co₂-metabolism hypothesis
P(HL) vs N, probability of a Lane-like proto-core
Nick Lane's hypothesis proposes that life originated not from a prebiotic soup but from the geochemistry of alkaline hydrothermal vents. Where hydrogen-rich fluids from serpentinizing rock met CO₂-laden ocean water, a natural electrochemical gradient drove carbon fixation, the same reaction that powers the acetyl-CoA pathway in modern methanogens and acetogens.
The 4-reaction Lane motif, a minimal proto-metabolic core:
A = activated intermediate, C = catalyst/cofactor, B = boundary precursor. Reaction r4 is autocatalytic: the system can amplify itself.
This playground models a threshold question: molecules are randomly assigned roles (A, C, B) and distributed across vent pore compartments. Each compartment is checked for a Lane-like motif. The central parameter is catalytic density λ, the probability that any given molecular interaction actually catalyzes a reaction.
The expected motif count scales as:
As N grows, proto-cores become probabilistically inevitable, not guaranteed in any single compartment, but increasingly hard to avoid across the ensemble.
The threshold N* marks where P(HL) crosses the target probability. Empirically, N* scales with λ as a power law. The heuristic analysis predicts:
Higher catalytic density dramatically lowers the molecular threshold for proto-metabolic emergence.
Compartmentalization matters: vent pores act as natural reaction vessels. More pores (higher q) dilute molecules across compartments, pushing thresholds upward. But each pore provides an independent trial, a tradeoff between concentration and combinatorial opportunity.
This is a toy model with no stoichiometry, kinetics, or thermodynamics. It captures threshold behavior, not chemistry. The point is that the transition from "possible" to "nearly forced" can be sharp, and that sharpness does not depend on any one lucky compartment but on the statistics of the entire vent system.
Implementation
Calibration
Assumptions
Model changelog
- role model: each molecule is an activator A, catalyst C or boundary precursor B with tunable probabilities, placed uniformly across q vent-pore compartments.
- motif rule: a compartment closes a proto-core when it holds at least one A, one C and one B and all four catalytic edges fire, each edge succeeding with probability lambda.
- closed-form core: expected motif count mu = p_A p_C p_B (N^3 / q^2) lambda^4, derived from independent role assignment and independent placement.
- Monte-Carlo estimator: ensemble probability curve P(H_L) vs N for several lambda values, with a seeded mulberry32 generator for reproducibility.
- threshold analysis: N* read off where P crosses a target probability, then a log-log power-law fit recovering the N* ~ lambda^(-4/3) scaling.
- witness draw: a single successful compartment configuration is surfaced so the abstract motif is made concrete.
- calibration: the deterministic mu function is checked case-by-case against a hand-written re-derivation, plus the cubic-in-N and lambda^(-4/3) scaling exponents.
- scope kept honest: no stoichiometry, energetics or kinetics; the model quantifies the sharpness of the possible-to-likely transition, not the chemistry.