Thermodynamics of Statistical Inference by Cells

The deep connection between thermodynamics, computation, and information is now well established both theoretically and experimentally. Here, we extend these ideas to show that thermodynamics also places fundamental constraints on statistical estimation and learning. To do so, we investigate the constraints placed by (nonequilibrium) thermodynamics on the ability of biochemical signaling networks to estimate the concentration of an external signal. We show that accuracy is limited by energy consumption, suggesting that there are fundamental thermodynamic constraints on statistical inference.

Figure 1

Schematic of a cell receptor and our model of a receptor. (a) A chemical ligand at concentration $c$ binds to the receptor at rate $k_{+}c$ and unbinds at rate $k_{-}$. (b) Example time series of a receptor binding. While unbound, the receptor is in a nonsignaling state, but upon ligand binding it transitions to a signaling state. After a long time $T$, the receptor has a series of nonsignaling times ${\tau }_{NS}$ and signaling times ${\tau }_S$ from which to estimate the concentration. (c) Two-state and (d) three-state biochemical models of a receptor. Upon ligand binding the receptor undergoes a physical change (represented as a conformational change) that transmits signals to the downstream biochemical network. (e) Two-state, (f) three-state, and (g) $L$-state Markov models of a receptor, where the chain of states $3,4,\dots ,L-1$ has been suppressed.

Figure 2

Two signaling state estimator performance. For varying sampling rate $\overline{n}=\overline{N}/T$, the plot shows estimator performance ($\mathcal{E}$) versus entropy production ($e_p$, with units of $k_{B}T=1$). The symbols represent results from simulated annealing, where $k_{01}=1$ and $k_{02}=\epsilon =10^{-3}$, while the other four rates are optimized. The continuous lines represent our ansatz [24] for the global minima. At high entropy production, the estimators asymptotically approach 1.5. The inset shows the data collapse when the estimator performance ($\mathcal{E}$) is plotted versus entropy production per sampling rate ($e_p/\overline{n}$). The vertical dashed line corresponds to the approximate energy released by hydrolysis of a single ATP.

Figure 3

Illustrative example of $L$ signaling state estimator performance. For a varying number of signaling states $L$, the plot shows estimator performance ($\mathcal{E}$) versus energy consumption ($e_p/\overline{n}$). For an increasing $L$, at high energy consumption the estimator approaches the maximum likelihood limit of 1. The following parameters are fixed at $\overline{n}=0.99$, $k_{01}=1$, $k_{0L}=10^{-3}$, $\alpha =k_{10}/b=10^{-3}$, and $\omega =k_{L0}/f=1$, while $b$ was varied to keep $\overline{n}$ fixed and $\theta =f/b$ was varied to change the estimator and the energy consumption. These parameters were chosen for convenience and are not global optima. The vertical lines correspond to the approximate energy released by hydrolysis of a single ATP (dashes) or two ATPs (dot dashes).