The spontaneous emergence of leaders and followers in a mathematical model of cranial neural crest cell migration

TL;DR

This study addresses how leader and follower behaviors can emerge in cranial neural crest cell migration without predefining cell identities. It introduces a polarity-based agent-based framework where all cells share identical rules and a polarity vector that evolves from time-averaged VEGF exposure, coupled with spring-like intercellular adhesion. The model demonstrates spontaneous emergence of leader-like cells at the migratory front and follower-like cells behind, while reproducing experimental perturbations from the chick embryo, such as transplantation and ablation. The work provides a mechanistic, experimentally consistent explanation for emergent leadership in CNCC streams and suggests broader applicability to other forms of collective cell migration.

Abstract

Many agent-based mathematical models of cranial neural crest cell (CNCC) migration impose a binary phenotypic partition of cells into either leaders or followers. In such models, the movement of leader cells at the front of collectives is guided by local chemoattractant gradients, while follower cells behind leaders move according to local cell-cell guidance cues. Although such model formulations have yielded many insights into the mechanisms underpinning CNCC migration, they rely on fixed phenotypic traits that are difficult to reconcile with evidence of phenotypic plasticity in vivo. A later agent-based model of CNCC migration aimed to address this limitation by allowing cells to adaptively combine chemotactic and cell-cell guidance cues during migration. In this model, cell behaviour adapts instantaneously in response to environmental cues, which precludes the identification of a persistent subset of cells as leader-like over biologically relevant timescales, as observed in vivo. Here, we build on previous leader-follower and adaptive phenotype models to develop a polarity-based agent-based model of CNCC migration, in which all cells evolve according to identical rules, interact via a pairwise interaction potential, and carry polarity vectors that evolve according to a dynamical system driven by time-averaged exposure to chemoattractant gradients. Numerical simulations of this model show that a leader-follower phenotypic partition emerges spontaneously from the underlying collective dynamics of the model. Furthermore, the model reproduces behaviour that is consistent with experimental observations of CNCC migration in the chick embryo. Thus, we provide an experimentally consistent, mechanistically-grounded mathematical model that captures the emergence of leader and follower cell phenotypes without their imposition a priori.

Figures (5)

• Figure 1: Schematics of previous leader–follower mclennan2012multiscalemclennan2023colec12mclennan2017danmclennan2015vegfjohnson2025mathematical and signal integration schumacher2019neural models, together with the spring–polarisation model of CNCC migration formulated here. In leader--follower models, a fixed subset of leader cells at the front of streams move according to VEGF gradients detected in the cranial micro-environment. Follower cells, in weaker gradients of VEGF, move according to cell--cell guidance cues, forming connected chains of follower cells led by a leader cell at the front. In the signal integration model, all cells migrate according to a combination of chemical and cell--cell guidance cues. The movement of each cell is determined by a linear combination of these two mechanisms of movement, with the relative contribution of each mechanism governed by the magnitude of the chemical gradient detected at the time of movement. In the spring--polarisation model, all cells interact with their neighbours via spring-like mechanical interactions and carry a cell-specific polarity vector that evolves in time according to VEGF gradient sensing. Movement occurs according to a combination of forces from spring-like interactions and a force determined by the orientation and magnitude of the polarity vector.
• Figure 2: A) Example model simulation ($t=24\rm{h}$) for the parameter set $\{f_0=1000\mu\rm{m\,h^{-1}}, k_{\mathrm{spring}}=100\rm{h^{-1}}, \lambda=300\rm{h^{-1}}\}$. Non-zero polarisation ($||\mathbf{m}_{i}|| > 0$) at the leading edge of the stream is maintained due to strong local VEGF gradients, driving collective migration towards ba2 over a biologically relevant distance. The statistic $x_{\rm{max}}$ denotes the maximum $x$-coordinate across all cells in the stream when migration ends. B) $\langle x_{\rm{max}}\rangle$ as a function of $\lambda$, $f_0$, and $k_{\rm{spring}}$. Yellow triangles denote regions of parameter space in which $>10\%$ of model simulations result in leader escape, where at least one cell lies more than three cell radii ahead of the bulk when migration ends. White circles denote regions of parameter space in which $\langle x_{\rm{max}}\rangle$ exceeds $800\mu$m. The blue star denotes the representative parameter set that is used to analyse model behaviour in Sections \ref{['emergence']} and \ref{['experiments']}. C) Scatter plots of $||\textbf{m}_{i}||$ as a function of $x$ at $t=24$h as for small, intermediate, and large values of $\lambda$ for the representative parameter set $\{f_0=1000\mu\rm{m\,h^{-1}}, k_{\mathrm{spring}}=100\rm{h^{-1}}\}$. The results for each parameter combination are averaged over $500$ simulations.
• Figure 3: Schematics of the relationship between $f_0$ and $k_{\rm{spring}}$ and their effect on CNCC migration in the model. A) When $f_0$ is low, the active force induced by VEGF gradients is insufficient to drive migration along the domain. B) When both $f_0$ and $k_{\rm{spring}}$ are sufficiently large, the force induced by VEGF gradients is sufficient to drive migration along the domain, leading to typical CNCC migratory formations. C) If $f_0$ is much larger than $k_{\rm{spring}}$, the active force induced by VEGF gradients leads to the escape of highly polarised cells at the leading edge of collectives, resulting in aberrant migration. Adjacent to each schematic are kymographs displaying the density of cells as a function of $x$ in numerical model simulations when migration ends at $t=24$h (500 simulation repeats).
• Figure 4: A) Typical model simulation for $\{f_0=1000\mu\rm{m\,h^{-1}}, k_{\mathrm{spring}}=100\rm{h^{-1}}, \lambda=300\rm{h}^{-1}\}$. Initially, CNCCs emerge into regions of high VEGF concentration, and thus, begin to generate large chemical gradients as VEGF is degraded. The generation of large VEGF gradients induces a strong chemical polarisation in CNCCs, which results in their movement along the domain. Cells emerging at later times emerge into regions of depleted VEGF, and therefore, remain largely unpolarised. As such, their movement is primarily governed by intercellular attraction-repulsion mechanisms. Throughout migration, CNCCs at the front of the collective remain highly polarised, thus creating a persistent subset of cells for which chemotaxis is the primary determinant of directional movement. B) Scatter plot of $||\mathbf{m}_{i}||$ for $\{f_0=1000\mu\rm{m\,h^{-1}}, k_{\mathrm{spring}}=100\rm{h^{-1}}, \lambda=300\rm{h}^{-1}\}$ at $t=8\rm{h}, 16\rm{h}$, and $24\rm{h}$. Cells at the leading edge of streams are highly polarised due to consistent exposure to large VEGF gradients. Cells behind the leading edge remain unpolarised. C) $\left<||\mathbf{m}_{i}(t=24\rm{h})||\right>$ as a function of $x$ for the same data set as B). Non-zero mean polarisations are restricted to the green region at the front of the stream. All other cells in the red region are largely unpolarised when migration ends. The data in Figures B) and C) were collected over 500 model simulations.
• Figure 5: The results of perturbations to numerical model simulations based on prior experiments conducted in the chick embryo mclennan2012multiscalerichardson2016leader. A) Upon the transplantation of tissue containing $n=15$ unpolarised cells ahead of the leading edge of CNCC streams $12$h after migration begins, migration transiently ceases, as there are no longer polarised cells at the leading edge to guide migration. After a short transient period, cell-induced VEGF gradients form in the region of transplantation. Subsequently, transplanted cells become polarised, such that migration resumes in a typical stream formation. B) Upon the ablation of $n=20$ cells at the leading edge of CNCC streams $12$h after migration begins, the remaining cells at the leading edge of the post-ablation stream are exposed to local gradients in VEGF, which drives their polarisation and a subsequent resumption in active migration. C) When polarised cells are transplanted into the region of the r4--ba2 pathway adjacent to the embryonic hindbrain $12$h after migration begins, they are positioned in a region of degraded VEGF and, therefore, cannot gain the polarisation required to invade ba2.