Source-linked AI summary
Efficiency of cellular information processing
Andre C. Barato, David Hartich, Udo Seifert
TL;DR
The paper addresses how to quantify cellular learning about fluctuating environments and its thermodynamic cost. It defines a conditional-entropy learning rate and informational efficiency, then analyzes three E. coli-inspired models. The results show that external chemical work can support learning without internal dissipation, while slow environmental changes can make adaptation inefficient and adaptation broadens the concentration range of substantial activity learning.
Problem
Existing studies considered cellular energy dissipation but did not provide an entropic rate characterizing information processing about a noisy external environment.
Method
The paper defines a conditional Shannon entropy reduction rate for an internal process learning about an unaffected external process and analyzes three E. coli-inspired models.
Results
The learning rate is bounded by thermodynamic entropy production, while external chemical work can support nonzero learning without internal dissipation and adaptation broadens the range of substantial activity learning.
Takeaways & Limitations
Cellular information-processing efficiency depends on both internal dissipation and work performed by the external environment.
Takeaways & Limitations
The examples assume an external ligand concentration jumping between two values, while more elaborate external and internal processes remain for future study.
Abstract
from arXiv · showhide
We show that a rate of conditional Shannon entropy reduction, characterizing the learning of an internal process about an external process, is bounded by the thermodynamic entropy production. This approach allows for the definition of an informational efficiency that can be used to study cellular information processing. We analyze three models of increasing complexity inspired by the E. coli sensory network, where the external process is an external ligand concentration jumping between two values. We start with a simple model for which ATP must be consumed so that a protein inside the cell can learn about the external concentration. With a second model for a single receptor we show that the rate at which the receptor learns about the external environment can be nonzero even without any dissipation inside the cell since chemical work done by the external process compensates for this learning rate. The third model is more complete, also containing adaptation. For this model we show inter alia that a bacterium in an environment that changes at a very slow time-scale is quite inefficient, dissipating much more than it learns. Using the concept of a coarse-grained learning rate, we show for the model with adaptation that while the activity learns about the external signal the option of changing the methylation level increases the concentration range for which the learning rate is substantial.
1. Introduction
The paper introduces a learning rate for cellular information processing and relates it to thermodynamic entropy production. It applies this framework to three increasingly complex E. coli-inspired models with a two-state external ligand concentration.
- Prior studies characterized cellular information-processing costs through entropy production but lacked an entropic rate for learning about a noisy external environment.
- The paper shows that conditional Shannon entropy reduction caused by internal dynamics is bounded by thermodynamic entropy production.
- The resulting learning rate supports an informational efficiency for studying cellular information processing.
- Three E. coli-inspired models examine learning when an external ligand concentration jumps between two values.
- The models progress from an ATP-consuming four-state system to a single receptor and an adaptive receptor model.
2. Bipartite systems with an external process
The framework models an unaffected external Markov process coupled to an internal process and defines learning through conditional entropy reduction. It establishes thermodynamic bounds, informational efficiency, and a coarse-grained learning rate for selected internal variables.
- Transition rates and thermodynamic entropy production: The bipartite Markov process separates an external variable unaffected by internal dynamics from an internal variable, with transitions changing only one variable at a time.
- Learning rate and informational efficiency: The learning rate is the rate at which internal dynamics reduce uncertainty about the external process.
- Learning rate and informational efficiency: The learning rate is nonnegative, while the internal subsystem obeys the second-law inequality σy − ly ≥ 0.
- Learning rate and informational efficiency: The learning rate is bounded by thermodynamic entropy production, enabling a definition of informational efficiency.
- Learning rate and informational efficiency: For equilibrium external processes, total entropy production equals internal entropy production and efficiency is η = ly/σ.
- Coarse-grained learning rate: The coarse-grained learning rate measures how one internal variable learns about the external process and cannot exceed the full learning rate.
- Coarse-grained learning rate: The coarse-grained learning rate and its associated discussion are introduced as novel in the paper.
3. Toy model
The toy model represents cellular learning as a four-state phosphorylation system driven by an external ligand concentration that switches between low and high values. Its learning rate and efficiency depend on the external switching rate and chemical potential, with optimal learning emerging between slow and fast limits.
- Learning and efficiency: The model’s efficiency is η = l_y/σ, while its learning rate l_y and maximal learning rate l* are evaluated against external transition rate γ_c and chemical potential ∆µ.Figure 1 organizes these quantities through transition rates, efficiency, learning rate, and maximal learning rate panels.
- Model: The four-state model couples external ligand switching to phosphorylation and dephosphorylation reactions driven by chemical potential difference ∆µ.High concentration permits dephosphorylation, whereas low concentration permits phosphorylation, producing a simplified E. coli chemotaxis model.
- Thermodynamic cost: A nonzero learning rate requires ATP consumption, and the learning rate is bounded by the rate of ATP consumption.The model has tight coupling because l_y and entropy production σ are proportional to the same probability current J.
- Limiting regimes: For γ_c ≪ κ_±, ω_±, the current approaches zero and efficiency approaches one, whereas for γ_c ≫ κ_±, ω_±, learning vanishes while ATP consumption is maximal.The slow limit is adiabatic; the fast limit is highly inefficient for fixed ∆µ.
- Optimal regime: At fixed ∆µ, an intermediate external time-scale γ*c(∆µ) maximizes learning, while efficiency at maximum power approaches 1/2 near equilibrium.The maximal learning rate increases with ∆µ, and the near-equilibrium efficiency result matches the stated tightly coupled benchmark.
4. Single Receptor Model
The single-receptor model represents ligand sensing through receptor binding and conformational activity, then uses coarse-graining to compare learning rates with chemical work and efficiency. Learning depends strongly on the relative time scales of environmental changes, receptor dynamics, and binding, while coarse-graining remains accurate under time-scale separation.
- Model: The model describes a single E. coli receptor whose ligand-binding state b and activity state a form a four-state internal process, while concentration c jumps externally between c1 and c2.The receptor's binding and conformational transitions obey detailed balance, with γb much faster than γa.
- Learning rates: The coarse-grained learning rate la measures how much the internal kinase activity a learns about the external concentration, while lab includes information carried by both a and b.The activity-based rate is treated as the central quantity for downstream chemical reactions influenced by kinase activity.
- Learning rates: When γc ≫ γa, activity cannot track concentration changes and la approaches zero, although lab remains nonzero because the faster binding variable b still contributes.For sufficiently rapid environmental changes, even b eventually fails to track the signal.
- Efficiency: For γc ≪ γa, efficiency η approaches 1 while lab approaches zero; around γc ≃ 0.1, η ≃ 0.5 and lab ≃ la > 0.015.Efficiency then decays as γc increases further and tends to zero for γc ≫ γb.
- Concentration dependence: For each concentration c, an optimal conformational free-energy difference ∆E maximizes learning, while lab and la become similar in the intermediate range K0 ≪ c ≪ K1.The binding and activity states are highly correlated in this range; b contributes more substantially at very low and very high c.
- Coarse-graining: Integrating out the faster binding variable b yields a two-state coarse-grained model, whose chemical-work lower bound is close to the full work when K0 ≪ c ≪ K1.With γa/γb → 0, the coarse-grained expression can calculate the full chemical work under time-scale separation.
5. Model with adaptation
The adaptation model combines fast activity changes with slower methylation dynamics, while chemical driving sustains internal cycles. Its learning and dissipation depend strongly on environmental switching, chemical potential, and concentration range.
- Chemical work and internal dissipation: The chemical potential difference drives internal cycles out of equilibrium, and adaptation occurs only when ∆µ exceeds the internal-cycle conformational free-energy difference ∆µ∗.For 0 < ∆µ < ∆µ∗, the system dissipates without adapting; adaptation requires ∆µ > ∆µ∗.
- Model definition: The 20-state model uses activity a and methylation level m as its internal variables, while ligand concentration jumps between c1 and c2.The external concentrations are defined through c1 = c/3 and c2 = 3c.
- Chemical work, SAM consumption and learning rate: When external concentration changes very slowly, internal SAM-consumption dissipation is much larger than the learning rate, making the bacterium highly inefficient.SAM consumption is nearly independent of the external jumping rate, whereas the external contribution to dissipation grows with that rate.
- Chemical work, SAM consumption and learning rate: The learning rate can exceed internal SAM-consumption dissipation because chemical work from the external process also contributes to the learning cost, although it saturates at high ∆µ.Internal dissipation increases with ∆µ while the learning rate approaches a saturation regime.
- Model definition: Fast activity responses trigger slower methylation changes that regulate the conformational free energy and restore average activity toward 1/2.A decrease in concentration rapidly increases activity, followed by slower methylation dynamics that reduce activity toward its adapted level.
- Chemical work, SAM consumption and learning rate: Adaptation lowers the coarse-grained learning rate relative to the fine-tuned nonadaptive maximum but broadens the concentration range where learning remains non-negligible.The broadened range is roughly K0 < c < K1, while learning approaches zero outside the receptor’s concentration range.
6. Conclusion
The paper introduces a learning rate for internal processes tracking unaffected external processes and bounds it by thermodynamic entropy production. Across E. coli-inspired models, energy may come from ATP, external chemical work, or adaptation-related dissipation, with slow environments producing marked inefficiency.
- Framework: For bipartite Markov processes, the learning rate measures conditional Shannon entropy reduction about an external process and is bounded by thermodynamic entropy production.This bound supports defining an informational efficiency for cellular information processing.
- Three models: In the toy tracking model, the learning rate is bounded by ATP consumption inside the cell.The learning rate is compared with extracted mechanical work in molecular motors.
- Three models: For an equilibrium single-receptor model without internal dissipation, external chemical work can account for a nonzero learning rate.The external work comes from ligand binding at one concentration and unbinding at another.
- Three models: With adaptation, slow environmental changes can make SAM-consumption dissipation much higher than learning, while adaptation broadens the concentration range with non-negligible learning.The framework assumes an external ligand concentration jumping between two values and motivates studying more elaborate processes in future work.