"Many cellular processes are carried out by molecular ‘machines’—assemblies of multiple differentiated proteins that physically interact to execute biological functions1, 2, 3, 4, 5, 6, 7, 8. Despite much speculation, strong evidence of the mechanisms by which these assemblies evolved is lacking. Here we use ancestral gene resurrection9, 10, 11 and manipulative genetic experiments to determine how the complexity of an essential molecular machine—the hexameric transmembrane ring of the eukaryotic V-ATPase proton pump—increased hundreds of millions of years ago. We show that the ring of Fungi, which is composed of three paralogous proteins, evolved from a more ancient two-paralogue complex because of a gene duplication that was followed by loss in each daughter copy of specific interfaces by which it interacts with other ring proteins. These losses were complementary, so both copies became obligate components with restricted spatial roles in the complex. Reintroducing a single historical mutation from each paralogue lineage into the resurrected ancestral proteins is sufficient to recapitulate their asymmetric degeneration and trigger the requirement for the more elaborate three-component ring. Our experiments show that increased complexity in an essential molecular machine evolved because of simple, high-probability evolutionary processes, without the apparent evolution of novel functions. They point to a plausible mechanism for the evolution of complexity in other multi-paralogue protein complexes."

Also: how about an inverse Doomsday algorithm. "We propose a simplification of a key component in the Doomsday Algorithm for calculating the day-of-the-week of any given date.…"

"Building a repository of proof-checked mathematical knowledge is without any doubt a lot of work, and besides the actual formalization process there also is the task of maintaining the repository. Thus it seems obvious to keep a repsoitory as small as possible, in particular each piece of mathematical knowledge should be formalized only once. In this paper, however, we claim that it might be reasonable or even necessary to duplicate knowledge in a mathematical repository. We analyze different situations and reasons for doing so and provide a number of examples supporting our thesis."

Find myself wondering if one could develop an algorithm to "invent" a taxon that would maximally disrupt the resulting tree structure if added…