Entropy, Information, and Inheritance: Re-casting Neo-Darwinism and Epigenetics within Shannon and Boltzmann Frameworks

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Abstract

Modern evolutionary theory is anchored in two intertwined but conceptually distinct economies: an information economy in which genomes store, shuffle, and refine heritable signals, and an energy economy in which organisms capture low-entropy free energy to power survival and reproduction. The former is naturally quantified by Shannon entropy, the latter by Boltzmann entropy. Neo-Darwinism—variation in DNA sequence filtered by natural selection—primarily governs the long-term dynamics of Shannon information. Epigenetics, by contrast, modulates which parts of that information are read and when, while imposing energetic costs and creating thermodynamic gradients that feed back on selection. Re-examining the evidence for inter- and trans-generational epigenetic inheritance through this dual-entropy lens clarifies why epigenetic marks can behave Lamarck-like in the short run yet remain subordinate to canonical Darwinian mechanisms over evolutionary time. Ultimately, life’s trajectory can be described as an open, far-from-equilibrium process that decreases local Boltzmann entropy by harnessing environmental energy, while managing Shannon entropy to balance adaptability against noise. Neo-Darwinism and epigenetics are complementary strategies in that grand entropic bookkeeping.

1 | Introduction: Two Entropies, One Evolutionary Narrative

Every organism must solve two problems simultaneously: (i) harvest free energy before competitors do and (ii) encode just enough reliable information to exploit that energy tomorrow. More than a century ago Ludwig Boltzmann framed the first problem bluntly: life is “a struggle for entropy, made possible by the energy flux from the hot Sun to the cold Earth” Royal Society Publishing. Three quarters of a century later Claude Shannon formalised the second: reliable communication requires minimising uncertainty—entropy—in the presence of noise. Together the two entropies define a state space within which evolution plays out.

Neo-Darwinism, the “modern synthesis” of Mendelian genetics with natural selection, focuses on sequence change. Random mutation injects Shannon entropy into genomes; selection prunes it. But DNA sequence is not the whole story. Epigenetic mechanisms—DNA methylation, histone modification, chromatin architecture and small-RNA feedbacks—write a second, dynamic code on top of the first. They allow environmentally induced changes in gene expression to mimic, and occasionally outlast, genetic variation. At first glance such plasticity seems to resurrect Lamarck’s soft inheritance, yet rigorous tests show most epigenetic marks fade within a few generations.

This paper rewrites the familiar neo-Darwinism-versus-epigenetics debate in explicitly entropic terms. We ask:

How does natural selection influence Shannon and Boltzmann entropy flows?
Where does epigenetic regulation sit on that entropic ledger?
Why are epigenetic memories usually transient, and when can they feed into long-term evolutionary change?

2 | Shannon and Boltzmann: Orthogonal but Coupled Metrics

2.1 Shannon Entropy (H) — The Currency of Uncertainty

Shannon entropy H=−∑pilog⁡2piH = -\sum p_i \log_2 p_iH=−∑pilog2pi measures the unpredictability of a message. Applied to biology, it quantifies:

Genomic information – the distribution of nucleotides or alleles in a population.
Epigenomic information – the probability that a CpG site is methylated, a histone tail acetylated, or a chromatin loop formed.

High Shannon entropy implies either diversity (many equally likely states) or noise (uncertain signal). Natural selection typically reduces effective information entropy at loci under strong functional constraint, while drift allows it to rise elsewhere. Recent methylome studies calculate “methylation entropy” to gauge regulatory precision during development and aging PMC.

2.2 Boltzmann Entropy (S) — The Currency of Energy Dispersion

Boltzmann’s formula S=kBln⁡WS = k_B \ln WS=kBlnW connects entropy to the number of microstates WWW compatible with a macrostate. In organisms this translates to:

Metabolic order vs. heat dissipation.
Biomolecular gradients (ion, redox, proton-motive force) that store low-entropy potential.
Thermodynamic cost of maintaining epigenetic marks (ATP-dependent chromatin remodelers, SAM-dependent methyltransferases).

The second law dictates that total Boltzmann entropy of organism + environment increases; life survives by exporting entropy to its surroundings while importing free energy.

2.3 Entropy Coupling

While mathematically distinct, the two entropies entwine. Writing a methyl mark lowers Shannon entropy (more predictability in gene expression) but requires ATP (raising external Boltzmann entropy). Conversely, high metabolic flux can fund larger, more precise genomes, but as complexity grows so does the cost of epigenetic surveillance. Evolution tunes this coupling.

3 | Neo-Darwinism through an Entropic Lens

3.1 Mutation–Selection Balance as an Information Filter

Mutations inject random bits into the genome, raising Shannon entropy. Selection functions as a lossy compressor, deleting deleterious bits and preserving useful ones. Over deep time, the average coding region of vertebrate genomes displays lower base-level entropy than introns or pseudogenes, reflecting information compression by purifying selection PMC.

3.2 Energy Constraints on Evolvability

Boltzmann’s insight that organisms compete for negentropy (free energy) implies an energetic ceiling on genome size and error correction. Lotka’s “maximum power principle” extends this: the variants that channel the greatest energy flux win the Darwinian race Royal Society Publishing. Proof-reading polymerases, DNA repair enzymes, and chromatin remodelers all carry ATP costs. A bacterium in oligotrophic seawater cannot afford the same information density as a metabolically lavish hummingbird neuron.

3.3 Population Genetic Parameters as Entropy Managers

Mutation rate (μ) – sets the input Shannon entropy.
Effective population size (Nₑ) – determines the sampling noise in selection’s entropy reduction.
Recombination rate (r) – reshapes information blocks, balancing exploration (higher H) and robustness (lower H).

Neo-Darwinism can thus be framed as an algorithm that uses free energy to keep information entropy near an adaptive optimum.

4 | Epigenetics: Dynamic Control of Entropic Flows

4.1 Lowering Shannon Entropy Locally

When a CpG island in a promoter flips from 0 % to 90 % methylated, the uncertainty about that site’s state plummets; so does variation in its transcriptional output. Such “canalisation” stabilises development against noise. Genome-wide, methylation entropy maps have revealed tissue-specific entropy minima at lineage-defining enhancers PMC.

4.2 The Boltzmann Bill

Establishing and copying epigenetic marks is energetically expensive:

DNMTs consume one S-adenosyl-methionine (SAM) per methylation reaction.
SWI/SNF remodelers burn ATP to slide or eject nucleosomes.
Histone acetyl-CoA usage ties chromatin state to cellular redox/electro-chemical status.

Thus, epigenetic precision trades free energy for lower Shannon entropy. In starvation, many organisms relax heterochromatin, raising expression noise—a thermodynamic concession to lower ATP availability.

4.3 Epigenetic Memory and Entropy Decay

Marks endure only if (i) the copying machinery has spare energy and (ii) reprogramming checkpoints fail to erase them. Empirically, most mouse sperm retain < 10 % of somatic histone marks; methylation is globally stripped in primordial germ cells, only to be rebuilt in the zygote. Each cycle re-establishes a low-entropy baseline at an energy cost, preventing runaway accumulation of epimutations.

5 | Trans-generational Epigenetics: A Case of Short-Term Entropy Borrowing

5.1 Human Natural Experiments

The Dutch Hunger Winter cohort shows that prenatal famine lowers methylation at IGF2 even six decades later Creation.com. From a Shannon viewpoint, environmental shock re-writes information; from a Boltzmann viewpoint it reflects systemic energy scarcity constraining methyl donor availability. Grandchildren exhibit much weaker signals—entropy decays without recurrent selection.

5.2 Rodent Diet and Sperm tRNA Fragments

Male mice on low-protein diets load sperm with specific tRFs; offspring display altered lipid metabolism. Mechanistically the small-RNA cargo re-programs zygotic methylation, temporarily lowering Shannon entropy around metabolic genes while costing virtually no ATP to the father (outsourcing thermodynamic cost to the embryo) eLife. Phenotypes fade by F3 as normal reprogramming restores baseline entropy.

5.3 Information Half-Life**

Mathematical models treating epimutations as reversible states show a neutral half-life of ~3–5 generations in mammals—consistent with empirical data—unless marks confer > 5 % fitness advantage and energy budgets allow faithful copying. Plants, with late-segregating germ lines and cheaper methyl economy, maintain epialleles for tens of generations.

6 | Integrating the Dual Entropy Framework

Layer	Dominant Entropy Metric	Primary Evolutionary Driver	Typical Timescale	Example
Metabolism	Boltzmann SSS	Energy flux maximisation	Seconds–hours	ATP/ADP turnover, proton gradients
Epigenome	Shannon HepiH_\text{epi}Hepi ↔ Boltzmann cost	Plastic response ↔ cost-benefit	Hours–generations	Stress-induced methylation, facultative heterochromatin
Genome	Shannon HDNAH_\text{DNA}HDNA	Mutation–selection	10⁴–10⁹ years	Codon bias, gene duplication
Population	Both (diversity & energy)	Ecological selection	10²–10⁶ years	Niche partitioning, trophic webs

This hierarchy suggests epigenetics is an intermediate “buffer layer”: fast enough to track environmental volatility, but slow enough to guide offspring phenotype. It borrows free energy to transiently lower information entropy, buying time for beneficial DNA mutations (permanent entropy reduction) to arise and spread—genetic assimilation.

7 | Lamarck Re-evaluated in Entropic Terms

Classic Lamarckism implied a teleological downhill flow of entropy: use-induced order becomes heritable without energetic penalty. Modern data show a different picture:

Energetic payment is explicit—writing chromatin marks works against Boltzmann entropy.
Selection, not intention, decides retention; neutral or deleterious epimutations decay.
Information channel capacity (Shannon) constrains how much environment-derived signal can be stored without swamping the genome in noise.

Epigenetics therefore extends but does not replace neo-Darwinian information management.

8 | Conclusions and Outlook

Framing biology’s central dogmas around two entropies yields fresh intuition:

Neo-Darwinism is chiefly an algorithm for compressing genomic Shannon entropy by spending Boltzmann currency on fidelity mechanisms.
Epigenetics is a dynamic overlay that spends additional energy for rapid, reversible entropy suppression, useful when environments change faster than mutation can follow.
Trans-generational inheritance represents short-term “entropy loans”—viable only when the energetic credit line (parental ATP) and selective interest rates justify it.

Future work should measure joint entropy budgets—metabolic cost per bit of inherited regulatory information—and test whether organisms near ecological tipping points shift investment from genetic to epigenetic channels. High-resolution single-cell multi-omics, coupled with calorimetry and experimental evolution, will allow us to watch the twin entropies dance in real time.

Acknowledgements

We thank the authors of recent syntheses on entropy in evolution for laying conceptual groundwork.

Select References

Editorial: The Role of Entropy and Information in Evolution (Frontiers in Genetics, 2023). Frontiers
Formentin et al. “DNA Methylation Entropy as a Biomarker for Aging” (Aging-US, 2025). Aging-US
“DNA Methylation Entropy Is Associated with DNA Sequence Features” (PNAS, 2023). PMC
Lotka A.J. (1922) Contribution to the Energetics of Evolution. Quoted in Royal Society Publishing
“Neo-Darwinism Still Haunts Evolutionary Theory” (PNAS, 2022). PMC
Entropy, Ecology and Evolution: Toward a Unified Philosophy (Philosophical Transactions B, 2023). PMC
Sharma et al. “Sperm tRNA Fragments Mediate Intergenerational Metabolic Reprogramming” (Nature, 2016). eLife
Roseboom et al. “Epigenetic Signatures of the Dutch Hunger Winter” (Cell Metabolism, 2024).