Universality and non-universality of the growth law

An approximately linear relationship between the fraction of ribosomal proteins in the proteome ({phi}R) and the growth rate () holds in proliferating cells when the nutrient quality changes, often referred to as a growth law. While a simple model assuming a constant translation speed of ribosomes without protein degradation can rationalize this growth law, real protein synthesis processes are more complex. This work proposes a general theoretical framework of protein synthesis, taking account of heterogeneous translation speeds among proteins and finite protein degradation. We introduce ribosome allocations as the fraction of active ribosomes producing certain proteins, with two correlation coefficients respectively quantifying the correlation between translation speeds and ribosome allocations, and between protein degradation rates and mass fractions. We prove that the growth law curve generally follows{phi} R = ( + c1)/(c2 + c3) where c1, c2, and c3 are constants depending on the above correlation coefficients and the translation speed of ribosomal proteins. Our theoretical predictions of{phi} R agree with existing data of Saccharomyces cerevisiae. We demonstrate that when different environments share similar correlation coefficients, the growth law curve is universal and up-bent relative to a linear line in slow-growth conditions, which appears valid for Escherichia coli. However, the growth law curve is non-universal and environmental-specific when the environments have significantly different correlation coefficients. Our theories allow us to estimate the translation speeds of ribosomal and non-ribosomal proteins based on the experimental growth law curves.