Analyzing longitudinal single-cell data with mixed-effect models

Abstract

Advances in time-lapse microscopy enable us to observe many individual cells for a long duration of time at high temporal resolution. The increasing availability of such data makes analysis an important open problem, for instance, to interpret underlying mechanisms leading to the observed cell-to-cell variability in the dynamic behavior of cells. Nonlinear mixed-effects models (NLMEs) enable us to quantify the different sources of variability observed in data by augmenting mechanistic models with a model for intra-cell and cell-to-cell variability. We model intra-cellular variability through a measurement error model and cell-to-cell variability through a multi-dimensional distribution from which every cell’s (kinetic) parameters are drawn. However, inferring and interpreting NLMEs from longitudinal single-cell data is challenging. The mechanistic models used are complex with many unknown parameters, and NLME estimation scales poorly with the number of parameters that vary between cells and with the number of cells. In this thesis, we show how to infer and interpret NLMEs by relying on the cell-specific (or individual) parameters. In chapter 2, we propose a simple and flexible pipeline to infer and interpret detailed dynamic NLMEs from time-lapse imaging data. By leveraging the quality of time-lapse microscopy data, we infer detailed NLMEs with the global two-stage method (GTS). We benchmark the method with a published model and dataset, studying the dynamics of an osmo-responsive reporter in individual budding yeast cells. We also show the scalability of the approach with a mechanistic model and a corresponding dataset for endocytosis of an amino acid transporter (Mup1) in budding yeast. While the state-of-the-art method is sensitive to the initial values used in the optimization and encounters numerical difficulties, the GTS gives reliable parameter estimates and requires weaker assumptions on the distribution of the cell-specific parameters and measurement errors. Furthermore, to interpret NLMEs, we propose variation-based sensitivity analysis to identify time-dependent causes of cell-to-cell variability, highlighting relevant sub-processes in endocytosis. Next, we turn to the task of model diagnosis, more specifically, identifying misspecifications in the assumed distribution of (individual) cell-specific parameters. For this, empirical-Bayes estimates (EBEs) of the individual parameters are often used, if the individual parameter estimates are certain. However, traditionally, NLMEs are used in sparse datasets, for instance, in clinical data, where the use of EBEs is often ineffective for model diagnosis. Datasets from time-lapse microscopy, meanwhile, are rich in the number of cells observed and the number of observations obtained in each cell. Since the mechanistic models we use may possess unidentifiable cell-specific parameters, we ask whether the EBEs can be used for model diagnosis in applications encountered in single-cell analysis. In chapter 3, by using a published model and corresponding dataset on mRNA translation after transfection in single-cells, we show that the uncertainties of parameter estimates at the single-cell level can result in spurious correlations between parameters in the population-level. Guided by the distribution of EBEs and by identifying the kinetic parameters that contribute most to the non-identifiability at the individual cell-level, we modify our assumption on the distribution of cell-specific parameters. With the modified model, we use linear discriminant analysis on the EBEs and parameter-covariate model-selection to identify mechanisms that explain the differences between two experiments. Finally, we ask if we can detect differences between the two experiments with EBEs while using a more complex model, where some cell-specific parameters are unidentifiable. Even with the complex model, we detect known differences between the two datasets; however, due to a considerable uncertainty of the cell-specific parameter estimates, our assumption on the distribution of cell-specific parameters influences the estimates of the NLME model and the EBEs. We, therefore, demonstrate and recommend using the EBEs and uncertainties of cell-specific parameter estimates to identify and rectify misspecifications in the distribution of cell-specific parameters. In chapter 4, we extend the NLME framework to account for lineage dependencies between cells using a bifurcating-autoregressive process. Given the cell-specific parameter estimates and their uncertainties, we extend the GTS to quantify the extent to which kinetic parameters (processes) are inherited from mother to daughters, and are conditionally correlated between sisters given their mother. Using in silico simulations based on the osmo-responsive gene expression system, we show that our framework performs better than the standard approach.