A Nonlinear Mixed Effects Approach for Modeling the Cell-To-Cell Variability of Mig1 Dynamics in Yeast

Abstract

The last decade has seen a rapid development of experimental techniques that allow data collection from individual cells. These techniques have enabled the discovery and characterization of variability within a population of genetically identical cells. Nonlinear mixed effects (NLME) modeling is an established framework for studying variability between individuals in a population, frequently used in pharmacokinetics and pharmacodynamics, but its potential for studies of cell-to-cell variability in molecular cell biology is yet to be exploited. Here we take advantage of this novel application of NLME modeling to study cell-to-cell variability in the dynamic behavior of the yeast transcription repressor Mig1. In particular, we investigate a recently discovered phenomenon where Mig1 during a short and transient period exits the nucleus when cells experience a shift from high to intermediate levels of extracellular glucose. A phenomenological model based on ordinary differential equations describing the transient dynamics of nuclear Mig1 is introduced, and according to the NLME methodology the parameters of this model are in turn modeled by a multivariate probability distribution. Using time-lapse microscopy data from nearly 200 cells, we estimate this parameter distribution according to the approach of maximizing the population likelihood. Based on the estimated distribution, parameter values for individual cells are furthermore characterized and the resulting Mig1 dynamics are compared to the single cell times-series data. The proposed NLME framework is also compared to the intuitive but limited standard two-stage (STS) approach. We demonstrate that the latter may overestimate variabilities by up to almost five fold. Finally, Monte Carlo simulations of the inferred population model are used to predict the distribution of key characteristics of the Mig1 transient response. We find that with decreasing levels of post-shift glucose, the transient response of Mig1 tend to be faster, more extended, and displays an increased cell-to-cell variability.

What Population Reveals About Individual Cell Identity: Single-Cell Parameter Estimation of Models of Gene Expression in Yeast

“…Such models assume that the different observations arise from different realizations of the same stochastic process and, therefore, are still based on the notion of a virtual mean-although noisy-cell. In comparison, and despite recent methodological developments [27,28], few attempts have been made to infer extrinsic noise models from data, see [4,10,23,29,30] and our previous work [31]. We refer the reader to Karlsson et al [24] for a detailed discussion of these works.…”

Multi-Experiment Nonlinear Mixed Effect Modeling of Single-Cell Translation Kinetics after Transfection

“…(I) The standard two-stage approach (STS) estimates single-cell parameters and population distribution parameters sequentially (Almquist et al 2015;Karlsson et al 2015). First, parameters for every single cell are estimated independently by fitting an ODE to the respective trajectory.…”

“…Then, a population-wide parameter distribution is reconstructed according to the single-cell parameter estimates. The STS approach enjoys great popularity Kalita et al 2011;Karlsson et al 2015;Almquist et al 2015), because it is easy to implement, as many methods and tools developed for bulk data can be applied. However, the STS approach fails to distinguish between cell-to-cell variability and uncertainty of the estimated single-cell parameters, resulting in the overestimation of cell-to-cell variability (Sheiner & Beal 1983).…”

“…The implementation of the NLME approach is more involved (Wang 2007;Kuhn & Lavielle 2005;Beal & Sheiner 1980) and it's application computationally more intensive. Originally developed in pharmacology (Beal & Sheiner 1980), the NLME approach has recently risen in popularity for the analysis of single-cell data (Zechner et al 2014;Karlsson et al 2015;Almquist et al 2015;Llamosi et al 2016). It has been reported that NLME is more robust than STS in settings with large parameter uncertainty, as it reduces uncertainty (Sheiner & Beal 1983;Karlsson et al 2015) and removes estimation bias (Almquist et al 2015).…”

New Insights into Tree Height Distribution Based on Mixed Effects Univariate Diffusion Processes

“…In this study, to project height distribution for a given diameter, a one-dimensional SDE with mixed effects was employed. The main feature of mixed effects models is that they allow parameter vectors to vary from plot to plot by splitting regression coefficients into a fixed part, common to the population, and random components, specific to each plot [9]. Mixed effects models allow fixed and random parameters to be estimated simultaneously and evaluate the value of the random parameters for a location not present in the original estimation dataset.…”