Bayesian inference of agent-based models: a tool for studying kidney branching morphogenesis

Abstract

The adult mammalian kidney has a complex, highly-branched collecting duct epithelium that arises as a ureteric bud sidebranch from an epithelial tube known as the nephric duct. Subsequent branching of the ureteric bud to form the collecting duct tree is regulated by subcellular interactions between the epithelium and a population of mesenchymal cells that surround the tips of outgrowing branches. The mesenchymal cells produce glial cell-line derived neurotrophic factor (GDNF), that binds with RET receptors on the surface of the epithelial cells to stimulate several subcellular pathways in the epithelium. Such interactions are known to be a prerequisite for normal branching development, although competing theories exist for their role in morphogenesis. Here we introduce the first agent-based model of ex vivo kidney uretic branching. Through comparison with experimental data, we show that growth factor-regulated growth mechanisms can explain early epithelial cell branching, but only if epithelial cell division depends in a switch-like way on the local growth factor concentration; cell division occurring only if the driving growth factor level exceeds a threshold. We also show how a recently-developed method, \"Approximate Approximate Bayesian Computation\", can be used to infer key model parameters, and reveal the dependency between the parameters controlling a growth factor-dependent growth switch. These results are consistent with a requirement for signals controlling proliferation and chemotaxis, both of which are previously identified roles for GDNF.\n\nAuthor SummaryA number of important congenital disorders arise due to incomplete development of the mammalian kidney. Elucidating the cause of these conditions requires an understanding of the mechanisms that contribute to kidney morphogenesis. Whilst experimental work has suggested several candidate mechanisms, their importance is still not well understood. Here we develop a computational model of kidney morphogenesis at the individual cell level to compare these different hypotheses. Guided by existing experimental evidence we propose that a generic growth factor, that we term \"GDNF\", produced from the mesenchyme surrounding the epithelium, can drive a number of cellular responses. Simulations of our agent-based model reveal that diffusion of GDNF, coupled with GDNF-stimulated epithelial cell division, can generate the branching patterns seen in ex vivo kidney explant experiments. We also find that branching depends on the sensitivity of cell proliferation to changes in GDNF levels. In particular our model only generates realistic branching when there is significant variation in GDNF levels along the boundary of the epithelium, and most cells divide only if the local concentration of GDNF exceeds a threshold value. We conclude that feedback between mesenchymal cells that produce GDNF, and epithelial cells that consume it, is vital for normal kidney organogenesis.