A spatially-averaged mathematical model of kidney branching morphogenesis.

Kidney development is initiated by the outgrowth of an epithelial ureteric bud into a population of mesenchymal cells. Reciprocal morphogenetic responses between these two populations generate a highly branched epithelial ureteric tree with the mesenchyme differentiating into nephrons, the functional units of the kidney. While we understand some of the mechanisms involved, current knowledge fails to explain the variability of organ sizes and nephron endowment in mice and humans. Here we present a spatially-averaged mathematical model of kidney morphogenesis in which the growth of the two key populations is described by a system of time-dependant ordinary differential equations. We assume that branching is symmetric and is invoked when the number of epithelial cells per tip reaches a threshold value. This process continues until the number of mesenchymal cells falls below a critical value that triggers cessation of branching. The mathematical model and its predictions are validated against experimentally quantified C57Bl6 mouse embryonic kidneys. Numerical simulations are performed to determine how the final number of branches changes as key system parameters are varied (such as the growth rate of tip cells, mesenchyme cells, or component cell population exit rate). Our results predict that the developing kidney responds differently to loss of cap and tip cells. They also indicate that the final number of kidney branches is less sensitive to changes in the growth rate of the ureteric tip cells than to changes in the growth rate of the mesenchymal cells. By inference, increasing the growth rate of mesenchymal cells should maximise branch number. Our model also provides a framework for predicting the branching outcome when ureteric tip or mesenchyme cells change behaviour in response to different genetic or environmental developmental stresses.

Modelling cell turnover in a complex tissue during development.

The growth of organs results from proliferation within distinct cellular compartments. Organ development also involves transitions between cell types and variations in cell cycle duration as development progresses, and is regulated by a balance between entry into the compartment, proliferation of cells within the compartment, acquisition of quiescence and exit from that cell state via differentiation or death. While it is important to understand how environmental or genetic alterations can perturb such development, most approaches employed to date are descriptive rather than quantitative. This is because the identification and quantification of such parameters, while tractable in vitro, is challenging in the context of a complex tissue in vivo. Here we present a new framework for determining cell turnover in developing organs in vivo that combines cumulative cell-labelling and quantification of distinct cell-cycle phases without assuming homogeneity of behaviour within that compartment. A mathematical model is given that allows the calculation of cell cycle length in the context of a specific biological example and assesses the uncertainty of this calculation due to incomplete knowledge of cell cycle dynamics. This includes the development of a two population model to quantify possible heterogeneity of cell cycle length within a compartment and estimate the aggregate proliferation rate. These models are demonstrated on data collected from a progenitor cell compartment within the developing mouse kidney, the cap mesenchyme. This tissue was labelled by cumulative infusion, volumetrically quantified across time, and temporally analysed for the proportion of cells undergoing proliferation. By combining the cell cycle length predicted by the model with measurements of total cell population and mitotic rate, this approach facilitates the quantification of exit from this compartment without the need for a direct marker of that event. As a method specifically designed with assumptions appropriate to developing organs we believe this approach will be applicable to a range of developmental systems, facilitating estimations of cell cycle length and compartment behaviour that extend beyond simple comparisons of mitotic rates between normal and perturbed states.

Expression and functional analysis of Dkk1 during early gonadal development.

WNT signalling plays a central role in mammalian sex determination by promoting ovarian development and repressing aspects of testis development in the early gonad. Dickkopf homolog 1 (DKK1) is a WNT signalling antagonist that plays critical roles in multiple developmental systems by modulating WNT activity. Here, we examined the role of DKK1 in mouse sex determination and early gonadal development. Dkk1 mRNA was upregulated sex-specifically during testis differentiation, suggesting that DKK1 could repress WNT signalling in the developing testis. However, we observed overtly normal testis development in Dkk1-null XY gonads, and found no significant upregulation of Axin2 or Sp5 that would indicate increased canonical WNT signalling. Nor did we find significant differences in expression of key markers of testis and ovarian development. We propose that DKK1 may play a protective role that is not unmasked by loss-of-function in the absence of other stressors.