Energetic costs of cellular computation

Abstract

Cells often perform computations in order to respond to environmental cues. A simple example is the classic problem, first considered by Berg and Purcell, of determining the concentration of a chemical ligand in the surrounding media. On general theoretical grounds, it is expected that such computations require cells to consume energy. In particular, Landauer's principle states that energy must be consumed in order to erase the memory of past observations. Here, we explicitly calculate the energetic cost of steady-state computation of ligand concentration for a simple two-component cellular network that implements a noisy version of the Berg-Purcell strategy. We show that learning about external concentrations necessitates the breaking of detailed balance and consumption of energy, with greater learning requiring more energy. Our calculations suggest that the energetic costs of cellular computation may be an important constraint on networks designed to function in resource poor environments, such as the spore germination networks of bacteria.

Fig. 1.

A cellular network for the computation of an external ligand concentration. External ligands are detected by a receptor that can exist in two conformations: A high-activity on state and a low-activity off state. Receptors switch between states at rate k_off and k_on. Receptors in state s = {on,off} can post-translationally activate (i.e., phosphorylate) a downstream protein at a rate . The protein is deactivated (i.e., dephosphorylated) at a constant rate k₁.

Fig. 2.

Top Slow switching regime, , with a bimodal distribution of activated proteins. Probability of having n activated proteins at steady-state (black solid line), probability of having n activated proteins when receptor is in the on state (blue dash-dot line), probability of having n activated proteins when receptor is in the off state (red dashed line). Middle Fast switching regime, , where the distribution of activated proteins is unimodal. Total probability (black solid line), probability when receptor is in the on state (blue dash-dot line), probability when receptor is in the off state (red dashed line). Bottom The uncertainty in ligand concentration, as a function of k₁ with mean number of active proteins (dashed red line) and . This can be compared to the Berg–Purcell result (solid black line). Parameters: , .

Fig. 3.

Upper The probabilistic Markov process underlying the circuit in Fig. 1 takes the form of a two-legged ladder. Any nonzero cyclic flux (depicted in red) results in entropy production and power consumption. Lower Power consumption (solid black line) and uncertainty (dashed purple line) as a function of when , and .

Fig. 4.

Total energy per independent measurement () as a function of k₁ when . (Inset) Power consumption as a function of k₁ over the same parameter range. Note that although the system’s power consumption decreases with decreasing k₁, the energy per measurement increases.