Elsevier

Journal of Theoretical Biology

Volume 338, 7 December 2013, Pages 66-79
Journal of Theoretical Biology

Modelling cell turnover in a complex tissue during development

https://doi.org/10.1016/j.jtbi.2013.08.033Get rights and content

Highlights

  • We develop a model of cell cycle and proliferation in an organ sub-compartment during development.

  • The model combined with experimental imaging predicts factors driving compartment dynamics.

  • The method is demonstrated on experimental data from the cap mesenchyme of the developing kidney.

  • Significant changes in cell cycle length and cell differentiation are detected by the model.

Abstract

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.

Introduction

Being able to determine the magnitude and dynamics of a cell population over time is invaluable for analyzing tissue development, homeostasis, and the origins of disease. Most cell populations do not occupy a steady state but instead are in flux, notably so in developing tissues. For any given cell population within a tissue, the size of the population will depend upon the rate of proliferation, entry into or exit of cells from that population where entry can include inward migration or other cells adopting that particular fate and exit can include cell death or differentiation into a distinct cell type (Fig. 1D). Proliferation involves the passage of a cell through the phases of the cell cycle to generate two daughter cells (Fig. 1E). While markers are available to identify what phase of the cell cycle a given cell is passing through, the length of each phase can vary and cells within a given population may be actively passing through the cell cycle or resting in G0 (quiescence) (Fig. 1E) (Cooper, 2000). For those cells actively proliferating, a variety of regulators of gap phases G1 and G2 can affect the rate at which a given cell passes through the cell cycle and cell division within a population is usually asynchronous (Cooper, 2000). Hence, the average proportion of cells within any one phase of the cell cycle can only yield relative proliferation comparisons and not absolute measures of cell cycle length.

The passage of any cell through the cell cycle can also yield different outcomes. Many developing organs contain progenitor populations responsible for generating more differentiated cell types. These populations are often self-renewing, however this can include both asymmetric self-renewal (parent cell replaces itself and generates one distinct progeny) and symmetric self-renewal (parent cell replaces itself with two other cells of the same type) (Fig. 1D). Even symmetric cell division can result in the formation of two differentiated cells as opposed to doubling the number of the initial cell type (Fig. 1D). Without a marker for the differentiated state and/or a lineage tracing method for showing what cell gave rise to what progeny, it may not be possible to document exit via differentiation. Finally, cell death can result in a decline in population that may artificially appear as a reduced level of proliferation (Fig. 1D). As a result, modelling the turnover of a cellular population, even in vitro, is complicated. This is compounded when the population of interest is within a complex tissue or developing organ. Note that in the current paper we use exit to refer to exit from the compartment, rather than exit from the cell cycle via quiescence or death (Fig. 1).

In order to fully understand defects in development or disease, we require a capacity to morphometrically and quantitatively model the rate of cell turnover within complex tissues. Early work (Steel and Bensted, 1965, Steel, 1967, Steel, 1977) in this area dealt with tumour cell dynamics. A more recent paper is this area (Bertuzzi et al., 1997) provides a detailed analysis of cell cycle dynamics in tissue affected by non-uniform loss, but does not consider cumulative S-phase marker experiments. A large body of more recent work (see (Nowakowski et al., 1989, Takahashi et al., 1993, Takahashi et al., 1995, Caviness and Takahashi, 1995, Cai et al., 1997, Kornack and Rakic, 1998, Calegari et al., 2005, Charvet and Striedter, 2008) and many other papers citing (Nowakowski et al., 1989)) addresses cell cycle length in the developing neocortex and other brain tissue, using cumulative cell cycle phase labelling strategies as well as multiple labelling techniques, whilst in (Alexiades and Cepko, 1996), quiescent cells are also considered and multiple markers used inside the developing rat retina. These studies involve isolated populations and often ignore the effect of entry and exit, important considerations particularly during development and repair. However the most critical issue is that to our knowledge these methods all use the linear model given in (Nowakowski et al., 1989) for the interpretation of cumulative S-phase marker experiments, which assumes a steady state homogenous non-quiescent population in which the total volume or number of cells in a tissue is assumed to be constant with proliferation simply replacing those cells that have exited though processes such as apoptosis. In a developing organ such as the kidney, there is often rapid proliferation with cell numbers increasing by orders of magnitude in a matter of days (see Section 3.1 below). Other studies of cell populations over time involve mathematical models with many parameters, as in (Baker et al., 1998), not all of which may be accessible. Fitting such models to experimental data can be problematic in that solutions may not be unique or are sensitive to initial conditions and care must be taken when fitting them to data. Multiple sources of information are often necessary to reduce the number of free parameters.

When studying development in the mouse kidney it became apparent that there was a need to develop an understanding of and explicitly model the full range of cell behaviours inside self-renewing compartments over time. Our goal was to understand what drives change in population size during organogenesis and to be able to mathematically model the outcome for the organ, in particular when direct markers for certain behaviours are absent.

Here we develop a cell cycle and proliferation model appropriate to the timescales that occur in development in an identifiable compartment of an organ. The compartment represents a progenitor population within a developing organ which is growing in size as well as generating new cell types via differentiation. The method brings together mathematical modelling and quantified experimental data to provide a flexible framework to study cell turnover. Experimentally, a cumulative marker of cell division is central to the approach to generate data on proliferation within a cellular compartment. In this study we use the thymidine analog EdU. A mathematical model of the expected proportion of marked to unmarked cells over time is developed and from parameter fitting to the experimental data, cell cycle length and S-phase duration are determined. Care is taken to specify assumptions clearly and not introduce parameters unnecessarily. Key elements of the model are that homeostasis is not assumed and cells are asynchronous in their cell cycles.

The strategy is then to combine the cell cycle information so determined with timed, static measurements of compartment volume and proportion of cells in mitosis. As we show, this enables the extrapolation to earlier and later time points and provides information about changes in proliferation or exit from that population across development without the requirement for explicit markers of differentiation.

These combined modelling and experimental methods are demonstrated using the nephron progenitor population inside the developing kidney. This population, also referred to as the cap mesenchyme, is defined by the presence of Cited1 and Six2 proteins and is a self-renewing population of progenitor cells that differentiate into the epithelial nephrons of the kidney through a mesenchyme to epithelial transition(Georgas et al., 2009; Boyle et al., 2008; Kobayashi et al., 2008; Hendry et al., 2011). Previous studies have shown using lineage tracing that all differentiated cells within the nephrons, aside from the collecting duct, arise from cap mesenchyme cells and that the starting size of this population is determined at approximately 11.5 days of development (days post coitum; dpc) in mice (Kobayashi et al., 2008). This defines the population as one in which there is no entry, leaving only exit (death and differentiation) and proliferation to regulate the size of the population over time. See, for example, (Hendry et al., 2011) for a detailed description of the biological context and key developmental role of the cap mesenchyme. Using the methods developed in this paper, we define key parameters of cap mesenchyme morphogenesis such as cell cycle length and S-phase duration, and we have quantified the level of cell exit via differentiation from the compartment during subsequent development.

Section snippets

Methods

The aim of this work is to understand and model cell turnover in a marked compartment by quantifying each of the cell behaviours that contribute to change in the population size over time. In the following, we begin by describing the biological assumptions about the proliferating populations under study (Section 2.1). Primarily we will be concerned with the cap mesenchyme in the developing kidney, though the assumptions made should be applicable to a wide range of developing organs and cell

Results

Here, the methods will be used to demonstrate two results in the cap mesenchyme, which yield insight into the turnover of cells in the compartment over the experimental time window. These results will show how the different experimental strategies may be combined within the mathematical framework described.

Conclusion

A novel method for quantifying and modelling cell cycle length, population heterogeneity and developmental exit via differentiation within an organ compartment has been demonstrated. A key element has been to use a model of the cumulative S-phase marker uptake over time that is appropriate to the assumptions and timescales typically occurring in the rapidly changing environment of a developing organ and to be able to estimate the affect of potential confounding factors such as cell cycle

Author contributions

JL and DJM developed the mathematical models of the paper and substantially wrote the manuscript. NAH, MHL and ANC contributed to the intellectual design of the research and also wrote substantial parts and contributed to the figures. ANC designed and performed the bulk of the experimental work together with ALJ. All authors read and approved the final manuscript before submission.

Acknowledgements

ML is an NHMRC Senior Principal Research Fellow. This work was supported by the National Health and Medical Research Council (APP1002748) and the Human Frontiers Science Program (RGP0039/2011). Confocal microscopy and optical projection tomography were performed at the Australian Cancer Research Foundation Cancer Biology Imaging Facility. We wish to acknowledge the facility manager, James Springfield, for designing and constructing the OPT hardware and software.

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