Critical Sequence Identity: An analytical threshold using the incomplete gamma function

Let p denote the per-base mutation probability during genome replication, and let n denote the genome length. In a primitive genome model possessing only self-replication capability, we derive the critical sequence identity Q required to maintain replication function as follows. Using the Poisson approximation to the binomial mutation count, we obtain Q = max{q ∈ {0, 1/n , 2/n , ..., 1} | Γ (1 + n − ⌊nq⌋, np) /Γ (1 + n − ⌊nq⌋) = 1/2} where ⌊ x ⌋ denotes the floor function. Notably, in the regime where the expected number of mutations np is sufficiently large, this analytical threshold simplifies to Q ⋍ 1 - p, which aligns with the intuitive understanding that higher mutation rates necessitate stricter sequence identity.