Bye bye, linearity, bye: quantification of the mean for linear CRNs in a random environment

Cell populations display variability due to both internal and external factors. Traditional bulk data captures average expression levels but overlooks underlying heterogeneity. Single-cell techniques have improved understanding, yet bulk data remains widely used. Heterogeneity may still influence mean expression through interactions with rare environmental species. This effect is not captured in bulk measurements directly. Prior research has shown that environmental heterogeneity can alter mean expression levels. No prior work had resolved how this occurs in linear reaction networks. This gap motivated the need to quantify the impact of environmental heterogeneity on mean expression.

The study aimed to investigate how environmental heterogeneity affects mean expression levels in linear chemical reaction networks. Specifically, the researchers sought to quantify deviations from the quasi-steady state caused by a random environment. They focused on linear reaction systems embedded in a discrete-state Markov environment. The goal was to derive an analytical expression for the stationary mean. This would clarify how reaction rate constants and environmental dynamics influence the mean. The study also aimed to demonstrate the effect of a slow environment on the system's behavior. By comparing heterogeneous and quasi-steady-state cases, the researchers sought to reveal hidden heterogeneity fingerprints in bulk data.

The researchers used a mathematical framework involving linear chemical reaction networks and discrete-state Markov environments. They derived an analytical expression for the stationary mean of the system. This expression depends on reaction rate constants and the Markov environment's generator and stationary distribution. They analyzed how the environment influences the mean expression levels. The study focused on bimolecular interactions with low-abundance species in the environment. They compared the heterogeneous case with the quasi-steady-state scenario. The approach involved rigorous mathematical modeling and simulation. The results were validated through demonstrations of environmental effects on the stationary mean.

The researchers derived an exact analytical expression for the stationary mean of linear reaction systems in a random environment. This expression incorporates both reaction rate constants and the Markov environment's properties. The study showed that environmental heterogeneity can significantly affect the mean expression levels. When the environment is slow compared to the system, deviations from the quasi-steady state become apparent. The results revealed that low-abundance environmental species can influence the mean through bimolecular interactions. The stationary mean depends on the generator and stationary distribution of the environment. The effect is most pronounced when the system is fast relative to the environment. These findings highlight how heterogeneity is encoded in bulk data even when not directly measurable.

The study provides a rigorous framework for quantifying how environmental heterogeneity affects mean expression levels in linear reaction networks. The derived analytical expression clarifies the role of reaction rate constants and environmental dynamics. The researchers demonstrated that deviations from the quasi-steady state occur when the system is fast compared to the environment. These findings suggest that heterogeneity can be detected in bulk data through its impact on the mean. The study confirms that environmental factors influence mean expression levels via bimolecular interactions. The results support the idea that heterogeneity fingerprints are present in bulk measurements. The authors propose that this framework can help interpret bulk data more accurately. Future work may explore extending the model to non-linear systems.