Over the last half-billion years, squid, octopuses and their kin have evolved much like a fireworks display, with long, anticipatory pauses interspersed with intense, explosive changes. The many-armed diversity of cephalopods is the result of the evolutionary rubber hitting the road right after lineages split into new species, and precious little of their evolution has been the slow accumulation of gradual change.

They aren’t alone. Sudden accelerations spring from the crooks of branches in evolutionary trees, across many scales of life — seemingly wherever there’s a branching system of inherited modifications — in a dynamic not examined in traditional evolutionary models.

That’s the perspective emerging from a new mathematical framework published in Proceedings of the Royal Society B that describes the pace of evolutionary change. The new model, part of a roughly 50-year-long reimagining of evolution’s tempo, is rooted in the concept of punctuated equilibrium, which was introduced by the paleontologists Niles Eldredge and Stephen Jay Gould in 1972.

“Species would just sit still in the fossil record for millions of years, and then all of a sudden — bang! — they would turn into something else,” explained Mark Pagel, an evolutionary biologist at the University of Reading in the United Kingdom.

Punctuated equilibrium was initially a controversial proposal. The theory diverged from the dominant, century-long view that evolution adhered to a slow, steady pace of Darwinian gradualism, in which species incrementally and almost imperceptibly developed into new ones. It opened the confounding possibility that there was a discontinuity between the selection processes behind the microevolutionary changes that occur within a population and those driving the long-term, broad-scale changes that take place higher than the species level, known as macroevolution.

In the decades since, researchers have continued to debate these views as they’ve gathered more data: Paleontologists have accumulated fossil datasets tracing macroevolutionary changes in ancient lineages, while molecular biologists have reconstructed microevolution on a more compressed timescale — in DNA and the proteins they encode.

Now there are enough datasets to more fully test the theories of evolutionary change. Recently, a team of scientists blended insights from several evolutionary models with new methods to build a mathematical framework that better captures real evolutionary processes. When the team applied their tools to a selection of evolutionary datasets (including their own data from research into an ancient protein family), they found that evolutionary spikes weren’t just common, but somewhat predictably clustered at the forks in the evolutionary tree.

Their model showed that proteins contort themselves into new iterations more rapidly around the time they diverge from each other. Human languages twist and recast themselves at the bifurcations in their own family tree. Cephalopods’ soft bodies sprout arms and bloom with suckers at these same splits.

The new study adds to previous support for the punctuated equilibrium phenomenon, said Pagel, who wasn’t involved in the project. However, the rapid evolutionary behavior isn’t a unique process separate from natural selection, as Eldredge and Gould suggested, but rather the result of periods of extremely rapid adaptation propelling evolutionary change.

“This is really a rather beautiful story in the philosophy of science,” Pagel said.

Jordan Douglas, an evolutionary biologist at the Australian National University in Canberra, is fascinated by the origins of the genetic code. To understand those first stages of life’s evolution, he studies aminoacyl-tRNA synthetases (aaRSs), a family of enzymes essential to building proteins. The aaRS enzymes appear to predate the last universal common ancestor for all life on the planet.

“These enzymes are responsible for creating that kind of reflexive logic that nature uses to build itself, by helping to translate RNA into proteins which copy RNA, which build more proteins, which copy more RNA,” Douglas said.