Balancing information and dissipation with partially observed fluctuating signals

TL;DR

This study introduces a general chemical model in which a sensor, coupled to a signaling pathway activated by hidden signals, can allosterically tune the production of a readout molecule, paving the way for an implementable design principle underpinning biological and biochemical adaptation.

Abstract

Biological systems sense and extract information from fluctuating signals while operating under energetic constraints and limited resolution. We introduce a general chemical model in which a sensor, coupled to a signaling pathway activated by hidden signals, can allosterically tune the production of a readout molecule. We propose viable strategies for the sensor to estimate, and eventually balance, information gathering on the hidden process and the associated dissipative cost relying solely on counting statistics of observed trajectories. We show that these strategies can be successfully implemented to adapt the readout production even with finite-time measurements and limited dynamic resolution, and remain effective in the presence of inhibitory regulatory mechanisms. Our study provides a plausible mechanism to actively balance information and dissipation, paving the way for an implementable design principle underpinning biological and biochemical adaptation.

Figures (4)

• Figure 1: (a) Schematic of the sensing architecture. A sensor attempts to gather information about the fluctuations $\bm{\eta}(t)$ of a hidden network, through a readout molecule $X$ coupled to an intermediate signaling molecule $Z$ through receptor binding dynamics. (b-c) Example trajectories of the fluctuations of the intermediate signal $z(t)$ and the corresponding readout $x(t)$. (d) Empirical joint distribution $p^\mathrm{emp}_{xz}$ constructed from the observed trajectories to estimate information (Eq. \ref{['eqn:pointwise_mutual']}). (e) Estimation of the traffic $\mathcal{T}_x$ from the short-time curvature of the autocorrelation function, providing an operational measure of the energetic cost of sensing (Eqs. \ref{['eqn:traffic_x']} and \ref{['eqn:autocorr']}).
• Figure 2: (a) Optimal readout coupling $a_\mathrm{opt}$ for infinite resolution as a function of the strategy $\lambda$ and signaling rate $\sigma$. At fixed $\sigma$, we find a transition from a regime where $a_\mathrm{opt}$ vanishes at low $\lambda$ and increases for $\lambda > \lambda_c$ (grey dashed line). (b-c) Entropy production, $\overset{\,\bm.}{S}{\newline}_a$, and information between $x$ and $\eta$, $I_{x\eta}$, as a function of the strategy $\lambda$ for different values of $\sigma$, when $a = a_\mathrm{opt}$. (d) Estimate of $\mathcal{T}_x$ from trajectories of finite duration $T_\mathrm{obs}$. As $T_\mathrm{obs}$ increases, the estimated value approaches the theoretical one (gray dashed line). (e) Same, but for the information between the readout and the signal, $I_{xz}$. (f) Adapted readout coupling after $10^4$ steps of adaptive dynamics, where the sensor attempts to tune $a$ by estimating $\mathcal{T}_x$ and $I_{xz}$ from measurement windows of duration $T_\mathrm{obs}$. For large enough $T_\mathrm{obs}$, the readout achieves a near-optimal coupling. Distributions are obtained over $64$ repetitions of the adaptive dynamics. (g) Example of the readout input signal, both for infinite resolution (gray trajectory) and for a resolution $r = 1$ (blue trajectory). (h) Comparison of the functional $\mathcal{L}$ at different resolution values. The maximum (triangles) moves at higher values of $a$ as $r$ decreases. (i) Optimal readout coupling as a function of the resolution. As the resolution increases, $a_\mathrm{opt}$ decreases until it reaches the $r\to\infty$ limit (dashed lines). (j) Same as (f), but for a finite resolution ($r=1$). In this figure, unless otherwise specified, $D_i = 1/\tau_i$ for $i=\eta,z,x$, $W_\eta = -1$, $\sigma = 1.5$, $\lambda = 0.9$, $\tau_\eta = 3\tau_z$, $\tau_z = \tau_x = 1$.
• Figure 3: (a) Optimal readout coupling $a_\mathrm{opt}$ as a function of the strategy $\lambda$. Without an inhibitory feedback coupling ($\alpha = 0$), it is enough to estimate the $x$-component of the traffic $\mathcal{T}_x$, which contains all relevant dependences of $\overset{\,\bm.}{S}{\newline}$ on $a$. When $\alpha < 0$, both $\mathcal{T}_x$ and $\mathcal{T}_z$ are needed, since the functional with the $x$-component alone overestimates $a_\mathrm{opt}$. As before, there is a critical $\lambda_c$ below which $a_\mathrm{opt} = 0$. (b) Information between readout and hidden dof, $I_{x\eta}$, as a function of the energy consumption $\overset{\,\bm.}{S}{\newline}^*$ of the system, which is the entropy production at the corresponding optimal coupling $a_\mathrm{opt}$. Here, $\lambda \in [0.1, 0.9]$. Since $a_\mathrm{opt}$ decreases with more negative values of $\alpha$, so does the energy consumption. Thus, the maximum value of $\overset{\,\bm.}{S}{\newline}^*$ (squares) is larger for $\alpha = 0$. The gray dashed line represents the baseline energy consumption when the readout does not interact with the signal, for $a = 0$. In this figure, $D_i = 1/\tau_i$ for $i=\eta,z,x$, $W_\eta = -1$, $\sigma = 1.5$, $\tau_\eta = 3 \tau_z$, $\tau_z = \tau_x = 1$.
• Figure 4: Scheme of the chemical model. Internal species $\bm{H}$, readout $X$, and signaling molecule $Z$ (whose fluctuations are in lowercase letters, $\bm{\eta}(t)$, $x(t)$, and $z(t)$) are coupled together only through intermediate fast complexes.