Chemical Carnot cycles, Landauer’s Principle, and the Thermodynamics of Natural Selection
Abstract • It has seemed inescapable to many investigators from Brillouin and Schroedinger onward, that life should be understood as a chemical system in which the flow and storage of energy are related to the flow and storage of information. However, the appropriate definition of information, and the manner in which its storage or flow may be limited by energetics, are not nearly understood even today. The separation of timescales from metabolic to evolutionary processes, and the flow of constraint and control between them, make an analysis from the details exceedingly difficult, and leave unanswered the question what options may be open to evolutionary innovation. The second law of thermodynamics gives some universal albeit limited constraints on information flow, but these are difficult to interpret directly in chemical terms. In this talk, I show how a decomposition similar to the Carnot decomposition for heat engines may be performed in chemistry, to give a general calculus and interpretation of chemical information limits. I then show that the decomposition corresponds, step for step, to the familiar Landauer derivation of the limits on computation, giving us an operational interpretation of life as a computational process, and using chemistry to clarify certain assumptions (appropriately) made by Landauer. The elementary application of such results is to metabolism and growth, but for variety I show that it also provides an illuminating analysis of a clever application of Shannon’s theorem to the problem of reliable sequence recognition, proposed by Tom Schneider.
Outline • The subtle task of asking sensible questions about information in the biosphere • Sample questions, difficulties, paradoxes • What role for equilibrium reasoning? • The chemical Carnot construction • The relation to computation
Big and little questions • (Big) how does energy flow limit the informational state of the biosphere? Requires theory of biological decay • (Little) how does energy flow limit the change in information in the biosphere? Can get from equilibrium thermodynamics (Similar questions can be asked about individuals, species, etc., as about the whole biosphere)
The obvious (little) answer dW = dQ = − TdS ≡ Tk B d I • Follows from dimensional analysis and the definition of temperature • Information gain should be entropy loss • Heat is entropy carried by energy • Work is an entropy-less energy source In what senses is such an answer useful? wrong? irrelevant?
� Intuition about energy and information Configuration Space Volume: V = e description length: L V sys. + res. = V sys. × V res. If independence L sys. + res. = L sys. + L res. and If exchanged energy is the constraint overall ∂ Energy = ∂ L res. ∂ L sys. max L sys. + res. at ∂ Energy 1 ≡ k B T
Entropy and information (about units) ∂ S 1 ≡ ∂ U T Traditionally, chemists recognized ∂ Desc. Length 1 = description length as entropy: ∂ Energy k B T S Desc. Length : L = k B k B T ≡ τ Simplify our notation more natural S X and sensible energy units ≡ σ X k B TS X τσ X = Gain information by reducing description length d I = − d σ Relation between energy and information dW = dQ = − τ d σ = τ d I then takes a simpler form? Traditionally, chemists recognized
� � Moving information around Suppose you want to go from: to: ∂ Energy ∂ Entropy × d Description length Heat = dQ = k B T × dS But if these variables didn’t have thermal energy to give: Work that must be d Work = Heat brought in from outside dW = dQ
I. The complex problem of thinking about information in the biosphere • Many levels, separation of timescales, and flow of constraint and control make assembling from the molecules very hard • Which information? Genes? Heats? • Which building process? Metabolism? Natural selection? • What level? Individuals? Ecosystems? Biosphere?
How I think about these talks • I am not mainly concerned with any one application • In many ways, this work will fall short of answering any of them adequately • I want a framework that is at least compatible with answering these questions • I will try to use examples to identify useful ways of thinking
������� ��������� Control flows and error correction reproduction, transcription /translation death • Long-lived states ∼ 10 1 − 10 2 s ∼ 10 3 − 10 8 s “control” faster processes • “Errors” removed catalysis ∼ 10 − 6 s by both control and selection allosteric • References are regulation ∼ 10 − 3 − 10 0 s assembly, contained in both interactions ∼ 10 − 3 − 10 2 s system and environment regulation, placticity ∼ 10 1 − 10 6 s
A paradox: What price for evolution? ∆ W ∆ F pure − ∆ F mixed = � � − ∆ S (comb) ∆ S (comb) k B T ∼ pure mixed � M � (extensive) k B T ∼ 10g / Mol whereas: genome entropy � 10 6 − 10 8 (intensive)
Extensive and intensive entropies? • Scaling relations suggest that physiology limits memory systems • Heat of formation is like a heat of phase transition • Adaptive (species- level) information behaves like a global order parameter Cavalier-Smith, Annals of Botany 95 : 147-175 (2005)
The motivation to think about bounds rather than models • Bounds from reversible processes also constrain irreversible ones • Reversible-process bounds can be aggregated through state variables ; irreversible models usually cannot be • Bounds supersede models, unknown innovations, and ignorance of details
�������� � � � �� �� �� � � �������� �� � � � �������� ��� ��� ��� � The challenge of using equilibrium information for the biosphere • Life involves kinetics as well as energetics • Our biosphere could (?) be a “frozen accident” • Only if barriers are small enough that energy flow is limiting is information a relevant constraint But such limits can be suggested in surprising places...
Allometric scaling of growth Energy balance in ontogenetic growth B 0 m 3 / 4 = B c m + E c dm m c m c dt Consequence: scale-invariant growth trajectories � m � m � 1 / 4 � 1 / 4 d = 1 − d τ M M � m � 1 / 4 = 1 − e − τ M West G.B., Brown J.H. & Enquist B.J. (2001) A general model for otogenetic growth. Nature, 413, 628-631
Informational consequences of allometric scaling Q: Does life history depend on energy or information? • Energy/mass used by any � τ D E lifetime = E c 1 − e − τ � 3 � d τ stage of life is an invariant M m c 0 • What minimal energy would E M ∼ k B T M we expect is needed to put 10g N A “information” into biomass? • Energy/ideal by any life � τ D E lifetime E c 10g 1 − e − τ � 3 � = d τ stage is an invariant E M k B TN A m c 0 • Formation of biomass is E c 10g clocked by information , not ≈ 30 k B TN A m c directly by energy
Curious consequences • No direct evidence from growth that there is a cost to maintaining the living state • Even decay seems to be created in proportion to growth and repair processes • Living system scale as if they were on the energy/information bound, even though they deviate from it by an “inefficiency” factor
II. Instantiating chemical measures of information • Would like a model that is as equivocally metabolic and evolutionary • A literal sub system is more intuitive than an abstract vision of “life” • Consider cycles to leverage the Carnot construction from engines
Thermodynamics of chemistry How improbable is a chemical state? e − G X / τ = e − H X / τ × e σ X Extensive systems and G X = N X µ X the chemical potential Often convenient to work N X = V N A [ X ] with concentrations Probability to form a state � � [ X ] (Often choose _ comes from internal and � ¯ µ X = ¯ µ X + τ log [X] to refer to � X external context an equilibrium ), but not always = G X H X − τσ X 1 = τ ( H X − N X µ X ) σ X � � Chemical entropy satisfies 1 [ X ] � ¯ = τ ( H X − N X ¯ µ X ) − N X log an informational chain rule � X
Toy model for metabolism & evolution N +1 Phosphate-driven � N ATP + M α i ⇋ N AMP + 2 N P i + Π � α polymerization i =1 http://www.cem.msu.edu/ ~reusch/VirtualText/nucacids.htm ATP regeneration AMP + 2P i ⇋ ATP (Possibly sequence-dependent) equilibrium relations � N � [ATP] [ Π � α ] = α ( T ) z K � [AMP] [P i ] 2 z =1 [M z ] ν � α � Z http://www.rpi.edu/dept/bcbp/ molbiochem/MBWeb/mb1/part2/f1fo.htm
Can one model be representative? • Polymer degradation (digestion) and re- synthesis (anabolism) account for much of the energy of physiology • (and we can generalize to other reactions once we see how the answer looks) • Saw in the evolution example that genomic information behaves like a global information difference between species • Sidenote : the RNA-world idea for origin of life identifies these two, by equating self- replicating RNA with individuals
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