# The costs and benefits of biochemical noise

The importance of protein concentration fluctuations for the growth rate In recent years, the noise characteristics of biochemical networks have been characterized in great detail, both experimentally and theoretically. Yet, how important protein concentration fluctuations actually are for the fitness of an organism is much less understood. We have developed a quantitative model for how protein concentration fluctuations affect the growth rate of a population of genetically identical cells. The model predicts that fluctuations will lower the population’s growth rate when the average protein composition is close to the optimum, simply because fluctuations will drive the system away from its optimum. However, when the average protein composition is suboptimal, then fluctuations will enhance the growth rate, even when the growth rate of an individual cell depends linearly on the protein composition. This non-linear affect arises at the population level, and is a consequence of the fact that cells that happen to grow faster because of fluctuations, will generate more offspring, and consequently become overrepresented in the population. Our analysis also reveals that the importance of protein concentration fluctuations depend on their correlation time; the longer the correlation time, the more they will affect the growth rate.

Our model relies on three main assumptions: 1) The statistics of protein concentration fluctuations are Gaussian; experiments and simulations indicate that this assumption is surprisingly accurate; 2) The growth rate of a single cell λ can be described by expanding it to second order around the steady state Xs: λ = λ0 (Xs) + a x + b x2; 3) The instantaneous growth of a cell is given by the instantaneous protein composition of the cell; this assumption is made for simplicity but arguably justified by the experimental observation that concentration fluctuations are often slow, with correlation times γ-1 that are on the order of the cell cycle time. This model yields the following prediction for the growth rate g of the population:

g = λ(Xs) + a2 υ2 / (γ – 4 b υ2) + b σ2 with σ2 = 2 υ2 / (1+√(1 – 4 b υ2 / γ).

Here υ2 is the time average of the variance of the fluctuations in the concentration X of protein X as obtained from the time trace of X of a given cell and its descendants. In contrast, σ2 is the ensemble average of the variance of the fluctuations in the concentration of protein X as measured over the population at a given moment in time.

The following points are worthy of note:

1) The time average need not equal the ensemble average;

2) When the average protein composition is optimal, a = 0, and g = λ(Xs) + bσ2. Since at the optimum the curvature b of the growth rate as a function of protein concentration is negative, it follows that fluctuations lower the growth rate – they drive the system away from the optimum.

3) When the curvature b is positive, fluctuations enhance the growth rate. In fact, when b is larger than γ / (4 υ2), growth can induce a bifurcation with rapidly growing cells coexisting with slowly growing ones.

4) When the average protein composition is suboptimal, fluctuations can enhance the growth rate. If b = 0, then g = λ(Xs)+ a2 υ2 / γ. In this regime, fluctuations always enhance the growth rate, with an amount that depends on how strongly the growth rate of a single cell depends upon the concentration, characterized by the value of a, the magnitude of the fluctuations, characterized by υ2 and the correlation time of these fluctuations, characterized by γ; if concentration fluctuations are very rapid, then cells will become identical on the time scale of the cell cycle time, and no benefit from the fluctuations can be obtained.

**Regulatory control**

Most, if not all, biological processes are regulated by biochemical networks. A fundamental question is what determines the optimal design of a regulatory network. We argue that there is a fundamental trade-off between the benefit of accurately regulating the target system, and the cost of making the regulatory machinery. We have studied this trade-off for gene regulatory control using the model described above. More specifically, we have calculated the optimal concentration of the LacI repressor, which controls the expression of the lac operon. Dekel and Alon (Nature, 2005) have measured the benefit of expressing the lac operon, defined as the increase in the growth rate, as well as the cost of expressing it, which is defined as the reduction in the growth rate when the lac operon is expressed when there is no benefit (i.e. in the absence of lactose). Using these cost-benefit functions and assuming that the average expression level of the lac operon is close to its optimum at each lactose concentration, we have computed the change in the growth rate as a function of the concentration of the lac repressor, using the model developed above. The result is shown in Fig.1. It is seen that there exists an optimal expression level of the lac repressor. This optimum results from the trade-off between the cost of making the regulatory protein and the benefit of reducing fluctuations in the target system, the lac operon – as the concentration of lac repressor increases, fluctuations in the expression of the lac operon decrease, which enhances the growth rate when the average expression level is close to its optimum, as discussed above. It is seen that the predicted optimum is between 10 and 50 copies, remarkably close to the value measured experimentally, which is around 10.

**Fig.1** The change in the growth rate of the bacterium E. coli as a function of the lac repressor concentration, which controls the expression of the lac operon. The plotted growth rate is an average over the growth rates at different lactose concentrations. Since we do not know the distribution of lactose in the environment, we show the predictions for two different scenarios: one with a uniform distribution of lactose levels between 0 and 6 mM and one in which the lactose distribution is bimodal, with peaks close to the minimum and maximum concentration. Both scenarios predict an optimal repressor expression level. The optimum arises from the interplay between the cost of making the repressor and the benefit of reducing fluctuations in the expression of the lac operon, which decrease as the lac repressor concentration increases.

#### Publication

- Sorin Tanase-Nicola and Pieter Rein ten Wolde , Regulatory control and the costs and benefits of biochemical noise , PLoS Computational Biology 4, e1000125: 1–13 (2008)