Figure 1.
A time-varying environmental signal, the concentration of a nutrient, is read by the cell through a noisy process. Through regulation, the cell chooses an enzyme level, which interacts with the true nutrient concentration to produce product. In this work we focus on the optimization of the regulatory strategy, the choice of enzyme level as a function of the imperfect readout of nutrient concentration.
Figure 2.
Cost convexity relative to benefit produces preference for either thresholding or for graded response.
For a benefit function that is linear in nutrient concentration (purple curves in top panel) and a simple polynomial cost function
, concave cost (
, left column) implies an optimal enzyme expression level
of either zero or the maximal enzyme level
(thresholding), whereas convex cost (
, right column) implies an optimal enzyme expression level that varies continuously with the cellular readout (graded response). Top row: costs (green curves) and benefits (purple curves) associated with an enzyme expression level for a given nutrient concentration. Bottom row: optimal regulatory strategy specifying a enzyme expression level for a given cellular readout.
Figure 3.
Increasing measurement noise shifts the optimal strategy from naive response to constitutive response.
For a quadratic cost function () and relatively slow environmental dynamics, the dimensionless ratio
of the measurement imprecision
(middle row) and the environmental variation
(top row) determines the preference for different regulatory strategies [see Eq. (5)]. Low relative measurement noise (
, left column) leads to a preference for naive response; high relative measurement noise (
, right column) produces a preference for constitutive response; and the intermediate case (
, middle column) leads to a preference for more sophisticated inference incorporating both prior knowledge and the current measurement of the environment. Top row: distribution of possible environmental nutrient concentrations around the mean
. Middle row: distribution of cellular readouts given a particular nutrient concentration (red dotted line).
Figure 4.
In a bimodal environment, increasing measurement noise shifts the optimal strategy from classification to constitutive response.
For a quadratic cost function, tight distribution within each environmental mode (such that ), and relatively slow environmental dynamics between distinct environmental modes (with mode separation
), the dimensionless ratio
determines the preference among regulatory strategies [see Eq. (8)]. High relative measurement noise (
, right column) leads to a preference for constitutive response; low relative measurement noise (
, left column) produces a preference for classifying the environment into the most likely among the two modes; and the intermediate case (
, middle column) produces a preference for non-degenerate Bayesian inference.
Figure 5.
In a rapidly changing environment, the value of memory peaks at intermediate measurement noise.
For a quadratic cost function and environmental changes on timescales comparable to cellular response, the dimensionless ratio determines the preference among different regulatory strategies [see Eq. (11)]. High relative measurement noise (
, right column) leads to a preference for constitutive response; low relative measurement noise (
, left column) produces a preference for naive response to the present measurements; and the intermediate case (
, middle column) produces a preference for dynamic Bayesian inference that takes into account both present and past measurements. In the heat maps (bottom row), blue represents high levels of enzyme and green represents low.