Efficiency of cellular information processing

We show that a rate of conditional Shannon entropy reduction, characterizing the learning of an internal process about an external process, is bounded by the thermodynamic entropy production. This approach allows for the definition of an informational efficiency that can be used to study cellular information processing. We analyze three models of increasing complexity inspired by the E. coli sensory network, where the external process is an external ligand concentration jumping between two values. We start with a simple model for which ATP must be consumed so that a protein inside the cell can learn about the external concentration. With a second model for a single receptor we show that the rate at which the receptor learns about the external environment can be nonzero even without any dissipation inside the cell since chemical work done by the external process compensates for this learning rate. The third model is more complete, also containing adaptation. For this model we show inter alia that a bacterium in an environment that changes at a very slow time-scale is quite inefficient, dissipating much more than it learns. Using the concept of a coarse-grained learning rate, we show for the model with adaptation that while the activity learns about the external signal the option of changing the methylation level increases the concentration range for which the learning rate is substantial. PACS numbers: 05.70.Ln, 87.10.Vg, 02.50.Ey