  The length distribution of proteins measured in amino acids follows the CoHSI
(Conservation of Hartley-Shannon Information) probability distribution. In
previous papers we have verified various predictions of this using the Uniprot
database but here we explore a novel predicted relationship between the longest
proteins and evolutionary time. We demonstrate from both theory and experiment
that the longest protein and the total number of proteins are intimately
related by Information Theory and we give a simple formula for this. We stress
that no evolutionary explanation is necessary; it is an intrinsic property of a
CoHSI system. While the CoHSI distribution favors the appearance of proteins
with fewer than 750 amino acids (characteristic of most functional proteins or
their constituent domains) its intrinsic asymptotic power-law also favors the
appearance of unusually long proteins; we predict that there are as yet
undiscovered proteins longer than 45,000 amino acids. In so doing, we draw an
analogy between the process of protein folding driven by favorable pathways (or
funnels) through the energy landscape of protein conformations, and the
preferential information pathways through which CoHSI exerts its constraints in
discrete systems.
  Finally, we show that CoHSI predicts the recent appearance in evolutionary
time of the longest proteins, specifically in eukaryotes because of their
richer unique alphabet of amino acids, and by merging with independent
phylogenetic data, we confirm a predicted consistent relationship between the
longest proteins and documented and potential undocumented mass extinctions.
