  Any organism is embedded in an environment that changes over time. The
timescale for and statistics of environmental change, the precision with which
the organism can detect its environment, and the costs and benefits of
particular protein expression levels all will affect the suitability of
different strategies-such as constitutive expression or graded response-for
regulating protein levels in response to environmental inputs. We propose a
general framework-here specifically applied to the enzymatic regulation of
metabolism in response to changing concentrations of a basic nutrient-to
predict the optimal regulatory strategy given the statistics of fluctuations in
the environment and measurement apparatus, respectively, and the costs
associated with enzyme production. We use this framework to address three
fundamental questions: (i) when a cell should prefer thresholding to a graded
response; (ii) when there is a fitness advantage to implementing a Bayesian
decision rule; and (iii) when retaining memory of the past provides a selective
advantage. We specifically find that: (i) relative convexity of enzyme
expression cost and benefit influences the fitness of thresholding or graded
responses; (ii) intermediate levels of measurement uncertainty call for a
sophisticated Bayesian decision rule; and (iii) in dynamic contexts,
intermediate levels of uncertainty call for retaining memory of the past.
Statistical properties of the environment, such as variability and correlation
times, set optimal biochemical parameters, such as thresholds and decay rates
in signaling pathways. Our framework provides a theoretical basis for
interpreting molecular signal processing algorithms and a classification scheme
that organizes known regulatory strategies and may help conceptualize
heretofore unknown ones.
