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Open Access

Peer-reviewed

Research Article

Heavy-tailed jumps induce intermittent patterns and gradual transitions in interacting cell populations

Abstract

It has been widely accepted, that a key step in cancer invasion and metastasis is the so-called epithelial-mesenchymal transition (EMT), where cells lose contacts with their neighbors and start moving freely, colonizing new tissues in the process. However, it has been found that cancer cells may retain cellular contacts and become highly metastasizing without undergoing EMT. Furthermore, single cancerous cells may reach far away tissues by displaying superdiffusive Lévy dynamics. Here, we study a system of collectively migrating cells with attracting (adhesive) interactions and heavy-tailed distributed jump lengths. To this end, we develop a multi-speed lattice-gas cellular automaton model, and analyze it both computationally and analytically. To study the effect of Lévy dynamics on collective behavior, we consider three migration scenarios: (classic) single-jump migration, corresponding to a “healthy” scenario, where cells move a single site per time step, block migration, where cells that move together remain together for the total duration of the jump but jump lengths are random and power-law distributed, corresponding to the case where leader cells can guide cell clusters through strong cell junctions, and cell-wise migration, where each cell in a group may perform jumps of different lengths, independently of other cells, corresponding to the case where cell junctions are dynamic enough to allow single cells to escape clusters. We show that, while in the classic case, destabilization of the homogeneous steady state only depends on cell density and/or adhesion strength, in the multi-speed case, increasing the probability of long jump lengths leads to a gradual order-disorder transition, as opposed to the sharp transition with increasing cell density and/or adhesiveness. Furthermore, observed spatial patterns formed are stable when jumps are preferentially small, unstable when long jumps are highly probable, and metastable at intermediate regimes.

Author summary

Living organisms, as well as some man-made objects, move while influencing the movement of one another along the way, in what is referred to as collective migration. Much research has focused on the effect of particular types of individual interactions on the formation of patterns at the population level. For example, attractive interactions (also called adhesive in the cellular context), promote the formation of groups with a large number of individuals. Most models, however, assume that when isolated, individuals will move a small distance in a fixed time window. However, current research has found that some cells perform “Lévy movement”, a type of movement where cells tend to move great distances, even when their direction may be random. In this work, we construct a model to study a system of individuals with attractive interactions, but which may randomly move very far away individually. We find that group formation can still happen, as long as large displacements are not too frequent, and that high odds of large displacements coupled to high attractive strength may result in the formation of short-lived clusters, rather than stable clusters.

Citation: Nava-Sedeño JM (2025) Heavy-tailed jumps induce intermittent patterns and gradual transitions in interacting cell populations. PLOS Complex Syst 2(6): e0000048. https://doi.org/10.1371/journal.pcsy.0000048

Editor: John Edward Green, The University of Adelaide - North Terrace Campus: The University of Adelaide, AUSTRALIA

Received: October 14, 2024; Accepted: April 20, 2025; Published: June 2, 2025

Copyright: © 2025 Josué Manik Nava-Sedeño. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: The code used for producing the simulation results reported in the manuscript can be freely accessed at https://github.com/tailswalker/Zeta-adhesive.

Funding: The author(s) received no specific funding for this work.

Competing interests: The author has declared that no competing interests exist.

Introduction

Several biological systems [1–4], as well as some artificial ones including robot swarms [5] and active colloids [6], consist of individuals which use energy to propel themselves, and when close enough to one another, may interact in some way to affect each other’s behavior. Commonly, it is assumed that individuals in isolation perform random walks and diffuse normally. The study of such systems generally falls under the field of “active matter” [7, 8].

In the particular case of groups of biological cells, interactions may result in the coordinated movement of individuals in a defined direction [9], or alignment of their body shape [10], to name a few. Animal cells, for example, interact strongly through attractive, or adhesive, interactions, caused by the presence of molecules such as cadherins [11]. Under physiological conditions, cells adhere strongly and form stable tissues and other structures. It is generally accepted that cells which become malignant become less adhesive, and thus can escape their original tissue to colonize other far away regions during the metastatic cancer phase [12], undergoing what is known as epithelial-mesenchymal transition, or EMT[13]. Additionally, it has been recently found that metastatic cells may not only have a decreased adhesiveness, but also behave superdiffusively in isolation [14, 15], meaning that they tend to move further distances than normally diffusive cells. Furthermore, some metastatic cell populations may be significantly more invasive when their adhesiveness is high, and remain in cell clusters, than when their adhesiveness is low, and move as individual cells [16], as highly motile cells may carry other tumor cells along through intact cell junctions [17].

Though collective migration, pattern formation, cluster formation, and self-organization has been actively studied using a plethora of mathematical models, so far most models assume cells perform random walks or Brownian motion when cell-cell interactions are absent [18–22]. The study of active matter with Lévy flight-like movement dynamics has only started to be studied recently [23–29], and, to the best of our knowledge, has never been considered in combination with attractive interactions. In some cases, the study of such systems skips the microscopic, individual-level description, and instead consider a macroscopic, partial-differential equation (PDE) model, where the effect of the individual-level superdiffusive dynamics is assumed to simply result in the exchange of local differential operators for non-local fractional pseudodifferential operators (see for example [30]).

In this work, we study the effects of non-Brownian jump dynamics in a population of interacting cells, by comparing the behavior of a population of adhesive cells with power-law distributed jump lengths, to that of of a population of equally interacting cells, but whose jump lengths are constant, thus performing a random walk in isolation. We start by defining a discrete lattice-gas cellular automaton (LGCA) model with random jump lengths in one dimension, which is akin to heavily confined cells in a stiff extracellular matrix. Afterward, we derive a mean-field, macroscopic description of the system which we analyze both analytically and numerically, to theoretically predict the behavior of the model with respect to cell density, adhesion sensitivity, and jump-length distribution parameters. Finally, we corroborate our theoretical results using computer simulations, and we discuss the spatio-temporal patterns observed in our simulations.

Materials and methods

Model definition

We model the system of adhesive cells by a lattice-gas cellular automaton. Like all true cellular automata, the LGCA consists of a discrete spatial lattice , a discrete state space , a local neighborhood , and a local rule [31].

Spatial lattice .

For simplicity, we consider a one-dimensional lattice . Due to computational restraints, we consider a finite lattice . Cells can only be located within lattice nodes . Unlike regular cellular automata, nodes in LGCA posses a substructure: every node has exactly two velocity channels, c1 = 1, c2 = −1. Cells moving to the right are said to occupy channel c1, while cells moving to the left occupy channel c2.

State space .

Traditionally, channels can be occupied by one cell at most, which introduces an exclusion principle. In this work, we instead use an exclusion-free LGCA extension [32]. Thus, channel occupation, , indicates how many cells are located within the velocity channel cj at node r at time step k, i.e. how many cells in the node are moving in a certain direction, and any number of cells can move in the same direction at the same spatial location. Thus, the state of a node r at time step k is given by

Note that the total number of cells in a node, irrespective of their direction of motion, is given by

Neighborhood .

The neighborhood of a node is a subset of the lattice , such that only cells within this interaction neighborhood can affect the behavior of cells at r. In this case, we consider only the first neighbors of r. Furthermore, since the lattice used for simulations is finite, we define periodic boundary conditions. Therefore, the interaction neighborhood of the node is defined as

Rule .

In cellular automata, time advances in discrete time steps . A rule is applied to every node in the lattice simultaneously, which changes the state of every node to a new state, after which we consider that a single time step has elapsed. In LGCA, the rule consists of two sequential steps, the interaction step , followed by the migration step . We will not consider cell birth/death, thus the interaction step rearranges cells within velocity channels without changing the number of cells per node.

The interaction step models the change in the cells’ direction of motion due to adhesive interactions. We assume that the probability that a single cell occupies a certain velocity channel is independent of the velocity channel occupation of other cells within the same node. Then, the probability that a cell occupies the j-th velocity channel after the interaction step is

(1)

where Z is the normalization constant, is called the sensitivity, which controls the randomness of the interaction, and the local discrete adhesive gradient is given by

which points in the direction of maximum cell density. Note that, when , states s change uniformly among all states with the same number of cells as in the original state, while in the limit , states change deterministically to the state where the cell’s velocity forms the minimum angle with respect to the adhesive gradient. Furthermore, the transition probability is uniform, irrespective of , whenever the adhesive gradient vanishes.

We consider three different migration modes. First, as in traditional LGCA, the migration step is deterministic, and cells move to the nearest neighboring node in the direction of the velocity channel they occupy after interaction. We refer to this mode of migration ’single step’. Additionally, we consider a multi-velocity extension, where cells move in the direction of their velocity channel, but into a node at a random distance M away from the original node. To model the possibility that cells travel very long distances, we define M as a Zeta-distributed random variable, with probability mass function (pmf) given by

(2)

where is the Riemann zeta function, which is the distribution’s normalization constant, and the parameter s>1 is inversely proportional to the weight of the distribution’s tail. The Zeta distribution was selected because power-law-distributed displacements () have been observed in the motility of several cell types[14, 33, 34]. Additionally, the Zeta distribution incorporates the exponent s as a tunable parameter, enabling the systematic study of the effect of varying jump lengths on collective behavior. While continuous heavy-tailed distributions, such as the Lévy distribution, have been employed in prior studies, the discrete nature of our model necessitates a discrete heavy-tailed distribution, for which the Zeta distribution is the most widely recognized. We consider two cases for the multi-velocity migration step:

  • Cell-wise jumps. In this case, each cell within a velocity channel chooses a node to land into, following Eq 2, independently of all other cells.
  • Block migration. In this case, all cells within the same channel move into the same node, according to Eq 2.

The full dynamics of the LGCA with block migration can be summarized as a stochastic finite difference equation, known as the microdynamical equation, which reads

(3)

where sj(r,k) and are the occupation of the j-th channel before and after interaction, respectively, of node r at time step k. This equation states that cells occupying the j-th velocity channel at the next time step, are cells that occupy the same channel at the current time step, after the interaction step has changed cell velocities, a distance M away, traveling in the direction of their velocity channel, as seen in Fig 1.

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Fig 1. Dynamics of the multi-velocity LGCA.

During the interaction step , cells change their velocities stochastically according to Eq 1, occupying new velocity channels within their original node. Afterwards, during the migration step , we consider three migration modes. 1. Single-jump migration: All cells within a channel are transported into the same channel in the neighboring node in the direction of the velocity channel. 2. Block migration: All cells within a channel are transported into the same channel in a node a random distance M away, distributed as Eq 2, in the direction of the velocity channel. 3. Cell-wise migration: Every cell is transported to a different random node M sites away, in the direction of their velocity channel. The time step number is increased after the interaction and migration steps have been applied to every node in the lattice simultaneously.

https://doi.org/10.1371/journal.pcsy.0000048.g001

Computational implementation

The lattice size was set at L = 300 nodes, which allowed to maintain computational efficiency while enabling a proper sampling of the long displacements introduced by the Zeta distribution. Periodic boundary conditions were used in all cases. To simulate a disordered, spatially homogeneous initial condition, each velocity channel of every lattice node was assigned a single particle randomly following a binomial process with probability . To characterize the degree of aggregation of cells, we measured the spatial entropy after 500 time steps had elapsed, defined as

(4)

where

is the total number of cells in the entire lattice. When cells are uniformly distributed among all nodes, entropy reaches its maximum value .

Results

Mathematical analysis in the local case

First, we consider single-step migration. To analyze this case, we start with Eq 3, setting M = 1. Next, we calculate the expected value of this equation conditioned to , the filtration of the stochastic process up to time step k, which yields

since the interaction step is binomial due to the independence between cells in the same node during interaction. Now, we take the total expected value of the previous expression. We use a mean-field approximation, which considers that the effect of cell interactions on any cell can be approximated by the effect of the average cell interaction on the cell. Mathematically, this approximation considers that all cell variables are independent random variables with zero standard deviation. Therefore, for any function F, we may approximate . Such an approximation results in

where Pj(r,k) is the mean-field channel occupation probability

(5)

and is the mean occupancy of the j-th velocity channel of node r at time step k. Afterward, we add over j and denote by

the expected number of cells in node r at time step k, and obtain the finite difference equation

(6)

We could perform the analysis on this discrete equation; however, we will instead derive an approximate partial differential equation, and analyze the model’s dynamics at the continuous level.

To this end, we now rescale the time by a constant time step length, , and a constant lattice spacing . We define the continuous time and space variables and , respectively and, using that ,we obtain the equation

Now, we perform a Taylor series expansion up to first order in t, and up to second order in x. We define the cell speed and diffusion constant to obtain the approximate partial differential equation

(7)

where the discrete adhesive gradient becomes the continuous gradient and therefore the channel occupation probability can be written as

where we have expanded the normalization constant Z, and .

The spatially homogeneous steady state, , (or equivalently, , ) is an equilibrium solution of Eq 7, since in this case and all derivatives vanish. Since the PDE is nonlinear in both u and , we linearize around the homogeneous steady state to obtain

(8)

which is just the diffusion equation with diffusion constant . Thus, cells stop spreading and start aggregating when the diffusion constant becomes negative, i.e. when

(9)

Therefore, the model predicts spontaneous aggregation if either the sensitivity or cell density are sufficiently high, where both sensitivity and cell density contribute equally to the instability of the homogeneous steady state. These results agree with previous research on cell populations with adhesive interactions [35, 36].

Mathematical analysis in the non-local case

In the case of Zeta-distributed jumps, we restrict ourselves to the block-migration case, due to ease of analysis, and since the qualitative differences between both cases are mostly restricted to jump distributions with extremely heavy tails and low cell-cell interaction strength.

We start by finding the continuous PDE similarly to the single jump case, following [37]. We use Eq 3 and the total expected value, as well as a mean-field approximation as before. However, due to the random jump lengths, we obtain

(10)

where pM(m) is the probability mass function of the Zeta-distributed jump length, given by Eq 2. Adding over n, rescaling time and space, subtracting u(x,t) on both sides of the equality, and using probability axioms, we obtain

We again use that , we add a zero to the right-hand side of the equation, and rearrange, to obtain

Now, we substitute Eq 2, define the new constant since , divide everywhere by , multiply and divide the right hand side by , and multiply and divide the second term in the right hand side by two, so that for small and , the equation is approximated by the integro-differential, fractional PDE

(11)

where the constants are defined as

and is the fractional Laplacian proportionality constant [38], and the censored fractional Laplacian is defined as [37]

(12)

Since the nonloncal terms (with the exception of the censored fractional Laplacian) do not correspond to pseudodifferential operators, the analysis of the solutions to Eq 11 is difficult to perform analytically. Thus, to analyze the model, rather than focusing on the continuous approximation Eq 11 as done previously, for simplicity we instead turn to the mean-field, discrete, finite difference equation. We start by considering Eq 10, noting that, due to the normalization of the probabilities pM(m) and Pn(r,k), the disordered, spatially homogeneous steady state , , , , is a solution to Eq 10. We linearize the equation around this steady state and, using the definition of the mean-field probabilities Eq 5, the linearized system reads

Finally, we perform a discrete Fourier transform, using the shift theorem

where , , and recalling the definition of the characteristic function of the random variable M,

we obtain the linear FDE system

where the matrix is called the Boltzmann propagator, whose entries are given in this case by

(13)

In this case, time is discrete and the number of cells is conserved, and thus the spatially homogeneous, disordered steady state is stable if and only if all eigenvalues of the Boltzmann propagator are such that for all . Therefore, patterns with spatial wavelength grow exponentially as if .

Since the the entries of the Boltzmann propagator do not depend on , its columns are not linearly independent, thus one eigenvalue is . This eigenvalue predicts the almost instantaneous decay of modes with different channel occupation within a single node [39], since the adhesive interaction does not promote velocity polarization, but rather enhances the formation of regions with high cell density.

Note that, when we have single-step migration, M = 1, and its characteristic function is simply , the second eigenvalue has a closed, real form

(14)

Differentating and equating to zero, we find that it always has two extrema, one at q = 0 where , corresponding to mass conservation, and another at where , corresponding to a checkerboard artifact due to lattice regularity. However, when , has another maximum greater than one, which destabilizes the spatially uniform steady state. This agrees with the result obtained from the linearization of the PDE approximation in the single-step migration case.

In the long-jump case, the remaining eigenvalue spectra depends on the Zeta distribution parameter, s, as well as on the product of the interaction sensitivity and , as in the single-jump case. This means that, linearly, changes in have the same effect as changes in .

While the non-trivial eigenvalue spectra in the single-jump and long-jump cases are qualitatively very similar in the low sensitivity, low density regime, important differences arise when considering the strongly interacting and/or high density cases. As seen in Fig 2, in the highly adhesive case, the dominant wavenumber q corresponding to the global maximum of the eigenvalue spectrum, , does not change noticeably between the short and long jump cases. Therefore, the wavelength of emergent spatial patterns is largely consistent, independently of jump lengths. However, the height of the maximum does decrease monotonically with decreasing s, dipping well below the mark for low enough values of s. Thus, while patterns may still form when long jumps become increasingly more probable, they do so more slowly, and stop forming altogether once the tails of the jump distribution become too heavy.

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Fig 2. Eigenvalue spectra for Zeta-distributed jump lengths.

The spectra shown were calculated numerically for . The blue line corresponds to the single-jump case, where the eigenvalue spectrum is given by Eq 14. Other colors correspond to spectra corresponding to the non-trivial eigenvalue of Eq 13, with s = 3 (yellow line), s = 2 (green line), and (red line). The characteristic function of the Zeta distribution was used in these cases. The critical value, is shown as a dashed line. As long jumps become more probable (deceasing s), the rate of divergence from the steady state decreases. The steady state becomes linearly stable at sufficiently low values of s when .

https://doi.org/10.1371/journal.pcsy.0000048.g002

Computer simulations

We performed computer simulations of the model with all three migration modes, and compared the results with the theoretical predictions discussed earlier.

First, we performed a parameter sweep, varying both the sensitivity , and the mean initial channel occupation of the initially homogeneous steady state, with fixed jump distribution parameter s = 6. This value was chosen such that long jumps are probable enough to have a substantial impact in the dynamics of the model, but not so common that patterns are destroyed, as found theoretically. We measured the global entropy (Eq 4) after simulating the model for 500 time steps, and averaged over 50 realizations. We find that the order-disorder transition for both cell-wise and block migration modes roughly follow the hyperbolic profile Eq 9 predicted theoretically for the single-jump migration mode (see Fig 3a and 3b). This suggests that the instability threshold resulting from Eq 13 with corresponding to the Zeta distribution would be qualitatively similar to that of the single-jump case, so long jumps do not significantly affect the critical sensitivity and density values at the transition.

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Fig 3. Simulations of the model with block (left column) and cell-wise (right column) migration.

(a,b): Global spatial entropy (Eq 4) for varying and . The Zeta distribution parameter was fixed at s = 6. The color bar indicates the value of the entropy. The red line indicates the stability threshold given by Eq 9 for the classic single-jump case. (c,d): Global spatial entropy as a function of sensitivity for several values of s. The mean initial channel occupation was set at . As long jumps become less probable, the phase transition becomes steeper. (e,f): Kymographs (spatiotemporal plots) of single model realizations. The color bar indicates the total number of cells at node r at time step k, . Parameter values were set at , , and either s = 6 (e) or s = 5 (f). In both cases, cell aggregates constantly break down and reform. In the cell-wise migration case, aggregates appear more long-lived than in the block migration case. Average entropy values were obtained by averging 50 model realizations.

https://doi.org/10.1371/journal.pcsy.0000048.g003

Then, we fixed , and varied both and s, and observed the mean entropy (see Fig 3c and 3d). First, we observe that, in both cases, the entropy profile is practically identical to that of the single-jump case for . Second, in both cases, the critical value does not change significantly for varying s. However, the entropy of the ordered state steadily increases with decreasing s, until the transition is barely noticeable at . This indicates that long jumps increase the global spatial disorder of cells, up to a point where cell-cell adhesion cannot prevent spatial disorder, independently of the adhesive strength. Finally, we notice that entropy in the block migration case reaches lower values than in the cell-wise migration case, even in the low-adhesion regime. This is to be expected, since even groups of cells which are located in the same velocity channel out of complete chance, will be forced to remain together during the migration phase, maintaining the spatial entropy at lower levels.

We also observed the spatial patterns formed in individual model realizations for high adhesiveness and moderate odds for long jumps (intermediate values of s), to make sense of the higher values of the entropy after the transition discussed earlier. As observed in Fig 3e and 3f, for both block migration and cell-wise migration modes, aggregation patterns at these highly adhesive, highly motile regimes do not longer form and remain stable and stationary. Rather, cells start aggregating at high-density zones, and then randomly dissociate after some time, and reassemble at a different spatial point, and so on and so forth. Thus, at these parameter regimes, patterns form intermittently and are metastable. This behavior may be due to cells within an aggregate suddenly leaving and destroying it when randomly performing a long jump. This may also explain why aggregation patterns in the cell-wise migration case are more long lived than in the block migration case: the loss of a few cells performing long jumps is less deleterious than the loss of a big group of cells moving together to a node far away.

Robustness and generalizations of the model

To assess the robustness and generality of our model, we explored the dependence of the model’s behavior on dimensionality, lattice geometry, and the number of neighbors. Additionally, we compared our model to an analogous non-local PDE model to highlight differences between this naïve approach versus our bottom-up model definition.

Model dimensionality

The two-dimensional local model has already been thoroughly studied in [35]. In the non-local case, for a two-dimensional lattice, Eq 10 would read

(15)

where is the node position vector, and we have four velocity channels, corresponding to the relative positions of the four nearest neighbors, c1=(1,0), c2=(0,1), c3=(−1,0), and c4=(0,−1), with mean-field channel occupation probabilities

Linearizing this equation around the spatially homogeneous steady state (since there are now four velocity channels, each occupied in average by particles) results in

We now perform a two-dimensional Fourier transform of the system of equations, which again yields a Boltzmann propagator, with entries given by

(16)

where qx and qy are the first and second components of the wavevector q, respectively. Again, we find that the propagator entries do not depend on , and thus it has only one nonzero eigenvalue. It is evident that the Boltzmann propagator for the two-dimensional non-local model is completely analogous to the one-dimensional case. Thus, the two-dimensional model can be considered as two one-dimensional models for each spatial dimension. Looking at Eq 13, however, one notices that in the one-dimensional case, there is a factor , while in the two-dimensional case, the factor is . This is due to the fact that, in one dimension, cells feel a gradient defined only by two neighboring nodes, while in two dimensions, cells sense two times as many nodes to determine the adhesive gradient. In fact, it is straightforward to see that, if cells are able to sense more neighbors, this will only result in a different constant multiplying the second term in the right side of Eqs 13 and (16). The effect of this constant is not a qualitative change in patterns and interactions, but rather, an earlier onset in the order-disorder transition for smaller values of β and , since cells can now sense a greater number of cells within more nodes. Simulations of the model in two dimensions can be observed in Fig 4a and 4b. Two-dimensional simulations were performed for a total of 500 time steps, due to computational efficiency. Note that clusters are more defined in the single migration case, as opposed to the block migration case, just as in one dimension, since clusters are more likely to fall apart and reform in the block migration case.

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Fig 4. Realizations of two-dimensional and volume exclusion extensions.

(a,b): Snapshots of model realizations on two-dimensional space after 500 time steps. The color bar indicates the total number of cells within a node at a given time point. Parameter values were set at , β=0.25, and s = 5. Cells migration was either via (a) block migration or (b) cell-wise jumps. (c,d): Kymographs (spatiotemporal plots) of single model realizations with volume exclusion. The color bar indicates the total number of cells at node within a node at a given time point. Parameter values were set at , β=0.25, and s = 5. Cells migration was either via (c) block migration or (d) cell-wise jumps.

https://doi.org/10.1371/journal.pcsy.0000048.g004

Cell volume exclusion

For simplicity, and due to the high deformability of cancer cells, we have not imposed volume exclusion interactions between cells, thus allowing for the accumulation of cells. There is, however, a physical limit to how many cells a single lattice site may accumulate, even when taking into account that cells may pile up in a three-dimensional arrangement. To take this into account, we could adopt the approach used in a previous study on cancer dynamics [40]: instead of defining the adhesive gradient as done previously, we could use a logistic adhesive gradient defined as

where H(x) is a logistic function

where A>0 is a new parameter indicating the critical number of cells such that they stop adhering and start repelling one another. With this approach, the homogeneous steady state is still a solution to the discrete mean-field equation. Since we have replaced u(r,k) in the adhesive gradient by H[u(r,k)], we can straightforwardly apply the chain rule, such that the linearized system is

where the only difference to the case with no exclusion is the factor accompanying . For single-step migration (M = 1), the non-trivial eigenvalue takes the form

leading to the instability condition

When the mean initial density is small, this condition suggests that pattern formation occurs as β increases. However, for large , the left-hand side of the inequality can become negative, stabilizing the spatially homogeneous steady state regardless of β. This stabilization arises because high cell densities cause repulsion, preventing aggregation.

The size of the aggregation patterns is also affected. The dominant wavenumber qmax, which corresponds to the maximum of the eigenvalue spectrum λ(q), determines the pattern wavelength . By differentiating Eq 14, we find that the dominant wavenumber for the model without exclusion is

while for the model with exclusion, it is

Since for all , and the arccosine function is monotonically decreasing, it follows that

Thus, aggregation patterns in the exclusion model are larger than those in the non-exclusion model.

In the non-local case, the form of the eigenvalues is more complicated, however the predictions are qualitatively similar: no patterns are formed for high , and patterns in the exclusion model are larger than in the model without exclusion. Simulations with volume exclusion can be observed in Fig 4c and 4d. Note that, for the same parameter values a with no volume exclusion, cell clusters appear more unstable. This is due to the fact that , so for a given density , the pattern is less stable.

Limitations of a naïve fractional approach

We have obtained the integro-PDE, Eq 11, bottom-up starting from our cellular automaton model. The resulting equation is markedly different from the local PDE, Eq 7. Naïvely, we could have defined the non-local model by replacing the ordinary derivatives in Eq 7 by fractional derivatives, claiming that such a replacement would take into account the superdiffusive nature of cells. Doing so, would result in the fractional PDE

(17)

where

and is the Riesz fractional derivative. Since both the fractional Laplacian operator and the Riesz fractional derivative are linear, and applying them to a constant yields the zero function, the spatially homogeneous steady state is a solution of this equation. Linearizing around this solution, and considering that applying the Riesz derivative twice is equivalent to applying the fractional Laplacian operator, we obtain

Using the Fourier transform of the fractional Laplacian,

we obtain the ordinary differential equation in Fourier space

Thus, we see that the whole eigenvalue spectrum is negative when and positive otherwise. This means that, in this naïve fractional model, there is no gradual stabilization of the homogeneous steady state with increasing s, nor is there a dominant eigenvalue resulting in pattern formation, unlike our model.

Discussion

In this paper, we studied an LGCA model of adhesive interacting cells, with Zeta-distributed jump lengths. We considered three migration modes: single-jump (classic) migration, where cells deterministically move by one lattice site every time step, cell-wise cell migration, where cells jump to other lattice sites independently of other cells within the same velocity channel, and block migration, where all cells within the same velocity channel travel to the same node. Mathematically, we derived a conventional, non-linear PDE for the single jump case, as well as a non-linear, non-local, fractional PDE for the block migration case. We found that, in the classic single jump case, cells stop diffusing and start aggregating as soon as either the adhesion sensitivity or the cell density is sufficiently high. Conversely, in the long-jump case, we found that long jumps mainly stabilize the homogeneous steady state, when the distribution tail is sufficiently heavy. Computationally, we found that all models behave qualitatively the same when the probability of performing long jumps decays fast enough. Furthermore, we observe that increasingly probable long jumps gradually inhibit pattern formation. Finally, we found that, for highly adhesive cells and jump distributions with moderately heavy tails, pattern formation is intermittent and patterns appear metastable.

Mathematically, our study has confirmed that, contrary to what some models assume, changing the movement strategy of agents from “diffussive” to “superdiffusive” does not simply result in exchanging local differential operators to non-local operators in the continuous PDE descriptions. In this regard, our observations agree with those found previously by other authors using physical arguments [41]. Indeed, our model shows much richer behavior than the naïve fractional model. Furthermore, our work highlights the potential of agent-based models as bottom-up tools to define novel partial differential equation models, among other macroscopic models[32, 42].

Biologically, we have found that highly-adhesive cells can still become invasive as long as they are highly motile, which agrees with experimental observations [43, 44]. Furthermore, our simulations show that even if cells seem to form aggregates, these could be metastable and the aggregate could spontaneously become malignant if cells have a considerable probability to move far away. We can rescale the automaton so that the time step length and lattice sizes reflect experimental obsevations. For example, for B16-F1 cells, by adjusting experimental mean run lengths and mean run times from [14] to the mean jump length and time step length in our model for corresponding parameter values, we obtain that the time step length in our simulations corresponds to approximately 9 minutes, while the length between adjacent nodes should be between 2 to 15 μm. Therefore, our 1000 time step simulations would correspond to about 9000 minutes, or about 6 days, enough time to observe metastable behavior. This could explain the sudden malignancy of apparently benign tumors in vivo after xenografting [45]. This observation also agrees with experimental findings which have found that clustering tumor cells are orders of magnitude more efficient in establishing metastases than single cells [46], since several cells could travel together in a single metastable cluster, and spontaneously breaking apart releasing hundreds of cells far away from the main tumor body. Our model could be experimentally tested by using metastatic breast cancer cells, which show Lévy dynamics [14], and regulating their E-cadherin expression, which controls cell-cell adhesivity, and comparing the rate of divergence from a homogeneous initial state, to the model predictions.

Although the model we have proposed in this work has been simplified to ease its mathematical analysis, it can be expanded to account for other types of interaction, volume exclusion effects, correlations between cell velocities during interaction, among others. In the future, we plan to enhance the physical realism of the model by considering superdiffusive behavior originating from non-Markovian time correlations, rather than from power-law jump-length distributions. We also plan to study the qualitative behavior of Eq 11 as well as potential deviations from the LGCA model for different weights of the Zeta distribution tail.

Acknowledgments

The author thanks Andreas Deutsch, Simon Syga, and Alberto Salda na for fruitful discussions which shaped this paper, and Zuriel Yescas for his input in the integro-PDE model derivation.

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