Early embryonic vascular patterning by matrix-mediated paracrine signalling: a mathematical model study

Abstract

During embryonic vasculogenesis, endothelial precursor cells of mesodermal origin known as angioblasts assemble into a characteristic network pattern. Although a considerable amount of markers and signals involved in this process have been identified, the mechanisms underlying the coalescence of angioblasts into this reticular pattern remain unclear. Various recent studies hypothesize that autocrine regulation of the chemoattractant vascular endothelial growth factor (VEGF) is responsible for the formation of vascular networks in vitro. However, the autocrine regulation hypothesis does not fit well with reported data on in vivo early vascular development. In this study, we propose a mathematical model based on the alternative assumption that endodermal VEGF signalling activity, having a paracrine effect on adjacent angioblasts, is mediated by its binding to the extracellular matrix (ECM). Detailed morphometric analysis of simulated networks and images obtained from in vivo quail embryos reveals the model mimics the vascular patterns with high accuracy. These results show that paracrine signalling can result in the formation of fine-grained cellular networks when mediated by angioblast-produced ECM. This lends additional support to the theory that patterning during early vascular development in the vertebrate embryo is regulated by paracrine signalling.

Figure 1. Paracrine chemotaxis model for vasculogenesis.

Angioblasts (endothelial progenitor cells) are derived from mesodermal cells and assemble into polygonal networks under instructive paracrine signalling provided by the endoderm. Endodermal cells express pro-vascular growth factors such as VEGF. Angioblasts are located in the space between endoderm and mesoderm, surrounded by extracellular matrix (ECM). Angioblasts produce ECM molecules (such as heparan sulphates and fibronectin) with VEGF binding domains (depicted in yellow). This matrix thus acts to store chemotactic growth factors, which provides spatial cues for cell migration.

Figure 2. In vivo vascular network in the quail embyro.

Laser confocal microscope reconstruction of the extra-embryonic (A) and intra-embryonic (B) vasculature of early quail embryos (36–40 hours of incubation). Embryonic blood vessels are identified by the QH1 antibody (red). Cell nuclei have been counterstained with DAPI (blue). The inserts depict extra-embryonic (1) and intra-embryonic (2) vascular networks in more detail; the former are used in analysis and validation of the mathematical model.

Figure 3. Morphometric comparison.

Comparison between experimental (in blue) and simulated (in red) vascular networks (after 3000 MCS). (A) Binary images over cellular structures (green) overlayed with skeletonized network (red), detected branching points (blue points) and corrected nodes (blue circles). (B) Morphometric statistics. Boxes show average values (n = 2 for experiments; n = 10 for simulation) and error bars indicate standard deviation. (C) Distributions of morphometric properties. Lines show average values; filled areas indicate standard deviations.

Figure 4. Sensitivity analysis of PDE parameters.

(A) Sensitivity to rates of binding (, VEGF and ECM production ( and ), and degradation of soluble VEGF (). Changes in various morphometric properties were measured for simulations (n = 3) in which each parameter was independently varied by a 10-fold decrease (blue) and a 10-fold increase (red). (B) Sensitivity to VEGF diffusivity. Morphometric quantities are shown for simulations (n = 3) with non-diffusive VEGF (), with normal VEGF diffusion (), and with well-mixed VEGF ().

Figure 5. Sensitivity analysis of chemotactic signal strength.

Sensitivity of morphometric parameters to relative strength of bound () and soluble () VEGF. Red points (lacunae) and blue points (nodes) show averages of measured quantities in simulations (n = 3), half-transparent regions represent standard deviations. Insets show portions of networks (200×200) where the relative signalling strength () is set to soluble-VEGF-only (, left), bound-VEGF-only (, right) and equal strengths (, center). Arrowhead indicates the reference value.

Figure 6. Dynamics of vascular network formation.

(A) Detail of simulated tissue at various time points, showing the cells (top) and relative concentrations of bound VEGF with isolines (bottom). (B) Dynamics of number of isolated cellular structures (blue) and number of lacunae (red), half-transparent regions indicates standard deviation (n = 10). (C) Inset depicts remodeling in a small region. This occupancy map is constructed by averaging over binary images in the interval between 1000 and 2000 MCS; lines show cell boundaries at 2000 MCS. Grey/white pixels are cells/lacuna which remained unchanged over this period; colored pixels indicates how long a pixel has been occupied by cells. It shows the creation of new connections (arrows) increasing the number of lacunae.

Figure 7. Cell elongation.

Distribution of cell lengths at different time points during the simulation. Red line and inset depict early, blue depicts late in development. Lengths are normalized to isotropic cells given the target area (, where is the scaling factor per pixel). Cells become increasingly anisotropic and elongated during vascular patterning. Filled regions represent standard deviation.

Figure 8. Morphometric dependence on cell density.

A number of morphometric properties of simulated networks are presented as a function of both cell density (number of cells per area) and coverage (the ratio of angioblasts-covered pixels to the total number of pixels). Averaged over 10 simulation runs, transparency indicates standard deviation. Three optima are shown from top to bottom, at increasing cell densities. Parameters as in table 1.