Recurrence in Sequence Space: How Epistasis Mimics Shared Ancestry
Sequences in a protein family do not explore the sequence space freely: folding, packing, and binding couple residues, so the set of viable sequences is irreducibly correlated, a phenomenon referred to as epistasis. We show that such correlations lower the \emph{intrinsic dimension} $d_{\rm IN}$ of sequence space, counteracting the concentration of measure and broadening the distribution of pairwise distances. Consequentially, they make it likely that independently evolving lineages \emph{recur}, approaching one another by chance long after they last shared an ancestor. Using Kac's lemma we show that the logarithmic mean recurrence time grows linearly with the intrinsic dimension, $\log τ_{\rm rec} \propto {d_{\rm IN}}$ for typical sequences, so strongly constrained (low-dimensional) families recur often and generate sequence similarity that carries no genealogical signal. We establish this first in a solvable Ising-like model with tunable correlation, then in a real protein family using Potts models and Markov chain Monte Carlo, where $d_{\rm IN}$ remains predictive of recurrence even in multi-basins landscapes. Because recurrence is a property of the evolutionary process rather than of any estimator, it bounds the recoverability of deep history for distance-based and likelihood methods alike.