Problem 2.1: Cost of a steady state

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This problem is still a draft.

We worked out that for unregulated gene expression, the steady state level of gene product is \(\beta/\gamma\), where \(\beta\) is the production rate and \(\gamma\) is the degradation rate. A steady state may be achieved for low \(\beta\) and low \(\gamma\), for high \(\beta\) and high \(\gamma\), and everything in between, provided the ratio \(\beta/\gamma\) is the same.

A loss function, also known as a cost function, takes as input an operating parameter or set of parameters and returns the “cost” associated operating with the given parameter value(s). Sketch a loss function \(L(\beta)\) as a function of \(\beta\), assuming \(\beta/\gamma\) is held constant. Describe why the curve might have the shape it does. Note that there is no right answer, or even a quantitative one, for this problem. It is meant to get you thinking about what some of the costs may be that are associated with different operating regimes.