϕ_i} denote the set of all N positions and angles, respectively; μ_ij > 0 is a coupling constant that, for identical particles, simplifies to μ_ij = μ; and the sum is taken over all particle combinations i, j that are within one interaction distance r_c from each other. Note that this potential has a minimum when both particles point in the same direction (ϕ_i = ϕ_j), whereas it exhibits a maximum when particles are pointing in the opposite direction (ϕ_i = ϕ_j + π). Additionally, it has been demonstrated that in the limit of fast angular relaxation, the introduced continuum description of angular alignment given by Eqs. 7 and 8 is equivalent to the 2D Vicsek model, i.e., Eq. 5 (50,51). This allows us to draw a direct comparison between the CPM simulations and the obtained theoretical results for aligning ABPs.

We note that when the alignment between ABPs disappears, i.e., η = 0, Eqs. 6 and 7 represent a system of noninteracting ABPs. In this case, the particles will simply follow a PRW, and their individual MSDs are given by Eq. 3 (23,43, 44, 45).

To describe the motion of the cluster of particles as a whole, we may focus on the center of mass, i.e., R = $\left(1/N\right){\sum }_{i=1}^N{\mathbf{r}}_i$, which obeys the following stochastic differential equation

$$\frac{d\mathbf{R}\left(t\right)}{dt}=\frac{1}{N}\sum _{i=1}^N\left(v_0{\mathbf{e}}_i\left(t\right)+\sqrt{2D_t}{\mathit{\xi }}_i\left(t\right)\right)$$ (9)

To impose fast angular relaxation, we assume that all the particles within the cluster align rapidly with each other (μη ≫ τ⁻¹); hence, the difference between each pair of angles will remain small, i.e., |ϕ_i − ϕ_j| ≪ 1 for all i, j. Because the particles travel with equal speeds v₀, this implies that they remain close together, and we expect the cluster of particles (like a CTC cluster) to travel as a whole. Moreover, it allows us to simplify the sums in Eq. 9. Because all involved angles are almost equal, the directors e_i will point in roughly the same direction. We can therefore approximate the sum of the N directors by N vectors that all point in the average direction of the particles. In other words, we can replace ${\sum }_{i=1}^N$e_i → Ne_cm in Eq. 9. where e_cm = [cos(ϕ_cm), sin(ϕ_cm)] and ϕ_cm $\equiv \frac{1}{N}{\sum }_{i=1}^N{\varphi }_i$. A visualization of this approximation for N = 2 is shown in Fig. 7.

Figure 7.

Visualization of the approximation for N = 2 particles in which we replace the sum of both particles’ directors e_{1, 2} = [cos(ϕ_{1, 2}), sin(ϕ_{1, 2})] by two times the director with the average angle of both. This director is denoted e_cm = [cos(ϕ_cm), sin(ϕ_cm)] with ϕ_cm = (ϕ₁ + ϕ₂)/2. We have used (ϕ₂ − ϕ₁) = π/9; it can be seen that the approximation is still reasonably accurate. To see this figure in color, go online.

Additionally, because the zero mean stochastic processes ξ_α are independent and δ correlated, we can, in an exact manner, replace a sum of these variables by a single one via ${\sum }_{i=1}^N{\xi }_{{\alpha }_i}\to \sqrt{N}{\xi }_{{\alpha }_{cm}}$. The factor $\sqrt{N}$ is added to ensure that the correlation remains consistent, i.e., $\langle {\sum }_{i=1}^N{\xi }_{{\alpha }_i}\left(t^{\prime }\right){\sum }_{i=1}^N{\xi }_{{\alpha }_i}\left(t\right)\rangle =N\langle {\xi }_{{\alpha }_{cm}}\left(t^{\prime }\right){\xi }_{{\alpha }_{cm}}\left(t\right)\rangle$. Overall, this allows us to simplify Eq. 9 as

$$\frac{d\mathbf{R}\left(t\right)}{dt}=v_0{\mathbf{e}}_{cm}\left(t\right)+\sqrt{\frac{2D_t}{N}}{\mathit{\xi }}_{cm}\left(t\right),$$ (10)

where ξ_cm = $\left[{\xi }_{x_{cm}},{\xi }_{y_{cm}}\right]$ represents a new vector of independent stochastic processes with zero mean and δ correlations.

The time evolution of the average direction of the particles (and thus of the cluster) can be formulated using Eq. 7, i.e.,

$$\frac{d{\varphi }_{cm}\left(t\right)}{dt}=\frac{1}{N}\sqrt{\frac{2}{\tau }}\sum _{i=1}^N{\xi }_{{\varphi }_i}\left(t\right),$$ (11)

where, because of symmetry, all alignment terms exactly cancel against each other. As already mentioned, we can replace a sum of the stochastic noise terms by a single one. Introducing a new stochastic process ${\xi }_{{\varphi }_{cm}}$ with zero mean and δ correlations, we arrive at

$$\frac{d{\varphi }_{cm}\left(t\right)}{dt}=\sqrt{\frac{2}{N\tau }}{\xi }_{{\varphi }_{cm}}\left(t\right)$$ (12)

Interestingly, these resulting equations that govern the motion of the center of mass (and thus of the entire cluster of N particles) (Eqs. 10 and 12), are identical in form to the equations that describe a single noninteracting ABP, i.e., Eqs. 6 and 7 with η = 0. The only difference lies in the fact that the center-of-mass persistence time has increased in proportion to the number of particles τ → Nτ, whereas the “thermal” diffusion coefficient has decreased as D_t → D_t/N. Summarizing, we conclude that the center-of-mass motion of a cluster of N fast-aligning ABPs follows a PRW (Eq. 3) that is characterized by a cluster persistence time τ^cm, thermal diffusion coefficient $D_t^{cm}$, and active diffusion coefficient $D_a^{cm}$ given by

$${\tau }^{cm}=N\tau ,D_t^{cm}=D_t/N,D_a^{cm}=v_0^{2}N\tau /2$$ (13)

Relating our ABPs to the CPM cells by interpreting N as N_cells, we see that these theoretical results are in good agreement with the ones from our CPM simulations (see Fig. 4) and thus provide a theoretical underpinning for the increased cluster persistence due to cell (particle) alignment observed earlier. The prediction of the “thermal” diffusion coefficient scaling as N⁻¹ deviates slightly from the observed power of −0.8 seen in the CPM simulation. This was to be expected; the passive motion exhibited by a CPM cell is considerably more complex than that of a point particle, and in that light, it is rather striking that the single, passive diffusive process in the theory so closely resembles the CPM results.

Our analytical argument is not limited to the specific potential we have chosen in Eq. 8. In addition to the requirement that the particles must align sufficiently quickly (which most likely becomes more difficult for larger N (52)), we only require that all alignment contributions to the time evolution of ϕ_cm will cancel out. In other words, for all potentials that satisfy

$$\sum _{i=1}^N\frac{\partial U}{\partial {\varphi }_i}=0,$$ (14)

our argument and the results should be valid. Moreover, one could add a symmetric (adhesion-)interaction force between the particles and retrieve the same results, provided all these interactions exactly cancel with respect to the center of mass.

Nonidentical particles

To incorporate cluster heterogeneity, i.e., to account for the fact that single-cell properties within a CTC cluster are generally not the same (22), we can extend our theory analysis to a set of N quickly aligning nonidentical ABPs in several ways. One way would be to let all particles travel at different speeds v₀ → v_{0, i}; one could also make the degree of alignment explicitly particle dependent by substituting η → η_i and μ → μ_ij (with μ_ij = μ_ji) in Eqs. 7 and 8, respectively. However, we refrain from exploring these options in too much detail for several reasons. Firstly, letting the particles travel at different speeds will eventually lead to particle separations that are larger than the interaction range of the particles. This scenario is obviously incorrect for CTC clusters, in which the cells stick together and travel at approximately equal velocities, and furthermore would invalidate the assumption of fast alignment between particles. Secondly, introducing μ_ij will not change the equations governing the center-of-mass motion; all alignment terms are still canceling out, i.e., Eq. 14 still applies. Conversely, by introducing a particle-dependent relaxation constant η_i (e.g., to account for different cell sizes with different friction constants), Eq. 14 will not be valid anymore. In particular, the time evolution of ϕ_cm will then also contain terms that are proportional to (η_i − η_j)sin(ϕ_i − ϕ_j). Nonetheless, for strong enough alignment, |ϕ_i − ϕ_j| will remain sufficiently small, allowing us to neglect these terms and recover the same results as for identical particles. In other words, a larger variety in relaxation constants η_i will result in a narrower fast-alignment regime for the cluster, but within this regime, it does not qualitatively change its motion.

The most relevant unexplored option for introducing heterogeneity in our aligning ABP model is therefore the scenario we have also studied numerically: to have each particle move with a different persistence time, i.e., to replace τ → τ_i in Eq. 7. This implies that instead of Eq. 12, we have

$$\frac{d{\varphi }_{cm}\left(t\right)}{dt}={\left(\frac{2\left({\sum }_{i=1}^N\frac{1}{{\tau }_i}\right)}{N^2}\right)}^{1/2}{\xi }_{{\varphi }_{cm}}\left(t\right)\equiv \sqrt{\frac{2}{{\tau }^{cm}}}{\xi }_{{\varphi }_{cm}}\left(t\right),$$ (15)

where the cluster persistence time is now given by

$${\tau }^{cm}=N^2{\left(\sum _{i=1}^N\frac{1}{{\tau }_i}\right)}^{-1}$$ (16)

Note that again, we have replaced the sum of stochastic terms by a single one, but because of the particle-dependent τ_i, this is less straightforward, i.e., ${\sum }_{i=1}^N\sqrt{\frac{1}{{\tau }_i}}{\xi }_{{\alpha }_i}\to {\left({\sum }_{i=1}^N\frac{1}{{\tau }_i}\right)}^{1/2}{\xi }_{{\alpha }_{cm}}$. When all persistence times are equal (τ_i = τ), we recover the linear increase of the persistence time with the number of particles (τ^cm = Nτ).

It is now interesting to see how the behavior of a cluster with a distribution of persistence times compares with the case in which all particles are identical. In fact, from Eq. 16 it can be (straightforwardly) derived (see (53) for details) that for each set of N persistence times {τ_i}, we have

$${\tau }^{cm}\le N\langle \tau \rangle ,$$ (17)

with $\langle \tau \rangle =\frac{1}{N}{\sum }_{i=1}^N{\tau }_i$ the average persistence time of the set and the equal sign corresponding to a constant τ_i = τ. This shows that broadening the distribution of individual persistence times of particles (keeping a constant average) will always decrease the center-of-mass persistence time τ^cm. In other words, alignment is always less effective in terms of increasing τ^cm when individual particles travel with a different persistence.

Finally, let us verify one last numerical observation: that even a single less persistent particle among the aligning particles can cancel out the benefits of alignment of the total cluster. Suppose we have N fast-aligning particles, N − 1 of which have an individual persistence time τ_large, and one particle has a smaller persistence time τ_small < τ_large. The persistence time of the center-of-mass motion is then given by

$${\tau }^{cm}=\frac{N^2{\tau }_{\text{small}}{\tau }_{\text{large}}}{\left(N-1\right){\tau }_{\text{small}}+{\tau }_{\text{large}}},$$ (18)

which agrees very well with our CPM simulation results (see Fig. 6) and confirms that one particle can substantially decrease the mobility benefits of alignment. Particularly, the effect of collective alignment will be canceled when τ^cm = τ_large. In that case, we find

$${\tau }_{\text{small}}=\frac{{\tau }_{\text{large}}}{N^2-N+1}$$ (19)

This value for τ_small thus presents a critical value, below which the cluster moves with less persistence than the individual particles. Note that for large N, we have τ_small ∼$\left({\tau }_{\text{large}}/N^2\right)$ → 0, and a single particle is not able to disturb the collective motion of a large cluster. However, because N is typically not large for CTC clusters, this effect is not negligible, and the inclusion of a rapidly reorienting cell to a CTC cluster can, at least in principle, strongly suppress its directional movement.

Durotaxis

In the discussion above, we have demonstrated numerically and explained theoretically how mutual velocity alignment can increase the collective persistence of the motion of cell clusters. So far, however, we have assumed the environment of the cell clusters (typically, the ECM) to be homogeneous, and we have included its interactions with the cells only via constant values of the persistence time (τ_σ), the active speed (κ_σ), and the adhesion coefficient (J_{σ, 0}). To establish proof of principle for the effects of alignment in an inhomogeneous environment, we also extend the CPM simulation setup to include durotaxis (migration in a stiffness gradient), a behavior that may be closely linked to cell persistence. Experimental results suggest that cells on substrates with higher stiffnesses tend to exhibit greater persistence (longer persistence times) (19,20), and simulations of persistently moving point particles have shown that a gradient in persistence time in itself is sufficient to generate durotactic motion (18). Moreover, experiments on a larger length scale have indicated that in multicellular settings, entire cell monolayers may exhibit much stronger durotaxis compared with their isolated constituents under the same circumstances, an observation that has, in part, been attributed to the fact that a larger cellular collective experiences a larger stiffness differential between its leading and trailing edge (17). This suggests that enhanced durotaxis might also occur in a smaller aggregate like the typical CTC cluster.

Motivated by these experiments and following the approach in (18), we model durotaxis by implementing a position-dependent single-cell persistence time τ_σ = τ(x), which, for convenience, is the same for all simulated cells and only depends on their center-of-mass position along the x axis. We then let τ(x) increase linearly from a minimal value τ_min to a maximal value τ_max over a region x ∈ [−w, w] around the origin, with the cluster (or single-cell) center of mass always starting in the middle of the gradient. Beyond the gradient region, the parameters remain constant, such that τ(x) = τ_min for x ≤ w and τ(x) = τ_max for x ≥ w (see inset, Fig. 8 A). This means that the width w effectively controls the steepness of the gradient dτ/dx. The stiffness gradient is always along the positive x direction, and, as in most experimental setups, the gradient only occupies part of the system (18), connecting two regions of approximately constant stiffness or persistence time.

Figure 8.

Plots of (A) the average displacement in the x direction and (B and C) the x component of the durotactic vector index as a function of time for a single cell (A and B) and fast-aligning different-sized clusters (C) experiencing a linear gradient in persistence time from τ_min = 200 MCSs (∼0.2 h) to τ_max = 2000 MCSs (∼2.0 h). Results for (A) and (B) correspond to different gradients and (C) to a fixed gradient of dτ/dx = 20 MCSs/a. Gradients are controlled by the width w of the gradient region (see inset). Simulation parameters used were γ = 1 and κ = 5. Averages taken over 10⁴ trajectories. To see this figure in color, go online.

To first test whether our implementation of durotactic motion is consistent with earlier simulation work on point particles (18), we have studied single-cell CPM simulations for different gradients dτ/dx between τ_min = 200 MCS (∼0.2 h) and τ_max = 2000 MCS (∼2.0 h). Fig. 8, A and B show the calculated x components of the average cell (cluster) displacement $\langle x\left(t\right)\rangle$ and the durotactic vector index DI_x(t) $\equiv \langle x\left(t\right)\rangle$/v₀t, respectively. The latter provides the fraction of the average drift velocity of the cell (cluster) along the gradient relative to its absolute speed v₀ and allows us to quantify the drift up the stiffness gradient (18,54). Note that y components are not reported because there is no gradient along this axis and thus no drift. The results are consistent with the fact that a gradient in persistence time suffices to produce a single-cell flux toward the stiff region of the domain (positive $\langle x\left(t\right)\rangle$), leading to a form of durotaxis. We also observe an increase (on the investigated timescale) of the drift velocity for increasing gradients (larger values of DI_x(t)). Additionally, note that DI_x(t) peaks and afterwards seems to decrease in the long time limit, which is a result of cells leaving the gradient region. A mapping of the retrieved results to the ones obtained for point particles presented in (18) shows that they are also quantitatively the same, making our work fully consistent with literature and extending the point-particle results to cells with a finite area in the CPM.

Having confirmed our implementation of single-cell durotaxis, we now proceed by placing different-sized aligned clusters in a fixed gradient dτ/dx = 20 MCSs/a between the extremes τ_min and τ_max (that is, identical environments but different cluster sizes). As one can see in Fig. 8 C, larger clusters indeed show stronger durotaxis, in the sense that the maximal durotactic index is larger for clusters consisting of more cells. We also see that DI_x(t) takes longer to peak for larger clusters. Comparing cluster behavior with that of single cells, the enhanced durotaxis can be attributed to the enhanced persistence of clusters (due to fast alignment) and possibly to the fact that the cluster—simply because it is larger—spans a wider gradient region and thus experiences a larger persistence differential between its front and rear end. As a result, durotaxis of a strongly aligning cluster may also effectively be treated as that of a single, fixed-size particle that moves in an increasingly steep gradient as N increases.

A prediction that follows from this observation is that increasing the distance between the leading and trailing edge of our model cell clusters, keeping the persistence gradient the same, will generally enhance collective durotaxis. This is indeed what is seen in the experiments reported in (17) and suggests that even in the absence of long-range force transmission, any cluster (with sufficient cell-cell alignment and adhesion to promote persistence and maintain cluster integrity respectively) will show enhanced durotactic efficiency as they navigate stiffness gradients.

Conclusions

We have studied and compared the motility of single cells and small cell clusters in the context of tumor cell clusters using a combination of cellular Potts modeling and analytical active matter theory. CTC clusters have been suggested to pose a far more serious threat than single cells in terms of metastatic potential, and although metastasis is far more involved than motility alone, differences between the ways that small clusters and single cells navigate their environment might play an important part in this striking difference. Our primary aim has been to gain more insight into how the motile behavior of small clusters is affected by their size, and in particular, we have focused on whether the effect of cell-cell alignment provides a possible mechanism for enhanced directional motion of clusters. To primarily focus on the latter effect, several physiological aspects such as the complex environment encountered by cancer cells or other forms of cell-cell interactions (besides adhesion and alignment) have not been included in the model.

We have first carried out CPM simulations to study cell motion in a homogeneous environment. Our single-cell simulations show excellent agreement (evidenced by their MSD) with the persistent random walk and with experimental results. Moreover, the emergent cell speed and manifested persistence time are directly controlled by model parameters. Extending the simulations to small cell clusters, we have examined the effect of cell-cell alignment by adding a Vicsek-like term to the CPM. Our results demonstrate that alignment enhances the persistence time of cell clusters; the enhancement scales linearly with the number of cells. This allows the cluster to cover more distance than a single cell, which may play some part in its potential to invade the extracellular matrix in the early stages of the metastatic cascade. Within our CPM description, however, this advantage can be suppressed, partly or completely, by adding only one rapidly reorienting cell to the cluster. In addition, we have found that reducing the strength of alignment beyond a critical point results in a rapid decrease of the cluster persistence and speed and can even cause a disintegration of the cluster.

To explain the CPM results, we have proposed a theoretical model in which CTCs are represented by a cluster of fast-aligning ABPs. Our analysis reveals that fast velocity alignment increases the persistence time of the ABP cluster, yielding, consistent with the CPM simulations, a linear scaling with the number of particles. The added effect of alignment on the overall cluster mobility is strongest for identical particles and is also counteracted by adding a rapidly reorienting particle to the cluster.

As a first attempt, to our knowledge, to investigate the consequences of cell-cell alignment in a more biologically relevant, inhomogeneous environment, we have investigated durotaxis, i.e., the migration up a stiffness gradient. Within our CPM simulations, we have implemented such a stiffness gradient as a linear gradient in the persistence time. In this scenario, we have shown that in a fixed gradient, there indeed exists a durotactic benefit for larger clusters, which may be attributed to the overall persistence differential between the leading and the trailing edge of the cluster and the enhanced cluster persistence due to cell-cell alignment.

Overall, we have shown that in the presence of generic velocity alignment, single-cell and cluster migration can be significantly different. In particular, enhanced directional migration is exhibited by larger clusters when alignment is fast compared to a typical persistence time in the system. Because these persistence times for living cells are generally on the order of several hours, the condition of rapid alignment may be quite broadly met.

Our results offer specific predictions (keeping in mind that our model system represents only a simplified version of real cell clusters and the complex environment they encounter) for the scaling of both the persistence time as well as the random motion of clusters of cells as a function of cluster size. These results fill in a previously uncharted regime between single-cell behavior and large-scale collective motility in confluent cell sheets, a physiologically very relevant regime for which our predictions should be directly observable in experiments.

Author contributions

L.M.C.J. and C.S. conceived the project. V.E.D., L.M.C.J., and C.S. designed the research. V.E.D. developed the theory and numerical codes and performed all calculations. All authors discussed the results. V.E.D. wrote the manuscript with input from all authors.

Editor: Lisa Manning.

Appendix A: Aligned Cell Cluster Motion In 3d

To extend the CPM to a 3D system of fast-aligning cells, we require new updating rules for the polarity vector p_σ, which is now described by the spherical angles ϕ_σ(t) ∈ [0, 2π) and θ_σ(t) ∈ [0, π): p_σ = [sin(θ_σ)cos(ϕ_σ), sin(θ_σ)cos(ϕ_σ), cos(θ_σ)]. Discretizing the angular Langevin dynamics for a 3D active Brownian particle, we obtain (49)

$$\begin{array}{l}{\theta }_{\sigma }\left(t+\text{Δ}t\right)={\theta }_{\sigma }\left(t\right)+\sqrt{\frac{1}{{\tau }_{\sigma }}}\text{Γ}\left(\text{Δ}t\right)+\frac{\text{Δ}t}{2{\tau }_{\sigma }\text{tan}\left({\theta }_{\sigma }\right)},\\ {\phi }_{\sigma }\left(t+\text{Δ}t\right)={\phi }_{\sigma }\left(t\right)+\sqrt{\frac{1}{{\tau }_{\sigma }}}\frac{1}{\text{sin}\left({\theta }_{\sigma }\right)}\text{Γ}\left(\text{Δ}t\right),\end{array}$$ (20)

which in turn, as a result of Vicsek alignment, are extended to

$$\begin{array}{c}{\theta }_{\sigma }\left(t+\text{Δ}t\right)={\text{arg}}_{\theta }\left(\gamma {\mathbf{p}}_{\sigma }\left(t\right)+\sum _{{\sigma }^{\prime }}{\mathbf{p}}_{{\sigma }^{\prime }}\left(t\right)\right)+\sqrt{\frac{1}{{\tau }_{\sigma }}}\text{Γ}\left(\text{Δ}t\right)+\frac{\text{Δ}t}{2{\tau }_{\sigma }\text{tan}\left({\theta }_{\sigma }\right)},\\ {\varphi }_{\sigma }\left(t+\text{Δ}t\right)={\text{arg}}_{\varphi }\left(\gamma {\mathbf{p}}_{\sigma }\left(t\right)+\sum _{{\sigma }^{\prime }}{\mathbf{p}}_{{\sigma }^{\prime }}\left(t\right)\right)+\sqrt{\frac{1}{{\tau }_{\sigma }}}\frac{1}{\text{sin}\left({\theta }_{\sigma }\right)}\text{Γ}\left(\text{Δ}t\right).\end{array}$$ (21)

Here, arg_θ(a) and arg_ϕ(a) denote the spherical coordinates θ and ϕ of a vector a, respectively.

Using these updated rules, we have calculated the MSDs for 3D fast-aligning (γ = 1) clusters consisting of different numbers of N_cells identical cells. Realizing that the MSD of 2D and 3D ABPs are identical up to a change 4D_t → 6D_t, we have again fitted the results to a PRW. Plots for D_in = MSD/6t including these fits and the respective fit parameters (τ_p, D_a, and D_t) are shown in Fig. 9. Comparing these with the 2D results (Fig. 4), we see almost the same behavior, i.e., D_a increasing linearly with N_cells, D_t decreasing (almost) linearly with N_cells, and τ_p = N_cellsτ. This shows that the effect of our Vicsek alignment on the dynamics is qualitatively the same for 2D and 3D CPM cell clusters.

Figure 9.

(A) Plots of the instant diffusion coefficient D_in = MSD/6t for fast-aligning (γ = 1) 3D cell clusters consisting of a variable amount of N_cells identical cells (markers) are given. The results have been fitted with a PRW (lines). (B) Cluster persistence time τ_p, active diffusion coefficient D_a, and thermal diffusion coefficient D_t were obtained from the fits shown in (A). Red dotted lines show a power-law fit yielding D_t ∝ $N_{\text{cells}}^{-0.9}$ and a comparison τ_p = N_cellsτ and fit D_a ∝ N_cellsτ based on the derived results of the 2D fast-aligning active Brownian particle theory, i.e., Eq. 13. Simulation parameters used were κ = 30 and τ = 500 MCSs. Data were obtained by time-ensemble averaging over 20 trajectories, each consisting of 50,000 MCSs. To see this figure in color, go online.

Appendix B: Circular Shape Constraint

As shown in the Hamiltonian $\mathcal{H}$, i.e., Eq. 1, we model cell-cell attachment via the adhesion coefficient J_{σ, σ′}. In particular, by setting the adhesion coefficient between cells J_cell-cell to a sufficiently small value relative to the one between cells and the medium J_cell-medium, it becomes energetically more favorable for cells to form a surface with other cells instead of with the medium. However, when this difference becomes too large or the cell-cell adhesion becomes negative, the cells will be able to easily create interfacial area with the other cells, which can lead to a disintegration of the cell shape. This is clearly unphysical behavior. To prevent it from happening, we impose a shape constraint on the cells that forces the cells to have a circular or spherical shape. We can interpret the constraint as a bending rigidity of the cells and formulate it in the form of an energy bias given by (55)

$$\text{Δ}{\mathcal{H}}_r={\lambda }_r\left(r_{\sigma \left({\mathbf{x}}_i\right)}-\left|{\mathbf{x}}_i-{\mathbf{R}}_{\sigma \left({\mathbf{x}}_i\right)}\right|\right)\left(1-{\delta }_{\sigma \left({\mathbf{x}}_i\right),0}\right)-{\lambda }_r\left(r_{\sigma \left({\mathbf{x}}_j\right)}-\left|{\mathbf{x}}_i-{\mathbf{R}}_{\sigma \left({\mathbf{x}}_j\right)}\right|\right)\left(1-{\delta }_{\sigma \left({\mathbf{x}}_j\right),0}\right)$$ (22)

Here, λ_r denotes the relative strength of the constraint and r_σ is the preferred radius of cell σ so that its area or volume fits precisely in a circle or sphere, respectively. The scalar |x_i − R_σ| denotes the length of the vector that points from the center of mass of cell σ, i.e., R_σ, to the location of the candidate site x_i and can be seen as a local cell radius. Note that the function only applies to cells (σ > 0).

We can explain the form of the energy bias by noting that during each attempt, we want to replace the candidate site value σ(x_i) by the value of its randomly chosen neighboring site σ(x_j). This means that the candidate cell locally retracts at its location x_i, and the neighbor cell locally extends toward x_i. The bias checks whether or not the extension or retraction moves the local cell radius (|x_i − R_σ|) toward or from the preferred radius of the cell r_σ. It then gives a negative energy bias for moves toward the preferred radius, thus making them more favorable. The strength of the energy bias scales with the difference between the local and preferred cell radius; that is, when this difference is large, the cell is more deformed and is therefore more likely to move toward the preferred radius.