Modeling the early stages of vascular network assembly

Abstract

In vertebrates, networks of capillary vessels supply tissues with nutrients. Capillary patterns are closely mimicked by endothelial cells cultured on basement membrane proteins that allow single randomly dispersed cells to self-organize into vascular networks. Here we provide a model including chemoattraction as the fundamental mechanism for cell-to-cell communication in order to identify key parameters in the complexity of the formation of vascular patterns. By flanking biological experiments, theoretical insights and numerical simulations, we provide strong evidence that endothelial cell number and the range of activity of a chemoattractant factor regulate vascular network formation and size. We propose a mechanism linking the scale of formed endothelial structures to the range of cell-to-cell interaction mediated by the release of chemoattractants.

Fig. 1. Capillary network formation proceeds along three main stages. (A–D) Human microvascular ECs were plated (125 cells/mm²) on Matrigel and the time course of network formation recorded by time-lapse videomicroscopy of a 4 mm² wide portion of surface. (E–G) Position of cell centroids obtained by Model (1–3) using the same number of cells and physical values of the relevant parameters as in panels A–D. Similar results have been obtained with macrovascular human ECs from cord veins.

Fig. 2. Tracking trajectories of EC plated on Matrigel. (A) First photogram of the film included as Supplementary data, showing a 1 mm² area of Matrigel surface. (B) EC trajectories showing a displacement >30 µm (average cell diameter) have been tracked. Red circles: EC starting point. White circles: arrival point. In the background a simulated concentration field of a chemotactic agent (see text), is shown in false colors (red: high concentration; blue: low concentration; arbitrary scale). Most trajectories are directed toward zones of higher cell concentration. (C) Magnification of a single cell trajectory from panel B. Arrows in the background represent the simulated chemoattractant gradient. (D) Scheme showing the meaning of the angles φ; and θ described in the text.

Fig. 3. Cell migration analysis of ECs plated on Matrigel in the absence or presence of a saturating amount of VEGF-A. Histograms of φ;, cos φ;, θ and cos θ (see Figure 2D) for the trajectories of ECs plated on Matrigel either in control culture conditions (green) or in the presence of a saturating (20 nM) amount of VEGF-A₁₆₅ (light blue). The observed densities of cos φ; and cos θ were fitted with Beta distributions (red lines) by maximum likelihood. The corresponding parameter estimates are given in Table I. For convenience, the same data are also shown as histograms of φ; and θ and the corresponding fitted distributions are shown as red lines. The observed densities in VEGF-A₁₆₅ saturating conditions are markedly more symmetric than those observed in control conditions, showing loss of directionality in EC motility. Histograms of φ; indicate that also after extinguishing VEGF-A gradients EC movement on Matrigel maintains a certain degree of directional persistence. However, histograms of θ show that in the presence of saturating amounts of VEGF-A₁₆₅ EC movement is completely decorrelated from the direction of simulated VEGF gradients. We checked the hypothesis that θ values in saturating conditions are uniformly distributed by performing a χ² goodness-of-fit test (P = 0.397). The same test applied to the θ values in control conditions gives a P = 3 × 10^–8, which allows rejection of the hypothesis at any reasonable significance level.

Fig. 4. Saturating VEGF-A gradients inhibits capillary network formation. (A and B) Human ECs from cord veins were plated (125 cells/mm²) on Matrigel in the absence (A) or presence (B) of a saturating amount of VEGF-A₁₆₅ (20 nM). Capillary-like network formation was recorded after 3 h and it is evident that saturating VEGF-A inhibits the first step of capillary-like formations, where ECs start moving toward their neighbors to eventually adhere and form a continuous multicellular network. Heat-inactivated VEGF-A₁₆₅ was inactive and did not allow capillary network formation (not shown). (C and D) Cell organization patterns obtained by simulating Model (1–3) using the same number of ECs as in panels A and B in the presence (C) or absence (D) of the chemoattractant mechanism. Compared with the control condition (C), simulated extinction of chemoattractant gradients (D) resulted in an inhibition of pattern formation similar to that obtained by extinguishing them experimentally.

Fig. 5. Appropriate EC density is required for capillary network formation. Types of capillary-like networks formed by plating different numbers of cells (n̄ = 50, 100, 200 and 400 cells/mm²) on Matrigel surface. (E–H) Results of numerical simulations of Model (1–3) using the same number of cells and physical values of relevant parameters as in (A–D). (I) Mean ± SD values (51 ≤ n ≤ 110) of chord lengths of network structures obtained with varying cell density (100–200 cells/mm²) in four different experiments (filled circle, open circle, filled triangle, open triangle). Points corresponding to different experiments but to the same cell density value have been separated horizontally for better readability.

Fig. 6. The range of cell-to-cell interaction mediated by chemoattractant(s) controls capillary pattern dimensions. (A–C) Results of numerical simulations obtained with three different values of the interaction range ξ and otherwise constant parameters. (D) Radial part of the two-point correlator (arbitrary units) for network structures simulated with ξ = 100 µm. The thin lines correspond to 10 different random realizations, the thick line is their average. The position of the absolute minimum of the thick line is ℓ_o. (E) Graph of ℓ_o versus ξ; each point is computed as explained in (D); the slope of the interpolating line is ≈ 0.63.