Information-thermodynamic bounds on biochemical signal transduction beyond the linear-Gaussian model: nonlinear and time-delayed feedback networks, backward transfer entropy, and experimental verification

Ito and Sagawa (Nature Communications 6, 7498 (2015)) showed that the second law of thermodynamics with information bounds the robustness of biochemical signal transduction with feedback: in a two-variable Langevin model of E. coli chemotaxis, the heat-absorption-like robustness quantity is upper-bounded by the transfer entropy from the output signal to the regulatory memory, with a closed-form bound for the linearized Gaussian model and a tighter bound involving a conditional-backward transfer entropy. This problem is the extension beyond that model along the generalization directions indicated in the source: deriving explicit transfer-entropy and entropy-production-bound expressions for signal-transduction networks that go beyond the linearized two-variable Gaussian Langevin model of E. coli chemotaxis in one or more of the following respects: (a) strongly nonlinear dynamics; (b) non-Gaussian statistics, including the discrete low-copy-number description; (c) higher dimensionality, that is, multi-component feedback loops; (d) intrinsic time delays in the feedback loop; establishing the role and measurability of the backward transfer entropy; and experimentally verifying the inequality in a living signaling pathway under realistic finite-sampling, measurement-noise, and nonstationarity conditions.

Background

In living cells, signal transduction is the chain of biochemical reactions by which a cell senses its environment and produces an internal response. The canonical example is the chemotaxis of the bacterium Escherichia coli, which swims up gradients of attractant molecules. In the standard coarse-grained model, the kinase activity $a$ (the output that steers the flagellar motor) is regulated by the receptor methylation level $m$ (a slow chemical modification that acts as a memory), while $a$ feeds back on the dynamics of $m$, closing a negative feedback loop that implements adaptation — the return of $a$ to its steady value after an environmental change. Ito and Sagawa modeled this loop with two coupled Langevin equations, that is, stochastic differential equations driven by white Gaussian noise, linearized around the steady state.

Information thermodynamics is the extension of nonequilibrium thermodynamics to systems in which a feedback controller measures and exploits information about fluctuating microscopic variables — the setting of Maxwell's demon, the nineteenth-century thought experiment in which an intelligent being appears to violate the second law by sorting molecules using information. Its central result is a generalized second law: the entropy production of a system under feedback is bounded below by an information-theoretic quantity, so that information acquisition and consumption carry a thermodynamic cost. Applied to biochemical signaling, Ito and Sagawa identified the relevant quantity as the transfer entropy $dI_t^{\mathrm{tr}}$ — an information-theoretic measure of directed information flow, defined as the conditional mutual information between $a_t$ and $m_{t+dt}$ given $m_t$ — and proved the inequality

$dI_t^{\mathrm{tr}} + dS_t^{a\mid m} \ge J_t^a dt / T_t^a,$

where $dS_t^{a\mid m}$ is the change of the conditional Shannon entropy of $a$ given $m$, and $J_t^a$ is a heat-absorption-like quantity quantifying the robustness of signal transduction against environmental (ligand) noise of intensity $T_t^a$. The inequality states that the robustness of adaptation is bounded by the information the network transmits from the output back to its memory, in analogy with a demon exploiting measurement records. For the linearized Gaussian model they obtained a closed-form expression for $dI_t^{\mathrm{tr}}$ structurally analogous to the Shannon-Hartley channel-capacity formula, and, in Supplementary Note 3 of the paper, a tighter bound involving the conditional-backward transfer entropy $dI_t^{\mathrm{Btr}}$ — the conditional mutual information between $m_t$ and $a_{t+dt}$ given $m_{t+dt}$, that is, a transfer entropy computed on the time-reversed process.

The analytic tractability of these results rests on three limitations of the model: the Gaussian statistics of the linearized Langevin description, the restriction to two variables, and the absence of intrinsic time delays in the feedback; of these, the source explicitly poses as a forward-looking direction the generalization of the theory "to a broad class of signal transduction networks, including a feedback loop with time delay". The general second-law-like inequality itself holds for arbitrary Markovian causal networks via the formalism of Ito and Sagawa (Physical Review Letters 111, 180603 (2013)); what is model-specific is the explicit closed-form expression and the quantitative chemotaxis analysis. Real biochemical networks are strongly nonlinear, often operate at low copy numbers where molecule-number fluctuations are non-Gaussian and discrete (requiring Markov jump rather than Langevin descriptions, i.e. stochastic dynamics modeled as random jumps between discrete molecule numbers rather than continuous diffusion), contain multi-component feedback loops, and involve finite reaction delays that make the dynamics non-Markovian, that is, dependent on the system's history rather than only its present state; moreover, experimental estimation of $dI_t^{\mathrm{tr}}$, $dS_t^{a\mid m}$ and $J_t^a$ requires reconstructing joint probability distributions from finite, noisy, and typically nonstationary time series in living cells. Extending the quantitative information-thermodynamic characterization to this realistic regime, analytically and experimentally, is the open problem.

Problem Statement

Extend the information-thermodynamic characterization of biochemical signal transduction with feedback established by Ito and Sagawa beyond the linearized two-variable Gaussian Langevin model of E. coli chemotaxis, as one coherent objective — a quantitatively explicit and experimentally verified information-thermodynamic theory of realistic signal-transduction networks — comprising three linked components:

(1) Analytic extension. Derive explicit closed-form (or systematically convergent perturbative) expressions for the transfer entropy $dI_t^{\mathrm{tr}}$ and the information-thermodynamic inequality bounding the robustness quantity $J_t^a$ (the heat-absorption-like quantity measuring the robustness of the output signal against environmental noise, with $T_t^a$ the effective noise intensity) for signal-transduction networks that go beyond the linearized two-variable Gaussian Langevin model of E. coli chemotaxis in one or more of the following respects: (a) strongly nonlinear dynamics; (b) non-Gaussian statistics, including the discrete, low-copy-number Markov jump description; (c) higher dimensionality, that is, multi-component feedback loops; (d) intrinsic time delays in the feedback loop; the last case requires handling the non-Markovian dynamics that the original Markovian causal-network derivation does not directly cover. Any such derivation must reduce to the published linear-Gaussian formulas in the corresponding limit.

(2) Role of the backward transfer entropy. Establish the role, tightness benefit, and measurability of the conditional-backward transfer entropy $dI_t^{\mathrm{Btr}}$ that strengthens the inequality, including whether it can be reliably estimated and used as a diagnostic quantity in biochemical signaling networks.

(3) Experimental verification. Verify the inequality $dI_t^{\mathrm{tr}} + dS_t^{a\mid m} \ge J_t^a dt / T_t^a$ — where $dI_t^{\mathrm{tr}}$ is the transfer entropy from the output activity $a$ to the memory variable $m$, $dS_t^{a\mid m}$ is the change of the conditional Shannon entropy of $a$ given $m$, and $J_t^a dt / T_t^a$ is the heat-absorption-like entropy term with $T_t^a$ the effective intensity of the environmental (ligand) noise — in a living signal-transduction pathway by jointly estimating $dI_t^{\mathrm{tr}}$, $dS_t^{a\mid m}$ and $J_t^a$ from measured time series at sufficient time resolution, quantifying the effects of finite sampling, nonstationarity, measurement noise, and unobserved degrees of freedom on the estimated bound.

The three components are analytically linked: the analytic expressions define what the experiment should measure, and the experimental verification exposes which analytic regime real networks occupy. They correspond to forward-looking directions that the source states independently rather than as a single jointly posed program: the generalization of the theory "to a broad class of signal transduction networks, including a feedback loop with time delay", the tighter bound of Supplementary Note 3 involving the conditional-backward transfer entropy $dI_t^{\mathrm{Btr}}$, and the experimental validation of the inequality by measuring the transfer entropy and thermodynamic quantities from the probability distribution of the amount of proteins in a biochemical system. An answer to component (1) should state precisely which class of feedback networks it covers and demonstrate consistency with the original linear-Gaussian result.

References

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