Role of sufficient statistics in stochastic thermodynamics and its implication to sensory adaptation

A sufficient statistic is a significant concept in statistics, which means a probability variable that has sufficient information required for an inference task. We investigate the roles of sufficient statistics and related quantities in stochastic thermodynamics. Specifically, we prove that for general continuous-time bipartite networks, the existence of a sufficient statistic implies that an informational quantity called the sensory capacity takes the maximum. Since the maximal sensory capacity imposes a constraint that the energetic efficiency cannot exceed one-half, our result implies that the existence of a sufficient statistic is inevitably accompanied by energetic dissipation. We also show that, in a particular parameter region of linear Langevin systems there exists the optimal noise intensity at which the sensory capacity, the information-thermodynamic efficiency, and the total entropy production are optimized at the same time. We apply our general result to a model of sensory adaptation of E. coli and find that the sensory capacity is nearly maximal with experimentally realistic parameters.

Figure 1

Graphical representation of the CBN. A node represents a state and an arrow represents a causal relationship.

Figure 2

Schematic of signal transduction of E. coli chemotaxis. The receptor senses the external ligand concentration $l$ and changes the kinase activity $a$, which affects the methylation level $m$ of the receptor. The methylation level again affects the kinase activity, which makes a negative feedback loop.

Figure 3

Contour plot of the s-SC ${\overline{C}}_A$ (upper panel) and the information-thermodynamic efficiency ${\eta }_A$ (lower panel) of the kinase activity. The horizontal and vertical axes are the time constants ${\tau }_A$ and ${\tau }_M$, respectively. The noise intensity $T^A$ is chosen to be (a) $T^A=0.1$, (b) $T^A=0.3$, (c) $T^A=0.5$, and (d) $T^A=0.7$. The star marks represent the real experimental values of the time constants in E. coli (${\tau }_M=0.2,{\tau }_A=0.02$) [31, 32, 41, 43]. Other parameters are given by $T^M=0.005$ and $\alpha =2.7$, which are consistent with real experiments [31, 32, 41, 43].

Figure 4

(a) The s-SC ${\overline{C}}_A$ (blue line) and the information-thermodynamic efficiency ${\eta }_A$ (orange line) against the noise intensity $T^A$. The optimal noise intensity, at which ${\overline{C}}_A$ and ${\eta }_A$ take the maximum value, is indicated by the black bold line. (b) The dotted line represents the trade-off relation (40) between ${\overline{C}}_A$ and ${\eta }_A$ in the present setup. The cases of $T^A=0.135$ (black), $T^A=0.7$ (blue), and $T^A=0.9$ (red) are indicated by the colored circles, where $T^A=0.135$ (black) is the optimal noise intensity. (c) The entropy productions of the subsystems (${\sigma }_M$ and ${\sigma }_A$) and those of the total system (${\sigma }_{\mathrm{tot}}$). The noise intensity at which the total entropy production takes the minimum value is indicated by the black bold line. The other parameters are the same as in Fig. 3.

Figure 5

Non-steady dynamics under the time-dependent ligand density and noise (left). (Middle) The s-SC ${\overline{C}}_A$ and the information-thermodynamic efficiency ${\eta }_A$. (Right) The learning rates $l_M$, $l_M$, and the instantaneous mutual information (inset). The initial state is the steady state under $\beta l_t=0$, $T_t^A=0.1$. The external driving of $l_t$ is switched on at $t=0$, where the functional forms at $t>0$ are given by (a) step function: ${\beta }_t=0.01,T_t^A=0.5,$ and (b) sinusoidal function: $\beta l_t=0.01sin\left(400t\right),T_t^A=0.1+0.4|sin\left(400t\right)|$. The other parameters are the same as in Fig. 3. Since it is difficult to experimentally determine the functional forms of $l_t$ and $T_t^A$, our choice of them is rather arbitrary from the viewpoint of real biology. The behaviors of ${\overline{C}}_A$ and ${\eta }_A$ can easily be changed if we adopt a different choice of the functional form of $T_t^A$.