ODE Constrained Mixture Modelling: A Method for Unraveling Subpopulation Structures and Dynamics

Abstract

Functional cell-to-cell variability is ubiquitous in multicellular organisms as well as bacterial populations. Even genetically identical cells of the same cell type can respond differently to identical stimuli. Methods have been developed to analyse heterogeneous populations, e.g., mixture models and stochastic population models. The available methods are, however, either incapable of simultaneously analysing different experimental conditions or are computationally demanding and difficult to apply. Furthermore, they do not account for biological information available in the literature. To overcome disadvantages of existing methods, we combine mixture models and ordinary differential equation (ODE) models. The ODE models provide a mechanistic description of the underlying processes while mixture models provide an easy way to capture variability. In a simulation study, we show that the class of ODE constrained mixture models can unravel the subpopulation structure and determine the sources of cell-to-cell variability. In addition, the method provides reliable estimates for kinetic rates and subpopulation characteristics. We use ODE constrained mixture modelling to study NGF-induced Erk1/2 phosphorylation in primary sensory neurones, a process relevant in inflammatory and neuropathic pain. We propose a mechanistic pathway model for this process and reconstructed static and dynamical subpopulation characteristics across experimental conditions. We validate the model predictions experimentally, which verifies the capabilities of ODE constrained mixture models. These results illustrate that ODE constrained mixture models can reveal novel mechanistic insights and possess a high sensitivity.

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.…”

Bayesian inference of reaction kinetics from single-cell recordings across a heterogeneous cell population

“…A combination of ODEs and mixture models is treated in [56]. This allows flexible modeling, and has the advantage of incorporating not only the RRE, but also moment approximations and LNA models, as computing these also reduces to the solution of ODEs.…”

Elucidating Cellular Population Dynamics by Molecular Density Function Perturbations

“…For example, gene transcriptional processes have been shown to occur in stochastic (random) bursts [4][5][6]. Many modeling frameworks have been used to capture cellular heterogeneity-for example, by using ensemble models (EM) of ordinary differential equations (ODEs) [7][8][9], population balance models (PBMs) [10], stochastic ordinary differential equations (SDEs) [11,12], and chemical master equations (CMEs) [13][14][15]. In these models, the cell-to-cell variability is described by a probability density or distribution function of cell state variables.…”