Protein Concentration Fluctuations in the High Expression Regime: Taylor's Law and Its Mechanistic Origin

Abstract

Protein concentration in a living cell fluctuates over time due to noise in growth and division processes. In the high expression regime, variance of the protein concentration in a cell is found to scale with the square of the mean, which belongs to a general phenomenon called Taylor's law (TL). To understand the origin for these fluctuations, we measure protein concentration dynamics in single Escherichia coli cells from a set of strains with a variable expression of fluorescent proteins. The protein expression is controlled by a set of constitutive promoters with different strength, which allows one to change the expression level over 2 orders of magnitude without introducing noise from fluctuations in transcription regulators. Our data confirm the square TL, but the prefactor has a cell-to-cell variation independent of the promoter strength. Furthermore, distributions of the normalized protein concentration for different promoters collapse onto the same curve. To explain these observations, we use a minimal mechanistic model to describe the stochastic growth and division processes in a single cell with a feedback mechanism for regulating cell division. In the high expression regime where extrinsic noise dominates, the model reproduces our experimental results quantitatively. By using the mean-field approximation in the minimal model, we show that the stochastic dynamics of protein concentration is described by a Langevin equation with multiplicative noise. The Langevin equation has a scale invariance which is responsible for the square TL. By solving the Langevin equation, we obtain an analytical solution for the protein concentration distribution function that agrees with experiments. The solution shows explicitly how the prefactor depends on strength of different noise sources, which explains its cell-to-cell variability. By using this approach to analyze our single-cell data, we find that the noise in production rate dominates the noise from cell division. The deviation from the square TL in the low expression regime can also be captured in our model by including intrinsic noise in the production rate.