Parameter inference in dynamical systems with co-dimension 1 bifurcations

Sloppy model analysis identifies bifurcation parameters without normal form analysis

Bifurcation phenomena are common in multidimensional multiparameter dynamical systems. Normal form theory suggests that bifurcations are driven by relatively few combinations of parameters. Models of complex systems, however, rarely appear in normal form, and bifurcations are controlled by nonlinear combinations of the bare parameters of differential equations. Discovering reparameterizations to transform complex equations into a normal form is often very difficult, and the reparameterization may not even exist in a closed form. Here we show that information geometry and sloppy model analysis using the Fisher information matrix can be used to identify the combination of parameters that control bifurcations. By considering observations on increasingly long timescales, we find those parameters that rapidly characterize the system's topological inhomogeneities, whether the system is in normal form or not. We anticipate that this novel analytical method, which we call time-widening information geometry (TWIG), will be useful in applied network analysis.

Multistability in Macrophage Activation Pathways and Metabolic Implications

Macrophages are innate immune cells with a dynamic range of reversible activation states including the classical pro-inflammatory (M1) and alternative anti-inflammatory (M2) states. Deciphering how macrophages regulate their transition from one state to the other is key for a deeper understanding of inflammatory diseases and relevant therapies. Common regulatory motifs reported for macrophage transitions, such as positive or double-negative feedback loops, exhibit a switchlike behavior, suggesting the bistability of the system. In this review, we explore the evidence for multistability (including bistability) in macrophage activation pathways at four molecular levels. First, a decision-making module in signal transduction includes mutual inhibitory interactions between M1 (STAT1, NF-KB/p50-p65) and M2 (STAT3, NF-KB/p50-p50) signaling pathways. Second, a switchlike behavior at the gene expression level includes complex network motifs of transcription factors and miRNAs. Third, these changes impact metabolic gene expression, leading to switches in energy production, NADPH and ROS production, TCA cycle functionality, biosynthesis, and nitrogen metabolism. Fourth, metabolic changes are monitored by metabolic sensors coupled to AMPK and mTOR activity to provide stability by maintaining signals promoting M1 or M2 activation. In conclusion, we identify bistability hubs as promising therapeutic targets for reverting or blocking macrophage transitions through modulation of the metabolic environment.

Inferring kinetic parameters of oscillatory gene regulation from single cell time-series data

Gene expression dynamics, such as stochastic oscillations and aperiodic fluctuations, have been associated with cell fate changes in multiple contexts, including development and cancer. Single cell live imaging of protein expression with endogenous reporters is widely used to observe such gene expression dynamics. However, the experimental investigation of regulatory mechanisms underlying the observed dynamics is challenging, since these mechanisms include complex interactions of multiple processes, including transcription, translation and protein degradation. Here, we present a Bayesian method to infer kinetic parameters of oscillatory gene expression regulation using an auto-negative feedback motif with delay. Specifically, we use a delay-adapted nonlinear Kalman filter within a Metropolis-adjusted Langevin algorithm to identify posterior probability distributions. Our method can be applied to time-series data on gene expression from single cells and is able to infer multiple parameters simultaneously. We apply it to published data on murine neural progenitor cells and show that it outperforms alternative methods. We further analyse how parameter uncertainty depends on the duration and time resolution of an imaging experiment, to make experimental design recommendations. This work demonstrates the utility of parameter inference on time course data from single cells and enables new studies on cell fate changes and population heterogeneity.

Collocation based training of neural ordinary differential equations

The predictive power of machine learning models often exceeds that of mechanistic modeling approaches. However, the interpretability of purely data-driven models, without any mechanistic basis is often complicated, and predictive power by itself can be a poor metric by which we might want to judge different methods. In this work, we focus on the relatively new modeling techniques of neural ordinary differential equations. We discuss how they relate to machine learning and mechanistic models, with the potential to narrow the gulf between these two frameworks: they constitute a class of hybrid model that integrates ideas from data-driven and dynamical systems approaches. Training neural ODEs as representations of dynamical systems data has its own specific demands, and we here propose a collocation scheme as a fast and efficient training strategy. This alleviates the need for costly ODE solvers. We illustrate the advantages that collocation approaches offer, as well as their robustness to qualitative features of a dynamical system, and the quantity and quality of observational data. We focus on systems that exemplify some of the hallmarks of complex dynamical systems encountered in systems biology, and we map out how these methods can be used in the analysis of mathematical models of cellular and physiological processes.