A Mathematical Model of Lymphangiogenesis in a Zebrafish Embryo

Abstract

The lymphatic system of a vertebrate is important in health and diseases. We propose a novel mathematical model to elucidate the lymphangiogenic processes in zebrafish embryos. Specifically, we are interested in the period when lymphatic endothelial cells (LECs) exit the posterior cardinal vein and migrate to the horizontal myoseptum of a zebrafish embryo. We wonder whether vascular endothelial growth factor C (VEGFC) is a morphogen and a chemotactic factor for these LECs. The model considers the interstitial flow driving convection, the reactive transport of VEGFC, and the changing dynamics of the extracellular matrix in the embryo. Simulations of the model illustrate that VEGFC behaves very differently in diffusion and convection-dominant scenarios. In the former case, it must bind to the matrix to establish a functional morphogen gradient. In the latter case, the opposite is true and the pressure field is the key determinant of what VEGFC may do to the LECs. Degradation of collagen I, a matrix component, by matrix metallopeptidase 2 controls the spatiotemporal dynamics of VEGFC. It controls whether diffusion or convection is dominant in the embryo; it can create channels of abundant VEGFC and scarce collagen I to facilitate lymphangiogenesis; when collagen I is insufficient, VEGFC cannot influence the LECs at all. We predict that VEGFC is a morphogen for the migrating LECs, but it is not a chemotactic factor for them.

Can VEGFC Form Turing Patterns in the Zebrafish Embryo?

“…Regarding the biophysics, we did not model convection because diffusion is the dominant transport phenomenon in the trunk. This is supported by the Péclet number calculated in our previous work: for the diffusivity of VEGFC, the maximum Péclet number is 0.14909 (Wertheim and Roose 2017 ). VEGFC and MMP2 can diffuse in the interstitial space, but their diffusion rates depend on the abundance of collagen I (Lutter and Makinen 2014 ).…”

“…To keep our model simple, we built it on the ventral–dorsal axis of the zebrafish embryo’s trunk. We made a reasonable approximation because the lymphatic ducts are distributed along that axis and the biochemical gradients in the trunk are along the ventral–dorsal axis (Wertheim and Roose 2017 ).…”

“…Regarding the biochemistry, we drew on our previous work (Wertheim and Roose 2017 ). In that study, we concluded that VEGFC, MMP2, and collagen I form the axis of the biochemistry underlying lymphangiogenesis.…”

“…Inspired by our previous model (Wertheim and Roose 2017 ), we decided to use a set of reaction–diffusion equations to model the aforementioned biochemical and biophysical events.…”

“…We have previously proposed a mathematical model about lymphangiogenesis in the zebrafish embryo’s trunk (Wertheim and Roose 2017 ); it describes the biochemistry of the process. Specifically, the concentration dynamics of the vascular endothelial growth factor C (VEGFC), matrix metalloproteinase 2 (MMP2), tissue inhibitor of metalloproteinases 2 (TIMP2), and collagen I are described.…”