Biochemical Machines for the Interconversion of Mutual Information and Work

We propose a physically realizable information-driven device consisting of an enzyme in a chemical bath, interacting with pairs of molecules prepared in correlated states. These correlations persist without direct interaction and thus store free energy equal to the mutual information. The enzyme can harness this free energy, and that stored in the individual molecular states, to do chemical work. Alternatively, the enzyme can use the chemical driving to create mutual information. A modified system can function without external intervention, approaching biological systems more closely.

Figure 1

(a) Schematic showing an enzyme interacting with the $n$th pair of sites, catalyzing $X^{*}+\mathrm{ADP}\rightleftharpoons X+\mathrm{ATP}$ on tape $A$ due to the presence of activator $Y$ on tape $B$. Tapes extend into and out of page, and yellow circles indicate phosphorylation. (b) Two tapes, coordinated on a single polymer, being pulled past a tethered enzyme. We propose a DNA origami-based construction in Ref. [38], Sec. 1. (c) State space of the model. $E_T$ and $E_D$ correspond to ATP- and ADP-bound enzymes, respectively, and $E_{-}$ to a free enzyme. All unlabeled transitions have rate 1. The blue arrows denote the reaction $X+\mathrm{ATP}\rightleftharpoons X^{*}+\mathrm{ADP}$.

Figure 2

(a) Regimes of operation for $q=0.5$, $\tau \to \infty$ as a function of chemical load $\alpha$ and correlation strength $\psi$ [see Eq. (3)]: $W$ indicates $⟨w_{\text{cycle}}⟩>0$, and $\mathrm{\Delta }I$ indicates $\mathrm{\Delta }I(A_n;B_n)>0$; $⟨w_{\text{cycle}}⟩$ and $\mathrm{\Delta }I(A_n;B_n)$ are negative elsewhere. (b) Product of covariances ${\xi }_0(A_n,B_n){\xi }_{\tau }(A_n,B_n)$ for $q=0.5$, $\tau \to \infty$, showing correlation reversal. Covariances calculated by taking $X^{*}$ and $Y$ as 1 and $X$ and $\overline{Y}$ as 0. Negative contours are dashed lines, positive are dotted, and zero is solid (contours separated by units of 0.004). (c) Behavior of $⟨w_{\text{cycle}}⟩$, $\mathrm{\Delta }I(A_n;B_n)$, and correlation ${\xi }_{\tau }(A_n,B_n)$ as a function of $\tau$ for $q=0.5$, $\alpha =0.99$, and $\psi =0.69$, showing nonmonotonicity of information. (d) Efficiency $\eta$ of converting the initial mutual information into work. Contours run downwards from 0.225 with a separation of 0.025.

Figure 3

(a) Regimes of behavior for $q=0.7$, $\tau \to \infty$: $\mathrm{\Delta }F$ indicates $\mathrm{\Delta }\tilde{F}(A_n)>0$; $W$ and $\mathrm{\Delta }I$ are defined equivalently. The highlighted (blue) region has $\mathrm{\Delta }I(A_n;B_n)\ge 0$ and a greater $⟨w_{\text{cycle}}⟩$ than for a system with the same $q$, $\alpha$, but $I_0(A_n;B_n)=0$. (b) A completely autonomous device in which a freely diffusing enzyme interacts with correlated tapes. (c) Autonomous dynamics, with $q=0.5$, $\psi =0.1$, and $\alpha =0.5$. In addition to the rate constants in Fig. 1, it is necessary to define a rate at which the enzyme binds to activator $Y$ ($k_{\mathrm{bind}}=0.01$ per activator here) and an unbinding rate ($k_{\mathrm{off}}=1$ here). We plot the average work per site $⟨w⟩/kT$ and $I(A_n;B_n)$ obtained from 200 independent Gillespie simulations [41] of 1000 pairs of sites (dashed lines), showing that they converge to the values predicted in the $\tau \to \infty$ limit of the externally controlled device (solid lines). Dotted lines show information and work within three individual simulations (for subtleties in interpeting $I(A_n;B_n)$ within a single trajectory, see Ref. [38], Sec. 6).