Regulatory control and the costs and benefits of biochemical noise

Abstract

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.

Figure 1. A Sketch of the Instantaneous Growth Rate λ of a Single Cell as a Function of the Concentration X of Component X.

If the average expression level X _s is close to the optimal expression level X _opt, biochemical noise will always decrease the growth rate. If, however, the average expression level deviates sufficiently from the optimal expression level (i.e. if ax>bx ² in Equation 11), then fluctuations can enhance the growth rate of the population, even when the growth rate λ of a single cell is linear in X, i.e. if b = 0. The reason is that fast growing cells dominate the population.

Figure 2. Relative Change in the Growth Rate as a Function of the Average Repressor Concentration.

The growth rate is averaged over different lactose concentrations in the environment (see Equation 17), for two different lactose concentration distributions in the environment.

Figure 3. Different Regulatory Networks Can Yield the Same Optimal Enzyme Expression Level as a Function of Inducer Concentration.

This is illustrated for two regulatory networks of the lac system, which differ in the dissociation constants of lactose-repressor binding and repressor-operator binding. Panels (A) and (B) show the response functions at two different stages of the lac regulatory network, while panel (C) shows the resulting optimal enzyme expression level as a function of lactose concentration. (A) The fraction of repressor that is not bound by lactose, X _free/X, as a function of lactose concentration for two different lactose-repressor binding constants. (B) The corresponding response curves of the enzyme expression level as a function of the fraction of free repressor. The total expression level of repressor is chosen to correspond to the optimal growth rate (see Figure 2). (C) The resulting optimal enzyme expression level as a function of the lactose concentration, as predicted by Equation 20 .

Figure 4. The Optimal Design of the lac Regulatory Network Is Determined by the lac Repressor Copy Number and the Repressor–Lactose Binding Constant.

Contour plot of the growth rate as a function of the repressor copy number X and repressor-lactose binding constant K _D. The weighting of the lactose levels is nonuniform. Lower binding constants allow for higher optimal growth rates at lower optimal expression levels for the repressor.