where $s$ is a *stress* variable modeling the environment, $\phi$ and $w$ are the fraction of proteome devoted to constitutively expressed proteins and the proteome share to be allocated to ribosome-affiliated proteins per unit of growth rate respectively. There exists an optimal value $\hat{x}\left(s\right)$ such that the growth rate is maximized. However, being $s$ a random variable distributed according to $p(s)$, cells face a certain degree of uncertainty about the environment and, therefore, the strategy to adopt. We denote as $p(x|s)$ the stochastic rule to choose $x$ for a given $s$ which at optimality is given by
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