Dysnomia

This morning I was looking at a nice website that lets you zoom in or out and see objects of different sizes, ranging from the Planck length to the entire observable Universe:

• Cary Huang, The Scale of the Universe.

When I zoomed out to a size somewhat larger than Rhode Island, I saw something strange:

Huh? I recently saw a movie called Melancholia, about a pair of sisters struggling with fear and depression, and a blue planet of that name that’s headed for Earth. And indeed, dysnomia is another mental disorder: a difficulty in remembering words, names or numbers. Are astronomical bodies named for mental disorders catching on?

I don’t know—but unlike Melancholia, Dysnomia is real! It’s the speck at left here:

The larger blob of light is the dwarf planet Eris. Dysnomia is its moon. Both were discovered in 2005 by a team at Palomar led by Mike Brown.

Why the funny name? In Greek mythology, Dysnomia (Δυσνομία), meaning ‘lawlessness’, was the daughter of Eris, the goddess of strife and discord. You may remember the dwarf planet Eris under its tentative early name ‘Xena’, from the TV character. That was deemed too silly to stand.

Eris is 30% more massive than Pluto, and thus it helped lead to a redefinition of ‘planet’: both are now called dwarf planets, because they aren’t big enough to clear their neighborhood of debris.

Eris has a highly eccentric orbit, and it takes 580 years to go around the Sun. Right now it’s at its farthest from the Sun: three times as far as Pluto. So it’s very cold, about 30 kelvin (-243 °C), and its surface is covered with methane ice. But in about 290 years its temperature will nearly double, soaring to a balmy 56 kelvin (-217 °C). Its methane ice will melt and then evaporate away, giving this world a new atmosphere! That’s pretty amazing: an annual atmosphere.

The surface of Eris is light gray. So, it’s quite different than Pluto and Neptune’s moon Triton, which are reddish due to deposits of tholins—complex compounds formed by bombarding hydrocarbons and nitrogen with sunlight.

Remember the smoggy orange surface of Titan?

That’s due to tholins! It’s possible that on Eris the tholins are currently covered by methane ice.

I wish I knew more about what tholins actually are—their actual chemical structures. But they’re a complicated mess of stuff, and they vary from place to place: people talk about Titan tholins, Triton tholins and ice tholins.

Indeed, the term “tholin” comes from the Greek word tholós (θολός), meaning “not clear”. It was coined by Carl Sagan to describe the mysterious mix of substances he created in experiments on the gas mixtures found in Titan’s atmosphere… experiments a bit like the old Urey-Miller experiment which created amino acids from stuff that was present on the early Earth—water, methane, ammonia, hydrogen—together with lots of electrical discharges.

Might tholins become complicated enough to lead to life in the outer Solar System, at least on relatively peppy worlds like Titan? There’s a complex cycle going on there:

Here ‘Da’ means daltons: a dalton is a unit of mass equal to about one hydrogen atom, so a molecule that’s 2000 Da could be made of 2000 hydrogens or, more reasonably, 2000/12 ≈ 166 carbons, or various other things. The point is: that’s a pretty big complicated molecule—big enough to be very interesting!

On Earth, many soil bacteria are able to use tholins as their sole source of carbon. Some think that tholins may have been the first food for microbes! In fact, some scientists have speculated that Earth may have been seeded by tholins from comets early in its development, providing the raw material necessary for life. But ever since the Great Oxygenation Event, the Earth’s surface has been too oxidizing for tholins to exist here.

Tholins have also been found outside our Solar System:

• Red dust in disk may harbor precursors to life, Carnegie Institute news release, 5 January 2008.

I bet tholins often go hand-in-hand with PAHs, or polycyclic aromatic hydrocarbons. PAHs are also common in outer space. In Earth you can find them in soot, or the tarry stuff that forms in a barbecue grill. Wherever carbon-containing materials suffer incomplete combustion, you’ll find PAHs.

PAHs are made of hexagonal rings of carbon atoms, with some hydrogens along the edges:

Benzene has a single hexagonal ring, with 6 carbons and 6 hydrogens. You’ve probably heard about naphthalene, which is used for mothballs: this consists of two hexagonal rings stuck together. True PAHs have more. With three rings you can make anthracene:

and phenanthrene:

With four, you can make napthacene:

pyrene:

triphenylene:

and chrysene:

And so on! The game just gets more complicated as you get to use more puzzle pieces.

In 2004, a team of scientists discovered anthracene and pyrene in an amazing structure called the Red Rectangle!

Here two stars 2300 light years from us are spinning around each other while pumping out a huge torus of icy dust grains and hydrocarbon molecules. It’s not really shaped like a rectangle or X—it just looks that way from here. The whole scene is about 1/3 of a light year across.

This was first time such complex molecules had been
found in space:

• Uma P. Vijh, Adolf N. Witt, and Karl D. Gordon, Small polycyclic aromatic hydrocarbons in the Red Rectangle, The Astrophysical Journal 619 (2005), 368-378.

By now, lots of organic molecules have been found in interstellar or circumstellar space. There’s a whole "ecology" of organic chemicals out there, engaged in complex reactions. Life on planets might someday be seen as just an aspect of this larger ecology. PAHs and tholins are probably among the dominant players in this ecology, at least at this stage.

Indeed, I’ve read that about 10% of the interstellar carbon is in the form of PAHs—big ones, with about 50 carbons per molecule. They’re common because they’re incredibly stable. They’ve even been found riding the shock wave of a supernova explosion!

PAHs are also found in meteorites called "carbonaceous chondrites". These space rocks contain just a little carbon: about 3% by weight. But 80% of this carbon is in the form of PAHs.

Here’s an interview with a scientist who thinks PAHs were important precursors of life on Earth:

• Aromatic world, interview with Pascale Ehrenfreund, Astrobiology Magazine.

Also try this:

• PAH world hypothesis, Wikipedia.

This speculative hypothesis says that PAHs were abundant in the primordial soup of the early Earth and played a major role in the origin of life by mediating the synthesis of RNA molecules, leading to the (also speculative) RNA world.

Another radical theory has been proposed by Prof. Sun Kwok, author of Organic Matter in the Universe. He claims that instead of PAHs, complex molecules like this would do better at explaining the spectra of interstellar clouds:

Would this molecule count as a tholin? Maybe so: I don’t know. He says:

Our work has shown that stars have no problem making complex organic compounds under near-vacuum conditions. Theoretically, this is impossible, but observationally we can see it happening.

For more see:

• Sun Kwok and Yong Zhang, Mixed aromatic–aliphatic organic nanoparticles as carriers of unidentified infrared emission features, Nature 479 (2011), 80–83.

This paper isn’t free, but here’s a summary that is:

• Astronomers discover complex organic matter exists throughout the Universe, ScienceDaily, 26 October 2011.

However, I’d take this with a grain of salt until more confirmation comes along! They’re matching very complicated spectra to hypothetical chemicals, without yet any understanding of how these chemicals could be formed in space. It would be very cool if true.

Regardless of how the details play out, I think we’ll eventually see that organic life across the universe is a natural outgrowth of the organic chemistry of PAHs, tholins and related chemicals. It will be great to see the whole story: how much in common life has in different locations, and how much variation there is. It may be rare, but the universe is very large, so there must be statistical patterns in how life works.

It goes to show how everything is connected. Starting from a chance encounter with Dysnomia, we’ve been led to ponder another planet whose atmosphere liquifies and then freezes every year… and then wonder about why so many objects in the outer solar system are red… and why the same chemicals you find in the tarry buildup on a barbecue grill are also seen in outer space… and whether life on Earth could have been started by complex compounds from comets… and whether life on planets is just part of a larger interstellar chemical ‘ecology’. Not bad for a Saturday morning!

Dolphins and Manatees of Amazonia

No, these aren’t mermaids. They’re sirenians!

Sirenians or ‘sea cows’ are aquatic mammals found in four places in the world. The three places shown here are home to three species called ‘manatees’:

For example, the sirenians shown above are West Indian manatees, Trichechus manatus, which live in the Caribbean. There’s also a big region stretching from the western Pacific Ocean to the eastern coast of Africa that’s home to the ‘dugong’.

Right now there’s one different species of sirenian in each place. But once there were many more species, and it’s just been discovered that there often used to be several species living in the same place:

• Multiple species of sea cows once coexisted, Science Daily, 8 March 2012.

The closest living relatives of the sirenians are elephants! They kind of look similar, no? More importantly, they share some unusual features. They keep growing new teeth throughout their life, molars that slowly move to the front of the mouth as the teeth in front wear out. And quite unlike cows, say, the females have two teats—located between their front limbs.

Here’s an evolutionary tree of sirenians:

You’ll see they got their start about 50 million years ago and blossomed in the late Oligocene, about 25 million years ago. Later the Earth got colder, and they gradually retreated to their present ranges.

You’ll also notice that three branches of the tree seem to reach the present day:

• Trichechus, which includes all the manatees,

• Dugong, which (surprise!) is the dugong… and

• Hypodamilis, which is another name for Steller’s sea cow.

Steller’s sea cow was discovered in the North Pacific in 1741, and hunted to extinction shortly thereafter. Ouch! It took 24 million years of evolution to refine and polish the information in that species, and it was wiped out without trace in just 27 years.

The Amazonian manatee, Trichechus inunguis, is of special interest to me today because it lives in many branches of the Amazon river:

How did it get there? Why does it live in rivers? Its nearest living neighbor, the West Indian Manatee, likes coastal waters but can also go up rivers. Another clue might be the wonderful Amazon river dolphin, Inia geoffrensis.

It’s also called a pink dolphin. Here’s why:

Their are some interesting myths about it… one of which connects it with the manatee!

In traditional Amazon River folklore, at night, an Amazon river dolphin becomes a handsome young man who seduces girls, impregnates them, and then returns to the river in the morning to become a dolphin again. This dolphin shapeshifter is called an encantado. It has been suggested that the myth arose partly because dolphin genitalia bear a resemblance to those of humans. Others believe the myth served (and still serves) as a way of hiding the incestuous relations which are quite common in some small, isolated communities along the river. In the area, there are tales that it is bad luck to kill a dolphin. Legend also states that if a person makes eye contact with an Amazon river dolphin, he or she will have lifelong nightmares. Local legends also state that the dolphin is the guardian of the Amazonian manatee, and that, should one wish to find a manatee, one must first make peace with the dolphin.”

Indeed, the range of the Amazon river dolphin, shown here, is similar to that of the Amazonian manatee:

Dolphins and other cetaceans are not closely related to sirenians. Dolphins are carnivores, but sirenians only eat plants. But they both started as land-dwelling mammals, and both took to the seas at roughly the same time. And it seems the Amazon river dolphin became a river dweller around 15 million years ago. Why? As sea levels dropped, what once was an inland ocean in South America gradually turned into what’s now the Amazon! According to the Wikipedia article:

It seems this species separated from its oceanic relatives during the Miocene epoch. Sea levels were higher at that time, says biologist Healy Hamilton of the California Academy of Sciences in San Francisco, and large parts of South America, including the Amazon Basin, may have been flooded by shallow, more or less brackish water. When this inland sea retreated, Hamilton hypothesizes, the Amazon dolphins remained in the river basin…

So maybe the manatees did the same thing. I don’t know. But I find the idea of an inland sea gradually turning into a river-filled jungle, and life adapting to this change, very intriguing and romantic!

This shows what South America may have looked like during the early-middle Miocene, when the Amazon river dolphin was just getting its start. The upper Amazon Basin drained into the Orinoco Basin at left, while the the lower Amazon Basin drained directly to the Atlantic Ocean at fight. This is from a paper on megafans, which are huge regions covered with river sediment:

• M. Justin Wilkinson, Larry G. Marshall, and John G. Lundberg, River behavior on megafans and potential influences on diversification and distribution of aquatic organisms, Journal of South American Earth Sciences 21 (2006), 151–172.

Almost needless to say, we’ll need to work a bit to protect the dolphins and manatees of Amazonia if we want them to survive. Check out this Amazon river dolphin in action:

This guy is swimming in the Rio Negro, a large tributary of the Amazon. But there are also Amazon river dolphins in the Orinoco, another huge river in South America, not connected to the Amazon! You can see it just north of the Rio Negro:

Was it ever connected to the Amazon? If not, what’s the story about how the same species of dolphins live in both river basins?

By the way, my joke about mermaids comes from the etymology of the word ‘sirenian’. There’s a legend that lonely sailors—very lonely, it seems—mistook sea cows for mermaids, also known as ‘sirens’.

The Global Amphibian Crisis

There’s a fungus that infects many kinds of amphibians. Some get wiped out entirely—but it’s harbored harmlessly by others, so it’s impossible to eradicate. Over a hundred species have disappeared in the last 20 years!

You’ve got to read this:

• Joseph R. Mendelson III, Lessons of the lost, American Scientist 99 (November-December 2011), 438.

The fungus causes a disease called chytridiomycosis. The effects are gruesome: when spores land on a susceptible amphibian, they quickly sprout and form a vase-shaped structure that harvests energy from the animal’s skin. This produces more spores, which swim around using flagella and spread. The disease progresses as these reinfect the host. The victim may become lethargic, lose skin over its body, go into convulsions, and die.

Amphibian populations have been dropping rapidly worldwide since the 1980s. There were about 6500 species, but now 30% of these are endangered, about 130 are ‘missing’, and about 30 are extinct in the wild.

There were many theories about the cause of this decline, but now we know this disease is playing a big role. As Mendelson says:

Herpetologists and wildlife biologists began observing inexplicable disappearances of amphibians around the globe in the mid-1970s and especially by the mid-1980s but were at a complete loss to explain them. Finally, in the late 1990s, an insightful team of pathologists at the U.S. National Zoo, led by Don Nichols, collaborated with one of the few chytrid fungus scholars in the world, Joyce Longcore, and identified this quite unusual new genus and species.

Conservationists and disease ecologists were unprepared for the reality of a pathogen capable of directly and rapidly—mere months!—causing the elimination of a population or an entire species that was otherwise robust. Classical host-pathogen theory held that such dramatic consequences to the host population or species were only realized when the host population was already drastically reduced in size or otherwise compromised. The concept of a lightning extinction was foreign to researchers and conservationists, and we argued vehemently about it throughout the 1990s at symposia worldwide.

In retrospect, the scenario of a spreading pathogen is parsimonious and clear, but in the midst of the massacre we were entangled in logical quagmires along these lines: “The disappearances cannot be the result of disease; diseases are not capable of such.” Not to mention the fact that the smoking gun, the pathogen itself, was not described until 1999. While we were debating the issue, a terrible lesson was playing out for us around the world as an unknown disease decimated amphibian populations.

What are the ‘lessons’ that Mendelson is talking about? Here are some:

Our powerlessness in this terrible crisis must be balanced by increased efforts in realms that we can control, such as reducing carbon emissions to protect what habitat remains from chemical and physical disruption. We can go further and restore what has been wounded but can still be salvaged. We need to inspire and fund truly innovative research on pathogens in order to better predict and thwart emerging infectious diseases. The lessons we learn here will extend far beyond the amphibians. We must support funding for programs such as the Amphibian Ark and the Amphibian Survival Alliance. We must keep looking for species gone missing, and continue biodiversity surveys, despite the sometimes paralyzing depression that both activities can induce in this era. But especially, we need to pay close attention to the lessons that legions of dead amphibians are teaching us. I note with some satisfaction that our colleagues in bat research and conservation did not spend a decade arguing whether the fungus that causes white-nose syndrome could possibly eliminate entire colonies of bats in a single season. Our colleagues assumed that it was possible and reacted quickly. We can thank the amphibians for leaving us that lesson, but at such cost.

Yes, millions of bats in America have died from a new fungal disease called white-nose syndrome.

What role, if any, do people play in the spread of these new diseases? Why are they happening now?

In the case of amphibians, people helped spread American bullfrogs. These are resistant to the disease, but carry it. They’ve largely taken over here in Singapore.

Global warming seems not to be responsible, because the worst outbreaks happen at high elevations, where it’s cool: that’s where the fungus thrives.

As for the bats, the same fungus that’s killing bats in America is found in healthy bats in Europe, which suggests the disease spread from there. People might carry spores on their clothes from infected caves to not-yet-infected ones, so visitors to caves with bats are being asked to limit their activities, and disinfect clothing and equipment. It’s completely against the rules to visit some caves now.

There have been successful attempts to cure some amphibians of chytridiomycosis:

• Reid Harris of James Madison University has claims that coating frogs with Janthinobacterium lividum protects them from chytridiomycosis.

• A team of scientists published a paper claiming that Archey’s frog (Leiopelma archeyi), a critically endangered species in New Zealand, was successfully cured of chytridiomycosis by applying chloramphenicol topically.

• Don Nichols claims to have cured several species of frogs using a drug called itraconazole.

• Jay Redmond at WWT Slimbridge, Gloucestershire claims that raising poison dart frogs in water containing Rooibos tea (Aspalathus linearis) wards off chytridiomycosis.

The Amphibian Ark is trying to keep populations alive that have died in the wild. They have a list of suggestions on what you can do to help. For starters:

• Don’t ever release pet amphibians into the wild.

• Build a frog pond: here’s how. Even in arid places like Riverside California, our friends who built some ponds soon found them occupied by sweetly chirping frogs.

• Get involved in collaborations that promote sustainable breeding and management, like the Amphibian Steward Network.

• Figure things out. Zoos don’t even know how to breed common toads without using artificial hormone injections! If you could find a way, maybe the same technique could be used with threatened species.

• If you’re a student, go to James Madison University and work with Reid Harris:

or go to the University of Maine and work with Joyce Longcore:

or find a university closer to you with someone leading a group that studies chytridiomycosis!

(Click on the pictures for even more info.)

I thank Allen Knutson for pointing out the American Scientist article. This is the best popular science magazine in the English language, but I let my subscription lapse when I came to Singapore!

Wild Cats of Arizona

Here’s a quick followup to our discussion of the wild cats of Sumatra caught on camera by the World Wildlife Foundation. Recently there have been sightings of rare wild cats in Arizona!

• Marc Lacey, In southern Arizona, rare sightings of ocelots and jaguars send shivers, New York Times, 4 December 2011.

• Guide describes roaring, powerful jaguar, Arizona Daily Star, 23 November 2011.

For example, consider Donnie Fenn, who specializes in hunting and killing mountain lions (also known as cougars or pumas). He was taking his 10-year-old daughter out on her first hunt when his pack of hounds took off and cornered something in a tree. He then saw with the telephoto lens of his camera that it wasn’t a mountain lion—it was a jaguar, which is about twice as big!

“It’s the most amazing thing that’s ever happened to me,” said Fenn, who leads hunters to mountain lions with his dogs. “To be honest with you—I got to see it in real life, my daughter got to see it, but I hope never to encounter it again.

“I was nervous, scared, everything. It was just the aggressiveness—the power it had, the snarling. It wasn’t a snarl like a lion. It was a roar. I’ve never heard anything like it.”

Fenn was thrilled as well as scared. He had never expected to see such a large, endangered cat so early in his life, at age 32, he said. A lifelong hunter and Benson resident, he runs the mountain lion guide service as a sideline while working full time in an excavating business. He described his one-hour encounter with the jaguar as “a dream come true.”

He came away respectful of its power, speed and size.

“All my dogs took a pretty good beating. They had puncture wounds. … I got to see it in real life, and I’m glad, but I hope to never encounter it again,” he repeated.

He crept up close and took photos and a video of the jaguar:

He also notified state wildlife officials, who were later able to find hair samples left behind by the animal and a tree trunk that showed signs of being climbed by a large clawed animal. They believe he saw an adult male jaguar that weighed about 90 kilograms.

The jaguar, Panthera onca, is the third-largest cat in the world, only outranked by the lion and tiger. It’s the only surviving New World member of the genus Panthera. For example, there was once an American lion, but that went extinct 10,000 years ago, along with a lot of other large mammals, after people showed up. DNA evidence shows that the lion, tiger, leopard, jaguar, snow leopard, and clouded leopard share a common ancestor, and that this whole gang is between 6 and 10 million years old. (The so-called ‘mountain lion’, Puma concolor, is not in this group.)

Jaguars have mostly been killed off in the United States, but they survive from Mexico to Central and South America all the way down to Paraguay and northern Argentina. They are listed as ‘near threatened’ by the IUCN, or International Union for the Conservation of Nature.

The Arizona Fish and Game Department has also announced two reliable sightings of ocelots this year!

The ocelot, Leopardus pardalis, is a much smaller fellow, about the size of a domestic cat. Ocelots live in many parts of South and Central America and Mexico, and they’re listed as being of ‘least concern’ by the IUCN. Once their range extended up into the chaparral thickets of the Gulf Coast of south and eastern Texas, as well as part of Arizona, Louisiana, and Arkansas. But by now they are very hard to find in the United States. They seem to eke out an existence only in several small areas of dense thicket in South Texas… and, we now know, Arizona!

Lynn Margulis, 1938-2011

Lynn Margulis died on Tuesday, the 22nd of November.

She is most famous for her theory of endosymbiosis, which says that eukaryotic cells—the complicated cells of protozoa, animals, plants, and fungi—were formed in a series of stages where one organism swallowed another but, instead of digesting it, took it on as a symbiotic partner!

As a young faculty member at Boston University, she wrote a paper on this theory. It was rejected by about fifteen scientific journals, but eventually accepted:

• Lynn Sagan, On the origin of mitosing cells, J. Theor. Bio. 14 (1967), 255–274.

(At the time she was married to the well-known astronomer Carl Sagan; she later wrote a number of books with their son Dorion.)

While Lynn Margulis was not the first to suggest that endosymbiosis played a major role in evolution, her brilliant and endlessly energetic advocacy of this idea helped trigger a scientific revolution. By now, almost all biologists agree that chloroplasts, the little green packages where photosynthesis happens in plant cells, were originally free-living bacteria of their own. Here’s a model of a chloroplast:

As you can see, it’s quite a complex world of its own. Biologists also believe that mitochondria, the ‘powerhouses’ of eukaryotic cells:

were originally bacteria.

Indeed, both chloroplasts and mitochondria still have their own DNA, and they reproduce like bacteria, through fission. This is very useful in research on human populations, since mitochondrial DNA is passed down, not through the sperm cells, but only through the egg cells—which means it’s passed down from mothers to daughters, with sons as a mere sideshow. On the other hand, Y chromosomes go down from father to son.

There’s a lot more evidence for endosymbiosis as the origin of mitochondria and chloroplasts… and in fact, we can see this process going on today in various organisms. There are also many other ways in which organisms pass genes to each other.

This led Margulis to advocate a worldview in which the ‘tree’ of life is replaced by something more like a ‘thicket’:

It also suggests that competition between organisms is part of a bigger story where the formation of partnerships plays a major role. She has written many popular books on this, and they make great reading, even though some of her ideas are still controversial.

In 1995, the famous evolutionary biologist Richard Dawkins wrote:

I greatly admire Lynn Margulis’s sheer courage and stamina in sticking by the endosymbiosis theory, and carrying it through from being an unorthodoxy to an orthodoxy. I’m referring to the theory that the eukaryotic cell is a symbiotic union of primitive prokaryotic cells. This is one of the great achievements of twentieth-century evolutionary biology, and I greatly admire her for it.

I’ll end by quoting an excerpt from this essay, which you can read in its entirety online:

• Lynn Margulis, Gaia is a tough bitch, in The Third Culture: Beyond the Scientific Revolution, ed. John Brockman, Simon & Schuster, 1995.

I think it’s a good sample of how she wasn’t afraid to stir up controversy. Indeed, you’ll see her poking Dawkins with a sharp stick at the same time he was complimenting her!

More importantly, you’ll get a sense of how her work helped push evolutionary biology away from neo-Darwinism, also known as the Modern Synthesis. This approach tried to explain all evolution as a process of populations slowly changing through random mutation and natural selection. Now biologists are struggling to formulate an Extended Synthesis which takes many more processes into account.

Her remarks also some contain good warnings for mathematicians and physicists, such as myself, who want to dabble in biology. Physics may be defined as ‘the study of natural systems that can be accurately modeled by beautiful mathematics’. That’s why we can learn a surprising amount about physics just by sitting around scribbling. But living systems always surprise us, since they’re always more rich and complex than our models. So, biology requires more emphasis on experiment, observation, and perhaps a feeling for the organism:

I work in evolutionary biology, but with cells and microorganisms. Richard Dawkins, John Maynard Smith, George Williams, Richard Lewontin, Niles Eldredge, and Stephen Jay Gould all come out of the zoological tradition, which suggests to me that, in the words of our colleague Simon Robson, they deal with a data set some three billion years out of date. Eldredge and Gould and their many colleagues tend to codify an incredible ignorance of where the real action is in evolution, as they limit the domain of interest to animals—including, of course, people. All very interesting, but animals are very tardy on the evolutionary scene, and they give us little real insight into the major sources of evolution’s creativity. It’s as if you wrote a four-volume tome supposedly on world history but beginning in the year 1800 at Fort Dearborn and the founding of Chicago. You might be entirely correct about the nineteenth-century transformation of Fort Dearborn into a thriving lakeside metropolis, but it would hardly be world history.

By “codifying ignorance” I refer in part to the fact that they miss four out of the five kingdoms of life. Animals are only one of these kingdoms. They miss bacteria, protoctista, fungi, and plants. They take a small and interesting chapter in the book of evolution and extrapolate it into the entire encyclopedia of life. Skewed and limited in their perspective, they are not wrong so much as grossly uninformed.

Of what are they ignorant? Chemistry, primarily, because the language of evolutionary biology is the language of chemistry, and most of them ignore chemistry. I don’t want to lump them all together, because, first of all, Gould and Eldredge have found out very clearly that gradual evolutionary changes through time, expected by Darwin to be documented in the fossil record, are not the way it happened. Fossil morphologies persist for long periods of time, and after stasis, discontinuities are observed. I don’t think these observations are even debatable. John Maynard Smith, an engineer by training, knows much of his biology secondhand. He seldom deals with live organisms. He computes and he reads. I suspect that it’s very hard for him to have insight into any group of organisms when he does not deal with them directly. Biologists, especially, need direct sensory communication with the live beings they study and about which they write.

Reconstructing evolutionary history through fossils—paleontology—is a valid approach, in my opinion, but paleontologists must work simultaneously with modern-counterpart organisms and with “neontologists”—that is, biologists. Gould, Eldredge, and Lewontin have made very valuable contributions. But the Dawkins–Williams–Maynard Smith tradition emerges from a history that I doubt they see in its Anglophone social context. Darwin claimed that populations of organisms change gradually through time as their members are weeded out, which is his basic idea of evolution through natural selection. Mendel, who developed the rules for genetic traits passing from one generation to another, made it very clear that while those traits reassort, they don’t change over time. A white flower mated to a red flower has pink offspring, and if that pink flower is crossed with another pink flower the offspring that result are just as red or just as white or just as pink as the original parent or grandparent. Species of organisms, Mendel insisted, don’t change through time. The mixture or blending that produced the pink is superficial. The genes are simply shuffled around to come out in different combinations, but those same combinations generate exactly the same types. Mendel’s observations are incontrovertible.

So J. B. S. Haldane, without a doubt a brilliant person, and R. A. Fisher, a mathematician, generated an entire school of English-speaking evolutionists, as they developed the neo-Darwinist population-genetic analysis to reconcile two unreconcilable views: Darwin’s evolutionary view with Mendel’s pragmatic, anti-evolutionary concept. They invented a language of population genetics in the 1920s to 1950s called neo-Darwinism, to rationalize these two fields. They mathematized their work and began to believe in it, spreading the word widely in Great Britain, the United States, and beyond. France and other countries resisted neo-Darwinism, but some Japanese and other investigators joined in the “explanation” activity.

Both Dawkins and Lewontin, who consider themselves far apart from each other in many respects, belong to this tradition. Lewontin visited an economics class at the University of Massachusetts a few years ago to talk to the students. In a kind of neo-Darwinian jockeying, he said that evolutionary changes are due to the Fisher–Haldane mechanisms: mutation, emigration, immigration, and the like. At the end of the hour, he said that none of the consequences of the details of his analysis had been shown empirically. His elaborate cost-benefit mathematical treatment was devoid of chemistry and biology. I asked him why, if none of it could be shown experimentally or in the field, he was so wedded to presenting a cost-benefit explanation derived from phony human social-economic “theory.” Why, when he himself was pointing to serious flaws related to the fundamental assumptions, did he want to teach this nonsense? His response was that there were two reasons: the first was “P.E.” “P.E.?,” I asked. “What is P.E.? Population explosion? Punctuated equilibrium? Physical education?” “No,” he replied, “P.E. is `physics envy,'” which is a syndrome in which scientists in other disciplines yearn for the mathematically explicit models of physics. His second reason was even more insidious: if he didn’t couch his studies in the neo-Darwinist thought style (archaic and totally inappropriate language, in my opinion), he wouldn’t be able to obtain grant money that was set up to support this kind of work.

The neo-Darwinist population-genetics tradition is reminiscent of phrenology, I think, and is a kind of science that can expect exactly the same fate. It will look ridiculous in retrospect, because it is ridiculous. I’ve always felt that way, even as a more-than-adequate student of population genetics with a superb teacher—James F. Crow, at the University of Wisconsin, Madison. At the very end of the semester, the last week was spent on discussing the actual observational and experimental studies related to the models, but none of the outcomes of the experiments matched the theory.

This passage shows her tough side—these are the top names in evolutionary biology that she’s criticizing here, after all. But when I saw her speak, she was engaging and fun! You can see that yourself in these interviews. Hear how she started as a bad student in 4th grade, why her laboratory budget got cut to $0 in 2004… and get a sense of her career, personality, and ideas.

 

 

Network Theory (Part 17)

joint with Jacob Biamonte

We’ve seen how Petri nets can be used to describe chemical reactions. Indeed our very first example came from chemistry:

However, chemists rarely use the formalism of Petri nets. They use a different but entirely equivalent formalism, called ‘reaction networks’. So now we’d like to tell you about those.

You may wonder: why bother with another formalism, if it’s equivalent to the one we’ve already seen? Well, one goal of this network theory program is to get people from different subjects to talk to each other—or at least be able to. This requires setting up some dictionaries to translate between formalisms. Furthermore, lots of deep results on stochastic Petri nets are being proved by chemists—but phrased in terms of reaction networks. So you need to learn this other formalism to read their papers. Finally, this other formalism is actually better in some ways!

Reaction networks

Here’s a reaction network:

This network involves 5 species: that is, different kinds of things. They could be atoms, molecules, ions or whatever: chemists call all of these species, and there’s no need to limit the applications to chemistry: in population biology, they could even be biological species! We’re calling them A, B, C, D, and E, but in applications, we’d call them by specific names like CO₂ and HCO₃^–, or ‘rabbit’ and ‘wolf’, or whatever.

This network also involves 5 reactions, which are shown as arrows. Each reaction turns one bunch of species into another. So, written out more longwindedly, we’ve got these reactions:

If you remember how Petri nets work, you can see how to translate any reaction network into a Petri net, or vice versa. For example, the reaction network we’ve just seen gives this Petri net:

Each species corresponds to a state of this Petri net, drawn as a yellow circle. And each reaction corresponds to transition of this Petri net, drawn as a blue square. The arrows say how many things of each species appear as input or output to each transition. There’s less explicit emphasis on complexes in the Petri net notation, but you can read them off if you want them.

In chemistry, a bunch of species is called a ‘complex’. But what do I mean by ‘bunch’, exactly? Well, I mean that in a given complex, each species can show up 0,1,2,3… or any natural number of times. So, we can formalize things like this:

Definition. Given a set of species, a complex of those species is a function .

Roughly speaking, a reaction network is a graph whose vertices are labelled by complexes. Unfortunately, the word ‘graph’ means different things in mathematics—appallingly many things! Everyone agrees that a graph has vertices and edges, but there are lots of choices about the details. Most notably:

• We can either put arrows on the edges, or not.

• We can either allow more than one edge between vertices, or not.

• We can either allow edges from a vertex to itself, or not.

If we say ‘no’ in every case we get the concept of ‘simple graph’, which we discussed last time. At the other extreme, if we say ‘yes’ in every case we get the concept of ‘directed multigraph’, which is what we want now. A bit more formally:

Definition. A directed multigraph consists of a set of vertices, a set of edges, and functions saying the source and target of each edge.

Given this, we can say:

Definition. A reaction network is a set of species together with a directed multigraph whose vertices are labelled by complexes of those species.

You can now prove that reaction networks are equivalent to Petri nets:

Puzzle 1. Show that any reaction network gives a Petri net, and vice versa.

In a stochastic Petri net each transition is labelled by a rate constant: that is, a numbers in . This lets us write down some differential equations saying how species turn into each other. So, let’s make this definition (which is not standard, but will clarify things for us):

Definition. A stochastic reaction network is a reaction network where each reaction is labelled by a rate constant.

Now you can do this:

Puzzle 2. Show that any stochastic reaction network gives a stochastic Petri net, and vice versa.

For extra credit, show that in each of these puzzles we actually get an equivalence of categories! For this you need to define morphisms between Petri nets, morphisms between reaction networks, and similarly for stochastic Petric nets and stochastic reaction networks. If you get stuck, ask Eugene Lerman for advice. There are different ways to define morphisms, but he knows a cool one.

We’ve been downplaying category theory so far, but it’s been lurking beneath everything we do, and someday it may rise to the surface.

The deficiency zero theorem

You may have already noticed one advantage of reaction networks over Petri nets: they’re quicker to draw. This is true even for tiny examples. For example, this reaction network:

corresponds to this Petri net:

But there’s also a deeper advantage. As we saw in Part 8, any stochastic Petri net gives two equations:

• The master equation, which says how the probability that we have a given number of things of each species changes with time.

• The rate equation, which says how the expected number of things in each state changes with time.

The simplest solutions of these equations are the equilibrium solutions, where nothing depends on time. Back in Part 9, we explained when an equilibrium solution of the rate equation gives an equilibrium solution of the master equation. But when is there an equilibrium solution of the rate equation in the first place?

Feinberg’s ‘deficiency zero theorem’ gives a handy sufficient condition. And this condition is best stated using reaction networks! But to understand it, we need to understand the ‘deficiency’ of a reaction network. So let’s define that, and they say what all the words in the definition mean:

Definition. The deficiency of a reaction network is:

• the number of vertices

minus

• the number of connected components

minus

• the dimension of the stoichiometric subspace.

The first two concepts here are easy. A reaction network is a graph (okay, a directed multigraph). So, it has some number of vertices, and also some number of connected components. Two vertices lie in the same connected component iff you can get from one to the other by a path where you don’t care which way the arrows point. For example, this reaction network:

has 5 vertices and 2 connected components.

So, what’s the ‘stoichiometric subspace’? ‘Stoichiometry’ is a scary-sounding word. According to the Wikipedia article:

‘Stoichiometry’ is derived from the Greek words στοιχεῖον (stoicheion, meaning element) and μέτρον (metron, meaning measure.) In patristic Greek, the word Stoichiometria was used by Nicephorus to refer to the number of line counts of the canonical New Testament and some of the Apocrypha.

But for us, stoichiometry is just the art of counting species. To do this, we can form a vector space where is the set of species. A vector in is a function from species to real numbers, saying how much of each species is present. Any complex gives a vector in , because it’s actually a function from species to natural numbers.

Definition. The stoichiometric subspace of a reaction network is the subspace spanned by vectors of the form where and are complexes connected by a reaction.

‘Complexes connected by a reaction’ makes sense because vertices in the reaction network are complexes, and edges are reactions. Let’s see how it works in our example:

Each complex here can be seen as a vector in , which is a vector space whose basis we can call . Each reaction gives a difference of two vectors coming from complexes:

• The reaction gives the vector

• The reaction gives the vector

• The reaction gives the vector

• The reaction gives the vector

• The reaction gives the vector

The pattern is obvious, I hope.

These 5 vectors span the stoichiometric subspace. But this subspace isn’t 5-dimensional, because these vectors are linearly dependent! The first vector is the negative of the second one. The last is the sum of the previous two. And those are all the linear dependencies, so the stoichiometric subspace is 3 dimensional. For example, it’s spanned by these 3 linearly independent vectors: and .

I hope you see the moral of this example: the stoichiometric subspace is the space of ways to move in that are allowed by the reactions in our reaction network! And this is important because the rate equation describes how the amount of each species changes as time passes. So, it describes a point moving around in .

Thus, if is the stoichiometric subspace, and is a solution of the rate equation, then always stays within the set

Mathematicians would call this set the coset of , but chemists call it the stoichiometric compatibility class of

Anyway: what’s the deficiency of the reaction complex in our example? It’s

since there are 5 complexes, 2 connected components and the dimension of the stoichiometric subspace is 3.

But what’s the deficiency zero theorem? You’re almost ready for it. You just need to know one more piece of jargon! A reaction network is weakly reversible if whenever there’s a reaction going from a complex to a complex , there’s a path of reactions going back from to . Here the paths need to follow the arrows.

So, this reaction network is not weakly reversible:

since we can get from to but not back from to and from to but not back, and so on. However, the network becomes weakly reversible if we add a reaction going back from to :

If a reaction network isn’t weakly reversible, one complex can turn into another, but not vice versa. In this situation, what typically happens is that as time goes on we have less and less of one species. We could have an equilibrium where there’s none of this species. But we have little right to expect an equilibrium solution of the rate equation that’s positive, meaning that it sits at a point , where there’s a nonzero amount of every species.

My argument here is not watertight: you’ll note that I fudged the difference between species and complexes. But it can be made so when our reaction network has deficiency zero:

Deficiency Zero Theorem (Feinberg). Suppose we are given a reaction network with a finite set of species , and suppose its deficiency is zero. Then:

(i) If the network is not weakly reversible and the rate constants are positive, the rate equation does not have a positive equilibrium solution.

(ii) If the network is not weakly reversible and the rate constants are positive, the rate equation does not have a positive periodic solution, that is, a periodic solution lying in .

(iii) If the network is weakly reversible and the rate constants are positive, the rate equation has exactly one equilibrium solution in each positive stoichiometric compatibility class.
Any sufficiently nearby solution that starts in the same stoichiometric compatibility class will approach this equilibrium as . Furthermore, there are no other positive periodic solutions.

This is quite an impressive result. We’ll look at an easy example next time.

References and remarks

The deficiency zero theorem was published here:

• Martin Feinberg, Chemical reaction network structure and the stability of complex isothermal reactors: I. The deficiency zero and deficiency one theorems, Chemical Engineering Science 42 (1987), 2229–2268.

These other explanations are also very helpful:

• Martin Feinberg, Lectures on reaction networks, 1979.

• Jeremy Gunawardena, Chemical reaction network theory for in-silico biologists, 2003.

At first glance the deficiency zero theorem might seem to settle all the basic questions about the dynamics of reaction networks, or stochastic Petri nets… but actually, it just means that deficiency zero reaction networks don’t display very interesting dynamics in the limit as So, to get more interesting behavior, we need to look at reaction networks that don’t have deficiency zero.

For example, in biology it’s interesting to think about ‘bistable’ chemical reactions: reactions that have two stable equilibria. An electrical switch of the usual sort is a bistable system: it has stable ‘on’ and ‘off’ positions. A bistable chemical reaction can serve as a kind of biological switch:

• Gheorghe Craciun, Yangzhong Tang and Martin Feinberg, Understanding bistability in complex enzyme-driven reaction networks, PNAS 103 (2006), 8697-8702.

It’s also interesting to think about chemical reactions with stable periodic solutions. Such a reaction can serve as a biological clock:

• Daniel B. Forger, Signal processing in cellular clocks, PNAS 108 (2011), 4281-4285.

Measuring Biodiversity

guest post by Tom Leinster

Even if there weren’t a global biodiversity crisis, we’d want to know how to put a number on biodiversity. As Lord Kelvin famously put it:

When you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot measure it, your knowledge is of a meagre and unsatisfactory kind: it may be the beginning of knowledge, but you have scarcely, in your thoughts, advanced to the stage of science.

In this post, I’ll talk about what happens when you take a mass of biological data and try to turn it into a single number, intended to measure biodiversity.

There have been more than 50 years of debate about how to measure diversity. While the idea of putting a number on biological diversity goes back to the 1940s at least, the debate really seems to have got going in the wake of pioneering work by the great ecologist Robert Whittaker in the 1960s.

There followed several decades in which progress was made… but there was a lot of talking at cross-purposes. In fact, there was so much confusion that some people gave up on the diversity concept altogether. The mood is summed up by the title of an excellent and much-cited paper of Stuart Hurlbert:

• S. H. Hurlbert, The nonconcept of species diversity: A critique and alternative parameters. Ecology 52:577–586, 1971.

So why all the confusion?

One reason is that the word “diversity” is used by different people in many different ways. We all know that diversity is important: so if you found a quantity that seemed to measure biological variation in a sensible way, you might be tempted to call it “diversity” and publish a paper promoting your quantity over all other quantities that have ever been given that name. There are literally dozens of measures of diversity in the literature. Here are two simple ones:

• Species richness is simply the number of species in the community concerned.
• The Shannon entropy is , where our community consists of species in proportions .

Which quantity should we call “diversity”? Do all these quantities really measure the same kind of thing? If community A has greater than species richness than community B, but lower Shannon entropy, what does it mean?

Another cause for confusion is a blurring between the questions

Which quantities deserve to be called diversity?

and

Which quantities are we capable of measuring experimentally?

For example, we might all agree that species richness is an important quantity, but that doesn’t mean that species richness is easy to measure in practice. (In fact, it’s not, more on which below.) My own view is that the two questions should be kept separate:

The statistical problem of designing appropriate estimators becomes relevant only after the measure to be estimated is accepted to be meaningful.

(Hans-Rolf Gregorius, Elizabeth M. Gillet, Generalized Simpson-diversity, Ecological Modelling 211:90–96, 2008.)

The problems involved in quantifying diversity are of three types: practical, statistical and conceptual. I’ll say a little about the first two, and rather more about the third.

Practical  Suppose that you’re doing a survey of the vertebrates in a forest. Perhaps one important species is brightly coloured and noisy, while another is silent, shy, and well-camouflaged. How do you prevent the first from being recorded disproportionately?

Or suppose that you’re carrying out a survey, with multiple people doing the fieldwork. Different people have a tendency to spot different things: for example, one person might be short-sighted and another long-sighted. How do you ensure that this doesn’t affect your results?

Statistical  Imagine that you want to know how many distinct species of insect live in a particular area — the “species richness”, in the terminology introduced above. You go out collecting, and you come back with 100 specimens representing 10 species.

But your survey might have missed some species altogether, so you go out and get a bigger sample. This time, you get 200 specimens representing 15 species. Does this help you discover how many species there really are?

Logically, not at all. The only certainty is that there are at least 15 species. Maybe there are thousands of species, but almost all of them are extremely rare. Or maybe there are really only 15. Unless you collect all the insects, you’ll never know for sure exactly how many species there are.

However, it may be that you can make reasonable assumptions about the frequency distribution of the species. People sometimes do exactly this, to try to overcome the difficulty of estimating species richness.

Conceptual  This is what I really want to talk about.

I mentioned earlier that different people mean different things by “diversity”. Here’s an example.

Consider two bird communities. The first looks like this:

It contains four species, one of which is responsible for most of the population, and three of which are quite rare. The second looks like this:

It has only three species, but they’re evenly balanced.

Which community is the more diverse? It’s a matter of opinion. Mostly in the press, and in many scholarly articles too, “biodiversity” is used as a synonym for “species richness”. On this count, the first community is more diverse. But if you’re more concerned with the healthy functioning of the whole community, the presence of rare species might not be particularly important: it’s balance that matters, and the second community has more of that.

Different people using the word “diversity” attach different amounts of significance to rare species. There’s a spectrum of points of view, ranging from those who give rare species the same weight as common ones (as in the definition of species richness) to those who are only interested in the most common species of all. Every point on this spectrum of viewpoints is reasonable. None should have a monopoly on the word “diversity”.

At least, that’s what Christina Cobbold and I argue in our new paper:

• Tom Leinster, Christina A. Cobbold, Measuring diversity: the importance of species similarity, Ecology, in press (doi:10.1890/10-2402.1).

But that’s not actually our main point. As the title suggests, the real purpose of our paper is to show how to measure diversity in a way that reflects the varying differences between species. I’ll explain.

Most of the existing approaches to measuring biodiversity go like this.

We have a “community” of organisms — the fish in a lake, the fungi in a forest, or the bacteria on your skin. This community is divided into groups, conventionally called species, though they needn’t be species in the ordinary sense.

We assume that we know the relative abundances, or relative frequencies, of the species. Write them as . Thus, is the proportion of the total population that belongs to the th species, where “proportion” is measured in any way you think sensible (number of individuals, total mass, etc).

We only care about relative abundances here, not absolute abundances: so . If half of a forest is destroyed, it might be a catastrophe, but on the (unrealistic) assumption that all the flora and fauna in the forest were distributed homogeneously, it won’t actually change the biodiversity. (That’s not a statement about what’s important in life; it’s only a statement about the usage of a word.)

This model is common but crude. It can’t detect the difference between a community of six dramatically different species and a community consisting of six species of barnacle.

So, Christina and I use a refined model, as follows. We assume that we also have a measure of the similarity between each pair of species. This is a real number between 0 and 1, with 0 indicating that the species are as dissimilar as could be, and 1 indicating that they’re identical. Writing the similarity between the th and th species as , this gives an matrix . Our only assumption on is that its diagonal entries are all 1: every species is identical to itself.

There are many ways of measuring inter-species similarity. Probably the most familiar approach is genetic, as in “you share 98% of your DNA with a chimpanzee”. But there are many other possibilities: functional, phylogenetic, morphological, taxonomic, …. Diversity is a measure of the variety of life; having to choose a measure of similarity forces you to get clear exactly what you mean by “variety”.

Christina and I are by no means the first people to incorporate species similarity into the model of an ecological community. The main new thing in our paper is this measure of the community’s diversity:

What does this mean?

• is what we call the diversity of order of the community. Here is a parameter between and , which you get to choose. Different values of represent different points on the spectrum of viewpoints described above. Small values of give high importance to rare species; large values of give high importance to common species.
• is shorthand for the relative abundances , and is the matrix of similarities.
• means .

The expression doesn’t make sense if or , but can be made sense of by taking limits. For , this gives

If you want to know the value at , or any of the other mathematical details, you can read this post at the n-Category Café, or of course our paper. In both places, you’ll also find an explanation of what motivates this formula. What’s more, you’ll see that many existing measures of diversity are special cases of ours, obtained by taking particular values for and/or .

But I won’t talk about any of that here. Instead, I’ll tell you how taking species similarity into account can radically alter the assessment of diversity.

I’ll do this using an example: butterflies of subfamily Charaxinae at a site in an Ecuadorian rainforest. The data is from here:

• P. J. DeVries, D. Murray, R. Lande, Species diversity in vertical, horizontal and temporal dimensions of a fruit-feeding butterfly community in an Ecuadorian rainforest. Biological Journal of the Linnean Society 62:343–364, 1997.

They measured the butterfly abundances in both the canopy (top level) and understorey (lower level) at this site, with the following results:

Species	Canopy	Understorey
Prepona laertes	15	0
Archaeoprepona demophon	14	37
Zaretis itys	25	11
Memphis arachne	89	23
Memphis offa	21	3
Memphis xenocles	32	8

Which is more diverse: canopy or understorey?

We’ve already seen that the answer is going to depend on what exactly we mean by “diverse”.

First let’s answer the question under the (crude!) assumption that different species have nothing whatsoever in common. This means taking our similarity matrix to be the identity matrix: if then (totally dissimilar), and if then (totally identical).

Now, remember that there’s a spectrum of viewpoints on how much importance to give to rare species when measuring diversity. Rather than choosing a particular viewpoint, we’ll calculate the diversity from all viewpoints, and display it on a graph. In other words, we’ll draw the graph of (the diversity of order ) against (the viewpoint). Here’s what we get:

(the horizontal axis should be labelled with a .)

Conclusion: from all viewpoints, the butterfly population in the canopy is at least as diverse as that in the understorey.

Now let’s do it again, but this time taking account of the varying similarities between species of butterflies. We don’t have much to go on: how do we know whether Prepona laertes is very similar to, or very different from, Archaeoprepona demophon? With only the data above, we don’t. So what can we do?

All we have to go on is the taxonomy. Remember your high school biology: for the butterfly Prepona laertes, the genus is Prepona and the species is laertes. We’d expect species in the same genus to have more in common than species in different genera. So let’s define the similarity between two species as follows:

• the similarity is 1 if the species are the same
• the similarity is 0.5 if the species are different but in the same genus
• the similarity is 0 if they are not even in the same genus.

This is still crude, but in the absence of further information, it’s about the best we can do. And it’s better than the first approach, where we ignored the taxonomy entirely. Throwing away biologically relevant information is unlikely to lead to a better assessment of diversity.

Using this taxonomic matrix , and the same abundances, the diversity graphs become:

This is more interesting! For , the understorey looks more diverse than the canopy — the opposite conclusion to our first approach.

It’s not hard to see why. Look again at the table of abundances, but paying attention to the genera of the butterflies. In the canopy, nearly three-quarters of the butterflies are of genus Memphis. So when we take into account the fact that species in the same genus tend to be somewhat similar, the canopy looks much less diverse than it did before. In the understorey, however, the species are spread more evenly between genera, so taking similarity into account leaves its diversity relatively unchanged.

Taking account of species similarity opens up a world of uncertainty. How should we measure similarity? There are as many possibilities as there are quantifiable characteristics of living organisms. It’s much more reassuring to stay in the black-and-white world where distinct species are always assigned a similarity of 0, no matter how similar they might actually be. (This is, effectively, what most existing measures do.) But that’s just hiding from reality.

Maybe you disagree! If so, try the the Discussion section of our paper, where we lay out our arguments in more detail. Or let me know by leaving a comment.