Regulatory control and the costs and benefits of biochemical noise

Experiments in recent years have vividly demonstrated that gene expression can be highly stochastic. How protein concentration fluctuations affect the growth rate of a population of cells, is, however, a wide open question. We present a mathematical model that makes it possible to quantify the effect of protein concentration fluctuations on the growth rate of a population of genetically identical cells. The model predicts that the population’s growth rate depends on how the growth rate of a single cell varies with protein concentration, the variance of the protein concentration fluctuations, and the correlation time of these fluctuations. The model also predicts that when the average concentration of a protein is close to the value that maximizes the growth rate, fluctuations in its concentration always reduce the growth rate. However, when the average protein concentration deviates sufficiently from the optimal level, fluctuations can enhance the growth rate of the population, even when the growth rate of a cell depends linearly on the protein concentration. The model also shows that the ensemble or population average of a quantity, such as the average protein expression level or its variance, is in general not equal to its time average as obtained from tracing a single cell and its descendants. We apply our model to perform a cost-benefit analysis of gene regulatory control. Our analysis predicts that the optimal expression level of a gene regulatory protein is determined by the trade-off between the cost of synthesizing the regulatory protein and the benefit of minimizing the fluctuations in the expression of its target gene. We discuss possible experiments that could test our predictions.

💡 Research Summary

The paper addresses a fundamental gap in our understanding of how stochastic gene expression influences the fitness of a population of genetically identical cells. While numerous studies have documented the high variability of protein concentrations, the quantitative link between this biochemical noise and the population growth rate has remained elusive. To fill this gap, the authors develop a mathematical framework that connects three key ingredients: (i) the dependence of a single‑cell growth rate on the concentration of a particular protein, expressed as a function g(c); (ii) the statistical properties of the protein concentration, modeled as a Gaussian random variable with mean μ, variance σ², and an autocorrelation time τ; and (iii) the dynamics of the noise, described by an Ornstein‑Uhlenbeck process. By expanding g(c) around the mean concentration, they derive an explicit expression for the population growth rate Λ:

Λ = g(μ) + ½ g″(μ) σ² τ.

This result reveals that the sign of the second derivative g″(μ) determines whether noise is beneficial or detrimental. If g(c) is convex (g″>0) near the operating point, fluctuations always reduce Λ; if it is concave (g″<0), fluctuations can increase Λ. Importantly, when the average protein level μ coincides with the growth‑optimal concentration c*, the curvature is positive, so any noise is harmful. However, when μ deviates sufficiently from c*, the linear term dominates and noise can raise the average growth rate even when g(c) is linear in c. Thus, stochasticity can paradoxically act as a source of fitness when the system operates away from its optimum.

The authors also emphasize that ensemble (population) averages differ from time averages obtained by following a single cell lineage. The time‑averaged growth rate depends on τ, whereas the ensemble‑averaged Λ does not, highlighting a subtle but crucial distinction for experimental design.

A major application of the theory is a cost‑benefit analysis of gene regulatory proteins. The synthesis of a regulator R incurs a metabolic cost proportional to its expression level p, while the benefit of R is the reduction of fluctuations in its target gene T. The benefit scales with the product σ_T²(p) τ_T, where σ_T²(p) is the variance of T’s expression as a function of p. Optimizing the total fitness yields an optimal regulator expression p* that balances the linear cost against the quadratic benefit of noise suppression. This formalism predicts that the optimal level of a regulatory protein is not simply the level that maximizes the target’s mean expression, but the level that minimizes the net cost of producing the regulator while sufficiently dampening target noise.

To validate the model, the authors propose three experimental strategies: (1) artificially vary the mean concentration of a protein and measure the resulting change in population growth rate; (2) manipulate the autocorrelation time τ by altering protein degradation rates; and (3) systematically adjust the expression of a regulatory protein and directly map the cost‑benefit curve. These experiments would test the counter‑intuitive prediction that, under certain conditions, increasing noise can improve growth.

In conclusion, the study provides a unified quantitative description of how biochemical noise propagates to population fitness, demonstrates that noise can be either a liability or an asset depending on the curvature of the growth‑rate function, and offers a principled framework for understanding the evolutionary pressures shaping gene regulatory architectures. The insights are broadly relevant to synthetic biology, industrial microbiology, and the study of heterogeneous cell populations such as tumors, where managing stochasticity is a key determinant of overall performance.