Parameter Identifiability Under Limited Experimental Data in Age-Structured Models of the Cell Cycle

TL;DR

This work considers how the ability to collate population summary measurements across the literature, from different cell lines and/or experimental set ups, affects identifiability of parameters for a cell cycle model, and presents an age-structured PDE model in which cell-cycle phase progression follows a delayed gamma distribution.

Abstract

The mitotic cell cycle governs DNA replication and cell division. The effectiveness of radiotherapy and chemotherapy depends on cell-cycle position, with increased resistance during DNA replication and mitosis. Thus, accurate mathematical models of the cell cycle are essential for understanding and predicting treatment response. However, mathematical modellers often face the problem of a lack of publicly available, sufficiently resolved, time-series datasets for parametrising models. In this work, we consider how the ability to collate population summary measurements across the literature, from different cell lines and/or experimental set ups, affects identifiability of parameters for a cell cycle model. Initially synchronised cell populations gradually desynchronise over successive cycles, converging to balanced exponential growth (BEG) which is characterised by exponential population growth and steady, time-independent phase proportions. These proportions can be obtained from fluorescence-activated cell sorting (FACS) data. The increasing use of the Fluorescent Ubiquitination-based Cell Cycle Indicator (FUCCI) provides higher-resolution information on phase dynamics, such as minimum phase durations and variability. We present an age-structured PDE model in which cell-cycle phase progression follows a delayed gamma distribution. We derive analytical expressions for BEG phase proportions and other FUCCI-observable quantities, and use them to assess how data availability influences parameter identifiability. When parameters are not uniquely identifiable, we determine identifiable parameter groupings, thereby determining the minimum amount of data that must be available for successfully fitting structured population models of the cell cycle.

Figures (7)

• Figure 1: Heatmaps showing the values of the (a) mean, (b) variance and (c) minimum phase delay $T_1$ of the $G_1$ phase length demonstrate small variance in mean value, and larger discrepancies in variance and $T_1$ across $(\alpha_1, \beta_1)$ parameter space. The value of $T_1$ is calculated from equation \ref{['eta_1']} using BEG proportions in Table \ref{['beg_prop_table']}, which is then used to calculate the mean and variance from equations \ref{['mean_and_var']}. We exclude unfeasible regions where $T_1<0$.
• Figure 2: Simulations of the full PDE system \ref{['X_pde']}-\ref{['Q_pde']} show that less variance in the values of the $G_1$ phase lengths delays the return to BEG significantly from cases of higher variance. In both cases, $(\alpha_i, \beta_i, T_i)$, $i = 2,3$ are listed in Table \ref{['unique_fit_table']}. (a) The $G_1$ phase proportion over time for the two different variance cases. (b) All four phase proportions over time for the two different variance cases. For the smaller variance case, $(\alpha_1, \beta_1, T_1) = (1, 5, 4.03)$, whilst for the larger variance case, $(\alpha_1, \beta_1, T_1) = (1, 0.22, 0.046)$. Both simulations had identical initial conditions, which had all cells starting in $G_1$.
• Figure 3: The percentage difference between the max and minimum mean $G_1$ phase lengths captured by the PDE decreases as the BEG $\bar{G}_1$ and $\bar{Q}$ proportions vary. See main text for the definition of percentage difference.
• Figure 4: The mean and variance of the $G_1$ phase lengths remain consistent across feasible parameter space, despite variations in $\alpha_1$, $\beta_1$ and $T_1$ when the CV and BEG values are fixed. (a) $\alpha_1(\beta_1)$, (b) $T_1(\beta_1)$, (c,d) mean and variance of $G_1$ phase length calculated via \ref{['mean_and_var']}, across all viable $\beta_1$ values.
• Figure 5: Varying the imposed minimum phase lengths $(T_1, T_2)$ in a reduced two-phase cell cycle model identifies a rectangular region in parameter space that produces an excellent unique fit to the prescribed BEG phase proportions. (a) A heatmap of the log-transformed value of $n(\vec{x})$ at the fitted parameters, and (b) classification of each point in the parameter space as producing an "unacceptable", "acceptable" or "optimal" fit to the BEG proportions, as defined in the main text. The black dashed lines represent the analytical bounds $T_1^*$ and $T_2^*$, defined by equations \ref{['T1_ub']} and \ref{['T2_T3_ub']}.