Analyzing the stability of gene expression using a simple reaction-diffusion model in an early Drosophila embryo

In all complex organisms, the precise levels and timing of gene expression controls vital biological processes. In higher eukaryotes, including the fruit fly Drosophila melanogaster, the complex molecular control of transcription (the synthesis of RNA from DNA) and translation (the synthesis of proteins from RNA) events driving this gene expression are not fully understood. In particular, for Drosophila melanogaster, there is a plethora of experimental data, including quantitative measurements of both RNA and protein concentrations, but the precise mechanisms that control the dynamics of gene expression during early development and the processes which lead to steady-state levels of certain proteins remain elusive. This study analyzes a current mathematical modeling approach in an attempt to better understand the long-term behavior of gene regulation. The model is a modified reaction-diffusion equation which has been previously employed in predicting gene expression levels and studying the relative contributions of transcription and translation events to protein abundance [10,11,24]. Here, we use Matrix Algebra and Analysis techniques to study the stability of the gene expression system and analyze equilibria, using very general assumptions regarding the parameter values incorporated into the model. We prove that, given realistic biological parameter values, the system will result in a unique, stable equilibrium solution. Additionally, we give an example of this long-term behavior using the model alongside actual experimental data obtained from Drosophila embryos.