Elongation, proliferation & migration differentiate endothelial cell phenotypes and determine capillary sprouting

Abstract

Background: Angiogenesis, the growth of capillaries from preexisting blood vessels, has been extensively studied experimentally over the past thirty years. Molecular insights from these studies have lead to therapies for cancer, macular degeneration and ischemia. In parallel, mathematical models of angiogenesis have helped characterize a broader view of capillary network formation and have suggested new directions for experimental pursuit. We developed a computational model that bridges the gap between these two perspectives, and addresses a remaining question in angiogenic sprouting: how do the processes of endothelial cell elongation, migration and proliferation contribute to vessel formation?

Results: We present a multiscale systems model that closely simulates the mechanisms underlying sprouting at the onset of angiogenesis. Designed by agent-based programming, the model uses logical rules to guide the behavior of individual endothelial cells and segments of cells. The activation, proliferation, and movement of these cells lead to capillary growth in three dimensions. By this means, a novel capillary network emerges out of combinatorially complex interactions of single cells. Rules and parameter ranges are based on literature data on endothelial cell behavior in vitro. The model is designed generally, and will subsequently be applied to represent species-specific, tissue-specific in vitro and in vivo conditions. Initial results predict tip cell activation, stalk cell development and sprout formation as a function of local vascular endothelial growth factor concentrations and the Delta-like 4 Notch ligand, as it might occur in a three-dimensional in vitro setting. Results demonstrate the differential effects of ligand concentrations, cell movement and proliferation on sprouting and directional persistence.

Conclusion: This systems biology model offers a paradigm closely related to biological phenomena and highlights previously unexplored interactions of cell elongation, migration and proliferation as a function of ligand concentration, giving insight into key cellular mechanisms driving angiogenesis.

Figure 1

Schematic of the three-dimensional model. Capillaries are represented by endothelial cells. An example of a growing network with four capillaries is shown in the gray inset. Cells are divided into segments. Each segment is represented by two nodes. Currently, cell segments are modeled as cylinders specified by a length and radius (gray inset); an activated segment's length and radius can change during a model run. The local environment surrounding a cell is defined in each voxel of the grid. In the present model, voxels contain values for the local VEGF concentration. All cell segments have the capability of sensing what is located in the 26 voxels surrounding each of its nodes. For every timestep of the current model, this sensing is restricted to the leading node of the tip cell (red) and the adjacent node (purple), shared by the tip and activated stalk segment. The local search for the highest growth factor gradient surrounding the leading node of a tip cell determines the direction the sprout tip moves.

Figure 2

Illustrations of cell movement represented by rules in the model. For clarity, movement is shown in two-dimensions. The tip cell is represented by a red node and segment; the node shared between the tip and stalk cells is yellow; and the blue node and segment is the adjacent stalk cell segment. Black segments and nodes represent quiescent vessels. Arrows represent direction of movement for nodes. (A) Schematic of a capillary with an activated tip cell. (B) Movement when there is no growth factor gradient. (C) Movement and the resulting cell segment positions when there is a VEGF gradient, and the effects of allowed stalk cell elongation. (D) Results when there is only elongation of the stalk cell occurring, and no additional migration of the tip cell. (E) Results when there is proliferation of the stalk cells.

Figure 3

Flowchart representing the main rules followed by tip and stalk cells throughout a run of the model. The rules are interconnected, as illustrated by the arrows.

Figure 4

Relative effect of [VEGF] on total vessel growth over time. (A) and (B) Effects of [VEGF] alone on total vessel length. Initial number of capillaries was three, and the number of initial sprouts varied from two to six, with branching allowed. Simulation sample size was five values for each concentration at a given time. Growth for this simulation was unrestricted in i- and j-planes, and the dimension of the k-axis was 400 μm. [VEGF] gradients and initial cell activation level ([VEGF] = 0.6 ng/ml) were held constant for all compared [VEGF] concentrations. (C) and (D) Comparison of sprout length changes as a function of VEGF (ng/ml) to experiments using human endothelial cell spheroids on 3D collagen gel. (C) shows fold increase compared to the control in each experiment, while (D) shows absolute changes in vessel length for the same experiments. Values are for growth from a single spheroid. Experiments in references [60-64] were for a mean of 10 spheroids, embedded in a matrix of collagen from rat-tails. Experiments in [60-62,64] used HUVEC alone in the spheroids, while reference [63] used a coculture of HUVEC and human umbilical artery smooth muscle cells. All experiments used 50 ng/ml VEGF₁₆₅ alone as the stimuli, except [61], where 25 ng/ml VEGF₁₆₅ and 25 ng/ml bFGF were added. Experimental data are shown by the purple bar [60], yellow bar [63], blue bar [61], orange bar [64] and red bar [62].

Figure 5

Results from the in silico experiments. Total vessel length from 2–72 hrs (A), a snapshot of vessel length 24 hrs after the onset of sprouting angiogenesis (B), and relative branching points over time (C) for the different in silico experimental configurations shown in Table 6. Dll4 = 1 corresponds to wild type, control conditions for this ligand.

Figure 6

Persistence comparisons. (A) through (D) use the default VEGF gradient (Table 3). Visual examples of how directional persistence affects capillary sprout morphology. (A) and (B) show two snapshots of random tip cell movement (A) and 20% intrinsic persistence weighting (B) in tip cell movement, at 48 hours. (C) Total vessel length from 2–72 hrs for the model, comparing 20% to 40% intrinsic persistence. (D) Total vessel length change from 2–72 hrs comparing intrinsic persistence of 20% with global persistence at 20% and 40% directional weighting. For (C) and (D), VEGF concentration is set as a uniform 0.6 ng/ml at each voxel point in the grid, i.e., there is uniform concentration and no gradient. (E) Maximum XY-plane distance reached beyond initial capillary structures for intrinsic persistence weighting of 20% compared to a random weight. (F) Total vessel length from 2–72 hrs for 20% intrinsic persistence weighting vs. random movement.

Figure 7

Analysis of the model's representation of branching for in vitro conditions in three dimensions. (A) Vessel length over time for stalk cell branching at a probability of 10% and 30%. (B) Corresponding number of branching points for (A). Insets show large scaled values for 2–24 hrs.

Figure 8

Effect of haploinsufficiency of Dll4 on blood vessel sprouting compared to control conditions. (A) Combined effect of [VEGF] and Dll4 haploinsufficiency (Dll4^+/-) on total vessel growth after 24 hours. (B) Number of sprout tips as a function of [VEGF] and Dll4 haploinsufficiency after 24 hours. For (B), [VEGF] represents both the initial [VEGF], and the [VEGF] used in migration and proliferation rates. Gridspace volume is 1.28 × 10⁶ μm³; initial tips were counted after two hours of stimuli. (C) and (D) Visual snapshots of the model output for control conditions (C) and Dll4^+/- (D) after 24, 40 and 200 hrs, with a mean local VEGF concentration above the activation threshold of 0.5 ng/ml. The 24 and 40 hrs runs are in 3D, while the 200 hrs run is shown in 2D.