Prediction and Power in Molecular Sensors: Uncertainty and Dissipation When Conditionally Markovian Channels Are Driven by Semi-Markov Environments

Sensors often serve at least two purposes: predicting their input and minimizing dissipated heat. However, determining whether or not a particular sensor is evolved or designed to be accurate and efficient is difficult. This arises partly from the functional constraints being at cross purposes and partly since quantifying the predictive performance of even in silico sensors can require prohibitively long simulations. To circumvent these difficulties, we develop expressions for the predictive accuracy and thermodynamic costs of the broad class of conditionally Markovian sensors subject to unifilar hidden semi-Markov (memoryful) environmental inputs. Predictive metrics include the instantaneous memory and the mutual information between present sensor state and input future, while dissipative metrics include power consumption and the nonpredictive information rate. Success in deriving these formulae relies heavily on identifying the environment's causal states, the input's minimal sufficient statistics for prediction. Using these formulae, we study the simplest nontrivial biological sensor model---that of a Hill molecule, characterized by the number of ligands that bind simultaneously, the sensor's cooperativity. When energetic rewards are proportional to total predictable information, the closest cooperativity that optimizes the total energy budget generally depends on the environment's past hysteretically. In this way, the sensor gains robustness to environmental fluctuations. Given the simplicity of the Hill molecule, such hysteresis will likely be found in more complex predictive sensors as well. That is, adaptations that only locally optimize biochemical parameters for prediction and dissipation can lead to sensors that "remember" the past environment.