Selection can preserve a message only if replication copies that message accurately enough. The error threshold turns this intuition into a sharp criterion: it separates a regime in which a quasi-species stays localized around its master sequence from a regime in which mutational pressure delocalizes the stored information entirely.

Whole-Sequence Fidelity, Not Per-Base Fidelity

• is the probability of correctly copying the entire sequence of length m in one replication round. Under independent-site errors it factorizes as, where q is the per-site fidelity.
• Because fidelity multiplies across sites, it falls exponentially with length. That single fact is the origin of the information limit.

The Threshold Condition

• The master sequence is maintained only if ⁻¹, where is the selective superiority of the master sequence relative to its mutant cloud.
• At the system is marginal; below it, selection can no longer hold the master sequence against mutational degradation.
• A non-trivial threshold requires . Higher selective superiority permits either lower fidelity or longer maintainable sequences.

Two Regimes

• Below the error threshold : the quasi-species remains localized around the master sequence, and the information is retained.
• Above the error threshold : the mutant cloud delocalizes — the error catastrophe — and the information content is lost.

The Information Limit

Setting equal to ⁻¹ and solving for the length gives the maximum amount of information that can be maintained reproducibly:

/ (−ln q)    or, for ,    .

Maintainable information therefore grows only logarithmically with selective advantage, while it responds inversely to the per-site error rate. Doubling adds only to the numerator.

Worked Example

With and , independent-site copying gives / (−ln 0.99) ≈ bases. Increasing either the fidelity or the selective superiority extends this limit.