Motility-induced mixing transition in exponentially growing multicellular spheroids

Abstract

Growth drives cellular dynamics in dense aggregates, including bacterial colonies, developing tissues, and tumors. However, its effects on other relevant activities have not received sufficient attention. Here, we investigate the underlying physical principles emerging from the interplay of unconstrained growth, steric repulsion, and motility in a minimal agent-based model of exponentially growing, three-dimensional spheroids. Our results reveal a motility-induced mixing transition: Despite cell-scale diffusion, cell lineages remain confined to their local environment until a certain motility threshold is reached, after which the system transitions to tangential superdiffusivity and global cell mixing, with a diverging timescale near the transition, reminiscent of glassy dynamics. Using a phenomenological model, we identify two effects governing this transition: Steric interactions that suppress active motion below a threshold, and the expanding nature of the system, which inhibits complete mixing. The results provide a baseline for identifying additional biological mechanisms in experiments and could be relevant for competition and heterogeneous tumor evolution.

Introduction

Biological systems composed of many cells are prime examples of active matter due to their inherent non-equilibrium activity, which is fueled by energy consumption at the microscopic scale. Self-organization in these systems can be driven by a number of different activities which are also present in non-biological systems, including chemical activity^1,2 and self-propulsion or motility^3,4.

Growth as an activity is a unique feature of living systems and, at the same time, ubiquitous. Due to its indispensable role in organizing processes as diverse as biofilm formation, embryogenesis, or tumor development, proliferating active matter has received increasing attention in recent years^5,6. Depending on the properties of the growing system and its constituents, proliferation can lead to distinct emergent phenomena. For example, for anisotropic particles with nematic properties, growth can lead to the formation of locally or globally aligned domains with peculiar mechanical properties^7,8,9,10 and to topological defects, which generically appear in active nematics and are important for the morphogenesis of tissues and bacterial colonies^7,11,12,13. Proliferation is also known to drive morphological changes such as the transition to the third dimension^14,15, patterning due to stress sensitivity¹⁶ or growth-induced entanglement of branching substrates¹⁷.

Growth often appears together with other processes such as chemical activity, either explicitly or implicitly due to inherent metabolism, whose effects on the environment may in turn regulate growth itself in addition to mechanical interactions^5,18,19,20. Together with cell removal, this feedback can lead to turnover of cells in well-defined homeostatic states, which has been shown to enable competition and mechanical fluidization^21,22. Although unbalanced growth with or without regulation has been considered in specific contexts^7,13,23, the effect of unconstrained exponential growth—as it occurs before the onset of regulation in tissue cultures, bacterial colonies, or tumors—on colony mechanics and its interaction with other activities has received less attention.

Therefore, here, we take a complementary approach and consider pure unbalanced growth in the absence of any regulation or removal. In three dimensions, any growing assembly of particles starting from a small seed will naturally develop into a spheroidal colony. Despite this simplistic morphology, it is important to note that there exists an analogous experimental model system: Already proposed decades ago^24,25,26,27, multicellular spheroids are highly reproducible, approximately spherical colonies of cells that can be produced in bioreactors²⁸ and whose complexity ranks between the ones of in vitro cellular monolayers and in vivo samples^29,30. They are becoming a standard model system for drug development^28,31,32, as well as for biophysical research, with the recent discovery of pressure-driven outbursts in 3D cancer aggregates³³, curvature-induced velocity waves in rotating spheroids³⁴, and the drive of cancer cell invasion in the surrounding extracellular matrix (ECM) by adhesive cancer-associated fibroblasts (CAFs) in spheroids³⁵, to only name a few examples. Consequently, spheroids have recently also been studied experimentally and theoretically from the perspective of growing active matter, establishing connections to several of the above-mentioned phenomena, such as alignment³⁶, heterogeneous and glass-like dynamics³⁷, and fluidization³⁸.

While relevant biological realizations such as tissue spheroids, tumors, or bacterial aggregates do indeed exist, our aim is not to model a specific one of these systems. Instead, by considering idealized physical limits, we hope to establish a clearly defined class of systems to study fundamental processes and to extract unique features compared to non-growing systems without volume expansion. The results could therefore be useful not just for cellular aggregates, but even for the future design of artificial self-organized growing materials.

To this end, we introduce an exclusively mechanical model where a single cell is represented by a dumbbell of spheres that separate over the course of the cell cycle. This idea has been employed before^22,39 and its opportunities have recently been discussed by Hupe and Pollack et al.⁴⁰. To establish a baseline, we first consider growth and division as the only drivers of activity, while steric repulsion mediates the intercellular interactions and working near the incompressible limit ensures homogeneous expansion. We characterize the resulting dynamics using directional mean squared displacements of cells, which separate expansion-flow driven motion from the remaining dynamics and reveal diffusive tangential motion. We then endow cells with a motility mechanism that ensures force balance and thus physical consistency by employing reciprocal intercellular forces. This allows us to investigate how the additional activity of cells actively moving through the colony alters the dynamics. We find several features, including superdiffusive tangential motion at a certain time scale and large-scale mixing, which only emerge above a critical motility at which the mixing time scale diverges. We build a simple phenomenological model that describes the mixing dynamics and allows theoretical insights into the interaction of different processes contributing to the mixing transition, and find two responsible mechanisms based on the dense and expanding nature of the spheroidal system. Some aspects of the phenomenology resemble a glass transition, which has been found in biological systems^38,41,42,43 or soft matter in general^23,44,45 before. Finally, to elucidate the interaction between the different kinds of activities, we investigate the influence of different growth rates on the mixing dynamics.

Methods

Our model can be understood as a 3D version of the disk cell model presented by Hupe and Pollack et al.⁴⁰. We model cells as dumbbells of two spherical nodes of unit diameter that continuously grow and divide, as illustrated in Fig. 1a. For each cell, this life cycle is parameterized by its growth progress g ∈ [0, 1], which increases with a constant rate γ = dg/dt. Node centers are connected by a spring of rest length b^eq along the backbone vector b (Fig. 1b). To make the cell expand, b^eq increases linearly with g: At g = 0, both nodes overlap completely; as the cell ages, they are pushed apart until finally detaching and forming two new cells at g = 1.

Fig. 1: Cell model and macroscopic results.

a Life cycle of a cell: growth and division in the 3D dumbbell model in one generation. b 2D representation of the internal structure and internal forces of a cell (see also Supplementary Note 1), R is the radius of a node, r^cm is the center of mass position of the cell, the vector b is the backbone axis. c 2D representation of external forces by steric interaction repulsion between nodes of two cells. F^Hertz is the repulsion force following a Hertzian law, proportional to d^3/2, where d is the overlap of nodes. d Temporal evolution of the radius of the spheroidal cell colony. The black line indicates expected exponential scaling as described in the text. α is the bulk doubling rate. e Snapshots of the spheroidal cell colony at different stages in time (see also Supplementary Movie 1).

To prevent global synchronization of cell cycles, the growth rates γ of these new daughter cells are drawn randomly from a uniform distribution over the interval $[\alpha -\frac{\alpha }{4},\alpha +\frac{\alpha }{4}]$. Here, α is the bulk doubling rate of the colony, corresponding to exponential growth with rate $\alpha \,\ln (2)$ (note that due to statistical effects discussed in detail by Isensee et al.⁴⁶, the actual average over cell growth rates can be smaller than α depending on the width of the distribution. Here, this statistical correction amounts to a factor of approximately 0.985.) For simplicity, we first set α to 1, such that 1 time unit corresponds to the average lifetime of a cell, or 1 generation. However, we will later also vary the growth rate, where this simple relationship will therefore not hold.

Analogous to γ, the orientations of the new cells’ backbone axes are chosen randomly on division to decorrelate cell orientations.

Intercellular interactions are limited to steric repulsion between individual nodes, based on Hertzian contact theory⁴⁷. Forces are computed from the node overlap d, as illustrated in Fig. 1c, with

$${{{{\bf{F}}}}}^{{{{\rm{ext}}}}}\propto \Upsilon \,H\,(d)\,{d}^{3/2}\,\hat{{{{\bf{d}}}}}$$

(1)

where $\Upsilon$ is the Young’s modulus of the cells, H( ⋅ ) is the Heaviside function, and $\hat{{{{\bf{d}}}}}$ is the normalized difference vector between the node centers. We choose $\Upsilon$ to be large enough to stay reasonably close to the incompressible limit (here: 10⁵, this choice is justified in Supplementary Fig. 2).

To keep forces at cell division continuous, we also use a Hertz-like force law with similar parameters for the internal spring, with

$$F\propto \Upsilon \,\,{{\mbox{sgn}}}\,(\Delta b)\,| \Delta b{| }^{3/2}$$

(2)

where Δb = b^eq − ∣∣b∣∣ is the current deviation of the backbone axis length from its rest length. Node forces are combined into a force F^cm acting on the cell’s center of mass, a torque T acting on the backbone axis orientation, and a force F^int acting on the internal degree of freedom.

The evolution of the cells’ degrees of freedom is described by overdamped equations of motion, where all forces are assumed to be instantaneously balanced by viscous effects. In this limit, velocities can be directly computed as the product of the relevant force with a mobility tensor, resulting in equations of motion such as ${\dot{{{{\bf{r}}}}}}^{{{{\rm{cm}}}}}={\underline{{{{\boldsymbol{\mu }}}}}}^{{{{\rm{cm}}}}}{{{{\bf{F}}}}}^{{{{\rm{cm}}}}}$ for the center of mass position r^cm with mobility tensor ${\underline{{{{\boldsymbol{\mu }}}}}}^{{{{\rm{cm}}}}}$, and analogously for the orientation and the internal degree of freedom. In our model, we use a numerical approximation for the mobilities of rods in a viscous fluid as derived by Tirado et al.^48,49,50,51.

Simulations are initialized with eight cells, whose center of mass positions r^cm are set to the corners of a cube with a side length of two node diameters, with the initial growth progress g and backbone orientation chosen randomly. Due to their growth, cells will come into contact and begin interacting after a short time, quickly forming a spherical colony. We integrate the system for 12 generations (corresponding to 12 time units in the case α = 1). Snapshots of the evolution of an initial condition are shown in Fig. 1e. As there is no cell removal, we expect about 8 × 2¹² = 32,768 cells to exist in the final state. However, this is a stochastic value since the growth rate and therefore also the times between divisions are stochastic.

All simulations are performed using InPartS⁵², an open-source framework for agent-based simulations of interacting particles developed in-house. Additional details on the model, its implementation, and the simulations used for this work can be found in Supplementary Note 1 and Supplementary Note 3.

Results

Radial expansion

We measure the radius of the spherical cell colony (Fig. 1e and Supplementary Movie 1) by computing the particle number density as a function of distance from the center and setting a threshold based on a reference bulk density (see Supplementary Note 4 for details). Figure 1d shows that this radius grows exponentially with time. This growth is directly linked to the particle doubling rate α in the limit of a radially expanding incompressible material, where volume grows ∝ 2^αt and therefore $R(t)\propto \exp (\alpha ^{\prime} t)$ with radial expansion rate $\alpha ^{\prime} =\frac{\ln (2)\alpha }{3}$. As shown in Fig. 1d, the numerical data are consistent with this scaling, confirming that we are working near the incompressible limit.

Cell-scale diffusive tangential motion

As a first characterization of cellular motion, we measure the velocities of individual cells in the spheroid. In the radial direction, we expect motion to be dominated by the outward expansion flow. In contrast, from symmetry considerations, we expect tangential velocities to vanish on average. To account for this fundamental anisotropy, we decompose the velocities into radial and tangential components. Details of the velocity decomposition are discussed in the Supplementary Note 4. Based on the same incompressibility assumption as above, we can easily calculate the global average of the radial velocity across all cells as a function of time, as well as the dependence of the radial velocity on radial position (independent of time):

$${\langle {v}_{{{{\rm{rad}}}}}\rangle }_{r}(t)=\frac{3}{4}\alpha ^{\prime} R(t)=\frac{3}{4}\alpha ^{\prime} {R}_{0}\exp (\alpha ^{\prime} t)$$

(3)

$${v}_{{{{\rm{rad}}}}}(r)=\alpha ^{\prime} r,$$

(4)

with $\alpha {\prime}$ as defined above. The average radial velocities extracted from the simulation are consistent with these theoretical expectations, as seen in Fig. 2a, b. The imperfect incompressibility of the particles is visible as a slightly reduced slope in Fig. 2a. Likewise, the initial growth phase of sparse seed cells before forming a compact spheroid (cf. Fig. 1e) is visible as a deviation from exponential growth for t ≲ 3 in Fig. 2b.

Fig. 2: Characteristic signatures of cell velocities in different directions.

a Distribution, average and theoretical expectation (4) of the radial velocity v_rad as a function of distance from the center. b Distribution, average and theoretical expectation (3) of the radial velocity v_rad against time. R₀ is determined from the last numerical data point. c Distribution and average of the tangential velocity ∣v_tan∣ against distance. d Distribution and average of the tangential velocity ∣v_tan∣ against time. In all panels, heatmaps represent histograms normalized for each column. The full spatio-temporal behavior of both velocity components is shown in Supplementary Fig. 6. See Supplementary Note 4 and Supplementary Fig. 3 for technical details on the decomposition.

To characterize fluctuations around the vanishing average velocity in the tangential direction, we compute the absolute tangential speed ∣v_tan∣. We find it to have a wide distribution throughout the colony (blue heatmap in Fig. 2c), with an average which is fairly independent of the distance from the center of the spheroid (red line in Fig. 2c) and which increases slightly with time (Fig. 2d). The initial steep increase before t ≲ 3 in Fig. 2d can again be associated with the very early compactification phase. We also checked that there is no correlation between radial and tangential velocities of individual cells (Supplementary Fig. 7). We conclude that tangential motion occurs everywhere in the colony, independent of position and radial velocities, as necessary rearrangements due to growth and division. This is also in line with 〈∣v_tan∣〉 being on the order of 1 cell node diameter per generation, which is much smaller than the radial velocity except very close to the center of the colony.

To quantify this non-trivial motion within the spheroidal cell colony, we calculate the mean squared displacement

$$\,{{\mbox{MSD}}}\,(\Delta t)={\left\langle {\left(\Delta {{{{\bf{x}}}}}_{c}({t}_{0},\Delta t)\right)}^{2}\right\rangle }_{{t}_{0},c}$$

(5)

by averaging the squared displacement ${\left(\Delta {{{{\bf{x}}}}}_{c}({t}_{0},\Delta t)\right)}^{2}$ after a lag time Δt over all trajectories of all cells c starting from all possible times t₀, only taking into account time intervals within each cell’s lifetime. This limits the MSD to (on average) one generation, but avoids choices regarding the treatment of splitting trajectories with shared history which can influence the statistics. While we are mostly interested in the characteristics of the non-trivial tangential motion, Δx_c for each cell can either be its tangential or total displacement.

Considering the total displacement, i.e., $\Delta {{{{\bf{x}}}}}_{c}({t}_{0},\Delta t)={{{{\bf{r}}}}}_{c}^{\,{{\mbox{cm}}}\,}({t}_{0}+\Delta t)-{{{{\bf{r}}}}}_{c}^{\,{{\mbox{cm}}}\,}({t}_{0})$, results in ballistic MSDs (Supplementary Fig. 8), as one might expect from the dominance of radial advective motion described above. In contrast, taking into account only tangential displacements sampled from 2D trajectories of tangential motion (see Supplementary Note 4 for details on their construction), we obtain the tangential MSD shown as a dark blue line in Fig. 3a (lowest curve, M = 0). It shows ballistic behavior only on short time scales, while the log-log slope κ decreases towards κ ≳ 1 for larger Δt, indicating long-term diffusive tangential motion (see Supplementary Fig. 9 for a plot of κ vs Δt). Even for Δt towards 1 gen., the absolute length scale $\sqrt{\,{{\mbox{MSD}}}\,}$ stays significantly below 1 cell node diameter, again consistent with motion caused by locally interacting, dividing cells.

Fig. 3: Cell motility enables tangential superdiffusion and loosening of the colony.

a Double logarithmic representation of the tangential MSD as a function of the lag time Δt for different motilities M. Insets depict slopes for comparison with ballistic and diffusive motion. b Motility mechanism as an interaction between the nodes of two cells. The dashed lines indicate backbone axis orientations. F^mot is the motility force that scales linearly with the node overlap d. c The slope κ of the tangential MSD obtained from the double logarithmic representation for different lag times Δt and different motilities M. d κ at a lag time of Δt = 0.25 gen. for different motilities indicates a transition at a motility M ≈ 1000. e A quarter of a slice through the center of the colony for different motilities at ${t}_{\max }$ shows the decrease in cell number density for high motility.

Cell motility enables tangential superdiffusion

So far, growth was the only provider of activity in our model, resulting in a strong radial expansion flow and cell-scale rearrangements to accommodate the growing and dividing matter. In this setting, all motion was caused by symmetrically expanding dumbbells of individual cells and volume exclusion interactions (cf. Fig. 1a, c). While the two separating compartments of a cell could be viewed as self-propelled particles, their motion in the reference frame of the cell is, by definition, coupled to (and therefore cannot, on average, overcome) overall local volume expansion. In addition, the antialignment of this dipole-like pseudo-motility along the cell’s backbone axis reduces the net effect on the cell’s center of mass position. Consistent with our results so far, these effects limit the magnitude and nature of cell motion. From a theoretical perspective, it is therefore natural to consider a model extension with active forces causing true directed motion and to study how they modify the ballistic and diffusive regimes found so far. In addition, such motility is of high biological relevance in many growing systems from bacterial colonies to tumors^{6,53,54,55,56}.

With cells being embedded in the dense three-dimensional aggregate, we assume that cell locomotion arises from cell-cell interactions, which aim to move cells in the direction given by their backbone axes. In contrast to a fixed inherent self-propulsion velocity, this allows us to maintain force balance in a physically consistent way. Following the scheme of steric interactions, these additional reciprocal interactions are implemented as forces between overlapping nodes of different cells i and j:

$${{{{\bf{F}}}}}_{i}^{{{{\rm{mot}}}}}\propto M\cdot d\cdot H(d)\cdot ({\hat{{{{\bf{b}}}}}}_{i}-{\hat{{{{\bf{b}}}}}}_{j}),$$

(6)

With this, each node contributes a force aligned with the backbone axis direction $\hat{{{{\bf{b}}}}}$ of the cell it belongs to, and an equal and opposite force is applied to its interaction partner (see Fig. 3b). Larger overlaps d represent closer contact between cells and are therefore assumed to result in a larger motility force. M is the main control parameter modulating the overall strength of motility. Note that we do not add any explicit dynamics of the cell orientation (given by the backbone axis). Instead, reorientations continue to be caused purely by interactions with other cells.

In this extended model, we now re-evaluate the tangential MSDs for different values of the motility parameter M (Fig. 3a). As expected, we generally observe overall larger tangential MSDs for higher motilities (even the originally larger total MSDs increase significantly, see Supplementary Fig. 8). A particularly noticable jump can be seen between M = 500 and M = 1000. In addition to the increase in magnitude, Fig. 3a also shows a change in the behavior over time, with the initial ballistic regime persisting for longer Δt compared to the non-motile case. A finer scan of the motility parameter M reveals a steep increase of the scaling exponent κ of the MSD for a large range of different Δt as soon as the motility parameter reaches, again, values around M = 1000 (Fig. 3c). This sudden onset of strongly superdiffusive tangential motion at intermediate sub-generational time scales can be seen more quantitatively in Fig. 3d for a lag time of Δt = 0.25 gen., where κ is virtually constant below M ≈ 500 before exhibiting a rather sharp transition towards higher values. Even higher motilities lead to visible dispersion of cells, manifesting as a slightly increased colony size with lower cell number density (Fig. 3e). This in itself will lead to less interaction (smaller overlaps d) and thus reduced motility forces for larger M, as seen in a slight decrease of κ in Fig. 3d. For now, we will focus on the regime of smaller M where the transition occurs and this effect is weak, such that the motility force will be roughly proportional to M itself, but further below, we will introduce a more suitable measure of motility force for larger values of M which takes into account this feedback on cell arrangement. Note that, whenever we compare data between independent simulations (e.g., the snapshots in Fig. 3e), we choose times relative to a well-defined time ${t}_{\max }$ at which a certain cell number is reached towards the end of each simulation (see Supplementary Note 4 for technical details). In this way, we always compare states corresponding roughly to the same number of cells, eliminating noise that is present simply due to small-number effects of randomly chosen cell cycle lengths in the initial stages of spheroid growth.

Motility-induced mixing of cell lineages

The apparent transition in the microscopic dynamics towards more directed motion at sub-generational time scales poses the question whether this goes along with changes in macroscopic and long-term dynamics. A first qualitative change in the dynamics we detect is shown in Fig. 4a: The radial speed abruptly loses the linear profile seen previously for the non-motile case (compare Fig. 2a), but only above a certain motility between M ≈ 700 and 800. In addition, a significant increase in the magnitude of radial speeds becomes visible only beyond this point. As another quantitative measure, we examine the fraction of cells f_in for which the radial velocity component v_rad < 0, i.e., which move inwards. For simple radial expansion due to growth, we expect v_rad > 0 and thus f_in ≈ 0 everywhere except very close to the center of the spheroid. As Supplementary Fig. 10 shows, this is indeed the case below a certain motility threshold. However, f_in quickly changes from 0 to about 1/2 in the range between M = 500 and M = 1000, depending on the radial position, indicating that the radial velocity becomes dominated and thus randomized by motility.

Fig. 4: Motility-induced mixing.

a Spatial profile of the radial speed after ${t}_{\max }$ for different motilities M. One can observe that for larger motilities, the radial speed is almost two orders of magnitude higher than for the non-motile case. Also, it is larger inside the colony. This goes along with cells moving both inwards and outwards. b Tracking cells from a sample volume and their descendants over time with respect to a growing reference volume serves as a measure to differentiate between confinement and mixing. c Mixing efficiency η_M at different motilities for three different lag times Δt after a starting time ${t}_{\max }-3\,$ gen. Data is averaged across all cells from six non-overlapping sample volumes (see Supplementary Note 4). See Supplementary Movies 2–5 for the spatiotemporal dynamics of the sample lineages at selected motilities. d Color-coded representation of the mixing efficiency as a function of motility and lag time. The black line is a contour at a mixing efficiency η_M = 1/2. The white dotted line is a shifted hyperbola fitted to the contour line. It exhibits asymptotic behavior at M_∞, indicated by the white dashed line.

The above analysis reveals a qualitative change of cell motion patterns in the spheroid at a specific motility threshold, pointing to a possible phase transition. However, it is not obvious how these global velocity changes translate into altered cell arrangements. This is further complicated by cell division, which makes the application of traditional measures like positional order parameters or neighbor tracking non-trivial. Since, for low M, the profiles in Fig. 4a and Supplementary Fig. 10 still align with our naive expectation for a homogeneously expanding spheroid, one could suspect that cells are still confined in their local neighborhood in this weak-motility regime and motility-driven rearrangements can only occur for strong motility.

To characterize how cells move relative to each other over long times, we therefore extend our analysis from single cells to lineages, i.e., cells and their descendants. More specifically, we detect all cells within a predefined spherical sample volume inside the colony at a certain sample time t₀ and then follow these cells and their descendants—the “sample lineages”—over some time interval Δt. Taking into account that the sample volume itself grows together with the entire colony, we can distinguish between cases in which the sample lineages stay roughly within the growing sample volume and cases in which they escape and mixing throughout the colony occurs, as illustrated in Fig. 4b. In particular, we can characterize the dynamics entirely based on the dynamics of the proliferating cells themselves, without having to use auxiliary particles such as tracers, which can potentially deviate in their characteristics⁵⁷.

We can formalize this idea by defining a mixing efficiency

$${\eta }_{M}({t}_{0},\Delta t)=2\frac{{\langle | | {{{{\bf{r}}}}}_{c}^{{{{\rm{cm}}}}}({t}_{0}+\Delta t)-\bar{{{{\bf{r}}}}}({t}_{0}+\Delta t)| | \rangle }_{c}}{R({t}_{0}+\Delta t)}-\frac{1}{2},$$

(7)

where R is the colony radius as before, ${{{{\bf{r}}}}}_{c}^{\,{{\mbox{cm}}}\,}$ is the center position of cell c, and the average is taken over all cells in the sample lineages which exist at time t₀ + Δt. $\bar{{{{\bf{r}}}}}={\langle {{{{\bf{r}}}}}_{c}^{{{{\rm{cm}}}}}\rangle }_{c}$ is the overall average position of these (possibly dispersed) descendants and serves as a reference point to measure the spread of the sample lineages. We expect the relative spread ${\langle | | {{{{\bf{r}}}}}_{c}^{{{{\rm{cm}}}}}-\bar{{{{\bf{r}}}}}| | \rangle }_{c}/R$ to stay constant if a compact sphere made up of the sample lineages simply expands together with the entire spheroid, whereas it will increase if mixing occurs. For simplicity, η_M is normalized here for a specific sample volume radius of R/3: The edge cases are a uniform distribution of cells within the sample volume (no mixing), resulting in a relative spread of $\frac{1}{4}$, and a uniform distribution of cells within the entire spheroid volume (perfect mixing), resulting in a relative spread of $\frac{3}{4}$. After normalization of the relative spread according to these limiting values, we obtain the mixing measure in Eq. (7), which starts out at η_M(t₀, 0) ≈ 0 at sample volume initialization, and where, in general, the edge cases now correspond to values of η_M = 0 and η_M = 1, respectively.

Figure 4c shows the mixing efficiency as a function of motility after different lag times Δt, all starting from ${t}_{\max }-3\,$ gen., respectively. These curves confirm the limiting cases of perfect confinement for small motilities and perfect global mixing for large enough motilities. Most importantly, however, mixing only seems to set in at a certain value of the motility parameter. Between the three values of Δt, only the steepness of the transition varies, whereas the onset remains fairly unchanged. This behavior can also be confirmed visually in Supplementary Movies 2–5, where no mixing of the sample lineages with the rest of the spheroid is observed both for M = 0 and M = 500, but M = 1000 and up show significant mixing. It should be emphasized that we have a homogeneous system in which all cells, both the marked cells of the spreading sample volume and those in the rest of the spheroid, have identical properties. Thus, the mixing measure, as well as its visual representation, indicates a global phenomenon.

Motility-induced mixing transition

This evidence suggests that waiting for a longer lag time Δt cannot compensate for low motility below a certain threshold, raising the question whether the existence of this regime can be substantiated quantitatively. To assess the characteristic time scale of mixing more directly, we therefore calculate the mixing efficiency η_M for arbitrary lag times Δt ∈ [0gen., 3gen.]. The heatmap in Fig. 4d shows η_M as a function of the motility and lag time. A contour line at an arbitrary value of η_M = 1/2 (black line) indicates the time scale of mixing as a function of motility. This timescale indeed appears to exhibit asymptotic divergence at a finite non-zero value M_∞ of the motility parameter. Such behavior would imply the existence of a motility threshold M_∞ below which mixing never reaches η_M = 1/2, irrespective of the lag time. To determine whether the scaling behavior of η_M is compatible with this, we fit a shifted hyperbola of the form

$$\Delta t{| }_{{\eta }_{M} = 1/2}(M)=A\cdot {(M-{M}_{\infty })}^{\beta }+C$$

(8)

to the contour line, shown as a white dotted line in Fig. 4d (see Supplementary Note 4 for details). The double-logarithmic plot in Supplementary Fig. 11 further confirms the appropriateness of this functional form, which indicates an asymptote at M_∞ ≈ 900. This is qualitatively reminiscent of a glass transition: A diverging time scale of mixing at a finite, non-zero motility M_∞.

Insights from a phenomenological model

These observations raise the question of how the mixing dynamics and, in particular, the suppression of mixing can be understood both conceptually and quantitatively. At what speed do individual cells actually move through their local environment at a given value of M, beyond proliferation-induced rearrangements? What role does the expanding nature of the system play in the ability of cells to spread over large distances? To answer these questions, we build a phenomenological model of the mixing dynamics that still allows us to extract relevant parameters that can be compared with direct measurements. In this effective description, we approximate the spreading of cells from the sample volume with a continuum model where a density, originally concentrated in the sample volume, undergoes a diffusion process. The two parameters are thus the radius Q of the initial uniform sphere and a diffusivity D. We first consider regular, non-expanding space. In this case, the probability density function P(r, t) can be found simply by a convolution of the initial condition with the heat kernel^58,59,60 (see Supplementary Note 4). From this distribution, the time course of the absolute spread can be computed as

$$\langle r\rangle (t) = {\int \! \! \! \iint}_{\!\!V}rP(r,t)\,{{{\rm{d}}}}V\\ = \frac{3}{8{Q}^{3}}\left[\frac{\sqrt{u}Q(u+2{Q}^{2})}{\sqrt{\pi }\exp \left(\frac{{Q}^{2}}{u}\right)}-\frac{{u}^{2}-4u{Q}^{2}-4{Q}^{4}}{2}\,{{\mbox{erf}}}\,\left(\frac{Q}{\sqrt{u}}\right)\right],$$

(9)

where u = 4Dt. Exponential growth of space can then be accounted for by rescaling time $t\to \frac{1-\exp (-2\alpha ^{\prime} t)}{2\alpha ^{\prime} }$, and consequently u (see Supplementary Note 4 for derivation). The result is a system in which the effective diffusivity decreases over time (while the domain does not grow), equivalent to a constant diffusivity in an expanding space where length scales have been rescaled to be kept constant. In particular, Eq. (9) with this rescaled u allows us to calculate the relative spread for a central initial sample volume with Q = $R_{*}$/3 using Eq. (7), where we can substitute $R_{*}$ = const. due to the nature of the rescaling and $R_{*}$ is the colony radius at sample volume initialization. We fit the resulting expression for η_M to data from simulations of the different motility values, where the only fit parameter is the diffusivity D (Fig. 5a). This provides a relation between the motility parameter M and the effective diffusivity D which is shown in Fig. 5b. Additionally, the transform itself contains useful information on the limiting behavior of this process: Taking t → ∞ in the transformed $u=\frac{2D}{\alpha ^{\prime} }(1-\exp (-2\alpha ^{\prime} t))$, we immediately see that u can reach a maximum value of $2D/\alpha ^{\prime}$, which also limits the maximum spread according to Eq. (9) and thus the maximum mixing efficiency that can be reached for a given diffusivity D. Let u_1/2 be the value of u which yields a mixing efficiency of η_M = 1/2, we can thus define

$${D}_{\infty }=\frac{{u}_{1/2}\alpha ^{\prime} }{2}$$

(10)

in analogy to M_∞, i.e., the critical diffusivity below which, geometrically, no mixing beyond η_M = 1/2 is possible. The necessary value of u_1/2 can be found numerically from Eqs. (9) and (7), and only depends on Q and thereby the radius R_* of the colony at sample volume initialization. The resulting limiting diffusivity D_∞ shown in Fig. 5b indeed indicates that mixing should not be able to surpass η_M = 1/2 below M ≈ 1000, close to the value we found before.

Fig. 5: Phenomenological model of a diffusing uniform sphere of non-interacting tumbling athermal ABPs in an exponentially growing domain.

a Time series of the mixing efficiency η_M for different motility values (transparent gray lines) with the respective fits of the mixing efficiency derived from a model of the diffusion of a uniform sphere on an exponentially growing domain. b Fit parameter D as a function of the motility. The black line indicates critical diffusivity D_∞ at which the expansion suppresses diffusion such that no mixing to η_M = 1/2 is possible. This is reached around a motility of M_∞ ≈ 1000. c Persistence time of rotational dynamics. Dark blue line is based on fits of exponential decays on the orientation autocorrelation, obtained from the time series of velocity directions between division events (Supplementary Fig. 12). Light blue line combines this with additional tumbles according to Eq. (12). d Tangential velocities from motility forces only, in the absence of steric interactions (yellow line, active velocity ${\bar{v}}_{{{{\rm{a}}}}}$) increase with the motility parameter, while tangential velocities from the simulation (purple line) begin to increase only above a certain motility threshold. The inferred self-propulsion velocity in the phenomenological model (light blue line) largely captures this.

Having described the dynamics in terms of an effective diffusivity D, the question remains how this diffusivity is related to the microscopic locomotion of cells through the spheroid. While our control parameter M modulates motility, the driving forces of motility also depend on the emerging cell configuration, such as the number and strength of cell-cell contacts, rendering the relationship of M and D (Fig. 5b) nontrivial. A more intuitive description for the source of diffusivity would be in terms of an effective self-propulsion velocity. Conceptually consistent with the previously described ballistic-to-diffusive MSDs, a description that captures the essential features of our motility mechanism is that of a free athermal active Brownian particle (ABP) with a constant active self-propulsion velocity v and rotational diffusivity D_r = 1/τ_r (caused by changes in cell-cell contacts). To take into account the randomization of orientation at cell division, we further assume tumbles after run times of fixed duration T corresponding to the cell cycle length. We can derive a long-term diffusion coefficient D associated with this type of motion by building on general results for intermittently self-propelled particles derived by Datta et al.⁶¹, resulting in a relation known from ABPs:

$$D=\frac{1}{3}{v}^{2}{\tau }_{{{{\rm{eff}}}}}$$

(11)

with an effective persistence time (see Supplementary Note 4)

$${\tau }_{{{{\rm{eff}}}}}={\tau }_{{{{\rm{r}}}}}\left(1+\frac{{\tau }_{{{{\rm{r}}}}}}{T}\left(\exp \left(-\frac{T}{{\tau }_{{{{\rm{r}}}}}}\right)-1\right)\right).$$

(12)

The intra-generational τ_r for each motility can be obtained from the initial decline of the orientation autocorrelation function (OACF, see details in Supplementary Fig. 12), for which the relevant orientation is given by the current direction of motion, consistent with the free ABP analogy. In contrast to this analogy, the intra-generational τ_r is not due to intrinsic rotational noise but emerges from mechanical interactions. Using a cell cycle length of T = 1 gen., the effective persistence time τ_eff resulting from Eq. (12) is shown for each value of the motility parameter in Fig. 5c (see Supplementary Note 4G and Supplementary Fig. 13 for the negligible effect of the true cell cycle distribution). Given these values of τ_eff and those of the diffusion coefficient D found in Fig. 5b, we can use Eq. (11) to calculate the effective self-propulsion velocity $v=\sqrt{3D/{\tau }_{{{{\rm{eff}}}}}}$ for each value of M. Assuming this velocity is isotropically distributed on the sphere, the corresponding tangential component will be πv/4, which we can compare with the actually observed tangential motion in the spheroid simulations (Fig. 5d). The excellent match between the self-propulsion velocity inferred from the ABP model and the true tangential velocities, particularly in the transition region, substantiates the validity of our phenomenological description and thus the conclusions drawn from it. Note that both velocities examined so far, as well as the diffusivity (Fig. 5b) characterize the motion actually performed by cells. Their relatively late onset with respect to M is a noteworthy feature, since abundant cell-cell contacts in this dense system should generate motility forces according to Eq. (6) for any finite value of M. To confirm this, we can compute the tangential velocity that would arise from motility forces only, in the absence of steric repulsion. This is done by simulating the system with motility interactions, only, for a marginal time step (starting from snapshots of the regular simulations within a time window from 10gen. before ${t}_{\max }$ until ${t}_{\max }$), and averaging over all cells. The result shows the immediate onset of this potential velocity, which we call the average active velocity ${\bar{v}}_{{{{\rm{a}}}}}$ (not to be confused with the effective self-propulsion velocity examined above), for any M > 0 (Fig. 5d). The nonlinearity of ${\bar{v}}_{a}$ in M is caused by the interplay between motility and density: A change in the parameter M scales the motility force, which affects the configuration of cells, which in turn affects the actual force generated. We saw above that larger M leads to a reduced cell number density (Fig. 3e), which reduces the motility interaction that is based on overlaps between cells. Judging by the absence or lower magnitude of observed tangential velocities (or, equivalently, inferred active self-propulsion velocities) compared to the average active velocity ${\bar{v}}_{{{{\rm{a}}}}}$ in Fig. 5d, steric interactions in the densely packed spheroid therefore appear to suppress locomotion to a certain extent. Note that ${\bar{v}}_{{{{\rm{a}}}}}$ can be seen as a parameter of the system, similar to M, only that it already accounts for other effects that modify the motility, Eq. (6). Using this effective parameter (which can also be more closely related to active velocities ordinarily considered in active matter) instead of M will allow us to compare scenarios where cell-cell contacts are modified due to changes in parameters unrelated to motility.

Interplay of growth and motility

The failure of a potential active velocity to actually manifest in the dynamics does not, in itself, reveal the physical origin of this effect. However, the other non-equilibrium activity, proliferation, is a plausible candidate due to the positive pressure it induces (i.e., on-average inward forces on cells), which might prevent lineages from overcoming their local confinement. This would mean that varying the growth rate should alter the strength of these confining forces and thereby the critical motility force to overcome them. To test this hypothesis, we performed simulations with different bulk doubling rates α between 0.1 and 1.0. According to the line of thought above, with smaller growth rates inducing a lower pressure, we expect that smaller motility forces are sufficient to actually observe motion. Since a reduced growth rate also leads to less contact between cells, it will also change the average motility force for the same value of M, such that comparisons of the mixing transition in terms of the parameter M (as seen in Fig. 4d) become meaningless. Therefore, we first measure the average active velocity ${\bar{v}}_{{{{\rm{a}}}}}$ (which corresponds to a certain effective motility force) as introduced above to enable a fair comparison between different scenarios. Plotting ${\bar{v}}_{{{{\rm{a}}}}}$ as a function of motility parameter M for the different bulk doubling rates α (Fig. 6a), we can then use these calibration curves (in reality, we use low-degree polynomial fits which capture the data extremely well but eliminate any noise that might lead to nonmonotonicity, see Supplementary Fig. 14) to convert our abstract motility parameter M into average active velocities for different growth rates, and examine the time necessary to reach η_M = 1/2 as a function of this more meaningful parameter for all growth rates (Fig. 6b). This shift between different curves already shows that the active velocities necessary to reach η_M = 1/2 within the same amount of time decreases with the growth rate, suggesting that it is easier for cells to move through a more slowly growing spheroid.

Fig. 6: Motility-induced mixing transition for different growth rates.

a Profiles of the average active velocity analogous to the corresponding yellow line in Fig. 5d for different bulk doubling rates α. b Contour lines representing the lag time (here scaled with α, such that one time unit corresponds to one true generation at the respective doubling rate) necessary to reach mixing efficiency η_M = 1/2, analogous to Fig. 4d, but in terms of ${\bar{v}}_{{{{\rm{a}}}}}$, for different bulk doubling rates α. Dots at the top indicate the critical active velocity ${\bar{v}}_{{{\mbox{a}}},\infty }$ at which the mixing time diverges, obtained from shifted hyperbola fits. c Critical active velocity ${\bar{v}}_{{{\mbox{a}}},\infty }$ from panel (b) as a function of the bulk doubling rate α (circles), compared to the minimum active velocity ${\bar{v}}_{{{\mbox{a}}},{{\mbox{onset}}}}$ at which the tangential speed v_tan begins to increase with motility (squares; see Supplementary Fig. 16 for details). Colors in all panels represent different bulk doubling rates α.

We can then fit shifted hyperbolas to these contours again as in Eq. (8), only now as a function of ${\bar{v}}_{{{{\rm{a}}}}}$, to determine the critical active velocity ${\bar{v}}_{{{\mbox{a}}},\infty }$ at which the time scale $\Delta t{| }_{{\eta }_{M} = 1/2}$ for reaching η_M = 1/2 diverges (see Supplementary Fig. 15 for details), as indicated by dots at the top of Fig. 6b. In a similar way, we can examine the initial onset of actually observed speeds as in Fig. 5d, but again as a function of ${\bar{v}}_{{{{\rm{a}}}}}$ instead of M, and thereby determine the minimum active velocity ${\bar{v}}_{{{\mbox{a}}},{{\mbox{onset}}}}$ necessary to induce appreciable tangential motion for different growth rates (see Supplementary Fig. 16). Both critical active velocities are plotted in Fig. 6c.

Two thresholds govern the transition

Already from Fig. 5 we learned that the mixing transition is in fact governed not by one but by two effects: First, active velocities below a certain threshold ${\bar{v}}_{{{\mbox{a}}},{{\mbox{onset}}}}$ are suppressed and manifest neither in increased actually observed velocities (Fig. 5d) nor an effective diffusion (Fig. 5b) on a lineage level. Second, even for ${\bar{v}}_{{{{\rm{a}}}}} > {\bar{v}}_{{{\mbox{a}}},{{\mbox{onset}}}}$, expansion prevents active locomotion from reaching arbitrarily large values of η_M, indicated by the existence of a minimum diffusivity D_∞ necessary to reach η_M = 1/2, which corresponds to a minimum active velocity ${\bar{v}}_{{{\mbox{a}}},\infty }$ as determined in Fig. 6b. Both effects act in concert to create the motility-induced mixing transition. While velocity suppression below ${\bar{v}}_{{{\mbox{a}}},{{\mbox{onset}}}}$ causes dynamical arrest on the lineage level at a finite active velocity, the expansion effect is responsible for the diverging mixing time scale when motility already exceeds this threshold. Therefore, velocity suppression shifts the necessary active velocity to larger values. In Fig. 6c we can see that, e.g., for the default bulk doubling rate α = 1.0, an active velocity of ${\bar{v}}_{{{{\rm{a}}}}}\approx 5$ is required before the observed onset of v_tan and an additional active velocity of ${\bar{v}}_{{{{\rm{a}}}}}\approx 4$ to reach a diffusivity sufficient to enable mixing up to η_M = 1/2.

The fact that ${\bar{v}}_{{{\mbox{a}}},{{\mbox{onset}}}}$ increases with the growth rate supports our hypothesis that cells need to overcome the confinement created by their environment due to growth-induced pressure. Interestingly, both critical active velocities increase roughly linearly with the growth rate, suggesting that both contributions remain essential across a wide range of growth rates instead of dominating each other in some limit. However, it should be noted that their relative contributions can vary with the extent of mixing (i.e., the specific value of η_M) that one considers successful mixing. This is because the critical diffusivity D_∞ (and thereby the actual velocity) required to overcome the expansion effect depends on the target value of η_M via u_1/2, with a lower required velocity for lower η_M. However, any actual velocity only manifests beyond ${\bar{v}}_{{{\mbox{a}}},{{\mbox{onset}}}}$. Therefore, the same active velocity is required to overcome velocity suppression, while the additional active velocity required to overcome the expansion effect varies with the extent of mixing.

Discussion

Our theoretical study set out to characterize the collective dynamics in a growing spheroidal cell colony. It can be seen in the context of a number of recent studies in the field of active matter, which have provided novel perspectives on cell motion in biological systems by relating cellular activity—proliferation or motility—to phenomena such as fluidization, glass or rigidity transitions^{21,37,38,41,62,63}. Here, we built an agent-based computational model that allowed us to study the dynamics and self-organization arising from minimal ingredients, namely growth, division, motility, and steric repulsion. We deliberately refrained from adding explicit growth regulation, resulting in simple scaling laws for volume expansion and radial velocity in the non-motile case. It is therefore consistent with the early stages of a biological colony when chemical gradients due to diffusion limitation or gene-regulatory-induced heterogeneities have not yet developed, but could also be applicable to future artificial systems in which such regulation is absent. However, more importantly, our model can serve as a baseline to delineate the effects of additional mechanisms in theory and experiments.

Even without motility, we observed tangential motion of cells in the spheroid. This is a purely emergent dynamical feature and may be understood as cell-scale rearrangements due to local growth and division, as evidenced by its small, position-independent magnitude and approximately diffusive nature in contrast to ballistic radial motion. Although we consider the cell center for our MSD calculation, which only moves due to the interaction with the bath of growing cells and not due to growth of the cell itself, tangential MSDs show ballistic scaling on very short sub-generational time scales, consistent with the deterministic single-cell growth process in this athermal system. Towards a timescale of one generation, the system randomizes sufficiently to yield approximately diffusive tangential motion. Our mixing analysis showed that lineages stay perfectly confined in their local neighborhoods without motility, which seems to contradict the locally diffusive motion, which indicates a fluidized system. However, this can be resolved by taking into account two aspects: First, the two characterizations apply to different entities, i.e., we have to differentiate between single-cell dynamics and lineage dynamics, which also means that analogies to glassy dynamics must necessarily remain on a conceptual level. Second, proliferation of cells creates additional space in which this motion can take place without leaving the (expanding) neighborhood. The coexistence of both features can therefore be considered a distinct feature of systems with proliferation and isotropic volume expansion. We also think that our results do not contradict the superdiffusive motion and fluidization seen in a tumor model with growth regulation³⁸, since our tangential MSDs exclude the more directed radial motion and the mixing analysis is agnostic to cell rearrangements within the sample lineages. Interestingly, tangential diffusion as passive translocation on the surface of tumors has been considered in mathematical reaction-diffusion models before⁶⁴.

On the sub-generational time scale, we revealed the emergence of a transient superdiffusive regime of tangential motion induced by motility. The sudden change in the scaling behavior of the tangential MSD at a finite motility is accompanied by equally abrupt changes in the instantaneous radial velocities. The additional directed motion on short time scales can be interpreted as a manifestation of motility, which is suppressed below the transition point due to the dense packing of cells. We were able to substantiate this interpretation by comparing the potential motility due to an average active velocity with the actually observed velocity, whose onset occurs around the same motility of M ≈ 500 (Fig. 5d) as the transition point in the tangential MSDs (Fig. 3d). Beyond the transition, cells are then able to use this directed motion to escape from their local neighborhoods. We could further underscore the crucial role of steric interactions by varying the growth rate and thereby modulating their strength, which led to changes in the critical active velocity in the expected direction. An interesting aspect of the velocity suppression is its reminiscence of contact inhibition of locomotion, which is an active biological regulatory process⁶⁵, whereas similar phenomenology here is caused by a purely mechanical mechanism. Related to our observations, density-dependent velocities have also been observed in other contexts such as motility-induced phase separation⁶⁶.

The suppression of motion at a non-zero motility is an emergent phenomenon that is similar to a glass transition and has previously been observed in biological systems^38,41,42,43, also explicitly in growing dense active matter^23,38. While Tjhung et al. found glass-like dynamics, manifested by different phenomena, such as aging, the regulation of growth by a total energy density played a critical role in their system²³. This can be contrasted with our model, which does not consider growth rate control mechanisms, and all local dynamics take place on a background of constant expansion. More specifically, an onset of motion beyond a critical motility has already been related to a glass transition in self-propelled Voronoi models⁴¹, and similarly, fluidization at a critical magnitude of a random traction force has been reported in a random traction vertex model⁶⁷. However, our baseline case differs in the sense that cell rearrangements due to proliferation fluidize the system even before the transition.

The most prominent feature of our system is the existence of a mixing transition characterized by a diverging timescale at finite motility. It is important to note that directed motion in the spheroid, once it happens, is still randomized on long time scales and does not result in collective motion or even rotation, which has been observed in other systems⁶⁸. This allowed us to quantitatively capture the mixing dynamics using a phenomenological model based on diffusion in an exponentially expanding space, where diffusion is imagined to result from the motion of a tumbling athermal active Brownian particle. It revealed that the exponentially expanding nature of the system, causing all points to continuously distance from each other, reminiscent of Hubble expansion^7,69, has a significant contribution to the existence of a diverging mixing time scale and leads to a critical diffusivity necessary to overcome expansion (Fig. 5b). The inferred diffusivity present at different motilities in the spheroid indeed reaches this critical value in the motility range that also exhibits the mixing transition (compare Figs. 4d and 5b). Both this and the quantitative match between the inferred self-propulsion velocities and measured ones (Fig. 5d) validate the description, which integrates all aspects of the dynamics, in particular the above-mentioned velocity suppression and the expansive growth of the spheroid—and can explain their mutual interactions.

Here, we did not consider apoptosis, i.e., removal of cells, which has been shown to cause fluidization on its own²¹, dependent on division and death mechanisms⁷⁰. In systems with apoptosis and additional adhesion, tissue fluidization also occurs independently of the existence of a glassy phase⁵⁷. It would therefore be interesting to systematically study how the specific, multi-stage, route to mixing found here is modulated by other processes such as apoptosis or cell adhesion, the latter of which has also been found to be important for cancer invasion³⁵.

Besides the analogy with rigidity and glasses, the mixing dynamics may also have practical consequences, as it enables transport within the colony, including between the bulk and its surface. This could be relevant for the escape of specific cells in the case of phenotypic or genetic heterogeneity. For example, in cancer research, such intratumor heterogeneity is a diligent subject^71,72. It might therefore be worthwhile to study how relevant collective effects are for the behavior found here, i.e., whether and how it translates to individual motile sub-populations or, in general, heterogeneous motility or growth.

Experimental investigations based on our findings could try to identify both the onset of (increased) tangential superdiffusive motion, which itself has already been found in similar systems³³, and the critical tangential velocity of cells required for mixing in growing multicellular spheroids. While calculating numbers characterizing the strength of activity, such as the Péclet number, is difficult in our system (due to the different levels of description and the presence of multiple activities), it is interesting to note that both critical velocities we detect are roughly proportional to the growth rate (Fig. 6c). Non-dimensionalizing the velocities with the growth rate therefore yields approximately constant velocities in units of cell diameters per generation. One additional avenue for future theoretical research could therefore be to determine which properties of the system determine this quasi-universal value. Other, more subtle interdependencies could also be of interest: For example, we expect the Young’s modulus to be able to partially compensate for changes in the bulk doubling rate α, potentially leading to similar motility forces. More generally, the quasi-universal velocity value on the order of 10cell diameters per generation is comparable to typical velocities observed in some experiments (see Supplementary Note 4 for estimation), e.g., (14 ± 8) cell diameters per generation before and (30 ± 15) cell diameters per generation after the epithelial-mesenchymal transition (EMT)⁷³, which is a known driver of cell motility. Therefore, the regime explored in our study seems relevant for real-world scenarios.