Background: Multi-cellular organisms as well as bacteria are faced with diverse, ever changing environments. To ensure survival and evolutionary success, bacterial populations exploit cell-to-cell variability, which increase the robustness against environmental changes. More complex organisms, such as mammals, evolved strategies to actively detect and respond to environmental changes. The building blocks for the necessary structures and functional units are cell types with distinct properties. Several statistical methods and modeling approaches have been introduced to analyze such structured populations, they are however either incapable of simultaneously analyzing different experimental conditions, computationally demanding and difficult to apply, or neglect information available in the literature.

Methods: To improve upon existing methods, we introduced ordinary differential equation (ODE) constrained mixture models. This hybrid modeling approach integrates ideas from statistics, namely mixture modeling, and mechanistic dynamical modeling. The mechanistic model enables the simultaneous analysis of multiple experimental conditions while the mixture components can exploit distributional information contained in the data. The fitting of different ODE constrained mixture models, representing different model hypotheses, to the available data allows for the assessment of the origin of cell-to-cell variability as well as subpopulation structures and dynamics.

To assess the proposed method we performed different simulation studies. Furthermore, ODE constrained mixture modeling has been used to study NGF-induced Erk1/2 phosphorylation in primary sensory neurons, a process relevant in inflammatory and neuropathic pain. We developed a simple pathway model for NGF-induced Erk1/2 phosphorylation and compared different hypothesis regarding the subpopulation structure and dynamics using the available time and dose response data. ODE constrained mixture modeling suggested that the population consists of two subpopulation with different NGF receptor abundances, which has been validated using additional experiments.

Discussion and conclusions: The ODE constrained mixture modeling introduced in this work bears great potential for the analysis of heterogeneous cell populations. The proposed method allows for a simple but pervasive analysis of heterogeneous cell systems and more profound, mechanistic insights. ODE constrained mixture models facilitate the reconstruction of subpopulation sizes and dynamics even in situations, where the subpopulations are hardly distinguishable. This is shown for simulation examples as well as for the process of NGF-induced Erk1/2 phosphorylation in primary sensory neurons.

ODE constrained mixture modeling is also highly flexible as the subpopulation dynamics can either be deterministic or stochastic. This only requires alterations in the ODE part, namely reaction rate equations or moment equations are used to model the dynamics.

cell-to-cell variability; structured population model; model selection