Exponential growth conditions for a coarse-grain model of ribosome synthesis

Ribosomes are fundamental cellular structures, playing a crucial role in the synthesis of proteins, the building blocks of life. They are composed of a large and a small subunit, each consisting of a combination of ribosomal RNA (rRNA) and numerous associated proteins. These subunits work collaboratively to translate messenger RNA (mRNA) into proteins through a process known as translation. Understanding the synthesis of ribosomal components is crucial for elucidating their function and role in cellular processes, including the setting of the cell growth rate. Recently, a coarsegrain mathematical model has been presented, accounting for the two distinct sub-units, their common precursor and the overall amount of proteins and ribosomes. The feedback of the ribosome-over-protein ratio on the ribosome synthesis was as well addressed. This note carries out the qualitative analysis on such a model, providing a condition ensuring the exponential growth of the cell as an emergent property of the network of the molecular players under investigation. The proposed analysis overcomes the drawbacks of a previous work, which provided a set of restrictive sufficient conditions for exponential growth; indeed, here a single necessary and sufficient condition is given, which is also easy to check from the knowledge of the model parameters. Besides, a sensitivity analysis is proposed, detailing which parameters the emergent growth rate and the ribosome-over-protein ratio are more sensitive to.