Collective cell migration plays an important role in development. Here, we study the posterior lateral line primordium (PLLP) a group of about 100 cells, destined to form sensory structures, that migrates from head to tail in the zebrafish embryo. We model mutually inhibitory FGF-Wnt signalling network in the PLLP and link tissue subdivision (Wnt receptor and FGF receptor activity domains) to receptor-ligand parameters. We then use a 3D cell-based simulation with realistic cell-cell adhesion, interaction forces, and chemotaxis. Our model is able to reproduce experimentally observed motility with leading cells migrating up a gradient of CXCL12a, and trailing (FGF receptor active) cells moving actively by chemotaxis towards FGF ligand secreted by the leading cells. The 3D simulation framework, combined with experiments, allows an investigation of the role of cell division, chemotaxis, adhesion, and other parameters on the shape and speed of the PLLP. The 3D model demonstrates reasonable behaviour of control as well as mutant phenotypes.
In this section, we describe additional biological background for our investigation. The PLLP consists of more than 100 cells (≈4-5 cells wide) when migration is initiated, which migrate about 2. 3 mm down the horizontal myoseptum during the time interval 22-48 hours post-fertilization [1]. Cells in the trailing region re-organize into rosettes, are periodically deposited and later develop into neuromasts, to form sensory organs of the posterior lateral line. The cells at the leading region are flatter and have a mesenchymal morphology [2]. The changes in cell morphology and behaviour across the PLLP is thought to be regulated by antagonistic Wnt-FGF signalling network [2–4]. The leading zone of the PLLP is characterized by high-levels of WntR activity while the trailing zone is dominated by FGFR activity. It is not known which Wnt ligand is involved although recent experimental work by members of our group (DDN and AC) implicates Wnt10a. More research has been done on the downstream Wnt signalling network [4]. WntR activity at the front of the PLLP promotes the expression of FGF3 and FGF10 ligands, as well as sef, considered to inhibit FGFR activity. Consequently, FGF pathway activation is inhibited in the leading cells, and the secreted FGF ligand can redistribute towards the rear. Activation of FGF signalling in the trailing domain subsequently drives the expression of Dkk1, a diffusible inhibitor of Wnt signalling [4]. In this way, Wnt and FGF signalling domains are established in the leading and trailing zones of the PLLP. Migration is dependent on chemokine signalling across the primordium, and is guided by a strip of CXCL12a, a chemokine that is produced by superficial muscle pioneer cells of the horizontal myoseptum, has limited diffusion on the surface of this layer, and determines the PLLP’s migration path [5–7]. Without CXCL12a the PLLP stalls. However, directional information is not encoded by CXCL12a, whose expression level is uniform. Instead, the differential expression of two chemokine receptors, CXCR4b and CXCR7b, across the length of the PLLP guides migration. Similar to the Wnt-FGF polarity, the expression of CXCR4b dominates in the leading end while expression of CXCR7b dominates in the trailing zone with some overlap [5,7]. Mechanisms that determine the differential expression of chemokine receptors in the PLLP are linked to the development of the polarized Wnt-FGF signalling, but are as yet incompletely understood [4,10]. Cells expressing CXCR4b respond to CXCL12a with protrusive and migratory behaviour. The cells expressing CXCR7b rapidly internalize and degrade CXCL12a. It has been proposed that this sets up a local gradient of CXCL12a that ensures unidirectional migration of the PLLP [9]. It was also suggested that the absolute level of CXCL12a serves as cue [8]. Moreover, recent laser ablation and cutting experiments in [8] revealed that the trailing cells undergo chemotaxis toward FGF ligand secreted by the leading PLLP cells. Evidence for this stems from several experiments described in [8]: (1) when trailing cells are physically separated from the leading cells by laser ablation, trailing cells move toward and eventually join with leading cells despite unpolarized expression of the chemokine receptor CXCR4b in the trailing fragment. (2) This movement is inhibited by treatment with the FGF inhibitor SU5402, suggesting that FGF signaling is essential for this process. (3) Furthermore, an isolated trailing cell cluster generated by laser ablation can be induced to migrate directionally by placing a bead soaked in recombinant FGF3 either ahead of or behind the cells. This suggests that while the leading and trailing cells coordinate migration by inducing a local CXCL12a gradient, the leader cells respond to chemokine cues and the trailing cells migrate toward the leading cells [8].
Our initial step was to explore mechanisms for the subdivision of the tissue into a Wnt-signalling leading domain and a FGF-signalling trailing domain. We specifically sought to determine how a spatially graded parameter, such as the steady state level of the Wnt receptors (W1), across the length of the primordium can set up this domain subdivision, and where the domain boundary would be positioned. We first explore this from a mathematical perspective, linking the outcome to underlying biochemistry, and then show the results numerically. We employ a 1D model simplification followed by the full 3D simulation for the PLLP. Having identified conditions for the formation of leading and trailing zones, we next asked what further conditions are required for collective migration of the PLLP. To probe this question, we assembled basic facts from the biology (surveyed above) supplemented, where needed, by reasonable minimal assumptions. In actual PLLP experiments, it is known that aside from the Wnt-FGF polarization of the tissue, the front and rear portions are also distinct in levels of chemokine receptors. Cells in the leading 60% of the PLLP express high levels of CXCR4b, whereas those in the remaining rear portion express high CXCR7b, [4]. This does not exactly coincide with the growth-factor zones: the leading 1/3 of the PLLP is the WntR activity zone [18]. However, here we made the simplification that from the standpoint of interactions with CXCL12a, the WntR-FGFR subdivision serves as reasonable proxy for the chemokine zone subdivision. One reason for this simplifying assumption is that we thereby avoid the further complications of untangling the signalling circuits that set up the zones of differential chemokine receptor expression. A second reason is that little is to be gained by this intermediate step currently, when the detailed biology of this step is largely unknown. It is well known that cell growth and division take place during the course of PLLP migration with the total number of cells doubling throughout the process [18,25]. This fact led us to inquire how such cell proliferation contributes to the zone formation, migration, and overall shape of the PLLP. While it was originally thought that leading zone cells proliferate faster [18,25], more recent experiments by two of us (AC and DDN) revealed a relatively constant and spatially uniform cell cycle length (539 ± 127 minutes) across the PLLP [26]. In the simulations the cell’s rate of growth in each time step is chosen from a uniform distribution between 0 and 0. 001 min−1 (corresponding to a cell cycle between 0 and 1000 minutes). Cell division occurs when the volume of a given cell has doubled. The daughter cells have the same parameter values as the mother cell. Results with a range of growth rate values are shown in Fig 10. For the higher growth rate, we find that as the PLLP migrates and elongates, the cells at the back no longer detect FGF, and eventually break away from the rest of the PLLP. The spatiotemporal FGF and Wnt ligand distribution is hard to quantify experimentally. Hence, direct comparisons of model predictions to experimental observations are not possible for the chemical distributions of FGF and Wnt ligands. For this reason, we sought simple experimental comparisons that could (in) validate the computational model. We consequently compared our model behaviour to the results of laser ablation (cutting) experiments, even though such experiments have already been successfully recapitulated in a computational model [8]. We further simulated several mutants to check whether our predictions agree with what has been observed. In the experiments, staining has been done for Wnt signalling cells, using lef1 as a marker, and for FGF signalling cells, using pea3 as a marker. Hence we color the cells in our simulations based on signalling cell behaviour.
Here, we investigated the lateral line primordium development and migration in zebrafish. In this intriguing system, a cluster of 100-200 cells maintains cohesion and directionality while migrating over a distance one order of magnitude longer than the cluster length. We first showed that the mutual inhibition of Wnt and FGF signalling can, on its own, set up the initial polarization of the PLLP into distinct leading and trailing domains under appropriate assumptions. Our model links ligand-receptor binding (in the Michaelis-Menten regime) to the inhibitory effect (of bound receptors on the antagonist receptors). Thereby, we could predict how the boundary between Wnt and FGF domains depends on parameters such as steady state receptor levels (in absence of inhibition) and sensitivity to inhibition by the antagonist bound receptors. The theoretical model discussed here has more universal applicability, and variations on this theme could be at play in other differentiation processes of tissue subdivision. In particular, “crossed gradients” (i. e. gradients of opposite slopes) in the two ratios of receptor-ligand properties that we identified as ϕ and ω could produce a tissue with a central zone of overlapped influence (transition across the bottom panels of Fig 4), or with a sharp boundary (transition across the top panels of this figure). For our model of the PLLP specifically, such crossed gradients do not exist because the spatial variation of ϕ and ω stem from the FGF and Wnt ligand distributions, which are both graded in the same direction (increasing towards the front). Nevertheless, tissue subdivision occurs, once the PLLP evolves into a bistable state, with WntR activity locked into the front, and FGFR activity overtaking WntR in the back. We have shown that the CXCL12a-expressing adhesive stripe provides directionality, as well as, coupled to active degradation by trailing cells, forms the guidance cue that directs PLLP migration along the lateral axis of the embryo. Given the viscous drag, it appears highly probable that all cells in the PLLP are actively motile. Simply having a “front engine” (a few cells at the front) pulling the entire mass would lead to migration stalling, and even fragmentation of the group, which is not observed under normal conditions. Chemotaxis is the simplest and most plausible mechanism for the active motion of trailing cells, though we do not excluded the possibility that trailing cells also actively follow leader cells through some other mechanism. The shape and speed of the collective migrating mass depends on numerous factors such as rate of cell growth (which determines cell division rate), active crawling and adhesion of cells to one another and to the underlaying adhesive substrate. When these factors are increased or decreased beyond certain ranges, shape and speed are abnormal, recapitulating a variety of mutants. Wnt signalling regulates cell growth rate, and FGF signalling is implicated in cohesion. Both signalling types determine differentiation and the distinct roles of leading and trailing cells. WntR active front cells sense the CXCL12a gradient and steer the cluster. Rear cells eventually express CXCR7b chemokine receptors and degrade CXCL12a. The model demonstrates that this could form the self-propagating gradient to which WntR cells orient. FGFR active cells eventually engender the initiation of neuromast deposition that we have not yet addressed in this paper. It is interesting to compare our results with those of previous modelling papers [8–12]. In Streichan et al. [9], the PLLP is modelled as a 1D rod, moving through a self-generated gradient of a single chemokine (CXCL12a), similar in flavor to “treadmilling” solutions in [27]. The rod binds the diffusible chemokine and moves with velocity assumed to be proportional to the gradient so created. The model for chemokine profile and rod position sustains a travelling wave solution, representing the steady-state long-range PLLP migration. The authors further considered deposition, assuming that below a threshold gradient of CXCL12a, the rear cells in a discrete 1D elastic array fall off. They derive an analytic expression for the migration velocity as a function of rates of ligand degradation and diffusion, the size of the PLLP and the rate of chemokine uptake. This model provides elegant analytic insights into minimal requirements for “self-generated” gradient chemotaxis in a general setting, while simplifying the biology and neglecting key features of the PLLP. For example, the authors do not consider interactions between CXCR7b and CXCL12a ligand, which is now well established in the literature. In comparison, our model begins to address this limitations by including some of these biochemical interaction, consequently limiting many of our results to numerical experiments. Two of us (AC, DDN) previously crafted a Netlogo model [8,10] as an aide to deciphering laser ablation experimental observations. The simulation uses “turtles” attached to one another with springs: when one moves, its neighbor moves with a slight lag, disallowing overlaps. This simulation had the advantage of close integration with experiments. While remaining a fairly elementary, it was able to capture key aspects of the biology, and explain experiments. A limitation was the absence of realistic forces of cell-cell interaction and friction. Consequently, a conclusion, based partly on the assumption of pulling exerted by few leading cells, was that cells respond to the CXCL12a concentration rather than CXCL12a gradient. In our simulation here, we endow a larger number of cells with the ability to sense, orient to, and migrate up the CXCL12a gradient to correct that feature. Furthermore, as we have argued, drag forces would stall the PLLP were it not for additional chemotaxis motion of the trailing cells. Allena and Maini [11] described chemical signalling coupled to cellular mechanics in the PLLP using a reaction-diffusion finite element model. Cells are depicted by 2D Maxwell viscoelastic elements inside an elliptical PLLP. The incorporation of additional signalling (FGF and Wnt ligands, CXCR4b and CXCR7b chemokines as “readouts” of the ligand distribution), coupling mechanics and biochemistry, and the successful recapitulation of mutant behaviours are all strengths of this model. As in Allena and Maini [11], we have adopted a hybrid model to describe the PLLP dynamics in 3D. Our discrete cell model builds on limitations of [11] by including interactions between the PLLP and CXCL12a, relative motion and rearrangement of cells, and more direct representation of intercellular forces. In contrast to [11], our model also explains the initial subdivision of the PLLP into leading and trailing zones. Another hybrid model [12] is based on cell-cell interaction, cell-substrate adhesion and cell differentiation. The model consists of discrete cells in 2D with a non-local sensing radius. The PLLP is described as a “flock” of cells accelerating via haptotaxis to CXCL12a, forces of Cucker-Smale alignment, attraction-repulsion, and chemotaxis to FGF gradient. The model captures the migration of the PLLP resulting from a graded distribution of CXCL12a. One issue with the representation of the PLLP as a “flock” is that macroscopic flocks and swarms generally operate in a high Reynold’s number regime, whereas we have chosen to neglect inertial forces and acceleration terms and to consider a low Reynold’s number regime for our cellular milieu. One interesting feature of [12] is that, by including cell-cell lateral inhibition in the trailing domain (and foci of FGF), their model can depict the formation of stable rosette structures within the PLLP (see their Figs 9 and 10). In contrast, we do not yet consider the rosette formation in our model but we base our cell motion and interactions on a closer concordance with recent biological results. In particular, we include downstream effects of Wnt/FGF signalling, i. e. , formation of receptor activity domains, chemotaxis of trailing cells to FGF, and degradation of SDF1a (CXCL12a) in the trailing domain. All models described above, including our own, would benefit from further experiments that could help to probe hypotheses or compare with predictions, in particular, the following experiments would be of primary interest: Primarily, our suggested experiments seek to determine the roles of the Wnt and FGF ligands in sustained organization and migration of the PLLP. Does an initial signalling event start the PLLP migratory process, or, as we have assumed in our model, is a continuous source of Wnt ligand required to maintain the organization and migration of the PLLP? In summary, we are not the first to model the PLLP migration, though ours is the first fully 3D simulation of its kind, with realistic cell-cell and cell-substrate forces. While the model is a drastic simplification of the biology, it nevertheless provides insights about the Wnt-FGF mutual inhibition, forces of propulsion and drag, and accounts for mutants. There are several opportunities for future work. First, there is the process of neuromast deposition that we consider in a future study. Second, there are many additional players in the signalling network responsible for segmenting the PLLP into leading and trailing zones, such as Dkk1 and sef, and including these players could aid in the understanding of their specific roles within the PLLP.
In these simulations we treat the system in one spatial dimension across the long axis of the PLLP. We then define the spatial variable x with values in 0 ≤ x ≤ L where x = L is the front of the primordium and x = 0 is the back. All boundaries were modelled as impermeable (no flux). We used the MATLAB pdepe function to approximate solutions to the system. The sharp boundary between the leading and trailing domains makes numerical approximation difficult. For this reason, we only approximate the solution over a short time interval. To obtain the position of the boundary between signalling domains in Fig 6, the 1D Wnt-FGF model was simulated to steady-state with 20 distinct parameter values (F0, F1, W0, pF, pW, b), no flux boundary conditions, and initial conditions W (x, 0) = 0. 01, F (x, 0) = 0. 005, WR (x, 0) = 0. 1, FR (x, 0) = 0. 01. From the steady-state concentration profiles, the boundary between the signalling domains was identified as the intersection between the Wnt (WR) and FGF receptor (FR) curves. For Fig 6, we plotted the numerically calculated boundary position versus p/pbaseline, the value of a given parameter, scaled by its baseline value in Table A in Supporting Information S6 Text. Parameter Estimation and Values. For the parameter sweeps, the following ranges were used: b ∈ [0,0. 05], F0 ∈ [0,0. 2], F1 ∈ [0,2], W0 ∈ [0. 0005,0. 0015], pF ∈ [15,35], pW ∈ [2,6]. The discrete-cell computational framework is modified from software developed by one of us (EP) for signalling and motility in multicellular systems [24]. (See Supporting Information S4 Text. Discrete cell model and [24,28] for technical details.) Cells are modelled as deformable ellipsoids in a three-dimensional (3D) domain. Cells move relative to one another, exert and respond to forces (chemotactic, adhesive, contact/exclusion, and random) in a low Reynolds number regime (neglecting inertia). FGF and Wnt ligand concentrations evolve in 3D based on a set of PDEs modified from Eqs. (19) in Supporting Information S4 Text. Discrete cell model to account for sources and sinks at individual cells. The ligands are depleted both through binding and internalization of the bound receptors as well as through nonspecific degradation in the domain. Cells can orient and move in the direction of a chemical gradient. In addition to chemotactic forces, the cells also adhere to other cells and to the surface on which they crawl. The strength of the cell-cell adhesion force is the same for all types of cells. However, cells adhere more strongly to the CXCL12a expressing surface, and, since CXCL12a forms a narrow stripe, this accounts for the elongated shape of the primordium. While direct data is not available, it is likely that some differences in adhesion exist between cell types. Here we arbitrarily assumed that WntR expressing cells have a 2-fold higher adhesion to the CXCL12a stripe than do other types of cells. This does not have a major qualitative impact on our results. The net force acting on each cell is calculated and the cells are moved according to the equation of motion (25). In a given simulation, the spatial concentration of the signalling molecules is calculated and updated, and then the cell positions are recomputed and updated each timestep. CldnB: lynGFP [6] embryos were treated from 32-36hpf with the indicated concentrations of SU5402 while shaking and then fixed in 4% PFA overnight. In situ hybridization for lef1 transcript was performed as previously described [29]. Following in situ hybridization, embryos were stained with anti-GFP antibody to visualize the CldnB: lynGFP. The lef1 ratio was computed by dividing the length of the lef1 domain by the total length of the PLLP.
Collective migration of a group of cells plays an important role in the development of an organism. Here we study a specific example in the zebrafish embryo, where a group of about 100 cells (the posterior lateral line primordium, PLLP), destined to form sensory structures, migrates from head to tail. We model the process from the initial polarization to the migration, with a focus on how tissue polarity could arise. Using a 3D deformable-ellipsoid cell-based simulation, we explore the effects of cell-cell, cell-substrate, and cell-chemical interactions. We discuss drag forces experienced by cells and what that implies about the inherent active motion of both leading and trailing cells. The model allows us to test how each of several biological parameters affects the shape, size, effective migration and speed of migration. A subsequent study will be aimed at understanding the formation and deposition of neuromasts.
signal transduction
signal inhibition
computer and information sciences
cell motility
developmental biology
cell biology
simulation and modeling
network analysis
cell migration
chemokines
biology and life sciences
signaling networks
wnt signaling cascade
cell signaling
chemotaxis
signaling cascades
research and analysis methods
Polarization and migration in the zebrafish posterior lateral line system